Conceptual problems in quantum electrodynamics: a contemporary historico-philosophical approach

Conceptual problems in quantum electrodynamics: a contemporary historical-philosophical approach Tesis doctoral Mario Bacelar Valente Universidad de Sevilla 2011 Problemas conceptuales en la electrodinámica cuántica: un enfoque histórico-filosófico contemporáneo Trabajo de investigación para la obtención del Grado de Doctor por la Universidad de Sevilla Conceptual problems in quantum electrodynamics: a contemporary historical-philosophical approach Dissertation submitted in fulfillment of the requirements for the Degree of Doctor by the Universidad de Sevilla Mario Bacelar Valente Supervisores: José Ferreirós Domínguez, Universidad de Sevilla. Henrik Zinkernagel, Universidad de Granada. Doctorado Ciencia y Cultura. Departamento de Filosofía y Lógica y Filosofía de la Ciencia CONTENTS 1 Introducción 1 Summary of the introduction 1 Comentarios iniciales 4 El papel de la historia en el trabajo 5 El trabajo 6 El capítulo 2 8 El capítulo 3 11 El capítulo 4 17 El capítulo 5 20 El capítulo 6 22 El capítulo 7 25 El capítulo 8 26 2 The Schrödinger equation and its interpretation 29 1 Introduction 29 2 The ‘pre-history’ of the Schrödinger equation 29 3 The coming to be of the Schrödinger equation 37 4 The interpretation and latter developments of the Schrödinger equation 40 3 The Dirac equation and its interpretation 47 1 Introduction 47 2 Before the Dirac equation: some historical remarks 47 3 The Dirac equation as a one-electron equation 50 4 The problem with the negative energy solutions 53 5 The field theoretical interpretation of Dirac’s equation 59 6 Combining results from the different views on Dirac’s equation 63 4 The quantization of the electromagnetic field and the vacuum state 67 1 Introduction 67 2 The historical emergence of the quantized electromagnetic field 68 3 The quantization of the electromagnetic field 77 4 The Casimir effect and formal aspects related to the vacuum state 80 5 The physical meaning of variance 86 6 Experimental results on the vacuum state using the balanced homodyne detection method 87 7 Conclusions 90 5 The interaction of radiation and matter 92 1 introduction 92 2. Quantum electrodynamics as a perturbative approach 93 3 Possible problems to quantum electrodynamics: the Haag theorem and the divergence of the S-matrix series expansion 102 4 A note regarding the concept of vacuum in quantum electrodynamics 111 5 Conclusions 112 6 Aspects of renormalization in quantum electrodynamics 114 1 Introduction 114 2 The emergence of infinites in quantum electrodynamics 115 3 The submergence of infinites in quantum electrodynamics 123 4 Different views on renormalization 135 5 Conclusions 142 7 The Feynman diagrams and virtual quanta 145 1. Introduction 145 2 The Møller scattering and Feynman’s fundamental equation 145 3 The description of interactions as space-time processes resulting from the exchange of virtual quanta 150 4 A quantum model of the delayed interaction between two bound electrons 158 5 Conclusions 160 8 The relation between classical and quantum electrodynamics 162 1. Introduction 162 2 Classical electrodynamics 163 3 The relation between classical and quantum electrodynamics 166 3.1 The quantization procedure and the so-called classical limit 167 3.2 The overall temporal description of physical processes in quantum electrodynamics 170 3.3 A tentative quantum electrodynamical model providing a temporal description of physical processes 171 4 Conclusions 173 9 Comentarios finales 175 Appendix. Bohr’s quantum postulate and time in quantum mechanics 177 1 Introduction 177 2 The quantum postulate 178 3 The concept of time in quantum theory and Bohr’s quantum postulate 183 4 Conclusions 192 Bibliography 194 Acknowledgements I want to express my gratitude to José Ferreirós Dominguéz and Henrik Zinkernagel for making possible this dissertation. I also want to thank Olivier Darrigol with whom I had a productive interchange of views regarding distinct aspects of this work. 4 Comentarios iniciales Un interés sistemático desde una perspectiva más filosófica respecto a la teoría cuántica de campos es algo reciente, de la década de los 80. Esto no significa que temas que se han tornado tópicos del debate filosófico no se hayan mencionado antes. Un ejemplo es el tratamiento por M. Bunge del estatus de las llamadas partículas virtuales en la electrodinámica cuántica (Bunge, 1970). Pero un tratamiento sistemático - una especie de programa de investigación filosófica de la teoría cuántica de campos – ganó ‘momentum’ en los 80 en particular con los trabajos de M. Redhead y P. Telller. Centrándome en la aportación de Teller y en particular en su libro “An interpretive introduction to quantum field theory” del 95, Teller, enfocando la electrodinámica cuántica, analiza una serie de aspectos de la teoría: trata de interpretar la teoría. ¿Qué es según Teller, en la práctica, interpretar la teoría? Por lo menos en parte es claramente un análisis conceptual de la teoría. Teller propone interpretar el concepto de partícula, específicamente de cuanto, que se tiene en la teoría tratando de distinguirlo de la concepción clásica; también el concepto de campo cuántico; la descripción de interacciones, y la cuestión de la renormalización. ¿Qué hace que este análisis conceptual sea algo filosófico y no simplemente algo hecho por un físico? Simplemente que el enfoque parta de una actitud filosófica. Así en este caso particular, asistimos, por ejemplo, a la propuesta por Teller de sustitución de la idea de sustancia por la de propensión (propensity) para hablar de los campos cuánticos (pues un campo cuántico se tiene que ver como ‘estando’ en un determinado estado cuántico al cual se pueden asociar distintas probabilidades para observar un número distinto de cuantos). En términos sencillos lo que hace el análisis conceptual-filosófico y no simplemente físico es la presencia implícita o explícita de preocupaciones filosóficas en particular de carácter ontológico (o ‘anti-ontológico’) y algunas veces de carácter epistemológico. La línea de trabajo desarrollada por Teller (y otros) se basa como antes he mencionado en el estudio de una teoría física: la electrodinámica cuántica. Este no es el único enfoque que encontramos en la filosofía de la física respecto a las teorías cuánticas de campo. A finales de los 90 empezó a ganar ‘momentum’ otro enfoque basado en versiones axiomáticas de teorías cuánticas de campo, en particular la teoría cuántica de campos algebraica (algebraic quantum field theory). En años recientes incluso empezó un debate respecto a que formulación de la teoría cuántica de campos es más meritoria para servir de base para la discusión filosófica. Un ejemplo reciente es un artículo de D. Fraser (2009), defendiendo que el trabajo filosófico de interpretación de las teorías de campos se debe basar en exclusiva en la versión algebráica. Me resulta extraña esta visión; el que se tenga que optar por un determinado enfoque, más aun cuando la opción es por las versiones axiomáticas que no tienen en palabras de Fraser ningún modelo físico realista (i.e. son versiones matemáticas sin aplicación empírica). Me resulta paradójico que en la filosofía de una ciencia empírica como la física se quiera utilizar como punto de partida del estudio filosófico no las teorías físicas que de hecho tenemos (como la electrodinámica cuántica) y sí formulaciones matemáticas que aún no han dado prueba de que puedan ser realmente teorías físicas (aunque eventualmente fallidas), pues estas versiones axiomáticas no están al nivel de presentar previsiones que se puedan contrastar con resultados experimentales. Aquí voy a tratar de la electrodinámica cuántica centrándome por lo tanto en una teoría física, pero tendré también en cuenta resultados formales que pueden ayudar a la 5 interpretación de la teoría. No se trata entonces de escoger una entre dos opciones excluyentes. Este trabajo se debe ver entonces dentro de la línea ‘iniciada’ por Redhead y Teller. Pero hay un aspecto en que se diferencia claramente de la filosofía de la física especializada en la teoría cuántica de campos que se ha desarrollado hasta la actualidad. Es en la opción, que se puede ver como metodológica, de usar la historia de la física como elemento esencial en el desarrollo del trabajo. Es por lo tanto un trabajo históricofilosófico. Aquí también existe un debate, en este caso respecto al papel de la historia en la filosofía de la ciencia (ver por ejemplo Schickore, 2009). No voy entrar en ello. Considero que se puede hacer un buen (o mal) trabajo en filosofía de la física con o sin aportación explícita de la historia de la física. Como referí, mi opción por la historia es metodológica, o sea, no tiene por qué conllevar una particular visión filosófica del papel de la historia en la filosofía de la ciencia. La opción por desarrollar un análisis conceptual de la electrodinámica cuántica desde una perspectiva histórica resulta del hecho personal e ‘intransmisible’ de que sólo a través del estudio histórico me es posible intentar comprender las teorías físicas. Es más bien una necesidad que una elección. Lo que resulta importante es que por lo menos parte de las ideas que voy a defender surgen precisamente de haber seguido un enfoque histórico. Así en este caso particular el enfoque histórico resultó ‘productivo’. No trato de sacar conclusiones más generales respecto al papel de la historia en la filosofía de la física. El papel de la historia en el trabajo Es importante referir que este no es un trabajo de historia y si un trabajo que parte de la historia interna de la electrodinámica cuántica 3 para hacer un análisis conceptual de la teoría, enfocando en particular lo que se pueden llamar anomalías conceptualesmatemáticas (i.e. anomalías que no conllevan problemas en lo que respecta a ‘salvar los fenómenos’). Debido al tema elegido, es posible reducir el ámbito del enfoque histórico a una historia conceptual. En particular considero solamente la evolución teórica sin tratar con detalle aspectos de la historia de la experimentación y su relevancia en el desarrollo conceptual de la teoría cuántica. Así no considero aspectos de historia cultural o material entre otras. 4 El objetivo aquí es hacer una presentación sintética y coherente del desarrollo conceptual de la electrodinámica cuántica centrándome en aspectos esenciales de la teoría y enfocando en particular aquellos relacionados con los problemas tratados en este trabajo. De este modo evito repeticiones y el tratamiento de líneas ‘secundarias’ que incluso han influido en el desarrollo conceptual. Por ejemplo en lo que respecta al cambio conceptual que llevó a la dualidad onda-partícula me centro solamente en la contribución teórico-conceptual de Einstein y no trato la importante contribución de los físicos experimentales (que influirán directamente en las ideas de de Broglie). 5 Eso porqué para tratar los desarrollos técnicos y conceptuales de la electrodinámica cuántica resulta más directa la conexión teórica entre el trabajo de Einstein y Jordan que el 3 Algunas de las principales referencias usadas en este trabajo son las siguientes: Jammer (1966), Mehra & Rechenberg (1982, 1987, 2000, 2001), Kuhn (1978), Kragh (1984, 1990, 1992), Darrigol (1984, 1986, 1992), Schweber (1994), Sánchez Ron (2001), Roqué (1992). 4 Algunos ejemplos en que se enfocan algunos aspectos que se pueden considerar de historia cultural o material de la teoría cuántica son, e.g., Kragh (1999), McCormmach y Jungnickel (1986), Galison (1987). 5 Una historia detallada de este tema se puede encontrar en Wheaton (1983). 6 origen de las ideas de de Broglie (ver capítulo 4). Tampoco considero la relevancia de este tema respecto a la interpretación de la teoría cuántica por ejemplo en la idea de complementariedad de Bohr. Hago siempre una presentación de los temas siguiendo solamente la ‘línea’ histórica que conecta el trabajo de los principales físicos teóricos que han hecho trabajos de primera importancia para el desarrollo de la teoría y en la medida que resulta necesario para la comprensión de aspectos conceptuales de la teoría tratados en este trabajo. Con este fin sigo una línea similar a la de Darrigol (2009). En su trabajo Darrigol busca presentar una historia coherente y simplificada de la teoría cuántica que por ejemplo sea útil a los filósofos: The present paper is an attempt at simplifying this history so as to make it more helpful to physicists and philosophers. (Darrigol, 2009, p. 151) Este objetivo lleva a la adopción de una metodología específica: The simplification I have in mind implies the selection of significant events and processes, as well as the occasional substitution of more direct reasoning for unnecessarily complicated reasoning. It does not imply any arbitrary invention, and it avoids common misconceptions I have selected a few important steps, in such a manner that any given step can be seen as a consequence of the anterior steps in a given situation. (Darrigol, 2009, p. 151) En este trabajo también busco una simplificación de la historia conceptual de la electrodinámica cuántica, en particular en la elección de etapas secuenciadas de forma lineal y presentadas de forma simplificada. Teniendo en cuenta el desarrollo simultáneo de dos de los elementos básicos de la teoría – la cuantización de la materia y del campo electromagnético – que comparten en gran parte la misma historia conceptual, busqué evitar, en los distintos capítulos, presentaciones históricas que fueran redundantes. Así hay elementos históricos que se mencionan de manera general en distintos capítulos por una cuestión de claridad y ‘secuencia histórica’ pero que solo se tratan en detalle cada uno una vez en distintas capítulos del trabajo. 6 Este enfoque histórico limitado cumple el objetivo específico de este trabajo, y es relativo a este que se debe analizar. Esto no significa que si el objetivo fuera por ejemplo el estudio de la interacción entre aspectos teóricos y experimentales en el cambio conceptual este enfoque sea suficiente. En este caso sería necesaria otro tipo de historia en que aspectos de instrumentación y experimentación (y más en general de historia material) sean considerados. Otro ejemplo sería un tratamiento simultáneo de los cambios conceptuales y la ‘retórica’ de la comunidad científica que les acompaña. En este caso sería esencial un enfoque teniendo en cuenta la historia cultural. El trabajo El trabajo se encuentra por supuesto dividido en capítulos perfectamente acotados que podrían dar la sensación de una progresión lineal en que exista una problemática central y algunos desarrollos derivados. No es así aunque lo presente así. De este modo, se podría ver el segundo capitulo (la ecuación de Schrödinger y su interpretación) y el tercero (la ecuación de Dirac y su interpretación) como la presentación de uno de los pilares de la electrodinámica cuántica, siendo el otro pilar la cuantización del campo 6 Por ejemplos en los capítulos 2 y 3 se hace referencia a trabajos de Einstein y Dirac publicados respectivamente en 1909 y 1927 que se analizan con más detalle en el capítulo 4. 7 electromagnético presentada en el capítulo 4 (donde también se trata la cuestión del concepto de vacío). Ya el capitulo 5 se podría ver como el central donde se trata la interacción del campo cuántico de Dirac (definido por la ecuación de Dirac) y del campo cuántico electromagnético. Aquí se tratan cuestiones claves para entender la teoría: la divergencia de la serie de expansión de la matriz-S y las posibles consecuencias del teorema de Haag en lo que respecta a la consistencia de todas las aplicaciones de la teoría. Partiendo de este punto central se explorarían algunas implicaciones y temas relacionados. En el capitulo 6 se trata la cuestión de la interpretación de la renormalización (en que se sigue una línea Bohriana también presente en el capítulo anterior) y, como preparación para el 7, aspectos de la renormalización de la carga eléctrica relacionadas con la descripción en el ámbito temporal de procesos físicos, tema tratado de forma más amplia en el capitulo 7. Aquí los resultados del capitulo central, el 5, serán útiles para el análisis de la descripción que la teoría nos da a nivel espacio-temporal de procesos físicos (resultados estos que también están presentes en parte en el tratamiento del vacío en el capitulo 4). En particular propongo una lectura del concepto de partícula virtual contra-corriente respecto a lo que es la comúnmente aceptada. Finalmente en el capitulo 8 desarrollo las consecuencias del anterior para la cuestión filosóficamente importante de la relación de la electrodinámica clásica y la cuántica. Al contrario de lo que esta presentación lineal y progresiva de la teoría (en la que la descripción de interacciones tiene un papel central) pueda dar que pensar, lo que me llevó a este trabajo es la comprensión de los conceptos de espacio y tiempo al nivel de la teoría cuántica de campos (así en realidad lo central sería el capítulo 7 y una serie de cosas que no elaboro aquí). Escogí la electrodinámica cuántica por ser la más sencilla de las teorías pertenecientes al llamado modelo standard, y por ser la versión cuántica de la mejor teoría clásica de que disponemos: la electrodinámica clásica. Con esto no quiero quitarle importancia a otras teorías clásicas, pero la electrodinámica es probablemente la teoría que tenemos mejor testada y mejor testable (tanto a nivel clásico como cuántico), o por lo menos así lo veo. Este estudio (‘in progress’) de los conceptos de espacio y tiempo va más allá de lo que presento aquí. En este trabajo analizaré solamente algunos aspectos de la descripción de procesos físicos en la electrodinámica cuántica. Esta parte más especifica del programa más general me llevó al estudio del concepto de intercambio de cuantos (quanta exchange) que aparece en la descripción perturbativa de la interacción de la radiación con la materia. Esto me obligó a considerar una serie de cuestiones: la cuestión del estatus de los llamados cuantos virtuales; la divergencia de la matriz-S (que me resultó ‘útil’ para las ideas que defiendo); un modelo, que resulta ser semi-clásico de interacción entre electrones de átomos (bounded electrons); el teorema de Haag que aparentemente hacía peligrar todo el edificio de la electrodinámica cuántica; la reinterpretación de la descripción hecha por la ecuación de Dirac del átomo de hidrógeno en términos de intercambio de cuantos entre el núcleo y el electrón; e incluso aspectos ‘no-temporales’ en el procedimiento de renormalización. Todo esto ha sido reelaborado y ‘linealizado’ en la presentación que sigue según el esquema presentado antes. En este proceso incorporé algunos temas relacionados como un estudio más detallado del vacío en la teoría y la relación entre la teoría cuántica y la clásica. Otro tema incluido es el de la interpretación de las funciones de onda en la teoría cuántica, que es un aspecto básico de cualquier análisis conceptual que se quiera hacer en la electrodinámica cuántica. Siguiendo a B. Falkenburg (2007) creo que la interpretación como colectividades (ensemble interpretation) es la que de modo más inmediato ‘conecta’ con el tipo de experimentos que se hacen en la física de altas energías. Esto me llevó a N. Bohr (al cual ya había llegado por el enfoque histórico 8 encontrando su crítica a la electrodinámica cuántica) y debido al programa más general que me interesa pronto llegué a la conclusión que los conceptos (clásicos) de espacio y tiempo que Bohr maneja en su interpretación de la cuántica son un marco fundamental para comprender esa interpretación. No exploro aquí las posibles conexiones que esa concepción del espacio y tiempo pueda tener con el caso de la electrodinámica cuántica ya que eso implicaría un tratamiento del espacio y tiempo en la teoría de la relatividad (especial). Eso pertenece al programa más ‘general’ de estudio de los conceptos de espacio y tiempo. Aquí me limito a algunos aspectos conceptuales de la electrodinámica cuántica. Paso ahora a un resumen ‘alargado’ en español de la versión linealizada y ordenada del trabajo en los aspectos que conciernen a la electrodinámica cuántica (habiendo dejado el tema de la interpretación Bohriana de la cuántica para un apéndice, que me permite especificar de forma más precisa la interpretación de la función de onda adoptada en este trabajo). El capítulo 2 Este capítulo es una preparación del siguiente. En el capítulo 3 enfocaré la cuestión de la interpretación de la ecuación de Dirac. Hay aspectos de esa historia que se pueden ver también en el caso ‘más sencillo’ de la mecánica cuántica no-relativista en la ecuación de Schrödinger. Así el capítulo 2 está dedicado a una presentación histórica del surgimiento de la ecuación de Schrödinger y de las tentativas iniciales para su interpretación. En 1926 apareció un artículo en Annalen der Physik en el cual su autor E. Schrödinger presentaba una nueva teoría atómica siguiendo las ideas de L. de Broglie en que este asociaba un fenómeno ondulatorio al electrón (la onda de de Broglie). Schrödinger obtuvo una ecuación de onda (no relativista) que permitía obtener los niveles de energía para el átomo de hidrógeno tal como Bohr había hecho en la llamada vieja teoría atómica (old quantum theory), pero desde un enfoque más fundamental. Se puede obtener una solución analítica de la ecuación en el caso del átomo de hidrógeno tratado como un electrón en un potencial central tipo coulombiano. Aprovechando la simetría del problema y usando coordenadas esféricas la función de onda (solución de la ecuación) tiene la forma ψ= ψ θ ψ φ ψ r . La parte radial de la ecuación tiene la forma 0 r 1)n(n rK me2 K mE2 rr 2 r r 22 2 2 r 2 r 2 =ψ        − −++ ∂ ψ∂ + ∂ ψ∂ , donde n tiene que ser un entero. La imposición de que n sea un entero resulta de que la parte de la función de onda ψ φ cuando φ aumenta de un múltiplo 2π permanece igual. Esto implica que ψ φ es dado por 2π −1/2 e imφ , donde m es un entero positivo, negativo, o cero. Para que la ecuación de la función ψ θ tenga soluciones es necesario que n sea un entero positivo de modo que nm ≤ . Un aspecto importante de la ecuación radial es que tiene puntos singulares en r = 0 y r = ∞. Teniendo en cuenta estos dos puntos singulares y las condiciones de frontera 9 (boundary conditions) que imponen a la ecuación de onda, esta sólo tiene solución cuando tenemos: 1nn'l 2mEK me2 2 ++== − . De aquí sigue que la energía de un electrón ‘dentro’ del átomo de hidrogeno sólo puede tener valores discontinuos: 22 42 l h me2π =Ε λ , donde l es el llamado número cuántico principal . Esta solución corresponde al típico problema de valor propio en el cual tenemos una ecuación dependiente de un parámetro, en este caso la energía E, y las soluciones tienen que satisfacer condiciones de frontera particulares (en este caso la función de onda tiene que ser finita en r = 0 y aproximarse a cero para r → ∞). Encontramos un problema similar en el caso clásico de una cuerda en vibración. Las extremidades están fijas y esto implica que la cuerda sólo puede tener una determinada frecuencia y sus harmónicos. Schrödinger consideró que de este modo la ecuación tenía en sí misma implícitas las condiciones cuánticas, que no resultarían de ninguna discontinuidad cuántica en la física y sí de tratarse de un fenómeno ondulatorio. Sea como sea, en sus primeros artículos Schrödinger no desarrolló demasiado la interpretación de la función de onda; se limitó a unos pocos comentarios, asociando la función de onda a algún tipo de proceso ondulatorio en el interior del átomo. Una limitación importante con la que Schrödinger se debatió en esta fase del desarrollo de su mecánica ondulatoria fue la imposibilidad de describir la intensidad y polarización de la radiación emitida por un átomo. En particular Schrödinger sólo pudo obtener la expresión hν = E´ – E´´ (donde ν es la frecuencia de la radiación emitida cuando un electrón pasa de un estado estacionario con energía E´ a otro con energía E´´) del modelo atómico de Bohr de forma aproximada. En un artículo posterior Schrödinger relacionó la función de onda que llamó de campo mecánico escalar (mechanical field scalar) con una distribución de electricidad en el espacio, que se podría ver como el término de ‘fuente’ (source) en las ecuaciones de Maxwell-Lorentz. Con este fin Schrödinger consideró que la densidad de electricidad ρ el sería dada por la parte real de t ∂ ψ ∂ ψ , en que ψ es el conjugado complejo de la función de onda ψ. Usando esta hipótesis en el tratamiento del efecto Stark, Schrödinger pudo obtener la ‘regla de frecuencias’ de Bohr de forma exacta, además de poder calcular la intensidad y polarización de la radiación emitida. Desarrollando más esta interpretación electromagnética de la función de onda Schrödinger extendió su formalismo para el caso de sistemas con un estado variable en el tiempo (time-dependent systems). Esto llevó a Schrödinger a adoptar una función de onda compleja y por lo tanto a redefinir su anterior expresión para la densidad de carga ahora dada por ψ ψ e , donde e es la carga eléctrica del electrón. 10 Schrödinger pronto reconoció las limitaciones de esta interpretación de la función de onda (describiendo una distribución de electricidad en el espacio) ya que incluso en el caso más sencillo de un único electrón es también necesario tener en cuenta la evolución espacio-temporal de la función de onda. Por una parte, la interpretación electromagnética permitía calcular intensidades y polarizaciones, pero por otra parte la interpretación como una onda propagándose en el espacio era necesaria para determinar sucesivas densidades de carga (y por lo tanto mediciones de intensidades), o sea, para conectar sucesivas observaciones. Así, en realidad la interpretación de la función de onda por Schrödinger tenía un aspecto doble. Pronto las tentativas iniciales de Schrödinger dieron paso a la interpretación probabilística (statistical interpretation), pero esto no lo voy a considerar. Aquí lo que me interesa es la ambigüedad que encontramos en la interpretación de Schrödinger que por lo menos en parte usa su ecuación de onda como una ecuación clásica describiendo la propagación de un campo mecánico escalar en el espacio. Es un punto interesante ya que en el caso de la ecuación de Dirac se va encontrar también una ambigüedad en lo que respecta a la interpretación de la ecuación, siendo incluso posible ‘escoger’ (como ha hecho P. Jordan) ver la ecuación como una ecuación de un campo clásico posteriormente cuantizado. Podemos imaginar una situación hipotética en la que no teniendo aún el concepto de electrón como partícula, un experimento como el de C. J. Davisson y L. H. Germen nos lleva a aceptar la idea de que existen unas ondas de materia que resultan tener las características propuestas por de Broglie (esto es doblemente imaginario ya que de Broglie claramente manejaba expresiones que implicaban al mismo tiempo una característica ondulatoria y corpuscular). ¿Podemos ver la ecuación de Schrödinger como la ecuación de una onda de materia ‘clásica’? Y en caso afirmativo ¿cómo se obtienen los aspectos corpusculares? Mirando el caso más sencillo de un átomo de hidrógeno queda claro hasta qué punto se puede usar la ecuación de Schrödinger como una ecuación describiendo un campo material ‘clásico’. Volviendo al enfoque inicial de Schrödinger, considerando una onda de de Broglie en un potencial central tenemos 0 r 1 h e mπ8∆ 2 2 =        ++ φνφ . De aquí resulta, en un tratamiento totalmente clásico, que las frecuencias posibles de la onda de de Broglie son dadas por 2 r 3 42 )1ln( 1 h meπ2 ++ −=ν , donde n r , l = 0, 1, 2, …. Por analogía con el caso del cuanto de luz, consideramos que la energía de una onda de de Broglie (en el interior de átomo) con una frecuencia ν es dada por E = hν. De esta ‘regla cuántica’ resulta que los niveles de energía en el átomo son dados por 2 r 2 42 1ln 1 h meπ2 )++( −=Ε , 11 un resultado de acuerdo con la derivación de la teoría atómica de Bohr. La diferencia con la derivación original de Schrödinger es que en ésta Schrödinger usó desde el inicio la relación entre energía y frecuencia adoptada en este caso al final. Así Schrödinger presentó realmente una ecuación ‘cuántica’ que se asemeja (en el caso de sistemas con un solo electrón) a una ecuación de onda clásica. Aquí tenemos un resultado que vamos a volver a encontrar en el capitulo 3. Cuando adoptamos como punto de partida una ecuación que interpretamos como clásica necesitamos una regla de cuantización de las variables físicas de la onda de manera que se pueda también describir el aspecto corpuscular asociado a los electrones. El capítulo 3 La primera tentativa de Schrödinger de obtener una ecuación de onda fue una ecuación relativista y no su conocida ecuación no-relativista. No resultó. La ecuación obtenida dio una descripción equivocada de los niveles de energía. Eso llevó Schrödinger a optar por un proyecto menos ambicioso pero que resultó más exitoso. La derivación de una ecuación de onda relativista para el electrón siguió desafiando a los físicos. Después del descubrimiento de que se podía asociar un momento angular interno al electrón – al cual se asoció un número cuántico específico, el spin – W. Pauli intentó incorporar el spin a la mecánica ondulatoria. Para eso consideró una función de onda de Schrödinger dependiente también del spin que resulta tener sólo los valores ± ћ/2. Así Pauli definió una función de onda con dos componentes, una correspondiendo a un electrón con el spin hacia arriba (spin up) ψ(x, +1/2) y otra a un electrón con el spin hacia abajo (spin down) ψ( x, –1/2). Esta función de onda de dos componentes será solución de dos ecuaciones acopladas (coupled equations) con la forma ψψs, qi HΕ=         ∂ ∂h La cuestión es ahora cómo definir en la ecuación de onda los operadores de spin s. Pauli los definió como s x = 1/2 σ x , s y = 1/2 σ y , s z = 1/2 σ z , donde σ x , σ y , σ z son las llamadas matrices de Pauli:         =σ 01 10 x ,        − =σ 0i i0 y ,         − =σ 10 01 z . Pauli tuvo que incorporar elementos en su ecuación de forma ad hoc para obtener un acuerdo con los resultados disponibles, pero sólo ha podido hacer correcciones relativistas de primer orden, o sea, no pudo llegar a una ecuación realmente relativista. A finales de 1927 P. A. M. Dirac llegó a una ecuación de onda relativista del electrón. Su primera tentativa de formular una ecuación relativista fue lo que se conoce como la ecuación de Klein-Gordon (obtenida por distintos físicos y que coincide con la ecuación relativista que Schrödinger exploró). En términos de un hamiltoniano relativista podemos ver la ecuación como una sustitución directa del momento y energía por los operadores cuánticos correspondientes: 0}ψcmcpp{p 222222 zyx =+/Ε−++ 12 0ψ h cm tc 1 zyx 2 22 2 2 22 2 2 2 2 2 =      − ∂ ∂ − ∂ ∂ + ∂ ∂ + ∂ ∂ . P. Ehrenfest hizo a Dirac interrogarse por la opción adoptada para el hamiltoniano, preguntándole si había diferencia entre el que Dirac había escogido y el siguiente: Ec)/mpp(p1mc 222 3 2 2 2 1 2 =++− Dirac consideraba que ninguna de las dos opciones servía como base para el desarrollo de una ecuación relativista. Necesitaba una ecuación lineal en la derivada temporal y que, para estar de acuerdo con la relatividad, las derivadas respecto a las variables espaciales fueran también lineales. Parece que lo que permitió a Dirac salir del bloqueo provocado por la expresión matemática del hamiltoniano fue constatar la siguiente identidad: 332211 2 3 2 2 2 1 ppppppp σ+σ+σ=++= r , donde σ 1 , σ 2 , σ 3 son las matrices de Pauli. Eso llevó a Dirac a buscar una expresión relativista análoga en que apareciera el término correspondiente a la masa de electrón: mcpppcmpppp 4332211 222 3 2 2 2 1 α+α+α+α=+++= r . Dirac consideró que de la ecuación obtenida usando la expresión anterior {p 0 – (m 2 c 2 + p 12 + p 22 + p 32 ) ½ }ψ = 0 debería ser posible obtener la expresión relativista {p 02 – m 2 c 2 – p 12 – p 22 – p 32 }ψ = 0. Esto implica un conjunto de relaciones matemáticas entre los coeficientes aún por determinar: α µ α ν + α ν α µ = 0 (µ ≠ ν); µ, ν = 1, 2, 3, 4, α µ2 = 1. No hay ningún conjunto de matrices 2 ä 2 que satisfagan las condiciones anteriores. Esto llevó a Dirac a considerar la siguiente posibilidad más simple, la de matrices 4 ä 4. De este modo los coeficientes propuestos por Dirac son:               =σ 0100 1000 0001 0010 1               − − =σ 0i00 i000 000i 00i 0 2               =σ 1000 0100 0010 000 1 3 13               =ρ 0010 0001 1000 0100 1               − − =ρ 00i0 000i i000 0i00 2               − − =ρ 1000 0100 0010 000 1 3 , con α 1 = ρ 1 σ 1 , α 2 = ρ 1 σ 2 , α 3 = ρ 1 σ 3 , α 4 =σ 3 . Así tenemos de inmediato la ecuación de Dirac: 0mc]ψρ)p,σ(ρ[p 310 =−− , donde p 0 = iħ ∂/(c∂t) y p = (p 1 , p 2 , p 3 ), con p r = –iħ ∂/(c∂x r ) y r = 1, 2, 3; σ σσ σ = (σ 1 , σ 2 , σ 3 ) es un vector formado por las anteriormente indicadas matrices 4 ä 4. Dirac generalizó su resultado para el caso de un electrón en un campo externo. Siguiendo una prescripción de la electrodinámica clásica Dirac hizo la sustitución p 0 → p 0 +e/c.A 0 y p → p + e/c.A, donde A 0 y A son el potencial escalar y vector. Esto resulta en la ecuación 0mc]ψρ) c e p,(ρ c e [p 3100 =−Α+σ−Α+ . Dirac desarrolló su ecuación partiendo del hamiltoniano para una partícula puntual sin tener en cuenta el spin. Pero resulta que el spin del electrón aparecía naturalmente en la descripción hecha usando esta ecuación. Dirac trató de explorar la relación entre su ecuación y la ecuación de Klein-Gordon que es la ‘esperada’ de acuerdo con el hamiltoniano clásico relativista. Tomando el cuadrado de su ecuación Dirac obtuvo una que incluye los términos de la ecuación de Klein-Gordon y dos adicionales 0ψ c e i c e cm c e p c e p 1 22 22 00 =           )Ε,σ(ρ+)Η,σ(−−       Α+−       Α+ hh . Dirac concluyo que el electrón en un campo externo se comportaba como teniendo un momento magnético eh/4πmc · σ σσ σ que es lo esperado para un electrón con un momento angular interno (spin). Así su ecuación implicaba que el electrón tenía un momento angular interno dado por ћ/2 · σ σσ σ. Como Dirac usó matrices 4 ä 4 en su ecuación, la función de onda tenia 4 componentes. Inicialmente Dirac pensó que podía retener sólo dos de los componentes que corresponderían a un electrón con carga negativa, energía positiva y dos posibles estados para el spin, siendo posible descartar los componentes correspondientes a energías negativas. Pronto Dirac reconoció que en el caso de un campo electromagnético externo intenso no era posible hacer una separación clara de soluciones correspondientes a energías positivas y negativas. En particular, O. Klein demostró que para el caso sencillo de una onda con energía positiva incidente en una barrera de potencial, podía además de una onda reflectada (correspondiente a una energía positiva) haber también una onda transmitida a través de la barrera (con energía negativa). Este resultado conocido como la paradoja de Klein será importante como analogía en la comprensión de dificultades matemáticas con las que se enfrenta la electrodinámica cuántica. 20 El capítulo 5 Los elementos más básicos de la electrodinámica cuántica ya están presentes en el artículo de Dirac de 1927. En este trabajo el campo electromagnético y la materia son descritos por hamiltonianos clásicos. Un tercer término describe la interacción entre el campo y la materia: H = H mat + H elect + H int . A través del procedimiento de cuantización el hamiltoniano total se convierte en un operador. Pero es importante notar que la radiación y la materia son cuantizados por separado. Por razones (aparentemente) prácticas Dirac usa un método perturbativo para desarrollar las aplicaciones de la teoría. Como ya comenté el enfoque de Dirac fue desarrollado, en particular, por Jordan, Pauli, Heisenberg, y Fermi. Mirando ahora a la electrodinámica cuántica desde el método lagrangiano establecido, tenemos dos campos clásicos descritos, uno por las ecuaciones de Maxwell-Lorentz y otro por la ecuación de Dirac. Como ya mencioné la ecuación de Dirac se puede ver como una ecuación describiendo un campo clásico. Usando el esquema habitual de la expansión en serie de Fourier de la función de onda, el campo se puede ver como un conjunto infinito de modos propios en que después de la cuantización los coeficientes se convierten en operadores. En el caso del campo electromagnético se usa un procedimiento equivalente. Hasta este momento se está considerando dos campos cuantizados independientes. La electrodinámica cuántica es una teoría que nos describe la interacción entre la radiación y materia representados por estos campos cuánticos. El lagrangiano de la electrodinámica cuántica se puede definir partiendo de los lagrangianos para el campo de Dirac libre, el campo electromagnético libre, y un término describiendo la interacción entre ellos: µ µ ⋅γ++= ψAψeLLL emm . De partida la cuantización se hace para los campos libres, no para el caso (que resulta imposible de tratar) de un sistema cerrado de campos en interacción. Se tiene en cuenta el término de interacción µ µ ⋅γ ψAψe por un procedimiento perturbativo que parte de los campos libres para describir procesos físicos. En las aplicaciones de la teoría se considera (siempre) un estado inicial (en t = −∞) correspondiendo a campos libres y un estado final (en t = +∞) que también corresponde a campos libres. La amplitud de transición desde el estado inicial φ A al estado final φ B es dada por S AB = (φ *B Sφ A ), donde S es la llamada matriz-S. La matriz-S es dada por )}ψ(xAψ),ψ(xAψP{dxdx n! 1 c e S n1n1 0n n //       = ∫ ∫ ∑ ∞ ∞− ∞ ∞− ∞ = , KKK h. Vemos que la matriz-S que depende solamente del término de interacción se escribe como una serie de potencias de e, donde e es la carga eléctrica: S = 1 + eS (1) + e 2 S (2) + … . Un elemento esencial de este método es el llamado ‘switching on’ y ‘switching off’ (conexión y desconexión) adiabática de la interacción entre los campos para poder calcular integrales que van de −∞ a +∞. Consideremos por ejemplo la aplicación de la electrodinámica cuántica a la descripción de la aniquilación en dos fotones de un par electrón-positrón: e + + e – → 2γ. El estado inicial corresponde a un campo de Dirac con dos cuantos, uno correspondiente al electrón y otro al positrón (el campo electromagnético se supone en el estado 21 fundamental). Después de la aniquilación entre el electrón y el positrón, el campo de Dirac se encuentra en su estado fundamental (i.e., sin que haya ningún cuanto) y el campo electromagnético se queda ahora en un estado con dos fotones. En segundo orden la amplitud de transición es dada por γδναβµνµ )γ()γ()(Α)(Αγ)−(= ∫ ∫ 0xx2xdxdeS 212 4 1 42 fi [ )−(θ)(Ψ)(Ψ)(Ψ)(Ψ× +)+( δ )+( α )−( γ )+( β21 - 2121 tteexxxx0 ] )−(θ)(Ψ)(Ψ)(Ψ)(Ψ− +)+( δ )+( α )−( β )+( γ12 - 2112 tteexxxx0 . No entraré en los detalles de esta expresión, sólo comentaré algunos aspectos más relevantes. En está expresión encontramos términos que corresponden a la ‘propagación’ de un cuanto (virtual) entre los puntos del espacio-tiempo x 1 y x 2 , y componentes relacionadas con los estados libres de cada campo (estados de Fock): + ee - , γ2 , y los estados de vacío de cada campo 0 . Así, sólo entran en el cálculo de interacciones estados pertenecientes al espacio de Fock de cada campo. Parece entonces que se tiene un procedimiento sencillo, basado en la matriz-S, para describir cualquier proceso físico que corresponda a una interacción entre el campo de Dirac y el campo electromagnético. El problema es que la conexión adiabática de la interacción (adiabatic switching) se supone que permite pasar de la situación descrita por los hamiltonianos (o lagrangianos) de los campos libres H a la situación de interacción descrita por el hamiltoniano (o lagrangiano) de los campos en interacción H I . Ahora, de acuerdo con el llamado teorema de Haag, en términos informales, los vectores de estado para los campos libres y los vectores de estado de los campos en interacción están en espacios distintos y no se pueden ‘conectar’. Por el teorema de Haag se puede concluir que si partimos de un campo libre en t = –∞ tendremos siempre un campo libre para todo t. Así resulta necesario desde el inicio considerar vectores de estado de campos en interacción. Ahora bien, como hemos visto, esto no es lo que se hace en la electrodinámica cuántica. Dicho en pocas palabras, el teorema de Haag implica que el método perturbativo usado en la teoría para describir la interacción entre la radiación y la materia es desde un punto de vista matemático inconsistente. Así tenemos el problema de comprender cómo es posible que un método matemáticamente inconsistente permita la obtención de resultados excelentes cuando los comparamos con datos experimentales. Trataré de mostrar que los buenos resultados de la teoría pese al teorema de Haag se pueden explicar si se tiene en cuenta la base física de la teoría y no solamente su estructura matemática. Como ya referí antes, la teoría se desarrolla partiendo de la cuantización independiente de dos campos clásicos. Describimos siempre su interacción como una pequeña perturbación a sus estados libres. En realidad no tratamos los campos como un sistema único cerrado. Pero desde un punto de vista matemático si consideramos todos los términos de la serie de expansión de la matriz-S, eso corresponde a describir un sistema cerrado. La cuestión es que la serie de expansión de la matriz-S es divergente. ¿Qué implicaciones tiene esto? Parece entonces que la situación es aún peor que la amenaza de inconsistencia matemática debido al teorema de Haag. Si la serie es divergente eso significa que si no calculamos sólo unos cuantos términos y tratamos de hacer un cálculo lo más exacto posible, a partir de un determinado orden, en vez de tenerse resultados cada vez más ajustados a los datos experimentales tendremos resultados cada vez peores. Así parece que la teoría sólo ‘funciona bien’ porque 22 hacemos unos cálculos limitados. Tal no es el caso. Si es cierto que desde un punto de vista matemático el que se tenga en cuenta sólo unos cuantos términos de la serie no tiene aparente justificación, cuando tenemos en cuenta la base física de la teoría sí se puede justificar el no considerar los términos de orden elevado que ‘estropean’ los cálculos. Como referí la teoría ha sido estructurada alrededor de la idea de que la interacción entre radiación y materia es débil (weak). Esto está relacionado con lo siguiente: (1) el término de interacción en el Lagrangiano es pequeño; (2) debido a esto cuantizamos los campos en separado como campos libres; (3) tomamos el término de interacción como un término perturbativo. Ahora, como ya mencioné, si intentamos calcular toda la matriz-S, correspondiendo a tratar la radiación y materia como un sistema cerrado, se obtiene un resultado divergente. Se puede concluir así que, en la práctica, la noción de que la interacción es débil implica que es imposible pasar de la descripción de campos libres (y su interacción) a la situación de campos (totalmente) en interacción. Desde una perspectiva formal se puede mostrar que un posible sistema de campos en interacción no se puede describir usando la representación de Fock para campos libres. Esto significa que desde un punto de vista formal para un sistema cerrado de campos en interacción no se puede usar la base física constitutiva de la electrodinámica cuántica que es la idea de interacción débil y su implementación en la teoría. Así, podemos ver que no se consideran los términos de orden elevado de la serie de expansión de la matriz-S debido a que su inclusión correspondería a un uso impropio de la estructura matemática más allá de la base física que se supone que ésta representa. Por tanto en la electrodinámica cuántica tenemos los conceptos de radiación y materia, y de una interacción débil entre ellos, pero no de campos en interacción (que corresponde a un sistema cerrado de radiación y materia). Así, hago una correspondencia entre la obtención de resultados matemáticos con significado (no divergentes) y la manera como se implementa la base física de la teoría. En este caso tenemos cálculos intrínsecamente aproximados de la interacción débil entre dos sistemas físicos distintos (radiación y materia). Creo que esta situación tiene semejanzas con las ideas de Bohr respecto a la paradoja de Klein. Como ya referí antes, para Bohr la paradoja de Klein resultaba de considerarse potenciales ‘matemáticos’ sin tenerse en cuenta la estructura atómica de la materia (parte constituyente de la base física de la teoría). La paradoja resultaba de usarse la matemática más allá de lo que los conceptos físicos de la teoría realmente permitían. Como en el caso de la paradoja de Klein la divergencia de la serie de expansión de la matriz-S resulta de aplicarse la estructura matemática de la teoría más allá de su contenido físico provocando un colapso de los cálculos Volviendo a la cuestión del teorema de Haag, resulta que en la electrodinámica cuántica evitamos las consecuencias del teorema porque ni siquiera intentamos describir un sistema cerrado de campos en interacción, solamente la interacción entre radiación y materia como sistemas distintos. Así no hay ningún conflicto de la teoría con el teorema de Haag. El capítulo 6 Cuando en 1929-30, Heisenberg y Pauli presentaron en dos artículos una teoría de campos relativista de la interacción entre un campo electromagnético cuantizado y un campo de Dirac cuantizado, cambiaron de una primera opinión en que la energía propia infinita del electrón no constituía un problema mayor a una postura más circunspecta en 23 que reconocían que este problema podía incluso hacer inaplicable la teoría. La situación de la electrodinámica cuántica en los años treinta no mejoró. Como hemos visto, para tratar de resolver inconsistencias en su teoría del electrón relacionadas con la existencia de soluciones con energía negativa, Dirac propuso su teoría de los agujeros (hole theory). De aquí surgió un nuevo problema con infinitos. En esta teoría/interpretación de la ecuación de Dirac se considera que todos los estados correspondientes a energías negativas están en su mayoría ocupados por electrones. Esto llevó a Dirac a considerar que el campo electromagnético es generado por la diferencia entre la densidad eléctrica actual y la que corresponde a tener todos los estados de energía negativa ocupados (el estado normal). Así la ecuación para el campo eléctrico es dada por Div E = – 4π (ρ – <ρ> vacío ). Explorando las consecuencias de esta definición Dirac estudió cómo un campo externo podría influir en la definición del estado normal, esto es, trató de calcular cúal era la distribución de electrones con energías negativas que se puede considerar como la normal en el caso en que estén sujetos a un campo externo. Considerando el caso sencillo de un campo eléctrico externo estático, Dirac concluyó que este cambiaba la distribución de electrones con energía negativa de una manera que correspondía a una neutralización parcial de la carga que producía el campo estático. Si se asumía que su ecuación era exacta eso llevaba a una cancelación total de la carga externa – un resultado sin sentido. Eso llevó a Dirac a suponer que la ecuación no se puede aplicar para energías ultra-relativistas del orden de 137mc 2 . Así no teniendo en cuenta el efecto del campo en electrones con energía más negativa que –137mc 2 , tenemos que la neutralización de la carga (que origina el campo externo) debido a los restantes electrones es pequeña, del orden de 1/137. Así Dirac concluye que todas las partículas cargadas tienen una carga efectiva menor que su carga real (del orden 136/137). Esta definición de un valor máximo de energía resulta fundamental para obtener un resultado finito. Así en un calculo posterior usando la llamada aproximación de Hartree- Fock Dirac obtiene un resultado infinito que sólo puede reducir a uno finito imponiendo una energía de corte (cut-off energy). Esto no es simplemente un truco matemático. Dirac presenta una justificación física para usarse este procedimiento. De acuerdo con Dirac (siguiendo ideas de Bohr) la teoría cuántica sólo se aplica a fenómenos en que las distancias son mayores que el diámetro clásico del electrón, esto es del orden de e 2 /mc 2 . Esto corresponde a energías del orden (ћc/e 2 )(mc 2 ) @ 137mc 2 , que es el valor adoptado por Dirac como límite para la energía. Esta aplicación está de acuerdo con la idea de Bohr de que no se pueden hacer integraciones tomando la energía con cualquier valor debido a que la teoría trata el electrón como una partícula puntual (point-charge). Así sería inconsistente en las aplicaciones de la teoría considerar energías que corresponderían a distancias menores que el ‘diámetro’ del electrón. Esta importante idea se va a quedar soterrada en el posterior desarrollo de técnicas para enfrentar los problemas de infinitos en la teoría. No voy a entrar aquí en los detalles de esta técnica provisoria aplicada por Dirac a la que se le dio el nombre un poco despectivo de física de substracción (subtraction physics); sólo refiero que durante los años treinta no ha sido posible resolver los distintos problemas de energía-propia que tenía la teoría. El único ‘desarrollo’ importante ha sido la comprobación por V. F. Weisskopf de que la energía-propia del electrón divergía ‘sólo’ de forma logarítmica. En la práctica sólo se podían hacer unas cuantas aplicaciones de la teoría en el orden más bajo de perturbaciones y nada más. En 1947 los resultados experimentales de W. Lamb respecto a un desvío del nivel de energía del estado 2 2 S 1/2 respecto al 2 2 P 1/2 en el átomo de hidrogeno llevó a una serie de físicos (incluyendo los más brillantes de una nueva generación) a enfrentar de nuevo la 24 cuestión de los infinitos en la teoría (la guerra había dejado estas cuestiones teóricas en segundo plano). El primer momento de esta nueva fase en la historia de la electrodinámica cuántica lo encontramos en un cálculo no-relativista realizado por H. A. Bethe que le permitió obtener de forma aproximada el desvío observado experimentalmente. Para eso Bethe hizo uso del método de ‘substracción’ de Dirac en el sentido en que impuso un limite de energía en las integrales (un ‘cut-off’ de energía). El otro aspecto fue tomar la parte que sería infinita como una corrección de la masa del electrón y considerar que la masa observable está constituida por una masa ‘mecánica’ y el efecto de la interacción del electrón con su propio campo (la parte que resulta infinita). Adoptó básicamente un procedimiento ya necesario en la electrodinámica clásica debido al mismo tipo de problema (una energía-propia infinita) que se denomina renormalización. Es importante notar que, al contrario de Dirac, Bethe no dio ninguna motivación física para el ‘cut-off’. Unos pocos meses después del cálculo de Bethe, un físico de la nueva generación J. Schwinger desarrolló un método relativista pero no-covariante para calcular el desvío de Lamb (Lamb shift), usando para ello la renormalización de la masa y de la carga. La visión de Schwinger respecto a la renormalización es que ésta indicaba que la teoría necesitaba cambios para energías ultra-relativistas. Schwinger esperaba que una futura teoría mejorada debería imponer límites a la energía que permitiera la existencia de un ‘cut-off’ natural. Pero, al contrario de Dirac, Schwinger pensaba en términos de una mejora más bien matemática y no vio el problema como una incompatibilidad entre los conceptos físicos de la teoría (e.g. el concepto de electrón puntual) y la inexistencia de un límite natural asociado en las aplicaciones matemáticas de la teoría. Otro aspecto del trabajo de Schwinger es que, como los cálculos fueron realizados de forma nocovariante, había muchas ambigüedades en el procedimiento adoptado para identificar los términos divergentes. Otro físico de la nueva generación que dio una contribución fundamental a la teoría fue R. P. Feynman. Partiendo de la propuesta de Bethe de buscar una modificación para frecuencias elevadas de forma claramente relativista (que permitiera evitar los infinitos por un procedimiento sin ambigüedades), Feynman buscó una modificación consistente de la electrodinámica. Feynman empezó con el caso de la electrodinámica clásica desarrollando un método llamado regularización, en que el uso de un término extra en las expresiones matemáticas permitía que todos los resultados fueran finitos. Feynman aplicó después este procedimiento para el caso en que los electrones (y positrones) se describen con la ecuación de Dirac. El método de regularización de Feynman resulta ser inconsistente desde el punto de vista de la interpretación física de la teoría. Eso significaba que en la práctica después de hacer la regularización de los términos que en otro caso serían infinitos, estos se tenían que ‘absorber’ en la masa y carga del electrón correspondiendo a hacerse una renormalización. Para que los resultados no dependieran del término de regularización, se tenía que considerar el límite en que este dejaba de aparecer en las expresiones. Como ya se había hecho la renormalización de los términos infinitos esto ya no causaba problemas. El método de Feynman era entonces un procedimiento equivalente al de Schwinger, su ventaja era ser un método covariante y por lo tanto sin ambigüedades en la manera como se manejaban los términos infinitos. Así el procedimiento de Feynman no fue una nueva teoría mejorada y fue solamente un procedimiento para manejar de forma clara integrales que resultan ser divergentes. Resulta claro que la nueva generación de físicos tenía una actitud mucho más pragmática que la anterior, tratando de poner la teoría en funcionamiento sin detenerse demasiado en los posibles problemas conceptuales asociados a la regularización y 25 renormalización. Una excepción es la visón de F. Dyson respecto a este tema. La concepción de Dyson resulta tener alguna semejanza con la de Dirac y Bohr, entonces ya olvidada. Dyson imagina dos tipos de observadores, el ‘ideal’ y el ‘real’. En el caso del observador ‘ideal’ este tendría instrumentos de medición sin estructura atómica pudiendo ‘medir’ la interacción puntual de los campos cuánticos (y así obtendría resultados infinitos); el observador ‘real’ tiene instrumentos de medida constituidos por átomos y por lo tanto se tiene que tener en cuenta la estructura atómica de la materia. Así las mediciones del observador ‘real’ están limitadas y eso le impide hacer observaciones submicroscopicas. De este modo solo obtiene resultados finitos. El capítulo 7 La descripción en orden más bajo de teoría de perturbaciones de los procesos físicos en la electrodinámica cuántica está libre del problema de infinitos. Esto puede permitir un análisis más claro de la descripción espacio-temporal de los procesos físicos que se tiene en la teoría. Me concentraré en la descripción de la dispersión entre electrones (electron-electron scattering). En 1948 Dyson demostró la equivalencia entre los métodos de Schwinger (y Tomonaga) y Feynman. Lo hizo usando la llamada matriz-S. Como ya mencioné la matriz-S se aplica por un procedimiento perturbativo y es divergente. Los términos de orden superior de la matriz-S no cambian la descripción espacio-temporal (cualquiera que sea) que se puede tener con este método, por eso consideraré sólo la expansión de la matriz-S hasta segundo orden. En este caso la interacción entre los electrones resulta de un intercambio (exchange) de fotones virtuales (ver figura 1). En la descripción espacio-temporal global (overall space-time approach) 7 de Feynman consideramos la propagación de un fotón virtual entre todos los puntos del espacio-tiempo de Minkowski. El propagador de Feynman es dado por { } )x'(xcDi0(x`)(x)AAT0 µν F νµ −= h Esta expresión significa que estamos considerando un fotón ‘creado’ en un punto del espacio-tiempo y ‘aniquilado’ en otro punto. El uso de T{ } significa que en esta expresión covariante incluimos, dependiendo del orden temporal, la propagación de un electrón a otro y viceversa, pues T{A µ (x)A ν (x’)}= A µ (x)A ν (x’) si t > t’, y T{A µ (x)A ν (x’)}= A ν (x’)A µ (x) si t’ > t. La amplitud de transición en segundo orden resulta de la contribución de todas las posibles interacciones entre los campos de Dirac y Maxwell ‘conectados’ por un propagador de fotónes: [ ] )x(xiD)ψγψ()ψγψ(Nxdxd 2! e )2e(2eS 21Fαx β x α 2 4 1 4 2 (2) 21 − − =→ ∫ +−+−−− β . Los fotones se denomina virtuales por dos motivos. Uno es que la relación de energía y momento (energy-momentum relation) para el fotón virtual no es k 2 = (k 0 ) 2 – 7 En este trabajo se va usar esta terminología pese a que teniendo en cuenta sus implicaciones respecto a la descripción temporal de los fenómenos físicos no parece la más adecuada. 26 k 2 = 0. Esto se debe a que en la expresión del propagador k y k 0 son independientes el uno del otro; también resulta que el propagador no es nulo para separaciones del tipo espacial (space-like separations) lo que corresponde a una ‘propagación’ con una velocidad superior a la de la luz. El segundo punto es que este fotón virtual no se puede en principio observar. Figura 1: La interacción entre electrones en segundo orden, que resulta del intercambio de un fotón virtual (Este es un ejemplo de un diagrama de Feynman). Una cuestión bastante debatida en la filosofía de la física es el estatus de las partículas virtuales. Por lo general la idea es que las partículas virtuales se tienen que ver como un artefacto de la expansión perturbativa sin aportar ninguna ‘imagen’ física de las interacciones. El argumento principal usado en la defensa de esta visión es el llamado argumento de la superposición. De acuerdo con este argumento la descripción por la matriz-S de las interacciones resulta en una serie infinita de términos de orden cada vez mayor, lo que corresponde a tener un número infinito de combinaciones de cuantos virtuales. Así los cuantos virtuales se tendrán que ver como descripciones pictóricas de términos matemáticos (pues se pueden asociar a distintos diagramas de Feynman). El problema es que este argumento no es válido en la electrodinámica cuántica. Como ya mencioné la serie de expansión de la matriz-S es divergente y tenemos argumentos físicos para sólo considerar los primeros términos. Así no tenemos un número infinito de cuantos virtuales en la descripción de las interacciones. En este caso el intercambio de fotónes virtuales es más que algo meramente matemático, es la manera en que la teoría nos describe (a nivel físico) las interacciones. Pero hay que tratar de comprender lo que es realmente la descripción global de las interacciones. El aspecto crucial es que la teoría no nos da una descripción en el tiempo de las interacciones. En el método de la matriz-S consideramos siempre límites de integración entre t = –∞ y t = +∞; no hay manera de evitarlo. Esto es consistente con las posibilidades experimentales. El caso es que no hay acceso experimental al ‘mecanismo’ de la interacción. Lo que sí se puede hacer es determinar (en el ámbito teórico y experimental) la llamada sección transversal (cross-section) que permite calcular/verificar la probabilidad de observar un determinado resultado final de la interacción. Se presenta la cuestión inmediata de saber si esta es una limitación de un método en particular o una característica de la teoría. En cuanto al método, la matriz-S es bastante general y otros métodos también perturbativos tienen el mismo tipo de problema. Lo que tal vez fuera posible es considerar otro tipo de situaciones que aún se puedan ver 27 como descritas por la teoría en las cuales se tenga una descripción temporal más en acuerdo con lo que se espera desde la física clásica: una interacción retardada debido a la velocidad finita de la luz. Dejaré esta cuestión para el siguiente capítulo. El capítulo 8 Aquí trataré la cuestión de la relación entre la electrodinámica clásica y la cuántica. Del capítulo anterior quedó por analizar una cuestión que trae implícita la que acabo de mencionar. En la electrodinámica clásica tenemos la posibilidad de considerar que la interacción electromagnética (dependiente del tiempo) resulta de la emisión y absorción de ondas electromagnéticas que se propagan a la velocidad de la luz. En el caso de la electrodinámica cuántica hasta el momento sólo se ha podido describir las interacciones a nivel temporal de forma global (entre t = –∞ y t = +∞). La cuestión está en saber si hay un término medio entre estos dos extremos y qué implicaciones tiene para nuestra visión de la relación entre las dos teorías. En un primer momento podría parecer que un modelo sencillo de la interacción entre dos electrones atómicos podría darnos este término medio que se busca. Aparentemente este modelo se desarrolla dentro de la electrodinámica cuántica. Así cuando un átomo inicialmente en un estado excitado emite un fotón, éste solo será absorbido por otro átomo después aproximadamente de un intervalo de tiempo r/c, donde r es la distancia entre los átomos y c la velocidad de la luz. Un aspecto fundamental de este modelo es la forma específica dada a la densidad bilineal (bilinear density) ψγψ µ que aparece en el término de segundo orden de la matriz-S usado en este modelo: (y))(y)y)((x(x))D(x)(ydxdconst c V 4 V 4 yx ψγψ−ψγψ µµ ∫∫ , con ψγψ µ ~ a 1 a 2* ρ µ (x) exp [iω 0 t – t 2 /T 2 ] , donde ω 0 > 0. Sin entrar en detalles se puede verificar que la expresión adoptada es la misma que se obtiene por consideraciones clásicas. Esto es importante pues significa que el modelo se está aplicando no en un ‘régimen cuántico’ sino en el límite de correspondencia entre las dos teorías donde por construcción la teoría cuántica tiene que dar resultados equivalentes a la teoría clásica. Así se trata más bien de un modelo semi-clásico. Vemos que no se consigue dentro de la electrodinámica cuántica tener una descripción en el tiempo de las interacciones. Si suponemos la electrodinámica cuántica como una teoría más fundamental a la cual se reduce la teoría clásica tenemos un problema. La teoría clásica nos permite asociar intervalos de tiempo (duraciones) a los procesos físicos. Se supone que ‘por debajo’ existe un mundo cuántico descrito por una teoría cuántica. Estos procesos cuánticos deberían estar en el ‘interior’ de los procesos macroscópicos a los cuales se asocian duraciones claras. El problema es que según la electrodinámica cuántica no podemos asociar ningún intervalo de tiempo, por más cualitativo que sea, a los procesos cuánticos. Tenemos así el problema de justificar la emergencia de los intervalos de tiempo asociados a procesos descritos por la física clásica desde una descripción cuántica. Así un elemento clave de la emergencia de lo clásico (classicality) se nos escapa. Por este motivo me resulta difícil aceptar la visión tradicional de la teoría cuántica como más fundamental y a la cual se reduce la teoría clásica. Teniendo en cuenta que la 28 electrodinámica cuántica se construye, en parte, por un proceso de cuantización de la electrodinámica clásica, resulta más aceptable la idea de que la electrodinámica cuántica depende de la clásica y no se puede considerar por separado de la versión clásica. Son las dos en conjunto lo que nos permite describir los fenómenos que ‘catalogamos’ como de interacción electromagnética entre radiación y materia. Esta visión está de acuerdo con la idea de O. Darrigol de una estructura modular de las teorías físicas. De acuerdo con Darrigol la práctica y la historia de la física revelan que las teorías no son un todo homogéneo y separado; por el contrario las teorías se aplican en conjunción unas con otras. Así una teoría se debe ver más bien como constituida por componentes o módulos que pueden ser ellos mismos teorías con distintos dominios de aplicación. 29 CHAPTER 2 THE SCHRÖDINGER EQUATION AND ITS INTERPRETATION 1 Introduction To set the ‘stage’ for the study of quantum electrodynamics, in particular of Dirac’s relativistic quantum-mechanical equation and its different interpretations, I will consider in this chapter the non-relativistic and quantum-mechanical Schrödinger equation. In section 2 I will consider some of the historical developments of the quantum theory previous to Schrödinger’s contribution. I will focus in particular on Albert Einstein’s work on light quanta (which as we will see in chapter 4 is a fundamental aspect of the quantized electromagnetic field) and Louis de Broglie’s work on wave aspects of electrons (which is an immediate precursor of Schrödinger’s approach). In section 3 I consider the early ‘stages’ of the development of the Schrödinger equation not addressing the interpretation problem, i.e. the physical meaning of the equation. Section 4 will be dedicated to this problem. I will focus on Schrödinger’s ‘electrodynamic’ interpretation. We will see that this interpretation has a two-fold aspect. On one side Schrödinger uses his electrodynamic interpretation of the wave function as giving rise to a charge density distribution in space. But on the other side Schrödinger also takes the wave function as a ‘real’ wave propagating in space. In fact, that there can be an ambiguity in the interpretation of the Schrödinger equation can be made clear by considering the simple case of the hydrogen atom. We will see that it is possible to consider the Schrödinger equation as the equation of a ‘classical’ de Broglie wave that is afterwards quantized. We will encounter similar ambiguities in the next chapter when looking into the details of the Dirac equation. 2 The ‘pre-history’ of the Schrödinger equation In 1900 Max Planck presented a derivation of the energy distribution law of the normal spectrum, which he had previously proposed with no rigorous theoretical justification but with a good agreement with observational data. In his derivation Planck considered a system of resonators (Hertzian oscillating dipoles) in thermodynamic equilibrium with electromagnetic radiation at a constant temperature (Klein, 1975, pp. 461-462). There are two main separate parts in Planck’s derivation. One consists in deriving the relation between the average energy of a resonator (having frequency ν) and the density of the (blackbody) electromagnetic energy at the same frequency. This Planck achieved by thermodynamics and classical electrodynamics. The other part of Planck’s derivation 36 expression with two terms. According to Einstein “only the first of these would be present if the molecules were independent of each other” (quoted Mehra & Rechenberg, 1982, p. 618). Regarding the second term Einstein wrote that “one can interpret it, also in the case of gases, by associating with the gas a radiation phenomenon in a suitable manner and computing its interference fluctuations” (quoted in Mehra & Rechenberg, 1982, p. 618). That is, like in the case of blackbody radiation Einstein interpreted the second term as resulting from an undulatory behavior, now associated with the monoatomic gas molecules. Following Louis de Broglie’s ideas (as presented in his 1924 thesis) Einstein associated a scalar wave field with the monoatomic gas and took the second term of the fluctuation formula as resulting from the “square fluctuation of this wave field” (quoted in Mehra & Rechenberg, 1987, p. 413). From his Brother, the experimental physicist Maurice de Broglie, it was clear to de Broglie, around 1920, that there is a need “of always connecting together the points of view of waves and corpuscles” (quoted in Mehra & Rechenberg, 1982, p. 584) in the interpretation of experimental results regarding different forms of radiation. 8 In a series of notes published between 1922 and 1923, de Broglie developed his undulatory mechanics, where he tried to explain the dual character of radiation, extending also this duality to the description of matter. It seems that the first step in de Broglie’s theoretical work came when he took the mass of the light-quanta to be very small but different from zero. This move made it possible to relate Einstein’s formulas E = m 0 c 2 and E = hν, where m 0 is the light quanta rest mass. The expression m 0 c 2 = hν 0 , suggested to de Broglie the existence of an internal periodic phenomenon with a frequency ν 0 associated with any massive particle not only the light quanta. An apparent paradox occurred when considering a particle moving with a uniform velocity v with regard to a stationary observer. The observer would take the energy of the moving particle to be E = m 0 c 2 (1– β) –1/2 , and the internal frequency to be ν 1 = ν 0 (1– β) 1/2 . In this way for a moving particle the internal frequency would not be related with the frequency given by E = hν. This ‘second’ wave appeared to be unrelated to the particle because it had a wave velocity superior to c. However, de Broglie noticed that this fictitious wave would remain in phase with the internal periodic phenomenon and took this ‘phase wave’ as guiding the motion of the particles. He considered that “any moving body may be accompanied by a wave and that it is impossible to disjoin motion of body and propagation of wave” (quoted in Jammer, 1966, p. 244). In the case of ‘matter waves’ (the phase waves associated with matter) de Broglie found an important result in the case of orbiting electrons. de Broglie considered that “the rays of the phase waves coincide with the trajectories [of the particles]” (quoted in Mehra & Rechenberg, 1982, p. 593). In simple terms we can say that a particle and its associated wave follow the same trajectory. In the case of an electron in a close circular trajectory, by taking the electron and the (fastest) wave to start motion at a particular point of the trajectory at an initial time t = 0, they will meet on the orbit again at a later time τ = Tβ 2 / (1– β 2 ), where T is the period of revolution of the electron. At this moment the internal phase of the electron is 2πν 1 τ = m 0 c 2 Tβ 2 / h(1– β 2 ) ½ . By requiring the phases of the phase wave and the internal periodic motion to be equal at this point, de Broglie obtained the expression m 0 ν 2 (1– β) –1/2 T = nh, where n is an integer. This expression corresponds to the Bohr- Sommerfeld quantum condition. In this way the quantization could be seen as a resonance condition on the phase wave associated with the electron when considering closed trajectories. 8 See Wheaton (1991) for an account of the importance of experimental work in conceptual developments in the quantum theory and in particular on de Broglie’s ideas. 37 3 The coming to be of the Schrödinger equation After reading Einstein’s remarks on de Broglie’s ideas, Erwin Schrödinger got hold of a copy of de Broglie’s thesis (where his ideas were developed in detail) and set to work on what would become known as wave mechanics. In a first moment Schrödinger tried to “visualize the phase wave of an electron on Kepler orbits. The ‘rays’ certainly correspond to neighbouring Kepler ellipses having the same energy. This, however, leads to horrible ‘caustics’ or the like, for the wave front” (quoted in Mehra & Rechenberg, 1987, p. 416). Having difficulties in exploring a ‘direct’ geometrical picture of the phase wave, Schrödinger turned to finding a wave equation that described the phase wave associated with the electron. The first attempt was the derivation of a relativistic wave equation that later would become known as the Klein-Gordon equation. Schrödinger used the equation to determine the eigenvibrations of the hydrogen atom, but got the wrong results (when compared with Arnold Sommerfeld’s 1924 results on the hydrogen atom obtained within the so-called old quantum theory). Only in the non-relativistic limit was Bohr’s original result recovered. This put in danger the whole program of a wave-theoretical approach since de Broglie’s phase wave was developed taking into account relativistic considerations. Schrödinger finally settled for the development of a non-relativistic wave equation as we can see from a letter to Wilhelm Wien in late December 1925: At the moment I am plagued by a new atomic theory … I believe that I can write down a vibrating system – constructed in a comparatively natural manner and not by ad hoc assumptions – which has as its eigenfrequencies the term frequencies of the hydrogen atom … at the moment I still have to learn the mathematics to handle the vibration problem fully – a linear differential equation, similar to Bessel’s, however less known and exhibiting strange boundary conditions; these are connected with it and not imposed from outside. (Quoted in Mehra & Rechenberg, 1987, 460 & 461) Within a few weeks the results of his ongoing research started to appear in Annalen der Physik. In his notebooks, besides the derivation of his relativistic equation, Schrödinger made a derivation of a non-relativistic wave equation by considering de Broglie’s results. Schrödinger considered a generalized de Broglie’s phase wave frequency relation, for an electron with mass m, charge e, and velocity υ, in a Coulomb potential (i.e. the electron of the hydrogen atom): r e 2 m mch 22 2 − υ +=ν . Using this result, Schrödinger took the non-relativistic momentum of the electron (mυ 2 ) into de Broglie’s relation for the phase velocity (in the non-relativistic limit) υ ν = m h u . Inserting the non-relativistic result for u into the relativistic equation for the phase wave he had previously put forward, Schrödinger obtained a non-relativistic wave equation for the hydrogen atom: 38 0 r e mchm2 h 4 2 2 2 2 =ψ        +−ν π +ψ∆ , or 0 r e K m2 2 2 =ψ        +Ε+ψ∆ , with 2 2 2 4 h π =Κ and 2 mch −ν=Ε , as it appears in the first Annalen paper, where Schrödinger presented the wave equation as resulting from a variational approach without giving much insight into the physical considerations that led him to the wave equation (Schrödinger, 1926a). Schrödinger presented some considerations regarding the adoption of a second-order differential equation as the basis for the development of his wave mechanics in his second paper on the subject (Schrödinger, 1926b). This type of equation had traditionally been used to describe wave phenomena. By ‘striving for simplicity’ Schrödinger settled for a secondorder differential equation: 0 tu 1 zyx 2 2 22 2 2 2 2 2 =ψ ∂ ∂ −ψ         ∂ ∂ + ∂ ∂ + ∂ ∂ . Then by assuming the wave to have a simple time dependency (i.e. a periodic behavior in time with frequency ν): ψ ~ exp (2π i ντ ), Schrödinger obtained a time-independent relativistic wave equation: 2 22 2 2 2 2 2 2 u 4 zyx νπ −=ψ         ∂ ∂ + ∂ ∂ + ∂ ∂ =ψ∆ . This was the equation that by taking into account de Broglie’s results led him to his wave equation for the hydrogen atom, as we have seen above. In the ‘derivation’ of the wave equation, Schrödinger considered, as we have just seen, the time-independent case corresponding to a situation where the parameter E (which in the case of a bound electron is the energy of the hydrogen atom) has a timeindependent value, i.e. E does not change in time. In the application of this wave equation to the hydrogen atom, an analytic solution can be found, by taking into account the spherical symmetry of the Coulomb potential, and making a separation of the wave equation in spherical coordinates: ψ= ψ θ ψ φ ψ r. It is in the radial equation that interesting things happen: 0 r 1)n(n rK me2 K mE2 rr 2 r r 22 2 2 r 2 r 2 =ψ        − −++ ∂ ψ∂ + ∂ ψ∂ , where n takes up only integer values. The limitation of n to integers comes about because in the solution of the angular equations the part of the wave function ψ φ dependent on the azimuthal angle φ must be single-valued (i.e. when φ is increased by 39 2 π or a multiple of 2 π , ψ φ must be unchanged). This implies that ψ φ is given by 2 π −1/2eimφ , where m is a positive or negative integer or zero. Also to have finite solutions in the equation for ψ θ it is necessary that n be a positive integer or zero so that nm ≤. An important aspect of the radial equation is that it has singular points at r = 0 and r = ∞. We will consider here the case of bound states for which E takes negative values. Taking into account that the Coulomb potential goes to zero more strongly than 1/r as r → ∞ the asymptotic solution of the radial equation can be obtained by disregarding the Coulomb potential in the radial equation, and by taking into account that the wave function must approach zero as r → ∞. This boundary condition implies that, in the asymptotic limit, ψ r ~ e–r. The other singular point r = 0 is taken care of by considering ψ r ~ Ure–r, where Ur can be expressed as a power series. By considering the two boundary conditions it turns out that the power series must terminate, and, being n’ the highest power of Ur in the finite order power series, there is a solution to the radial equation only when we have: 1nn'l 2mEK me2 2 ++== −. From this it follows that the energy for the bound states of the hydrogen atom can take only discrete values: 22 42 l h me2π =Ε λ , where l is the so-called principal quantum number. This type of solution to a wave equation can be seen as a proper value problem, i.e. there is a differential equation with a numerical parameter (in this case the energy E), and the solutions must satisfy particular boundary conditions (in this case the wave function has to be finite at r = 0 and approach zero as r → ∞). The values of the parameter (E) for which there are non-trivial solutions are called proper values (eigenvalues), and the corresponding solutions of the equation are called proper functions (eigenfunctions). In the case of Schrödinger’s wave equation further ‘physical’ conditions are imposed on the wave function to obtain the non-trivial or physical solutions (one example of such a condition is the assumption of the singlevaluedness of the wave function). We have a similar situation in the classical case of a vibrating string. The end points of the string are held fixed, and this entails that the string can only take a discrete set of wavelengths (Mandl, 1957, pp. 5-8). In a work previous to the publications on wave mechanics, Schrödinger addressed Einstein’s work on a quantum gas theory by applying an idea inspired on, but different from, de Broglie’s phase wave. Contrary to de Broglie who considered phase waves propagating in the field-free space, Schrödinger considered standing waves enclosed in a cavity (which provides natural boundary conditions exactly like in the classical case of a vibrating spring). In his first paper on wave mechanics Schrödinger called attention to this difference: I was led to these deliberations in the first place by the suggestive papers of M. Louis de Broglie, and by reflecting over the space distribution of those ‘phase waves’, of which he has shown that there is always a whole number, measured along the path, present on each period or quasi-period of the electron. The main difference is that de Broglie thinks of progressive waves, while we are led to stationary proper vibrations 40 if we interpret our formulae as representing vibrations. I have lately shown that the Einstein gas theory can be based on the consideration of such stationary proper vibrations, to which the dispersion law of de Broglie’s phase waves has been applied. (Schrödinger, 1926a, p. 9) Another essential difference with de Broglie is that, while de Broglie had a dualistic perspective associating a phase wave with a material particle, Schrödinger simply replaced the molecules by phase waves. According to Schrödinger “a molecule of restmass m, moving with velocity v = β c, constitutes nothing but a ‘signal’ – one might say, ‘the wave crest’ – of a wave system” (quoted in Mehra & Rechenberg, 1987, p. 439). Schrödinger’s interpretation based solely on wave characteristics is something that was new in Schrödinger’s work on wave mechanics (as can be seen in another earlier work on Einstein’s gas theory, where Schrödinger only makes some considerations related to the statistics of the molecules), it emerged with the developments of the work. We have then already at the early stages of the development of wave mechanics a turn towards an (apparently) more classical approach to the description of quantum systems using some sort of a yet very provisional wave interpretation. This is clear from the fact that also in the case of the energy levels of the hydrogen atom, Schrödinger sees it as a proper value problem. That is, in this case we consider also standing waves where the natural boundary conditions (due to the singular points) can be seen as resulting from the fact that the equation “carries within itself the quantum conditions” (quoted Mehra & Rechenberg, 1987, p. 575). We would be in a situation no different from that of a vibrating spring. This is for Schrödinger the most important aspect of his work. Schrödinger was able to “show that the customary quantum conditions can be replaced by another postulate, in which the notion of ‘whole numbers’, merely as such, is not introduced. Rather when integralness does appear, it arises in the same natural way as it does in the case of the node-numbers of a vibrating string” (Schrödinger, 1926a, p. 1). In this way the discrete energy levels of the hydrogen atom has its origin “in the finiteness and single-valuedness of a certain space function” (Schrödinger, 1926a, p. 9). 4 The interpretation and later developments of the Schrödinger equation In his first articles on wave mechanics Schrödinger did not develop much a physical interpretation of the wave function. Schrödinger had the idea that “the waves in the phase space are something real in a sense” (quoted in Mehra & Rechenberg, 1987, p. 536), but to make this ‘sense’ precise turned out to be a difficult task. Schrödinger’s initial idea was to “connect the [wave] function ψ with some vibration process in the atom” (Schrödinger, 1926a, p. 9), but he recognized that “it is only in the one electron problem that the interpretation [of the wave function] as a vibration in the real threedimensional space is immediately suggested” (Schrödinger, 1926b, p. 28). This situation had come out in Schrödinger’s correspondence with several physicists. In a letter to Sommerfeld, Schrödinger recognized this point: “since ψ in general depends on many more than three variables, the immediate interpretation as vibrations in the threedimensional space is made difficult in any desirable manner” (quoted in Mehra & Rechenberg, 1987, p. 542). Also his simple view of particles as ‘wave crests’ brought the sharp criticism of Hendrik Lorentz who called Schrödinger’s attention to the fact that a wave packet, which might according to Schrödinger represent a particle, “can never stay together and remain confined to a small volume in the long run … because of this unavoidable blurring a wave packet does not seem … to be very suitable for representing things to which we want to ascribe a rather permanent individual existence” (quoted in Jammer, 1966, p. 31; see also Bohm, 1951, pp. 60-66). 41 An even greater problem was that at this point Schrödinger was unable to account for the intensities and polarizations of the emitted radiation in an atomic transition. To be exact not even a clear relation between the emission frequency and the energies of the atomic levels involved in the emission process was available. Bohr had developed in 1912-1913 a model of the atom based on the idea of stationary states – not submitted to the consequences of the classical theory of radiation –, where the electron can jump from one stationary state to another by emission or absorption of radiation with a frequency ν given by the relation h ν = E´ – E´´, where E´ and E´´ are the energies of each stationary state (Bohr, 1913). A first attempt to address this line of problems was presented in the first paper on wave mechanics. In here Schrödinger tried to obtain Bohr’s frequency relation within his wave mechanics approach. For this Schrödinger considered a constant parameter C that would appear in an expression relating the proper frequency with the proper energy of a stationary proper vibration (representing in Schrödinger’s approach Bohr’s stationary states). By considering C “very great compared with all the admissible negative E-values” (Schrödinger, 1926a, p. 10), an expansion in terms of E of the expression was possible: K++=+=ν E C2 C' CC'ECC' According to Schrödinger this expression “permits an understanding of the Bohr frequency condition. According to the latter the emission frequencies are proportional to the E-differences, and therefore from [the previous expression] also to the differences of the proper frequencies ν of those hypothetical vibration processes” (Schrödinger, 1926a, p. 10). In this way Schrödinger considered that “the emission frequencies appear therefore as deep ‘difference tones’ of the proper vibrations themselves. It is quite conceivable that on the transition of energy from one to another of the normal vibrations, something – I mean the light wave – with a frequency allied to each frequency difference, should make its appearance” (Schrödinger, 1926a, p. 10). Schrödinger recognized that in this way he could only obtain Bohr’s relation in an approximate form, but expected that the development of a relativistic wave theory would solve this problem. An improvement over this situation was published in a subsequent paper where Schrödinger again addressed the problem of the coupling between the dynamic processes in the atom and the electromagnetic field. According to a letter to Sommerfeld, Schrödinger expected that a relation might be obtained between the matter wave functions and the Maxwell-Lorentz electromagnetic current: “The ψ - vibrations must then correspond to the four-current, i.e., the [Maxwell-Lorentz] fourcurrent must be replaced by something derived from the function ψ , say the fourdimensional gradient of ψ. But this is all pure fantasy; in reality I have still not thought about it at all” (quoted in Mehra & Rechenberg, 1987, p. 779). In the paper, a first approach to this idea was presented. According to Schrödinger “the mechanical field scalar (which I denote by ψ ) is perfectly capable of entering into the unchanged Maxwell-Lorentz equations between the electromagnetic field vectors as the ‘source’ of the latter; just as, conversely, the electromagnetic potentials enter into the coefficients of the wave function, which defines the scalar field” (Schrödinger, 1926c, p. 60). In this work Schrödinger considered the case of an electron in a central field (the hydrogen atom) and made the fundamental assumption that the space density of electricity ρ el is given by the real part of 42 t ∂ ψ ∂ ψ , where ψ is the complex conjugate of the wave function ψ . This implies that “the charge of the electron is not concentrated in a point, but is spread out through the whole space” (Schrödinger, 1926e, p. 1066). The fluctuation of the charge is given by the previous expression. From this it was simple to determine “the radiation, that by ordinary electrodynamics will originate from these fluctuating charges” (Schrödinger, 1926a, p. 1066). This was done by calculating the electrical dipole moment, using the above expression for ρ el. For example the dipole moment in the Cartesian z direction is given by dxdydz t z∂ ψ ∂ ψ ∫∫∫ . Despite Schrödinger’s recognition that this approach “is far from completely satisfactory and is in no way final” (Schrödinger, 1926c, p. 61), it was clear that a progress had been made as can be seen by his treatment of an atom in a constant electric field (the Stark effect). By applying the above mentioned hypothesis to the physical meaning of ψ , Schrödinger was able not only to obtain, exactly and not as an approximation, Bohr’s ‘frequency-rule’ that gives the frequency of the electromagnetic radiation as the difference between the (term) frequencies of two stationary wave functions, but also to calculate the intensities and polarizations of the emitted radiation, which were in agreement with experiments. A further development of this electrodynamic interpretation occurred due to Schrödinger’s extension of his wave mechanics to the case of time-dependent systems, which was necessary for example to treat the case of dispersion of radiation by atoms. The development of a general time-dependent wave equation (which Schrödinger referred to as the real wave equation) made it necessary to consider the wave function complex. This led Schrödinger to re-work his expression for the space density of electricity ρ el, which now was written as ψ ψ e , where e is the total electric charge of the electron and ψ ψ can be seen as a weight function. With this new expression Schrödinger was able to give a more clear ‘electromagnetic meaning’ to the ‘mechanical field scalar’: when integrating the space density of electrical charge ρ el over the whole of space Schrödinger obtained a time-independent finite value (while the previous expression yields zero). This implied the conservation of charge. Schrödinger extended this interpretation to the case of a system with several electrons: ψ ψ is a kind of weight-function in the system’s configuration space. The wave-mechanics configuration of the system is a superposition of many, strictly speaking of all, point-mechanical configurations kinematically possible. Thus, each point-mechanical configuration contributes to the true wavemechanics configuration with a certain weight, which is given precisely by ψ ψ . (Schrödinger, 1926d, p. 120) Schrödinger recognized the loose character of this provisory interpretation, as we can read in the final part of a set of lectures delivered in March 1928: We have hitherto avoided putting forward any definite assumption as to the physical interpretation of the function ψ(q 1 , q 2 , … , q n , t) relating to a system whose configuration in terms of ordinary mechanics is 43 described by the generalized co-ordinates q 1 , q 2 , … , q n . This interpretation is a very delicate question. As an obvious generalization of the procedure of spreading out the electronic charge according to a relative density function ψ ψ (which furnished satisfactory results in the one-electron problem; see section 5), the following view would present itself in the case of a general mechanical system: the real natural system does not behave like the picture which ordinary mechanics forms of it (e.g. a system of point-charges in a definite configuration), but rather behaves like what would be the result of spreading out the system, described by q 1 , q 2 , … , q n , throughout its configuration-space in accordance with a relative density function ψ ψ . This would mean that, if the ordinary mechanical picture is to be made use of at all, the actual system behaves like the ordinary mechanical picture, present in all its possible configurations at the same time, though “stronger” in some of them than in others. I maintained this view for some time. The fact that it proves very useful can be seen from the oneelectron problem (see section 5). No other interpretation of the ψ-function is capable of making us understand the large amount of information which the constants a kl [(related to the electric moment)] furnish about the intensity and polarization of the radiation. Yet this way of putting the matter is surely not quite satisfactory. For what does the expression “to behave like” mean in the preceding sentences? The “behaviour ” of the ψ-function, i.e. its development in time, is governed by nothing like the laws of classical mechanics; it is governed by the wave-equation. (Schrödinger, 1928, p. 205-206) We see that Schrödinger recognizes his view as not quite satisfactory.9 This came about from his recognition that even in the case of one electron it is not enough to see the wave function as meaning a continuous distribution of electricity in actual space. As Schrödinger mentions, it is necessary also to take into account the space-time behavior of the wave function as described by the wave equation. Michel Bitbol has stressed this point: Schrödinger became fully aware that [the electrodynamic interpretation] could not work alone, that somehow it had to be combined with the original wave interpretation. Whereas the electrodynamic interpretation enables one to make sense of the line intensities and polarizations, the wave representation was still needed to calculate the relevant charge distributions as well as the evolution between two measurements of line intensities. The problem Schrödinger had to face was the following: we need not one but two representations of the atomic phenomena. One of them, namely the electrodynamic representation, is directly related with the ‘observed facts’, but it does not provide the link between subsequent observed facts; in a word, it is ‘factual’ but not ‘effective’. The second one, namely, the wave representation, is perfectly able to provide a link between the observed facts, but not to account for all the aspects of the facts themselves; it is ‘effective’ but not ‘factual’. As long as Schrödinger wanted to merge the ‘effective’ and the ‘factual’ into a single representation, according to the classical ideal, the persistent duality of the models had to be considered as a symptom of failure. (Bitbol in Schrödinger, 1995, p. 3) The two-fold aspect of Schrödinger’s tentative interpretation is clearly present in Schrödinger’s article on the Compton effect10 published in 1927. In this work Schrödinger explained the Compton effect as resulting from the scattering of a light wave and a matter wave (a de Broglie wave). Even if Schrödinger uses his interpretation of the wave function as giving rise to a charge density distribution in space, the crucial aspect of his calculations is to consider a wave function as a ‘real’ wave propagating in space (Schrödinger even makes the analogy to a sound wave). A very interesting aspect 9 Following these statements showing the difficulty Schrödinger was having in giving a solid and clear meaning to his interpretation, Schrödinger mentioned that “an obvious statistical interpretation of the ψfunction has been put forward, viz. that it does not relate to a single system at all but to an assemblage of systems, ψ ψ determining the fraction of the systems which happen to be in a definite configuration” (Schrödinger, 1926c, p. 206). I will not look into Born’s well-known statistical interpretation or Schrödinger’s criticism of this interpretation (which like his own reformulation of his interpretation extended for several decades. For details on early interpretation issues see e.g. Jammer (1974). For an account of the adoption of Born’s statistical interpretation see e.g. Mehra & Rechenberg (2000), Mara Beller (1990), Pais (1982). For details on Schrödinger‘s views see Bitbol (1996). 10 Regarding the Compton effect see e.g. Sánchez Ron (2001, pp. 382-392). 44 of this two-fold use of the wave function is that, the de Broglie or matter wave (associated with an electron with rest mass m0) is described by Schrödinger’s (relativisitc) wave equation, which in the context of its use, is, in practice, taken to be an equation for a ‘classical’ wave function of a matter wave (Schrödinger, 1927). Later workers on quantum mechanics have explored and clarified this possibility of giving very different meanings to the wave equation, i.e. as an equation describing a probability wave in a configurational space or an equation describing a matter wave in actual space. In the next chapter we will look in particular into Pascual Jordan’s approach, fundamental in the development of a quantum theory of fields. In this chapter I shall follow Schrödinger’s tentative interpretation and see how much can we stretch out a reading of the equation as the equation for a classical matter wave. It will be enough to consider the simple case of a one-electron system. We will see that Schrödinger’s difficulties with the two-fold aspect of his tentative interpretation can be traced back to an ‘ambiguity’ in the physical interpretation (in terms of a particle equation or a matter wave equation) of the one-electron Schrödinger equation (Becker, 1964, Vol 2., pp. 92-107;Tomonaga, 1962, Vol 2., pp. 14-115). In 1927 Clinton J. Davisson and Lester H. Germen presented clear experimental evidence for matter waves propagating in space. The experiment consisted in the bombardment of a metal plate by an electron beam. Davisson expected that by analyzing the angular distribution of the electrons scattered by a metal plate it would be possible to obtain information regarding the shell structure of the electrons inside the atoms of the metal plate. In the early stages of the sequence of experiments (from 1921 to around 1925) there was no evidence of any wave properties of the electron beams. The first view on the results was that all the main features of the distribution curves so far observed for the scattering from nickel seem reasonably accounted for on the supposition that a small fraction of the bombarding electrons actually do penetrate one or more of the shells of electrons which are supposed to constitute the outer structure of the nickel atom and, after executing simple orbits in a discontinuous field, emerge without appreciable loss of energy. (Quoted in Jammer, 1966, pp. 249-250) The experimental finding of the diffraction of the electron beams by a metal plate resulted from an accident in Davisson’s laboratory in 1925. In their paper Davisson and Germen tell the story of the discovery of the diffraction of electrons: The investigation reported in this paper was begun as the result of an accident, which occurred in this laboratory in April 1925. At that time we were continuing an investigation, first reported in 1921, on the distribution-in-angle of electrons scattered by a target of ordinary (polycrystalline) nickel. During the course of this work a liquid-air bottle exploded at a time when the target was at a high temperature; the experimental tube was broken, and the target heavily oxidized by inrushing air. The oxide was eventually reduced and a layer of the target removed by vaporization, but only after prolonged heating at various high temperatures in hydrogen and in vacuum. When the experiments were continued it was found that the distribution-in-angle of the scattered electrons had been completely changed … This marked alteration in the scattering pattern was traced to a re-crystallization of the target that occurred during the prolonged heating. Before the accident and in previous experiments we had been bombarding many small crystals, but in the tests subsequent to the accident we were bombarding only a few large ones … It seemed probable from these results that the intensity of scattering from a single crystal would exhibit a marked dependence on crystal direction, and we set about at once preparing experiments for an investigation of this dependence. (Davisson & Germen, 1927, p. 706) The ‘marked alteration in the scattering pattern’ made Davisson and Germen change their research towards an investigation of a possible dependency of the scattered beams on the crystal direction. It was clear that, in this case of a single nickel crystal, the 45 intensity variation of the angular distribution was not simply related with the internal structure of the atoms of the plates, since in the older experiments they also used plates of nickel, which in that case consisted in ordinary (polycrystalline) nickel. An explanation for these results must be found elsewhere. Here is how Davisson and Germen addressed the new experimental findings: We must admit that the results obtained in these experiments have proved to be quite at variance with our expectations. It seemed to us likely that strong beams would be found issuing from the crystal along what may be termed its transparent directions − the directions in which the atoms in the lattice are arranged along the smallest number of lines per unit area. Strong beams are indeed found issuing from the crystal, but only when the speed of bombardment lies near one or another of a series of critical values, and then in directions quite unrelated to crystal transparency. The most striking characteristic of these beams is a one to one correspondence, presently to be described, which the strongest of them bear to the Laue beams that would be found issuing from the same crystal if the incident beam were a beam of x-rays. Certain others appear to be analogues, not to Laue beams, but of optical diffraction beams from plane reflection gratings − the lines of these gratings being lines or rows of atoms in the surface of the crystal. Because of these similarities between the scattering of electrons by a crystal and the scattering of waves by three- and twodimensional gratings a description of the occurrence and behavior of the electron diffraction beams in terms of the scattering of an equivalent wave radiation by the atoms of the crystal, and its subsequent interference, is not only possible, but most simple and natural. This involves the association of a wave-length with the incident electron beam, and this wave-length turns out to be in acceptable agreement with the value h/mυ of the undulatory mechanics, Planck’s action constant divided by the momentum of the electron. (Davisson & Germen, 1927, pp. 706-707) In this way Davisson and Germen found that they could make sense of the experimental results by resorting to the idea of de Broglie waves, finding that the de Broglie expression λ = h/p relating the wavelength λ and the momentum p of the electron was verified. We can choose to see Schrödinger’s equation as the equation for this matter wave, as we in part have in Schrödinger’s two-fold interpretation. In this way Schrödinger’s equation can describe the wave properties of the electrons as revealed for example in Davisson and Germen experiments or in Schrödinger’s account from a wave perspective of the Compton effect. But as we have seen this approach does not gives us a description of the electron’s corpuscular aspects as revealed in several other experiments, since Schrödinger was unable to provide an account of the electron’s particle-like characteristics as simplyj a wave-crest (more exactly a wave packet), and Schrödinger was forced to associate a charge (or mass) density to the wave function, which in the case of several electrons results in a highly abstract entity: a weight function. Looking at the simple case of the hydrogen atom it is clear to what point we can take this use of Schrödinger’s equation as an equation for a matter wave (Becker, 1964, Vol 2., pp. 92-107;Tomonaga, 1962, Vol 2., pp. 14-115). Returning to Schrödinger’s initial treatment of the hydrogen atom, considering the Schrödinger wave equation for the proper oscillation of a de Broglie wave in a central potential, now writing it in terms of the de Broglie wave frequency: 0 r 1 h e mπ8∆ 2 2 =        ++ φνφ , we obtain, in a totally classical treatment, for the proper frequencies of the de Broglie wave 52 In this way, Dirac had immediately the relativistic wave equation for a free electron: 0mc]ψρ),(ρ[p 310 =−− pσ , where p0 = iħ ∂ /(c ∂ t) and p = ( p1 , p2 , p3 ), where pr = –iħ ∂ /(c ∂ xr) with r = 1, 2, 3; σ σσ σ = (σ 1 , σ 2 , σ 3 ) is a vector formed with the above 4 × 4 matrices. Dirac generalized his equation to the case of an electron in an external electromagnetic field. Dirac followed the rule of replacing p0 by p0 +e/c.A0 and p by p + e/c. A (where A0 and A are the scalar and vector potentials). This gives us the equation 0mc]ψρ) c e ,(ρ c e [p 3100 =−+−Α+ Αpσ , which is the “fundamental wave equation of the relativistic theory of the electron” (Dirac, 1958, p. 257). Dirac had developed his equation by considering the relativistic Hamiltonian of a free point particle, that is, Dirac did not take into account in his Hamiltonian (as for example Pauli did in the derivation of his equation) any term related to the spin of the electron. It was a surprise to Dirac that “the simplest possible case did involve the spin” (quoted in Kragh, 1981, p. 55). Dirac set out to explore the relation between his wave equation with external potentials and the Klein-Gordon equation (based on a classical relativistic Hamiltonian), which was according to Dirac “the wave equation to be expected from analogy with the classical theory” (Dirac, 1958, p. 264): 0ψcm c e c e p 22 22 00 =           −       +−       Α+ Αp . By squaring his equation, Dirac obtained a differential equation that included the operator of the Klein-Gordon equation and two additional terms: 0ψ),( c e i),( c e cm c e c e p 1 22 22 00 =           +−−       +−       Α+ ΕσΗσΑp hh ρ . Dirac concluded that the electron will therefore behave as though it has a magnetic moment eh/4πmc · σ σσ σ and an electric moment ieh/4πmc · ρ 1 σ σσ σ. This magnetic moment is just that assumed by the spinning electron model. The electric moment, being a pure imaginary, we should not expect to appear in the model. It is doubtful whether the electric moment has any physical meaning, since the Hamiltonian … that we started from is real, and the imaginary part only appeared when we multiplied it up in an artificial way to make it resemble the Hamiltonian of previous theories. (Dirac, 1928, p. 619) Dirac showed how this internal magnetic moment resulted from the electron having a spin angular momentum. According to Dirac the spin angular momentum does not give rise to any potential energy and therefore does not appear in the result of the preceding calculation. The simplest way of showing the existence of the spin angular 53 momentum is to take the case of the motion of a free electron or an electron in a central field of force and determine the angular momentum integrals [of the motion]. (Dirac, 1958, p. 266) By setting A = 0 and A 0 a function of the radius r, the Hamiltonian in Dirac’s equation takes the form H = – eA 0 (r) + cρ 1 (σ σσ σ, p) + ρ 3 mc 2 . Considering, for example, the equation of motion of the x 1 -component of the orbital angular momentum m 1 = x 2 p 3 – x 3 p 2 , we have iћ 1 m & = iћcρ 1 {σ 2 p 3 – σ 3 p 2 }. We see that m 1 is not a constant of the motion. By considering also the equation iћ 1 σ & = 2icρ 1 {σ 3 p 2 – σ 2 p 3 }, Dirac found that 1 m & + ћ/2 · 1 σ & = 0. This means that the vector m + ћ/2 · σ σσ σ is a constant of the motion. According to Dirac “we can interpret this result by saying that the electron has a spin angular momentum of ћ/2 · σ σσ σ, which, added to the orbital angular momentum m, gives the total angular momentum M, which is a constant of the motion” (Dirac, 1928, p. 620). In this way Dirac found that his relativistic wave equation described an electron with a spin angular momentum (and corresponding magnetic moment). Dirac did not provide an exact solution of his equation for an electron in a central potential. He only made a first order calculation. In this treatment of the hydrogen atom, Dirac was able to obtain Pauli’s results on the energy levels, but without using any arbitrary assumptions: the spin angular momentum, gyromagnetic ratio (g 0 ) and Thomas factor all came out right. Just a few weeks after the publication of Dirac’s paper on the relativistic wave equation, it was shown independently by C. G. Darwin and W. Gordon that an exact solution of Dirac’s equation gave an expression for the discrete energy levels of the hydrogen spectrum which was identical to Sommerfeld’s original formula derived in 1915, which was in good agreement with experimental results. 4 The problem with the negative energy solutions Since Dirac was using 4 × 4 matrices in his equation, the wave function had four components (recall that Schrödinger’s original equation had one component, and Pauli’s equation had two components due to the spin degree of freedom). Initially Dirac thought that he could simply drop two of the components, since “half of the solutions must be rejected as referring to the charge +e on the electron” (Dirac, 1928, p. 618). This was possible in a first-order approximation. Looking at the exact solution we can see that the situation is far from that simple. Dirac’s equation can be written as a set of two coupled differential equations for a pair of two-component wave functions ψ Α and ψ Β , where Dirac’s wave function is given by         = B A ψ ψ ψ . These two wave functions are essential in the solution of Dirac’s equation and cannot be disregarded in the calculation of the energy levels. What happens is that the lower twocomponents (ψ Β ) are smaller that the upper two-components (ψ Α ), roughly by a ratio of υ/2c, where υ is the ‘velocity’ of the orbiting electron in Bohr’s theory. Now, when considering the solution of Dirac’s equation for a free electron with momentum p, we see that there are two solutions corresponding to electron states with momentum p and energy E p , and two solutions corresponding to states with momentum –p and energy – E p , or as Dirac mentioned to an electron with charge +e (and positive energy E p ). In the case of the exact solution for an electron in an external electromagnetic field, as we 54 have seen, we have a four-component wave function, which we can say, by resort to the free electron case, to ‘have’ positive and negative energy components, or as Dirac mentions, solutions referring to a charge –e and +e. 11 In this way we must take, as Dirac did, the reference to positive and negative energy components as a “rough one, applying to the case when such a separation is approximately possible” (Dirac, 1958, p. 274). When taking the non-relativistic limit of Dirac’s equation, the equation for the upper two-components, takes the form of the Schrödinger-Pauli two-component wave equation (corresponding to a negative energy state), and as mentioned, the lower components are smaller than the upper components and can be disregarded. This is the procedure taken by Darwin to ‘derive’ Schrödinger’s equation from Dirac’s equation (Mehra & Rechenberg, 2000, p. 302). That is we can only disregard two of the components of the four-component wave function in the non-relativistic limit. Now, Dirac’s results are not non-relativistic. The whole point of Dirac’s approach was to develop and apply a relativistic wave equation. Dirac soon recognized that there was a fundamental difficulty with his equation. Already in his early work with the Klein-Gordon equation Dirac had noticed the possibility of solutions corresponding to a charge +e (this is a general characteristic of any relativistic equation due to the relativistic formula for the Hamiltonian involving E 2 ). The problem is that when considering any small external electromagnetic field, “in general a perturbation will cause transitions from states with E positive to states with E negative” (quoted in Mehra & Rechenberg, 2000, 306). Accordingly Dirac considered that such a transition would appear experimentally as the electron suddenly changing its charge from –e to +e, a phenomenon which has not been observed. The true relativity wave equation should thus be such that its solutions split up into two non-combining sets, referring respectively to the charge –e and the charge +e. (Dirac, 1928, p. 612) As we have seen, also in the case of an electron in a central potential as described by Dirac’s equation, that is not the case. Dirac knew that; he recognized in the beginning of his paper on the relativistic wave equation that he was unable to remove this difficulty and considered that his theory “is therefore still only an approximation” (Dirac, 1928, p. 612). However Dirac expected “the probability of these transitions [to be] extremely small” (quoted in Mehra & Rechenberg, 2000, p. 306). That was not the case. Soon afterwards Werner Heisenberg showed that the probability for transitions in which an electron in a state corresponding to a charge –e goes into a state corresponding to a charge +e (a negative energy state) was much larger than Dirac’s estimation. Also, Heisenberg showed that the negative-energy states where necessary to obtain the correct dispersion formulae (Mehra & Rechenberg 2000, pp. 306-307). The problem with the negative-energy solutions was highlighted by O. Klein, when, in the end of 1928, he showed that the simple case of a positive-energy wave incident on a potential barrier could give rise to a transmitted negative-energy wave (Mehra & Rechenberg, 2000, pp. 309-311). By the end of 1929 Dirac had found a way to solve the ‘± difficulty’ of his electron theory. In late March 1929 Heisenberg wrote to Dirac mentioning that H. Weyl thought he had a solution to the ± difficulty and asked Dirac if he knew and could give him any 11 According to Dirac, “it is not possible, of course, with an arbitrary electromagnetic field, to separate the solutions of [the relativistic wave equation] definitely into those referring to positive and those referring to negative values [of energy], as such a separation would imply that transitions from one kind to the other do not occur”. (Dirac, 1958, p. 274) 55 details about it. Weyl’s idea consisted in suggesting that the two extra components might be ascribed not to the electron but to the proton (Dirac’s equation would be in this way describing simultaneously electrons and protons). Dirac worked on Weyl’s ideas and set forward a new interpretation of his equation that might provide a solution to the problem of the negative energy solutions. Dirac’s first written accounts of his new views were made in an exchange of letters with Bohr. In a letter to Dirac from November 24, Bohr had put forward the idea that there might not be a strict conservation of momentum and energy in some nuclear processes and that this might lead to a solution of the negative energy problem, resulting from the fact that, according to Klein, the potential that confines the electron in the nucleus induces transitions to negative energy states (by that time it was believed that there were electrons in the nucleus of the atoms). Dirac answered, on November 26, that “I should prefer to keep rigorous conservation of energy at all costs”, and that “there is a simple way of avoiding the difficulty of electrons having negative kinetic energy” (quoted in Kragh, 1990, p. 90). Dirac then elaborate on his views: Let us suppose the wave equation [w/c + e/c·A 0 (r) + ρ 1 (σ σσ σ·γ γγ γ + e/c·A) + ρ 0 me)] ψ = 0 does accurately describe the motion of a single electron. This means that if the electron is started off with a +ve energy, there will be a finite probability of its suddenly changing into a state of negative energy and emitting the surplus energy in the form of high-frequency radiation. It cannot then very well change back into a state of +ve energy, since to do so it would have to absorb high-frequency radiation and there is not very much of this radiation actually existing in nature. It would still be possible, however, for the electron to increase its velocity (provided it can get the momentum from somewhere) as by so doing its energy would be still further reduced and it would emit more radiation. Thus the most stable states for the electron are those of negative energy with very high velocity. Let us now suppose there are so many electrons in the world that all these most stable states are occupied. The Pauli principle will then compel some electrons to remain in less stable states. For example if all the states of –ve energy are occupied and also few of +ve energy, those electrons with +ve energy will be unable to make transitions to states of –ve energy and will therefore have to behave quite properly. The distribution of –ve electrons, will, of course, be of infinite density, but it will be quite uniform so that it will not produce any electromagnetic field and one would not expect to be able to observe it. It seems reasonable to assume that not all the states of negative energy are occupied, but that there are a few vacancies or “holes.” Such a hole which can be described by a wave function like an X-ray orbit would appear experimentally as a thing with +ve energy, since to make the hole disappear (i.e. to fill it up,) one would have to put –ve energy into it. Further, one can easily see that such a hole would move in an electromagnetic field as though it had a +ve charge. These holes I believe to be the protons. When an electron of +ve energy drops into a hole and fills it up, we have an electron and proton disappearing simultaneously and emitting their energy in the form of radiation. I think one can understand in this way why all the things one actually observes in nature have positive energy. One might also hope to be able to account for the dissymmetry between electrons and protons; one could regard the protons as the real particles and the electrons as the holes in the distributions of protons of –ve energy. However, when the interaction between the electrons is taken into account this symmetry is spoilt. I have not yet worked out mathematically the consequences of the interaction. It is the “Austausch” effect that is important and I have not yet been able to get a relativistic formulation of this. One can hope, however, that a proper theory of this will enable one to calculate the ratio of the masses of proton and electron. (Quoted in Kragh, 1990, pp. 90-91) Bohr answered a few days later (in December 5), after discussing Dirac’s letter with Klein: We do not understand, how it works out in detail. Before all we do not understand, how you avoid the effect of the infinite electric density in space. According to the principles of electrostatics it would seem that even a finite uniform electrification should give rise to a considerable, if not infinite, field of force. In the difficulties of your old theory I still feel inclined to see a limit of the fundamental concepts on which atomic theory hitherto rests rather than a problem of interpreting the experimental evidence in a proper way by means of these concepts. Indeed according to my view the fatal transition from positive to 56 negative energy should not be regarded as an indication of what may happen under certain conditions but rather as a limitation in the applicability of the energy concept. In the case of electrons impinging on a potential barrier examined by Klein we have, on the one hand, a striking example of the difficulties involved in an unlimited use of the concept of potentials in relativistic quantum mechanics. On the other hand, we have just in this case an example of the actual limit of applying the idea of potentials in connection with possible experimental arrangements. In fact, due to the existence of an elementary unit of electrical charge we cannot build up a potential barrier of any height and steepness desired without facing a definite atomic problem. In Klein’s example the critical height of the barrier is of order mc 2 , and the rise of potential shall take place within a distance of the order h/mc which is the order of magnitude of the wavelength of the electrons concerned. But if the dimensions of the barrier perpendicular to the electric force shall be large compared with this wavelength λ 0 , it claims the presence of a double layer of electricity of such a strength that a surface element of size λ 02 of the negative layer contains al least hc/e 2 electrons. It is therefore clear that the problem in question cannot legitimately be treated as that of one electron moving in a given potential field, but is essentially a many electron problem which falls outside the range of present quantum mechanics. On the whole it appears that the circumstance that hc/e 2 is large compared with unity does not only indicate the actual limit of the applicability of the quantum theory in its present form, but at the same time ensures its consistency within these limits. In fact the radius r 0 of the electron estimated on classical theory is e 2 /mc 2 = (h/me)(e 2 /hc), and we can therefore never determine the position of an electron within an accuracy comparable with r 0 without allowing an uncertainty in its momentum larger than mc, thus entailing an uncertainty of energy surpassing the critical value mc 2 . The idea that the reach of quantum mechanics is bound up with the actual existence of the electron would also seem to be in harmony with the fact that the symbols e and m appear in the fundamental equations of the present theory … As regards the transitions from positive to negative energy accompanied by radiation I am not sure that they present as serious a difficulty for your wave equation as it might appear. The question is, how much those features of the theory which claim the transitions in question are involved in the problems, where your theory has been found in so wonderful agreement with experiments. In this connection I must correct the statement in my former letter regarding the probability of these transitions which is not nearly so large as I believed. In discussing the problem more closely with Klein we convinced ourselves that the estimation of this probability did not take sufficient regard to the smallness of the wavelength of the radiation concerned compared with atomic dimensions. We have not made an actual calculation of any such probability, and if you have considered the problem in detail I should be very thankful for any information regarding this point. My hope is that it should be possible to defend all the successful applications of your wave equation, but I suspect that the natural limitation of these applications prevents an extrapolation of the kind you describe in your letter. (Quoted in Moyer, 1981, pp. 1057-1058 [my emphases]) It is important to notice that Bohr started seeing the Klein paradox as resulting from an unlimited mathematical application of the concept of potential (or more generally of field) in relativistic quantum mechanics. Bohr called attention to the necessity of taking into account the elementary unit of electrical charge in the determination of the actual potential barrier. Calculations made not taking into account this fact and the limits in the determination of the electron’s position (and its associated uncertainty in momentum, and according to Bohr also in energy) would be beyond the ‘actual limit of applicability’ of the theory and any ‘possible experimental arrangement’. Accordingly, Bohr considered that the elimination of Klein’s paradox passed through an essential limitation of the mathematical use of the concept of field (see also Darrigol, 1991, pp. 154-155). However Bohr considered that within the domain of applicability of the theory, the concepts (being used in a restricted context) and the results obtained were consistent (in chapter 5 I will return to Bohr’s view on relativistic quantum theory). Due to this, the problem of the transitions to negative-energy states would not occur in ‘all the successful applications of the theory’. In this way there would be no need for the hole theory. Dirac did not agree with Bohr’s views and in a letter sent to Bohr in December 9, stressed his differences with Bohr and gave a further elaboration of his views: 57 I do not completely agree with your views. Although I believe that quantum mechanics has its limitations and will ultimately be replaced by something better, (and this applies to all physical theories,) I cannot see any reason for thinking that quantum mechanics has already reached the limit of its development. I think it will undergo a number of small changes, mainly with regard to its method of application, and by these means most of the difficulties now confronting the theory will be removed. If any of the concepts now used (e.g. potentials at a point) are found to be incapable of having an exact meaning, one will have to replace them by something a little more general, rather than make some drastic alteration in the whole theory … There is one case where transitions of electrons from positive to negative energy levels does give rise to serious practical difficulties, as has been pointed out to me by Waller. This is the case of the scattering of radiation by an electron, free or bound. A scattering process is really a double transition, consisting of first an absorption of a photon with the electron jumping to any state and then an emission with the electron jumping to its final state (as in Raman effect) (or also of first the emission and then the absorption). The initial and final states of the whole system have the same energy, but not the intermediate state, which lasts only a very short time. One now finds, for radiation whose frequency is small compared with mc/h, that practically the whole of the scattering comes from double transitions in which the intermediate state is of negative energy for the electron. Detailed calculations of this have been made by Waller. If one says the states of negative energy have no physical meaning, then one cannot see how the scattering can occur. On my new theory the state of negative energy has a physical meaning, but the electron cannot jump down into it because it is already occupied. There is, however, a new kind of double transition now taking place, in which first one of the negative-energy electrons jumps up to the proper final state with emission (or absorption) of a photon, and secondly the original positive-energy electron jumps down and fills the hole, with absorption (or emission) of a photon. This new kind of process just makes up for those excluded and restores the validity of the scattering formulas derived on the assumption of the possibility of intermediate states of negative energy. I do not think the infinite distribution of negative-energy electrons need cause any difficulty. One can assume that in Maxwell’s equation div E = –4πρ, the ρ means the difference in the electric density from its value when the world is in its normal state (i.e when every state of negative energy and none of positive energy is occupied.) Thus ρ consists of a contribution –e from each occupied state of positive energy and a contribution +e from each unoccupied state of negative energy. I have not made any actual calculation of the transition probabilities from +ve to –ve, but I think they are fairly small. (Quoted in Kragh, 1990, pp. 92-93) A paper containing Dirac’s hole interpretation of his equation was published in early 1930. In this work Dirac first made clear his departure from Weyl’s original idea of associating the negative energy solutions directly to the protons. According to Dirac: One cannot, however, simply assert that a negative-energy electron is a proton, as that would lead to the following paradoxes: (i) A transition of an electron from a state of positive to one of negative energy would be interpreted as a transition of an electron into a proton, which would violate the law of conservation of electric charge. (ii) Although a negative-energy electron moves in an external field as though it has a positive energy, yet, as one can easily see from a consideration of conservation of momentum, the field it produces must correspond to its having a negative charge, e.g. the negative-energy electron will repel an ordinary positive-energy electron although it is itself attracted by the positive-energy electron. (iii) A negative-energy electron will have less energy the faster it moves and will have to absorb energy in order to be brought to rest. No particles of this nature have ever been observed. (Dirac, 1930, p. 362) Dirac then presented in more details his hole theory as described in the letters to Bohr. Dirac gave in particular an account of the scattering of radiation by an electron according to his hole theory. Dirac mentions that in a scattering process two types of intermediate states can occur. In one case we have a 58 transition process, consisting of first an absorption of a photon with the electron simultaneously jumping to any state, and then an emission with the electron jumping into its final state, or else of first the emission and then the absorption. (Dirac, 1930, p. 364) In the other case first one of the distribution of negative-energy electrons jumps up into the required final state for the electron with absorption (or emission) of a photon, and then the original positive-energy electron drops into the hole formed by the first transition with emission (or absorption) of a photon. Such processes result in a final state of the whole system indistinguishable from the final state with the more direct processes, in which the same electron makes two successive jumps. (Dirac, 1930, p. 365) In this second case we have an intermediate state with two electrons (or one electron and the hole). It is clear that the description of the light scattering by an electron involves more than just the original electron: we are faced with a many-body theory. Dirac with his hole theory changed the character of his equation from a single-electron equation into an equation describing a many-body (in reality infinite) electron system. Even before the publication of his paper on the hole theory, several physicists, knowing the general lines of Dirac’s ideas, had a sceptical reaction to it. In particular Heisenberg made a rough calculation of the electron-proton interaction in the new theory. Heisenberg concluded that the electron and the proton had to have the same mass (Kragh 1990, p. 94). Dirac had already recognized this problem in one of his letters to Bohr, but expected that a future detailed theory of the interaction between electrons and protons (holes) might solve this difficulty. A further objection was given a few months latter, in a note by J. R. Oppenheimer (1930a) in which the author calculated the transition probability for the annihilation of an electron and a proton that corresponds to the filling of a hole in the sea. The result was not very promising. Oppenheimer obtained a mean lifetime of a free electron in matter that was too low, and totally inconsistent with the observed stability of matter (Kragh, 1990, pp. 101-102). Oppenheimer’s proposition was to “return to the assumption of two independent elementary particles of opposite charge” (quoted in Kragh, 1990, p. 102). That is, to consider the electron and the proton as dissociated particles, each one being related to its ‘Dirac sea’ of negative-energy particles. In this way there would not be a problem of a possible proton-electron annihilation. More importantly, in 1931 Weyl published a paper in which he proved by symmetry properties of Dirac’s equation that the negativeenergy electrons must have the same mass as the positive-energy electrons. In this same year Dirac rethought his hole theory in face of the objections being made and presented a new view on the problem: It thus appears that we must abandon the identification of the holes with protons and must find some other interpretation for them. Following Oppenheimer, we can assume that in the world as we know it, all, and not merely nearly all, of the negative-energy states for electrons are occupied. A hole, if there were one, would be a new kind of particle, unknown to experimental physics, having the same mass and opposite charge to an electron. We may call such a particle an anti-electron. We should not expect to find any of them in nature, on account of their rapid rate of recombination with electrons, but if they could be produced experimentally in high vacuum they would be quite stable and amenable to observation. (Dirac 1931, p. 61) In 1932 a brief article by C. D. Anderson was published presenting experimental evidence for a new kind of positively charged particle with a mass much smaller than that of the proton. These experimental findings were presented without taking into account Dirac’s theory. By that time P. Blackett and G. Occhialini had independent evidence for the positrons and previous to publication they discussed their findings with 59 Dirac. They published their results after Anderson’s publication, mentioning Dirac’s theory and presenting some ideas regarding the origin of the positive particle. According to Blackett and Occhialini “one can imagine that negative and positive electrons may be born in pairs during the disintegration of light nuclei” (quoted in Schweber, 1994, p. 69). However they were cautious regarding their proposed mechanism: when the behavior of the positive electrons has been investigated in more detail, it will be possible to test these predictions of Dirac’s theory. There appears to be no evidence as yet against its validity, and in its favour is the fact that it predicts a time of life for the positive electron that is long enough for it to be observed in the cloud chamber but short enough to explain why it had not been discovered by other methods. (quoted in Schweber, 1994, p. 69) After a thorough revision of the experimental evidence for positrons Blackett and Occhialini published another article where their support to Dirac’s theory was stronger: these conclusions as to the existence and the properties of positive electrons have been derived from the experimental data by the use of simple physical principles. That Dirac’s theory of the electron predicts the existence of particles with just these properties, gives strong reason to believe in the essential correctness of his theory. (quoted in Schweber, 1994, p. 69) Even with the experimental evidence for the positron there was resistance to Dirac’s hole theory. In relation to this discovery, Bohr considered that “even if all this turns out to be true, of one thing I am certain: that it has nothing to do with Dirac’s theory of holes!” (quoted in Kragh, 1990, p. 112). And Pauli wrote to Dirac saying: “I do not believe on your perception of ‘holes’, even if the existence of the ‘antielectron’ is proved” (quoted in Kragh, 1990, p. 112). It turns out they were right and a different – field theoretical – approach was possible without any need for an infinite sea of negative-energy particles with some holes in it. 5 The field theoretical interpretation of Dirac’s equation To understand how this field theoretical interpretation of Dirac’s equation came to be, we need (at least) to go back to a work by Dirac published in 1927. In this work, Dirac presented a non-relativistic treatment of the interaction of electromagnetic radiation and atoms, which enabled him to give a dynamical derivation of Einstein’s laws for the emission and absorption of radiation, which Einstein had obtained by statistical considerations (Darrigol, 1986, p. 226). Dirac followed initially an approach in which he considered an assembly of classical particles (that would interact with an atom), which were described by the Schrödinger equation. For the case of light quanta, Dirac knew that he could not use all the available wave functions, but had to select only symmetrical wave functions, corresponding to Bose-Einstein statistics: “The solution with symmetrical eigenfunctions must be the correct one when applied to light quanta, since it is known that the Einstein-Bose statistical mechanics leads to Planck’s law of black-body radiation” (Dirac, 1926, p. 672). In his 1927 work, Dirac did not follow the simpler procedure of imposing symmetrical wave functions (Dirac, 1958, p. 225), but a procedure more complex and physically unclear. This so-called ‘second quantization’ turns out to be simply a method that guaranteed that the (quantized) particles, that is particles described by a Schrödinger equation, satisfied Bose-Einstein statistics (Schweber, 1994, p. 28). Dirac did not arrive at the method by some physical insight. According to his words: 60 I remember the origin of that work was just playing about with equations. I was intending to get a theory of radiation at the time. I was just playing about with the Schrödinger equation. I got the idea of applying the quantization to it and worked out what it gave and found out it just gave the Bose statistics. (Quoted in Darrigol, 1984, p. 461) In this work Dirac provided a different approach to the interaction of radiation and atoms, by considering the electromagnetic radiation as a classical wave (and not as constituted by particles), which after quantization satisfies Bose-Einstein statistics. Dirac made the bold move of quantizing not directly the electric and magnetic field but the vector potential (Dirac, 1927a, p. 262; Miller, 1994, p. 22; Kragh, 1990, p. 126), which is resolved into its Fourier components. Using a non-relativistic approximation consistent with the one adopted for the light quanta case, Dirac found it possible – by making a necessary reinterpretation of the state corresponding to zero light quanta as a state with an infinite number of unobservable light quanta with zero energy and momentum – to find a Hamiltonian for the system constituted by a (quantized) electromagnetic field interacting with an atom, that “takes the same form as in the lightquantum treatment” (Dirac, 1927, p. 265). This result led Dirac to consider that: Instead of working with a picture of the photons [light quanta] as particles, one can use instead the components of the electromagnetic field. One thus gets a complete harmonizing of the wave and corpuscular theories of light. One can treat light as composed of electromagnetic waves, each wave to be treated like an oscillator; alternatively, one can treat light as composed of photons, the photons being bosons and each photon state corresponding to one of the oscillators of the electromagnetic field. One then has the reconciliation of the wave and corpuscular theories of light. They are just two mathematical descriptions of the same physical reality. (Quoted in Schweber, 1994, p. 31) The fact that Dirac considers that there is “a complete harmony between the wave and light-quantum description” (Dirac, 1927a, p. 245), does not mean that he takes over this equivalence to the case of the electrons. In fact, as Dirac clearly states in this article, he makes a sharp distinction between the case of electromagnetic radiation and matter. For Dirac there simply is no real de Broglie wave that, after quantization, permits the description of the electrons (Dirac, 1927a, p. 247). Jordan’s reading of Dirac’s work was quite different. Going back to the cumbersome method of second quantization that for Dirac was “nothing but a convenient way to take Bose statistics into account” (Darrigol, 1986, p. 229), Jordan interpreted the scheme of second quantization as the quantization of a classical wave described by a classical wave equation that could be the Maxwell-Lorentz equations for the case of the electromagnetic field or a Schrödinger equation for the case of electrons seen not as particles but as de Broglie waves (Darrigol, 1986, pp. 229-230). This view of Jordan had the advantage to make it possible to treat the quantized waves in a threedimensional space (or four-dimensional Minkowski space-time) instead of using a 3n (multidimensional) space in the case of an n-particle system (Schweber, 1994, p. 36). In a paper sent to publication in July 1927, Jordan made a conceptual turnaround on Dirac’s approach, and applied his method to the quantization of quanta obeying Pauli’s exclusion principle, 12 that is to electrons. Contrary to the case of photons (light quanta), in which there is no limit to the number of particles that may occupy the same state, the electrons (according to Pauli’s exclusion principle) cannot be in the same state, that is, the occupation number for each possible state can only be 0 or 1 (another way of making this statement is to say that the electrons are fermions, that is, they obey Fermi- 12 Regarding Pauli’s exclusion principle see e.g. Sánchez Ron (2001, pp. 348-350). 61 Dirac statistics). Jordan’s approach was to take the method used by Dirac with the light quanta, and to apply it to the electron as a de Broglie wave – for which there was experimental evidence (Darrigol, 1986, p. 219) – that was a solution of a (classical) Schrödinger equation. In this way, Dirac’s ‘second quantization’ was from Jordan’s perspective a ‘first quantization’ of a classical wave. This was simply implemented by taking the coefficients of each normal mode to be matrices satisfying certain commutation relations. In this work, Jordan did not take correctly into account the phase factors in the matrices “necessary to guarantee that the creation operators for different energy states anticommute with one another” (Schweber, 1994, p. 37). This point was settled by the end of the year in a collaboration with E. Wigner, where the correct anticommutation relations were given. In any case, already in his first incomplete work Jordan was able, by his quantization of the wave, to obtain “an evolution identical to the one given by anti-symmetrical wave functions in configuration space” (Darrigol, 1986, p. 231). That is, Jordan showed the equivalence, for the electrons, of adopting as a classical starting point, not the electrons as particles but electrons as classical de Broglie waves. With this procedure there was no need for a multidimensional abstract configuration space, since it was possible with the wave approach to maintain the description solely in terms of a quantized wave described in a three-dimensional space. In this way, already in this first incomplete work, Jordan was able to conclude that “a quantum-mechanical wave theory of matter can be developed that represents electrons by quantum waves in the usual three-dimensional space” (quoted in Darrigol, 1986, p. 232). This conclusion made possible for Jordan to make the ‘programmatic’ assertion: “The natural formulation of the quantum theory of electrons will be attained by conceiving light and matter as interacting waves in three-dimensional space” (quoted in Darrigol, 1986, p. 232). In the abstract of the work done with Wigner similar considerations were made: The problem at hand is to describe an ideal or nonideal gas that satisfies the Pauli exclusion principle with the idea of not using any relation in the abstract (3N-dimentional) configuration space of the atoms of the gas, but using only three-dimensional space. This is made possible by representing the gas by a threedimensional quantized wave, for which the particular non-commutative properties for multiplying wave amplitudes are simultaneously responsible for the existence of corpuscular atoms of the gas and for the validity of Pauli’s exclusion principle. (Quoted in Schweber, 1994, p. 38) Heisenberg and Pauli adopted Jordan’s approach in the development of a quantum field approach to the description of the interaction of radiation and matter. In their case they took Dirac’s equation as a classical wave equation. The ‘de Broglie’ solution of Dirac’s equation is then quantized according to the procedure set forward by Jordan. The elementary excitations (quanta) of the field resulting from the quantization are the particles. In a sense this field theoretical approach as applied by Heisenberg and Pauli is incomplete. The point is that the energy of the field can be negative. This is so due to the presence of the negative energy quanta. This means it was still necessary to make use of Dirac’s hole theory to make sense of the field quantization. The negative energy states where taken to be full and a positron was identified with an empty negativeenergy state. That is, a positron in this field theoretical view was taken to be the lack of an elementary excitation in an infinite sea of negative-energy elementary excitations. In November 1933, V. Fock published a paper where he made a symmetrical treatment of free electrons and positrons without using negative-energy particles, following a procedure by Heisenberg from 1931 in which he explored “a far-reaching analogy between the terms of an atomic system with n electrons and those of a system in which n electrons in a closed-shell are lacking” (quoted in Pais, 1986, p. 379). In a 68 variance of the vacuum state. This implies that we can retain a notion of vacuum with experimental meaning that is consistent with an interpretation of the variance in terms of a statistical spread (distribution) in the results of independent measurements made on identically prepared systems. It seems then that we cannot recover the ‘nothingness’ of the classical notion, but, nevertheless, the physical properties we can really associate with the vacuum concept are much more subtle than usually thought, and do not present any experimental particularities that are not found in all quantum states corresponding to a defined number of quanta: the non-vanishing variance is a common characteristic of all these states, not only the ground state. In this way, I will be making a case for an empirically demonstrable notion of the vacuum in quantum electrodynamics independent of dynamical fluctuations. 2 The historical emergence of the quantized electromagnetic field Though the story of quantum theories began around 1900 with Planck’s theoretical work on the so-called blackbody radiation, a working quantum theory of the electromagnetic field (and its interaction with matter) was not available until the late 1940s. The quantum theory of radiation can be seen to have its first development not in Planck’s early and subsequent work, but in an article published in 1905 that its author referred to as “very revolutionary“ (quoted in Kragh, 1999, p. 66). As we have seen, in this article Einstein presented what he called a heuristic viewpoint on the nature of radiation, considering in particular the emission, absorption, and propagation of light in space. Einstein proposed that in the limit where Wien’s law is valid light shows atomistic features behaving in “a thermodynamic sense, as if it consisted of mutually independent energy quanta” (Einstein, 1905, p. 102). An important step forward was made by the end of 1908, when Einstein came up with a new argument in favor of the light quanta hypothesis. Einstein studied the energy fluctuations of radiation enclosed in a cavity (Einstein, 1909a). Let us consider radiation inside an isothermal enclosure at a constant temperature, and take the radiation system to be composed of two parts one of them with a small volume v (and the other with a volume V). The idea is to determine the (mean square) energy fluctuation in v of the instantaneous energy η from the equilibrium value η 0 . Denoting the entropies of the two parts as Σ and σ ( for the volume v ) , and taking Σ 0 and σ 0 to be the equilibrium values, the total entropy can be written as L 2 0 2 2 0 00 d )(d 2 1 d )d( Sε       ε σ+∑ +ε       ε σ+∑ +σ+∑=σ+∑= , where η = η 0 + ε. By taking V to be much larger than v, the deviation of the entropy from its equilibrium value is given by 2 0 2 2 d )(d 2 1 Sε       ε σ+∑ =∆ , to second order in ε . By combining this result with Einstein’s reinterpretation of the Boltzmann principle it is possible to obtain the probability of an energy fluctuation between ε and ε + d ε ( Klein, 1964, pp. 10-11): 69 de const. dW 2 0 2 2 d d R2 1 ε⋅⋅= ε           ε σΝ − . From this it is immediate to determine the mean value 2 ε of the square of the energy fluctuation of the radiation occurring in v 0 2 2 2 d d R 1       ε σΝ =ε . (Einstein, 1909a, p. 365) The essential point now is to consider the Planck’s law as empirically given and use it to calculate σ . After a simple calculation, Einstein obtained the result      η ⋅ νπν +ην κΝ =ε vd8 c h R 2 0 2 3 0 2 , for the mean value of the fluctuations of the radiation energy present in v. According to Einstein, from the classical Maxwell-Lorentz electrodynamics we would expect only to obtain the second term. The appearance of the first term was a new piece of evidence for the light quanta hypothesis. According to Einstein “the first term, if present alone, would yield a fluctuation of the radiation energy equal to that produced if the radiation consisted of point quanta of energy h ν moving independently of each other” (Einstein, 1909a, p. 366). Einstein made a first remark on his results, mentioning that it was as though there was two independent causes for the energy fluctuations: “the formula says that in accordance with Planck’s formula the effects of the two causes of fluctuation mentioned act like fluctuations (errors) arising from mutually independent causes (additivity of the terms of which the square of the fluctuation is composed)” (Einstein, 1909a, p. 369). In a conference held at Salzburg in September 1909, Einstein made further remarks related to his results. At this conference Einstein set forward the idea of a dual character of light: light possesses certain fundamental properties that can be understood far more readily from the standpoint of Newton’s emission theory of light than from the standpoint of the wave theory. It is my opinion that the next stage in the development of theoretical physics will bring us a theory of light that can be understood as a kind of fusion of the wave and emission theories of light. (Einstein, 1909b, p. 379) Einstein proposed a tentative model in which each light quantum could be regarded as a singularity in space surrounded by a field. This was meant simply as a suggestion to make plausible the idea that “the two structural properties (the undulatory structure and the quantum structure) simultaneously displayed by radiation according to the Planck formula should not be considered as mutually incompatible” (Einstein, 1909b, p. 394). A theory taking into account the dual nature of light would have to wait for the development of quantum mechanics. In two articles published in 1925, Heisenberg’s formalism – based on a particular law of multiplication of quantum-theoretical quantities derived by correspondence 70 arguments – was developed with more formal mathematics by using matrices (van der Waerden, 1967, pp. 36-57; Sánchez Ron, 2001, pp. 419-443). In these articles the question of the quantization of the electromagnetic field was considered. In the first article an unsuccessful attempt was made to treat a quantized electromagnetic field in interaction with a Hertzian dipole oscillator (Mehra & Rechenberg, 2000, p. 200; Darrigol, 1986, p. 220). The advance brought by this method was made clearer in the second article, where in the last section Jordan treated solely the situation of a free electromagnetic wave in a cavity. In a mathematically unrigorous way, Jordan was able to obtain Einstein’s formula of 1909 for the average squared energy fluctuations in radiation (Darrigol, 1984, p. 450; Darrigol, 1986, p. 221-222). To arrive at this result Jordan took advantage of the possibility to treat a wave as an infinite set of linear harmonic oscillators corresponding each to a normal mode of the wave. By using the recently formulated formalism of matrix mechanics it was a simple task to quantize a system of independent linear harmonic oscillators (Kramers, 1964, p. 422), which corresponds to imposing the quantum commutation conditions to each normal mode (Schweber, 1994, p. 11). To avoid calculational complications Jordan considered the simplified case of a onedimensional string fixed at both ends of the cavity. The lateral displacement u (x, t) of a string with length l is given, when expressed as a Fourier series, by x l sin t)qt)u(x, 1 k π κ(= ∑ ∞ =κ . The energy in a part (0, a) of the string is given by dx x l cos x l j cos l jkqq x l sin x l jsin qq 2 1 E a 0 2 k jkj 1j, ∫∑          π κ π      π + π κ π = ∞ =κ && . Using the previous expression, the fluctuation in the energy is given by 2a 2 2 Ε =∆ . This is exactly the term in Einstein’s derivation resulting form the interference of classical electromagnetic waves. Up till now this is a totally classical derivation applied to any classically described wave phenomena. Going over to quantum mechanics the previous expressions for u(x, t) and E become matrix equations. The coefficients of the Fourier expansion q k become matrix elements q k (n, m) which vanish except when n j – m j = 0 for j ≠ k, n k – m k = ±1. Following a straightforward quantum mechanical calculation Jordan now obtained the result Einstein had obtained for the fluctuation in the energy by purely statistical methods: 71 Vz h 2 2 ν Ε +Εν=∆ , where z ν is the number of characteristic frequencies in the interval d ν and V is the volume that contains the string. The most important aspect of this, otherwise straightforward, procedure was the conceptual implications brought with it. As said before, in the procedure followed by Jordan “ each individual eigenvibration [normal mode] is to be treated as a simple linear oscillator” (Born, Heisenberg & Jordan, 1925, p. 375). The fundamental conceptual change was that “Jordan saw that he could relate light quanta to wave excitations, and elucidate Bose’s counting as the natural counting for quantized waves … Furthermore, energy discontinuities in the field appeared as properties of radiation itself, and not of the material emitters” (Darrigol, 1986, p. 221). That is, Jordan found a relation between the light quanta concept of Einstein satisfying Bose-Einstein statistics – light quanta that cannot be seen as classical independent particles, but that show statistical dependence – with the quantized normal modes of the electromagnetic field. According to Jordan, “so strong an association between the eigenvibrations of a cavity and the light quanta postulated formerly can nonetheless be drawn that every statistics of cavity eigenvibrations corresponds to a definite statistics of light quanta and conversely” (Born, Heisenberg & Jordan, 1925, p. 376). Also we have that the states of the system of oscillators [corresponding to the normal modes of the electromagnetic field] can be characterized by ‘quantum numbers’ n 1 , n 2 , n 3 ,… of the individual oscillators, so that apart from a additive constant the energies of the individual states are given by ∑ ν=Ε k kkn nh . (Born, Heisenberg & Jordan, 1925, p. 377) Following Jordan we must also see n 1 , n 2 , n 3 ,... as the number of light-quanta (photons) inside the cavity with a frequency ν 1 , ν 2 , ν 3 ,…. Further developments were made by Dirac. In his ‘The quantum theory of the emission and absorption of radiation’, published in February 1927, Dirac developed a quantum treatment of the electromagnetic field from two different approaches, which at the quantum level gave the same mathematical result. In the final part of his paper he extended Jordan’s initial work on the quantization of the electromagnetic field (Schweber, 1994, pp. 9-11). However, the main aspect of Dirac’s work was not the quantization of a wave; on the contrary his paper is mainly a treatment of an assembly of identical quantized particles. As we have already seen, for Dirac this method, later called ‘second quantization’, turns out to be simply a different procedure to impose the Bose-Einstein statistics on the particles. He could instead have simply selected symmetrical wave functions as the physically admissible wave functions for this type of particle (Dirac, 1926, p. 672). Dirac started with an assembly of N similar independent particles (subject to an interaction with an atomic system), whose wave function, solution of the Schrödinger equation, is ψ = Σ r a r ψ r (where ψ r is the eigenfunction of a particle in the state r). Dirac took the expansion coefficients a r to be canonical conjugates. Working with the canonical variables b r = a r e -iwrt/ħ and b r * = a r * e iwrt/ħ (where w r is the energy of a particle in the state r), Dirac assumed that these variables were “q-numbers satisfying the usual quantum conditions instead of c-numbers” (Dirac, 1927a, p. 251). This gives the false impression that an additional quantization scheme is being used, but what is being done is changing from a configuration space representation to an occupation number representation (Cao, 1997, pp. 166-167). The commutation relation [b r , b s *]= δ rs holding 72 between b r and b r * serves to impose the symmetrization of the wave functions. This implies that the particles obey Bose-Einstein statistics (Dirac, 1927a, pp. 252-255; Schweber, 1994, p. 28). As mentioned previously, in the final part of his work Dirac considered the quantization of a classical electromagnetic wave. Starting with the classical Hamiltonian describing an atom in interaction with radiation, the field was described by canonical variables, which in the quantization procedure were taken to be q-numbers satisfying the usual quantum commutation relations. Dirac took the field to be described by “the canonical variables N r , θ r , of which N r is the number of quanta of energy of the component r, θ r , is its canonically conjugate phase, equal to 2 π h ν r times φ r [as determined by r φ & = ∂ Η / ∂E r = 1, where H is the field’s Hamiltonian, and E r is the energy of a component labelled r]” (Dirac, 1927a, p. 244). We see here Dirac following Jordan’s interpretation of the normal modes of the field in terms of the number of lightquanta associated to each eigenvibration. With this procedure Dirac obtained a Hamiltonian for the quantized system, which was, according to his view, consistent with the Hamiltonian obtained using the particle view (Dirac, 1927a, p. 263). Dirac considered that this work demonstrated the equivalence between a quantized electromagnetic wave and a system of bosons (lightquanta). Implicit in this conclusion is the identification of the quanta of energy with the particles (light-quanta). In order to get this result, a particle cannot cease to exist when it is apparently absorbed, or be created when it is emitted. It is therefore necessary to have an infinite sea of light-quanta (photons), in a state in which their momentum and energy are zero, from which the particle can jump from or into (Cao, 1997, pp. 163-164; Dirac, 1927a, p. 261). In his approach to the quantization of a classical electromagnetic wave, Dirac “resolved the radiation into its Fourier components, and supposed that their number is very large but finite. Let each component be labelled by a suffix r … each component r can be described by a vector potential k r chosen so as to make the scalar potential zero” (Dirac, 1927a, p. 262). That is, in modern term, Dirac quantized the vector potential in the Coulomb gauge. This meant to consider the vector potential (usually taken in classical electrodynamics to be a subsidiary concept when compared with the electric and magnetic fields) as an operator, and in this way more fundamental quantummechanically than the electric and magnetic fields strengths. This had never been tried before (Kragh, 1990, p. 126). Another important contribution in the development of a quantum theory of the electromagnetic field (and its interaction with matter) was made in 1928. In parallel with a line of work based on the quantization of de Broglie waves (mentioned in the previous chapter), Jordan worked out with Pauli a relativistic quantization of the free electromagnetic field. Dirac’s quantization method was not relativistically invariant (as was also the case with Jordan’s quantization of de Broglie waves). The fields, taken to be in a cavity of volume V, were expanded in Fourier series, each term corresponding (mathematically) to a harmonic oscillator E k = i(ћ ω κ /2 V) ½ ε εε ε α {a k,α exp (–i ω k t + ik.r) – a *κ, α exp (i ω k t – ik.r) }, B k = i(ћ /2 V ω κ ) ½ k × ε εε ε α {a k, α exp (–i ω k t + ik.r) – a *k, α exp (i ω k t – ik.r) } ( ε εε ε κ is a unit linear polarization vector), where the operators for the total electric and magnetic fields are given by 73 ∑ =Ε k k E , ∑ =Β k k B . The amplitude (normal mode) coefficients were taken to be operators satisfying the commutation relations: [a k, α , a *k’, α ] = δ kk’ δ αα , [a k, α , a k’, α’ ] = [a *k, α , a *k’, α’ ] = 0. From this it follows immediately (as in the simplified case taken by Jordan in 1925) that the energy of the quantized electromagnetic wave is given by ∑ ω=Ε α k kk, N h , where N k,α is the number of light quanta in a state with momentum k and polarization α . From these commutation relations, Jordan and Pauli determined the commutation relations for the electric and magnetic fields strengths themselves: [ ] )t't,r'D(r rrt'tc ci4t',r'tr, kj 2 jk kj −−⋅         ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ δ π=)(Ε),(Ε h , [ ] [ ] )(Β),(Β=)(Ε),(Ε t',r'tr,t',r'tr, kjkj , [ ] [ ] )t't,r'D(r t' z ci4t',r'tr,t',r'tr, yxyx −− ∂ ∂ ∂ ∂ π−=)(Ε),(Β−=)(Β),(Ε h , with cyclic permutations of x, y and z in this last expression. Essential to achieve the goal of an expression that was manifestly relativistically invariant, was the introduction by Pauli of a relativistic generalization D (r, t) = 1/4 π r { δ (r + ct) – δ (r – ct)} of the Dirac delta-function δ( r ) appearing in the non-relativistic commutation relations (Mehra & Rechenberg, 2000, p. 220). By that time, Pauli and Heisenberg were considering the possibility of trying to develop a relativistic quantum electrodynamical theory picking up Jordan’s idea on the quantization of matter and radiation fields. By taking into account correspondence ideas (Miller, 1994, p. 31; Mehra & Rechenberg, 2000, p. 318; Darrigol, 1984, p. 484), in 1928 Pauli and Heisenberg took as their theoretical starting point the classical Lagrangian field theory of continuous systems (that can be applied for example to classical sound waves). In general the Lagrangian density for a classical field can be written as L = L( ψ α , grad ψ α , α ψ & ). 74 By calculating the conjugate momentum of each field (described, as Dirac did, by the scalar and vector potentials), the quantization of the field followed from imposing commutation relations between field components and their canonically conjugate momentum. The canonical momentum π α conjugated to each field component is α α ψ∂ ∂ =π & L . The Hamiltonian density can be written as L ∑ α αααα −ψπ=)ψ,π(Η & . The canonical cuantization results from treating π α and ψ α as operators and imposing the commutation relations: [ π α , ψ α ] = ћ / i. This approach revealed itself inapplicable (as it stood) to the electromagnetic field, because the conjugate momentum of the scalar potential was identically zero. The Lagrangian density for the electromagnetic field is in terms of potentials defined as              ϕ∇− ∂ ∂ −)( π = 2 2 t A c 1 A curl 8 1 L . We do not have dA 4 /dt = id ϕ /dt in the expression for L (where ϕ is the scalar potential). This implies that its conjugate momentum π 4 = 0. From this we have the commutation relation [ π 4 , ψ 4 ] = 0, instead of ћ / i. This made impossible the quantization of the electromagnetic field by this method. In January 1929, Heisenberg came up with a formal trick that made it possible to circumvent this difficulty and quantize the electromagnetic field; it consisted in adding a new term to the electromagnetic field Lagrangian depending on a parameter ε that was taken to zero ( ε → 0) in the end of the calculations (Darrigol, 1984, p. 484; Schweber, 1994, p. 41): 2 x2 L'         ∂ ϕ∂ε = α α . In their work Pauli and Heisenberg demonstrated the equivalence of their quantization method, when applied to the free electromagnetic field, with the previous procedure adopted in the work of Pauli and Jordan (Mehra & Rechenberg, 2000, pp. 322-323). When Pauli and Heisenberg sent their work to publication in 19 march 1929, it was still very incomplete, in particular it was lacking clear applications (Mehra & Rechenberg, 2000, p. 326), and it did not deal with the problem of the infinite selfenergy of the charged particles, which was simply discarded (Miller, 1994, p. 34). 75 Independently of the Pauli and Heisenberg approach, in 1929, Enrico Fermi developed a more straightforward approach in which, contrary to Dirac’s 1927 work, not only the radiation field was quantized, but the full electromagnetic field described by the vector and scalar potentials, which can be seen as the components of a covariant four-vector electromagnetic potential (in modern terms Dirac worked in the Coulomb gauge, where only the two components of the transverse radiation field are quantized). To achieve this goal Fermi considered “an electromagnetic field of the most general type” (Fermi 1932, 125; see also Fermi, 1929), using the d’Alembert equation for the vector and scalar potentials: A µ = j µ . To make this equation equivalent to Maxwell’s equations, Fermi had to take into account the so-called Lorentz condition ∂ µ A µ = 0, which he regarded as a condition to be satisfied by the field operators. Making a Fourier expansion of the vector and scalar potentials, Fermi easily obtained a Hamiltonian depending on the charged particle variables and the coefficients of the Fourier series expansions of the electromagnetic potentials. From this a quantization procedure was immediately possible: “it is sufficient to consider the Hamiltonian expression as an operator in which, according to the usual rules, the momenta are equivalent to the operator of differentiation with respect to the corresponding coordinate and to multiplication by – h/2πi” (quoted in Schweber, 1994, p. 74). In the months following their first paper on quantum electrodynamics, Pauli and Heisenberg improved their approach, in particular avoiding formal tricks like the use of the ε parameter and others (Carson, 1996, pp. 111 & 113). The key was gauge invariance. According to Pauli: In the new paper of Heisenberg and myself it will be shown that exactly with the help of gauge invariance these ε-terms can be avoided. As gauge invariance I consider a substitution [of the matter fields ψ and ψ and the electromagnetic potentials µ Φ ] µ µµ λλ ∂ λ ∂ −Φ=Φ′ ,ψ=ψ′ ,ψ=ψ′x ee i-i , where λ may even be a q-number but one connecting with ψ and ψ on every space-like section t = const. (Quoted in Mehra & Rechenberg, 2000, p. 328) By taking into account gauge invariance, Pauli and Heisenberg were able to adopt a gauge that made the calculations simpler, the Coulomb gauge, already used (implicitly) by Dirac in 1927, in this way avoiding the use of artificial parameters in the quantization procedure. Since they dominated the aspect of proving the relativistic invariance of the formalism, they were able to show that the Coulomb gauge, which is not manifestly covariant, did not affect the relativistic invariance of the theory: “all statements about gauge invariant quantities satisfy the condition of relativistic invariance” (quoted in Mehra & Rechenberg, 2000, p. 329). They also showed that their formalism reduced to Dirac’s theory in the non-relativistic limit (Mehra & Rechenberg, 2000, p. 330). In this second work, Pauli and Heisenberg noticed Fermi’s note on quantum electrodynamics, and re-worked it using their Lagrangian formalism instead of the Hamilton formalism adopted by Fermi. They showed that by changing the electromagnetic Lagrangian by the term – 1/2 (∂A µ /∂x µ ) 2 the d’Alembert equation used by Fermi as his starting point could be derived. Also they noticed that the subsidiary condition ∂ µ A µ = 0 (which implied working in the Lorentz gauge) necessary to make the d’Alembert equation equivalent to Maxwell’s equations, could not, as Fermi initially 76 thought, be imposed as an operator identity but solely as a supplementary condition on admissible state vectors: (∂ µ A µ )Ψ = 0 (Schweber, 1994, p. 75). Fermi in part elucidated the physical meaning of the subsidiary condition in a subsequent note. Contrary to Dirac’s work of 1927, in Pauli and Heisenberg’s first paper they did not consider solely the radiation field (described by a transverse vector potential with two degrees of freedom, corresponding to two polarizations perpendicular to the direction of propagation of the wave), but like Fermi they considered the four degrees of freedom associated with the full vector and scalar potentials (Carson, 1996, p. 110). But they did not discuss the relation between the scalar and vector potentials adopted and the transverse vector potential of the radiation field (Carson, 1996, p. 112). Neither did Fermi address this point in his first note, he solely mentioned that he wanted to develop a more general theory than Dirac’s by taking into account the full electromagnetic interaction. Some elucidation on the role of the non-transverse components of the four-vector electromagnetic potential was given by L. Rosenfeld also in 1929 by showing (choosing a particular reference frame) that the four polarizations of each normal mode can be put in a simple relation to the wave vector: two components corresponding to transversely polarized light, one to a longitudinal polarization and another to a time-like polarization (Carson, 1996, p. 113; Källén, 1972, p. 19). 16 Adopting Pauli and Heisenberg’s view on the condition to be imposed on the available states, Fermi knew that this subsidiary condition “determine[s] the form of the dependence” (Fermi, 1932, p. 130; see also Fermi, 1930) of the system’s wave function on the scalar potential and the longitudinal component of the vector potential. Knowing this, Fermi also developed an approach where “the coordinates Qs and χs representing the scalar potential and one component of the vector potential are completely eliminated” (Fermi, 1932, p. 131; see also Fermi, 1930). By this procedure Fermi obtained a Hamiltonian whose electromagnetic part had a term corresponding to a transverse (radiation) field and a term corresponding to the Coulomb interaction between the charged particles (this procedure corresponds to going from the Lorentz gauge to the Coulomb gauge). 17 With this approach, Fermi showed that we can regard the scalar and longitudinal components of the field – that are not independent degrees of freedom of the field as implied in the subsidiary condition – as a representation of the Coulomb interaction between charged particles. At this point the problem of the quantization of the electromagnetic field in interaction with charges was basically settled. In more general terms the theory of quantum electrodynamics was still very imperfect; the problem of the infinite selfenergy of the charged particles was not solved. In fact, the theory faced even more problems with divergences in the calculations (Schweber, 1994, chapter 2). In practice, only a few second-order calculations using perturbation theory were available (Pais, 1986, pp. 374-376). The ‘serious’ divergences in the theory (Jauch & Rohrlich, 1976, pp. 174-175) were only circumvented in the late forties with the renormalization program, which made possible to obtain finite results in higher-order calculations. This was done by attaching the higher-order contributions – that from a physical point of 16 This situation already occurs (as should be expected) in classical electrodynamics. For each normal mode we can always choose a particular reference frame in which the electromagnetic potential is represented by two transverse, one longitudinal and one time-like components, and due to the Lorentz condition, the contribution of the longitudinal and time-like components compensate each other (Bogoliubov & Shirkov, 1959, pp. 55-57). 17 This result is not particular to the quantum theory of the electromagnetic field; it is also derived in the classical theory when considering the Fourier expansion of the field produced by charges, and not only the free radiation field (Landau & Lifshitz, 1971, pp. 124-125). 77 view should be smaller and smaller, but that in the mathematical calculations appear in integrals that are divergent – to the mass and charge of the electrons (and positrons), whose values are not determined by the theory but from experiments (Pais, 1986, pp. 374 & 462). The quantization of the electromagnetic field in the (manifestly covariant) Lorentz gauge as realized by Fermi, Pauli and Heisenberg, with the adopted Lorentz subsidiary condition, showed not to lead to any incongruence in practical applications, as can be seen in the work done in the forties by R. P. Feynman (Carson, 1996, pp. 128). In his approach Feynman always considered that a “photon can be polarized in any one of the four directions” (Feynman, 1961, p. 122). This was no fortuitous event. Feynman wanted a scheme that was relativistically invariant: “Everything I was computing was covariant. The way others had formulated everything, they had separated the Coulomb potential and the transverse waves … I knew which terms went together … and how to generalize to four dimensions from the two transverse dimensions” (quoted in Mehra, 1994, p. 229). Nevertheless when looking closely at the consequences of the subsidiary condition adopted in the quantum electrodynamics developed by Fermi, Pauli and Heisenberg, this approach is strictly speaking not consistent (Pais, 1986, p. 355). A solution to the inconsistency was given around 1950 by the development of a formalism based on the use of an indefinite metric for the Hilbert space (Schweber, 1961, pp. 245- 251). But in the usual way in which covariant calculations are done, where the four polarizations of the field are simultaneously taken into account – a typical case being the description of the electron-electron interaction –, the indefinite metric operator does not appear explicitly (Heitler, 1954, pp. 129-130). This gives an a posteriori justification for the procedure of Feynman and others. 3 The quantization of the electromagnetic field In simple terms, the quantization of the electromagnetic field can be presented as follows. In the case of an electromagnetic field in a region free of charges, the Maxwell Lorentz equations are: 0 = ⋅ ∇ E t c ∂ ∂ =×∇ E B 0 = ⋅ ∇ B t c ∂ ∂ −=×∇ B E . From the second pair of equations we see that the electric and magnetic fields (strenght) can be defined in terms of a scalar and vector potentials φ(x,t) and A(x, t): AB × ∇ = t c ∂ ∂ −φ−∇= A E . 84 we can conceive a single dipole in empty space. In this case “the only field acting on the dipole is its own radiation reaction field” (Milonni, 1994, p. 52). The difference from the classical case is that when considering the quantized electromagnetic field “there is an ‘external’ field, namely, the source-free or vacuum field” (Milonni, 1994, p. 52). Milonni shows that, in contrast to the classical case, the ground state of the free electromagnetic field cannot be disregarded as soon as a charged body (which can be seen as a source of electromagnetic field) is considered, as it is necessary to take into account the source-free field variables for the preservation of the commutation relations (Milonni, 1994, p. 53). Let us look at this in more detail. The Hamiltonian for the dipole oscillator in interaction with the quantized electromagnetic field can be written as F 22 0 2 m 2 1 c e 2 1 HΗ+ω+      −= xΑp , where H F is the field Hamiltonian, x is the operator corresponding to the classical coordinate of the oscillator, p is the operator for the dipole momentum, and ω 0 is the frequency of oscillation of the dipole. In the Heisenberg representation we have (in the electric dipole approximation): [ ] λ ∗ λλ / λ (−(      ωπ ==ω+ ∑kkk k Εxx et)at)a V 2 m e i m e 21 k 2 0 h && , t)(t'i t 0 21 k ti kk )et'edt' V 2 ie0)eaa −ω λ / ω− λλ (⋅         ω π +(= ∫ x kkk & h & (where a kλ and a *kλ are respectively the photon annihilation and creation operators for the field mode (k, λ), e kλ are the polarization vectors, and E is the electric field operator). In this way the Heisenberg equation for the operator x can be written as: )(+)(=ω+ t m e t m e RR0 2 0 ΕΕxx && , where we have [ ] λ ω ∗ λ ω− λ / λ (−(      ωπ =( ∑ kkk k Εe0)ea0)ea V 2 it) titi 21 k 0 kk h , [ ] x 3c e2 tt'cose)(t'edt' V e4 t) 3 k t 0 RR &&&& =)−(ω⋅ π −=( λλ λ ∑∫ kk k xΕ . According to Milonni, E 0 (t) is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e. the solution, at the position of the dipole, of the wave 85 equation [“ 2 – c –2 ∂ 2 /∂t 2 ]E = 0 satisfied by the field in the (source-free) vacuum. For this reason E 0 (t) is often referred to as the vacuum field … E RR (t) is the source field, the field generated by the dipole and acting on the dipole. (Milonni, 1994, p. 52) Considering the previous operator equation for x that can be written as )(=τ−ω+ t m e 0 2 0 Εxxx &&&&& (where, as mentioned, E 0 (t) is the vacuum electric field operator, and τ = 2e 2 /3mc 3 ), and the corresponding equation for the momentum operator p, we have the following commutation relation between the two operators [ ] h hh i mc3 e2ei x dx mc3 e2ei (t)pz(t), 3 0 3 3 0 2 6 0 22 3 0 3 2 z = τω π π ω = ωτ+ ω π ≅ ∫ ∞ ∞− , as is expected according to general quantum mechanics rules. Now if we had not considered the vacuum field E 0 (t), then in the operator equation for x, “the operator x(t) would be exponentially damped, and commutators like [z(t), p z (t)] would approach zero for t >> (τω 02 ) –1 ” (Milonni, 1994, p. 53). Because of this Milonni concluded that “the free field is in fact necessary for the formal consistency of the theory” (Milonni, 1994, p. 53). This result and Milonni’s derivation of the Casimir effect in the context of standard quantum electrodynamics have been questioned. In Rugh, Zinkernagel & Cao (1999, p. 129) two critical remarks are made on Milonni’s (and collaborators) approach. One point is that Milonni’s approach regarding the derivation of the Casimir effect as due only to source fields is not conclusive, because in higher orders of perturbative calculations we will have contributions to the vacuum energy from the so-called vacuum blob diagrams. But this is the case only when considering the ‘interacting’ vacuum (e.g. Rugh & Zinkernagel, 2002, p. 675), it has nothing to do with the quantized (source-free) electromagnetic field and its possible quantum states. The other point, also made in Rugh & Zinkernagel (2002, p. 683, footnote 50), is that Milonni uses the socalled fluctuation-dissipation theorem for linearly dissipative systems to arrive at his result that the source-free field is necessary for consistency reasons (i.e. the preservation of commutation relations), and this is not a sound approach. However, that is not the case, even if Milonni presents his results as an example of the theorem (see e.g. Milonni, 1988, p. 106). As we have seen the need for the source-free field for the preservation of the commutation relations is derived by Milonni without any need for the fluctuation-dissipation theorem (see e.g. Milonni, 1984, p. 342; Milonni, 1988, p. 106; Milonni 1994, pp. 50-54). 19 That is, according to Milonni, we must have simultaneously with a system consisting of charged matter a different system corresponding to the quantized free electromagnetic field, even if only in its ground state (when considering charged matter in empty space). However for a different reason than the one presented by Rugh & Zinkernagel I regard Milonni’s view regarding the 19 This, as mentioned, is the main difference between Zinkernagel’s views on the Casimir effect and my own. Zinkernagel considers that Milloni’s work is not sound and we must go beyond standard quantum electrodynamics to show that the Casimir effect is not a vacuum effect. My view is that Milloni’s work is sound and shows that we do not have to see the Casimir effect as a vacuum effect (see also Jaffe 2003, Jaffe 2005, Graham et al. 2004). 86 need of the free field for the formal consistency of the theory as not conclusive. The point is that Milonni arrives at his result by considering a quantum model of a dipole oscillator; it is not an ab initio derivation from the formalism of the theory. In this way we cannot conclude that Milonni’s consistency claim follows directly from the formalism of the theory. Nevertheless, as we will see in the next sections, this (proposed) formal aspect related to the vacuum adjusts well with experimental results. 5 The physical meaning of variance As is well known, in the ground state of the quantized electromagnetic field the expectation value of the electric and magnetic fields vanishes, that is 00000 == ΒΕ , but not its variance, because 00 2 Ε and 00 2 Β are non-zero in the ground state. What to make of this result? There is a tendency in the literature to refer to the non-vanishing variance as ‘fluctuations’ of the vacuum state (Sakurai, 1967, pp. 32-33; Aitchison, 1985, pp. 246-247). For example, according to I. J. R. Aitchison “the vacuum can now be thought of as a state in which the fields are all in their ground states, but executing random fluctuations (even at T = 0) about their zero average values” (Aitchison, 1985, p. 347). We must take some care in adopting this type of terminology. As has been noticed, we cannot associate the non-vanishing variance of the ground state of the quantized electromagnetic field to some sort of fluctuation in time: “there is no time evolution of this vacuum state” (Rugh & Zinkernagel, 2002, p. 673). I will defend here that the mathematical result of a non-vanishing variance of the quantized electromagnetic field can be given an interpretation that has a clear experimental meaning. According to the interpretation of quantum mechanics adopted here, the non-zero variance (or its square root, the standard deviation) is determined by considering a large (ideally infinite) number of measurements performed in similarly prepared systems (Isham, 1995, pp. 80-81; Peres, 1995, pp. 24-26; Ballentine, 1998, pp. 225-227, Falkenburg, 2007, pp. 205-207). According to what C. J. Isham called the minimal interpretation of quantum theory (which as other authors I refer to as the ensemble interpretation): Quantum theory is viewed as a scheme for predicting the probabilistic distribution of the outcomes of measurements made on suitably prepared copies of a system. The probabilities are interpreted in a statistical way as referring to the relative frequencies with which various results are obtained if the measurements are repeated a sufficiently large number of times. (Isham, 1995, p. 80) The view of A. Peres is that A quantum system is a useful abstraction … defined by an equivalent class of preparations. For example there are many equivalent macroscopic procedures for producing what we call a photon, or a free hydrogen atom, etc. The equivalence of different preparation procedures should be verifiable by suitable tests…. While quantum systems are somewhat elusive, quantum states can be given a clear operational definition, based on the notion of test. Consider a given preparation and a set of tests … if these tests are performed many times, after identical preparations, we find that the statistical distribution of outcomes of each test tends to a limit. Each outcome has a definite probability. We can then define a state as follows: A state is characterized by the probabilities of the various outcomes of every conceivable test…. Before we examine concrete examples, the notion of probability should be clarified. It means the following. We imagine that the test is performed an infinite number of times, on an infinite number of replicas of our quantum system, all identically prepared. This infinite set of experiments is called a statistical ensemble. 87 … In this statistical ensemble, the occurrence of event A has relative frequency P{A}; it is this relative frequency which is called probability. (Peres, 1995, pp. 24-25) Under this view we cannot associate for example the Schrödinger wave function to a single system. We must consider a large number of identical ‘quantum systems’ prepared in the same way and then subjected to the same measurement procedure. From the wave function we can calculate the relative frequencies (probabilities) of particular outcomes regarding physical observables of the system. 20 Now we can address the meaning of having 00 2 Ε ≠ 0. We must consider a particular experimental setup that will enable us to make measurements on a quantum electromagnetic field. We can consider successive measurements made on a field, which can be taken to be in the same initial state at each successive measurement, or we may think in terms of different identical experimental setups being used at the same time to make measurements of identically prepared fields (in this case all in the vacuum state). Then on each of a large number of similarly prepared systems a measurement is made of the electric field. According to the adopted interpretation of the theory, there will be a frequency distribution for the results of the independent measurements according to a standard deviation of 00 2 Ε from a measurement corresponding to no transverse photos. That is, sometimes an electric field will be measured different from zero even if the electromagnetic field is in its ground-state. Is this again a formal aspect of the theory, even if a formal aspect of its interpretation? Or is there any real experiments where a measurement is made on the vacuum field and results are obtained corresponding to a frequency distribution of results deviating from ‘nothingness’? 6 Experimental results on the vacuum state using the balanced homodyne detection method In experiments using the method known as balanced homodyne detection it is possible to determine what can be interpreted as quadrature fluctuations of the vacuum or simply vacuum noise which corresponds to the non-zero standard deviation predicted by the theory (Leonhardt, 1997, pp. 23, 47 & 84-88). In the balanced homodyne method, the signal under study is sent into one of the ports of a beam splitter. In the present case since we are making a measurement of the vacuum electromagnetic field, the port is left unused, that is, there is no external field present. The other port receives a strong coherent laser field (called the local oscillator), which will provide the phase reference for measuring the quadrature statistics of the signal field, which in this case is the vacuum. We can write the quantum field operator for a single-mode field as ( ) sin(kz)eaae Vε Eti*ti 21 0 x ωω− / +        ω =h , 20 In the appendix, I will give a presentation of part of Bohr’s interpretation of quantum mechanics, providing a Bohrian reinterpretation, or better, an integrated view on Bohr’s ideas, enlightening the internal consistency of Bohr’s ideas, and showing its compatibility with the so-called minimal or ensemble or statistical interpretation addressed in the main text. 88 where ω is the frequency of the mode and k is the wave number related to the frequency according to k = ω /c. Defining the quadrature operators )+(= aa 2 1 q*, )−(= aa 2 i p*, the field operator can be written as t))sin(pt)cos(sin(kz)(q Vε 2E 2/1 0 x ωω ω +         =h. The balanced homodyne detection makes it possible to make a measurement of these quadrature components of a quantum field. The beam splitter will combine the incoming fields. Taking, for simplicity, each incoming field to be described by the mode operators a and b, the output modes are )+(= iba 2 1 c , )+(= iab 2 1 d . After the optical mixing of the signal and the local oscillator, each beam is directed towards a photodetector, which enables a measurement of the field intensities. The photodetectors respond to the intensity of the incident light, generating the photocurrents Ic = cc * and Id = dd * . By assuming that the photocurrents are proportional to the photon numbers of the beams striking each detector we have that the difference I21 = I1 – I2 is proportional to the difference in the photon numbers of each beam θ / α=−= q2nnn 21 1221 , where 2 αis the intensity of the coherent field (local oscillator), and qθ is a quadrature component of the vacuum field (signal). By changing the phase of the coherent field it is possible to measure an arbitrary quadrature of the signal field. In particular if the phase was initially chosen so as to measure X1 by changing the phase by π/2 it is possible to measure X2 and in this way a balanced homodyne detector enables one to measure the quadrature components defining the quantum state of the field (Gerry & Knight, 2005, pp. 167-168). Let us see in more detail what the experiment tell us about the vacuum state. First we must recall the interpretation of the quantum formalism. If we use the balanced homodyne detector to make one measurement of a quadrature component that by itself does not give us any valuable result: 89 It must be distinguished between an individual (single) and an ensemble measurement (i.e. in principle, an infinitely large number of repeated measurements on identically prepared objects). Performing a single measurement on the object, a totally unpredictable value is observed in general. (Vogel & Welsch & Wallentowitz, 2001, p. 225) In this way we need an experimental procedure that may give us the relative rate at which a particular value for a quadrature component qθ is observed, i.e. the probability distribution pr(q, θ), where θ is the relative phase between signal and local oscillator. Thus, making a large number of measurements of the observable qθ yields pr(q, θ), i.e. the probability distribution of its eigenvalues. In general the experimental procedure goes as follows: The phase θ can be easily varied using a piezo-electric translator. To measure quadrature distributions, we may fix the phase angle θ and perform a series of homodyne measurements at this particular phase to build up a quadrature histogram pr(q, θ). Then the [local oscillator] phase should be changed in order to repeat the procedure at a new phase, and so on [, in such a way that we obtain results for a set of different phase angles between 0 and π]. (Leonhardt, 1997, p. 99) The probability distribution pr(q, θ) is equal to q)(U)U(q * θρθ , where ρ is the density operator (which provides the most general description of a quantum state), and U(θ) = exp(–iθn) is the phase-shifting operator (where n is the photon number operator). From the experimentally obtained probability distribution pr(q, θ), it is possible to reconstruct the so-called Wigner function W(q, p), which is closely related to the density operator. In reality both can be seen as “one-to-one representations of the quantum state” (Leonhardt, 1997, p. 40). The interesting part comes now. In the case of the vacuum field (like in all others), we have a good agreement between the experimentally reconstructed Wigner function and the theoretical Wigner function (Leonhardt, 1997, pp. 46-47). In particular, if we consider the quadrature wave function of the vacuum state        −π=)(ψ /− 2 q expq 2 41 0 , the measured quadrature probability distribution 2 0 q)(ψ “is approximately Gaussian and already follows the theoretical expectation” (Leonhardt, 1997, p. 23). In general there is a good agreement between the theoretical predictions for several quantum states of light (e.g. single-photon Fock states and squeezed states) and the experimental results (e.g. Breitenbach & Schiller & Mlynek, 1997; Bertet et al, 2002). This gives some assurance that the results obtained in the case of the vacuum state can be taken to be a property of the vacuum state and not as resulting from some other physical origin, for example, from the matter of the photodetectors. I mention this because in these experiments we are considering a high intensity field that is treated, by correspondence arguments, as a classical field that interferes with the vacuum field, producing two different beams. As I said, each beam is directed to a photodetector. Ideally each of the two photodetectors will produce a photocurrent that is proportional to the number of photons of the beams striking each one. It would seem that we are detecting individuated photons due to the vacuum field, which would imply that the photodetectors are receiving momentum and energy from the vacuum. However we 90 must recall that what is being detected are the beams resulting from the interference of (what can be considered) a classical field and the vacuum field. The possible ‘photons’ from the vacuum are not ‘differentiated’ from the ‘photons’ from the classical field. We must take into account the usual identification of a classical field with a quantum coherent field with a large expectation value of the photon number operator. The coherent state does not have a definite number of photons. In fact it can be defined by an infinite expansion in terms of photon number states, that is, by taking into account photon states corresponding to an infinite number of photons. Due care is needed in the physical interpretation of this situation, in particular in what concerns the possibility of detection of photons in the ground state, which I consider not to be possible (and theoretically nonsense), and this makes problematic the usual quantum theory of the photodetectors based on the idea of photon absorption (Vogel & Welsch & Wallentowitz, 2001, pp. 169-190). More than questioning the experimental results regarding the vacuum electromagnetic field, this points to a need of revision of the theoretical treatment of the interaction of photodetectors with the quantized electromagnetic free field. However, I believe that there is no final and conclusive approach to address this problem. In quantum electrodynamics we face an ‘intrinsic’ limitation in the description of the interaction of radiation and matter (since we are only able to make approximate calculations) that leads me to consider that we cannot go beyond a ‘model’ level of description of the interaction between the quantized electromagnetic field and the photodetectors. However, this problem is well beyond the scope of the present work. As I said, what gives me some confidence in the interpretation of the experimental results for the quantum electromagnetic vacuum is the coherence between the interpretation of 00 2 Εand the experimental results obtained for different quantum states of light, including the vacuum state. We see then that we can ascribe to the vacuum state of the quantized electromagnetic field a clear operational meaning in terms of the measurement of a non-zero variance, which is consistent with the adopted interpretation of the quantum formalism. 7 Conclusions In this chapter I am more interested in a tentative clarification (or at least in a contribution towards it) of the concept of vacuum in quantum electrodynamics than in possible philosophical ramifications of the view presented. The lengthy, but necessary, technical discussions presented show that there are intricate aspects that make it very difficult to have any strong metaphysical commitment regarding the concept of vacuum; more than exploring possible metaphysical consequences of the conceptual analysis being presented, the intention here is to provide a ‘frame’ within which to make clear what we should not attribute to the concept of vacuum and what we might attribute to it even if with some reservations. We have seen that there are subtle theoretical and experimental aspects related to the ground state of the quantized electromagnetic field, which represent a clear departure from the ‘nothingness’ of the classical concept of vacuum. However, we must be careful not to ascribe too much to the ground state of the quantized field. One difference with the classical theory is that (if accepting Milonni’s consistency claim to be generally valid) when considering a charged particle we must consider it to be at least in ‘interaction’ with the ground state of an external quantized electromagnetic field. In a certain sense the quantized radiation and matter need a more integrated description, while maintaining a clear distinction between radiation and matter. As Milonni stressed, 91 “without [the source-free field] the whole quantum theory of a charged particle in vacuum becomes inconsistent” (Milonni, 1994, p. 125). As we have seen the other aspect in which the quantum concept of vacuum presents a clear departure from the classical ‘nothingness’ is at the experimental level. We see experimental results that for their interpretation make it necessary to take into account the concept of quantum vacuum. The view defended here is that we can deflate the socalled experimental vacuum effects to a simple aspect common to any n-photon state of the quantum field: a non-vanishing variance of the electric and magnetic fields.21 That is, there are no dynamical effects of the vacuum state. In this way the formal considerations (pointing to the need of an external ‘independent’ quantized electromagnetic field even if in its ground state) and the experimental results (giving observational meaning to the mathematical expression of the variance even in the case of the vacuum state) are consistent with each other and point to a concept of vacuum that has theoretical and experimental relevance. However the only experimental results that we can attribute with relative security to the vacuum state relate to a non-vanishing variance and not to some spectacular dynamical effects. This implies that we cannot attribute any feature to the vacuum state that makes it ‘special’ when compared with other states of the quantized electromagnetic field. In this way, whatever metaphysical ramifications might be related to the concept of vacuum they do not go beyond the ones we might endorse in relation to any other state of the quantized electromagnetic field, which is not an unexpected result since the quantized electromagnetic field is a more fundamental concept than the particular state it might be in. Looking from this perspective, if it was the case that the vacuum state had no experimental relevance (not being possible to give an observational meaning to its variance) this would lead to an awkward situation regarding the concept of quantum field. We would be in a situation in which the physical-mathematical structure of the theory would be inconsistent with the available experimental results. On one side, for example a one photon state of the quantized electromagnetic field would have experimental meaning (and we could relate the variance to observed outcomes of measurements made on the electromagnetic field in this state) while on the other side, another state of the field, the ground state, would be a sort of a formal artifact (and in this case the variance would be a physically meaningless mathematical expression). That is not the case. Thus we see that even when not accepting dynamical fluctuations as a property of the vacuum state we can retain an experimentally meaningful notion of vacuum state that is consistent with the adopted interpretation of the variance within the ensemble interpretation of quantum theories. 21 We must recall that this variance is related with measurements made on identically prepared systems not one individual system. As mentioned, there is a tendency in the literature to refer to the non-vanishing variance as ‘fluctuations’ of the vacuum state (Sakurai, 1967, pp. 32-33; Aitchison, 1985, pp. 346-247). This view is misleading since the non-zero variance of the ground state of the quantized electromagnetic field cannot be related to a fluctuation in time: “there is no time evolution of this vacuum state” (Rugh & Zinkernagel, 2002, p. 673). Considering measurements made on equally prepared systems, they will show fluctuations in the results of the successive observations – according to the interpretation of the theory followed here (Isham, 1995, pp. 80-81; Peres, 1995, pp. 24-26; Ballentine, 1998, pp. 225-227, Falkenburg, 2007, pp. 205-207). The non-vanishing variance is not a temporal property of one single system. We observe a statistical fluctuation on the results of measurements made on equally prepared systems, not a temporal fluctuation of the same system. 92 CHAPTER 5 THE INTERACTION OF RADIATION AND MATTER 1 introducion The more basic and fundamental elements of quantum electrodynamics are already present in Dirac’s 1927 work. In it, the electromagnetic field and matter are described by classical Hamiltonians; a further term gives the interaction between the field and matter (Jordan’s reinterpretation of matter as waves or Pauli and Heisenberg’s Lagrangian formalism does not change the mathematical core of the theory and procedures used in its applications). All this can be developed within a correspondence approach with classical mechanics and field theory, that is, this type of Hamiltonian can be put to use in the Maxwell-Lorentz classical electrodynamics or a classical theory of fields in interaction (Barut, 1964, p. 138; Bogoliubov & Shirkov, 1959, p. 84). Then a second ‘layer’ is put on top of the classical description (in the fully developed theory, matter is described by the Dirac equation) through which the quantization of the individual fields is achieved (the so-called ‘second quantization’). That is, the generalized coordinates (and conjugate momenta) of each field are submitted to commutation or anticommutation relations, and the terms in the Hamiltonian for each field become operators, as is also the case for the term describing the interaction between the fields. But it is important to notice that the fields are quantized as free noninteracting fields, each by itself. Then we are into the game. For practical purposes Dirac makes use of perturbation theory to treat the interaction of radiation and matter.22 So it was then, and it still is now. In section 2 we will look into the details of setting quantum electrodynamics (as the theory that describes the interaction of matter and radiation) into ‘motion’. It turns out that quantum electrodynamics is a perturbative approach. Also quantum electrodynamics relies on the doubtful method of adiabatically switching on/off the interaction between radiation and matter. However, not looking too closely into the mathematical structure of the theory and considering only a few order terms perturbative calculations, quantum electrodynamics presents an astonishing agreement with experimental results. 22 The use of perturbative methods has a long history in celestial mechanics. One example is the development of an analytical perturbation theory for the three-body problem: the Sun-Earth-Moon system (Hoskin & Taton, 1995, pp. 89-107). From the planets, perturbative methods went to the planetary models of atoms, being a calculational tool present in the so-called old quantum theory (Darrigol, 1992, pp. 129 & 171). Also it became fundamental in the creation of matrix mechanics, as it was from the perturbative study of the anharmonic oscillator that Heisenberg developed his quantum-theoretical approach (Darrigol, 1992, pp. 266-267; Paul, 2007, pp. 4-5). Soon after, Heisenberg and Max Born put together a perturbation theory within the formalism of quantum mechanics recently developed (van der Waerden, 1967, pp. 43- 50; see also Lacki, 1998). 93 Unfortunately there appear to be severe problems with the previous view. These will be addressed in section 3. According to John Earman and Doreen Fraser, the Haag theorem seems to imply the mathematical inconsistency of the usual treatment of interactions in quantum electrodynamics using perturbative methods (Earman & Fraser, 2006), in which the concept of quanta is central. Fraser presents the idea that “once infinite renormalization counterterms are introduced … it is no longer possible to prove Haag’s theorem” (Fraser, 2006, p. 2). I shall argue that simply calling attention to the fact that renormalization “renders the theory mathematically not well-defined” (Fraser, 2006, p. 171), does not provide any answer to the question: “why perturbation theory works as well as it does” (Earman & Fraser, 2006, p. 307). This is not an unimportant question, since we are facing the puzzling situation that the predictions of quantum electrodynamics are so accurate, while the theory from which they are derived appears to be mathematically inconsistent. In this section I will try to provide an answer to this question. In the process I will try to show that Earman and Fraser’s conclusion that “Fock representations are generally inappropriate for interacting fields” (Earman & Fraser, 2006, p. 330) or Fraser’s related conclusion that “an interacting system cannot be described in terms of quanta” (Fraser, 2008, p. 842), are not valid in quantum electrodynamics: we do not have an interacting system but two weakly interacting fields (systems), and their interaction is described in terms of quanta. In a nutshell my argument will be as follows: it turns out that quantum electrodynamics, because the series expansion of the S-matrix is divergent, is unable to treat radiation and matter as one closed system. Rather the theory can only give an approximate description (using a perturbative approach) of the interaction between radiation and matter as distinct systems. If we try to close the gap and treat radiation and matter as one closed system, corresponding to an exact solution of the coupled nonlinear Maxwell-Lorentz and Dirac equations, our perturbative approach fails. My view is that there is a one-to-one correspondence between having meaningful mathematical results and being clear about the physical assumptions used to set up the theory (in quantum electrodynamics I take part of the input physical concepts to be radiation and matter taken to be independent systems that are independently quantized due to the weak interaction between then). This is so, because the physical concepts are defined in connection with a specific mathematical ‘support’, not beyond or independently of maths. In a situation where we have an infinity popping up we cannot have a welldefined use of the physical concepts involved. In the case of quantum electrodynamics this situation occurs when trying to give a full description of the (weak) interaction between matter and radiation, which would correspond to treating them as one closed system. I see the divergence of the S-matrix series expansion as a fingerprint of a tentative application of the mathematics of the theory beyond its physical content. By focusing on the basic physical concepts used to set up the theory, and not solely on its (ill-defined) mathematical structure, we can make sense of the success of quantum electrodynamics (independently of the Haag theorem). This provides a different perspective than the one adopted by Earman & Fraser that not only contradicts several of their conclusions, but also, in my view, makes quantum electrodynamics more intelligible. 2. Quantum electrodynamics as a perturbative approach In his 1927 paper Dirac dealt with a system consisting of an atom interacting with electromagnetic radiation. Dirac used a non-relativistic Hamiltonian and made his 100 quantity, it is the measured relative frequency of scattering events of a given type” (Falkenburg, 2007, p. 107) When calculating the S-matrix we must extend the initial time t 0 to –∞ and the final time t to +∞. As we have seen in Dyson’s presentation the initial (and final) state of the system is taken to be an eigenstate of H 0 the Hamiltonian for the non-interacting fields. This idea can be given a more formal presentation by considering that the interaction between particles in a scattering process is adiabatically switched on in the remote past and adiabatically switched off in the remote future (Lippmann & Schwinger, 1950, p. 473; Bogoliubov & Shirkov, 1959, p. 197). What this means is that at infinity the interaction term is taken to zero, that is, there is no interaction between matter and radiation. This has important implications. One is that since there is no interaction we are really considering two uncoupled systems, the Dirac field, and the electromagnetic field. Another point is that in this case the state vectors are taken to be a product of Fock states of each field: bare states. Looking at this in terms of scattering of particles (quanta of each field), the initial and final states of the scattered particles are states of the (freeparticle) Fock space. As we have seen the S-matrix is given as a series expansion in powers of the interaction term V(t). We can look at the description of an interaction process as described by the S-matrix as a perturbative approach in which only Fock states are considered. The use of the interaction representation in the S-matrix approach can be seen then as part of a perturbative approach in which the interaction term is leading to a perturbation of the free states of the fields. To see the importance of the adiabatic ‘switching on’ and ‘switching off’ of the interaction to this overall perturbative approach let us look into Dyson’s take on this subject. As we have seen Dyson gave an operational justification to the use of a Fock state function φ B to describe the initial and final state of the Dirac field, instead of Ψ B (t) the state-vector representing for example a self-interacting electron. Dyson asks the question: “what is the connection between Ψ B (t) and φ B ?”(Dyson, 1952a, p. 94). Let us look into his presentation of the adiabatic trick. Suppose t B to be a time so long in the future after the scattering process is over, that from t B to +∞ the state B consists of separated outward-travelling particles. Then the relation between ψ B (t) and φ B is simple. We imagine a fictitious world in which the charge e occurring in the radiation interaction decreases infinitely slowly (adiabatically) from its actual value at time t B to zero at time (+∞). In the fictitious world, the state ψ B (t B ) at time t B will grow into the bare-particle state φ B at time +∞. Thus φ B = Ω 2 (t B ) ψ B (t B ) (489) where )(tg)(t)}g(xA,),(xAP{dxdx n! 1 c e )(t nB1Bn t1n t1 0n n B2 B B KKKK h ψψψψ //       =Ω ∫ ∫ ∑ ∞ ∞ ∞ = (490) and g B (t) is a function decreasing adiabatically from the value 1 at t = t B to zero at t = ∞. Similarly, when t A is a time so far in the past that the state A consists of separated converging particles from t = −∞ to t = t A we have )(tg)(t)}g(xA),(xAP{dxdx n! 1 c e )(t nA1An t 1n t 1 0n n A1 A A KKKK hψ/ψ,ψ/ψ       =Ω ∫ ∫ ∑ ∞− ∞− ∞ = (492) where g A (t) is a function increasing adiabatically from t = −∞ to t = t A . 101 The scattering matrix element between states A and B is given exactly by ))(t)S(t( M A t tB B A Α ∗ Β ψψ= (493) )g(t))}g(t(xA),(xAP{dxdx n! 1 c e S n1n t t1n t t1 0n n t t A B A B B A KKKK hψ/ψ,ψ/ψ       = ∫ ∫ ∑ ∞ = (494) Of course (493) is independent of the times t A and t B . When t A and t B are chosen so far in the past that (489) and (491) are satisfied, then (493) may be written in the form (487), where now )(t)S(t S A2 t tB2 B A ΩΩ= )g(t))}g(t(xA),(xAP{dxdx n! 1 c e n1n1n1 0n n KKKK hψ/ψ,ψ/ψ       = ∫ ∫ ∑ ∞ ∞− ∞ ∞− ∞ = (495) and g(t) is a function increasing adiabatically from 0 to 1 for −∞ < t < t A , equal to 1 for t A ≤ t ≤ t B , and decreasing adiabatically from 1 to 0 for t A < t < ∞. Thus we come to the important conclusion that [the] formula [M = (φ Β∗ Sφ Α ) (487)] for the matrix element is correct, using the bare particle state-functions φ A and φ B , provided that [the] formula )}(xA,),(xAP{dxdx n! 1 c e S n1n1 0n n ψψψψ //       = ∫ ∫ ∑ ∞ ∞− ∞ ∞− ∞ = KKK h (488) for S is interpreted by putting in the slowly varying cut-off function g(t i ) to make the integrals converge at t i = ±∞. The cut-off functions are to be put in as they appear in (495), and then S is defined as the limit to which (495) tends as the rate of variation of g(t) is made infinitely slow. The main practical effect of this limiting process in the definition of S is to justify us in throwing away all terms in the integrals which oscillate finitely at t i = ±∞. There are however certain cases in which the integral (488) is in a more serious way ambiguous due to bad convergence at t i = ±∞. In these cases the cut-off functions have to be kept explicitly until a late stage of the calculations before going to the limit g(t) = 1. In all cases, if the limiting process is done in this way, the matrix element M is obtained correctly and unambiguously. The use of bare-particle wave-functions φ A and φ Β in (487) is thus justified. (Dyson, 1952a, pp. 94-95) In this way, in the S-matrix approach we will be calculating transition amplitudes between free-particle states (M = (φ Β∗ Sφ Α )). Since the interaction term is given by µ µ ⋅γ ψAψe , we can also see the series expansion of the S-matrix as given in powers of e, where e is the electric charge (or in powers of the so-called fine structure constant α = e 2 /4π): S = 1 + eS (1) + e 2 S (2) + … (Källén, 1972, p. 88). Let us consider, for example, the quantum electrodynamical treatment of the twophoton annihilation of an electron and a positron (Sakurai, 1967, pp. 204-208): e + + e – → 2γ. The initial state corresponds to a Dirac field with two quanta, one corresponding to the electron, and the other to the positron (the electromagnetic field is taken to be in the vacuum state). In the final state after the ‘scattering’, the Dirac field is in the ground-state, that is, with no quanta, and the electromagnetic field is in a state with two photons. The second-order transition or scattering amplitude is given by γδναβµνµ )γ()γ()(Α)(Αγ)−(= ∫ ∫ 0xx2xdxdeS 212 4 1 42 fi 102 [ )−(θ)(Ψ)(Ψ)(Ψ)(Ψ× +)+( δ )+( α )−( γ )+( β21 - 2121 tteexxxx0 ] )−(θ)(Ψ)(Ψ)(Ψ)(Ψ− +)+( δ )+( α )−( β )+( γ12 - 2112 tteexxxx0 . I will not go into the details of this expression but only address some relevant features. The expression is an example of the perturbative quantum electrodynamical treatment of the interaction between the Maxwell and Dirac fields. In it we have the description of the propagation of quanta between two space-time points x 1 and x 2 (corresponding to Dirac’s ‘third state’), and components related to the non-interacting states of each field (Fock states): + ee - , γ2 , and the vacuum states of each field 0 . That is, we use only the Fock space for each field to calculate the interactions in quantum electrodynamics. Also, each quantum field operates in different Fock spaces. 3 Possible problems for quantum electrodynamics: the Haag theorem and the divergence of the S-matrix series expansion It seems then that we have a simple procedure to calculate whatever situation of interaction between radiation and matter we might have by simply addressing each case as if it is a scattering problem and dealing with it using the S-matrix approach with the adiabatic switching trick, where we can use as initial and final states Fock states of each field. However the situation is not that simple. One crucial aspect of all this adiabatic switching trickery is that the following supposition (called the adiabatic theorem) is being made: “If a state is an eigenstate of the Hamiltonian and if a parameter in the Hamiltonian is adiabatically changed …, then the same state is also an eigenstate, after the Hamiltonian is changed, but with a different eigenvalue” (Källén, 1972, pp. 52-53). As we have seen in Dyson’s presentation this is being taken for granted: ‘the state ψ B (t B ) at time t B will grow into the bare-particle state φ B at time +∞. Thus φ B = Ω 2 (t B ) ψ B (t B )’. Is this really the case? No! From Haag’s theorem (Haag, 1955) we know that we cannot have a unitary transformation that relates the field operators corresponding to the free Hamiltonian H and the interacting field Hamiltonian H I . Considering that at t 0 the Heisenberg picture and the Dirac picture (interaction representation) coincide (Earman & Fraser, 2006, p. 320), it would seem that the state vector in the interaction representation, in the limit t Ø ±∞, corresponds to free particles due to the fact that the interaction part of the Hamiltonian is negligible. But from Haag’s theorem it seems that “at times t = ±∞, all the assumptions of the theorem hold for the Heisenberg representation, which represents an interaction, and for the interaction representation, which is a Fock representation for a free system” (Earman & Fraser, 2006, p. 322). In informal terms Haag’s theorem implies that the state vectors in the interaction representation, that for t Ø ±∞ are supposed to represent the free field, and the state vector in the Heisenberg representation for the interacting fields, are not in a common domain of both H and H I (Schweber, 1961, p. 416). From Haag’s theorem we can conclude that if we have a free field at t = –∞, the interaction representation describes also a free field at any time t 0 . This means that we need to have a state of the full interacting Hamiltonian from the start so that we can consistently give the interaction representation its usual interpretation as giving a different time dependency to the state vector and the operators (Schweber, 1961, p. 317). 103 Both the Heisenberg and Dirac pictures can hypothetically be used in either free or interacting systems, if we can separate the Hamiltonian in two parts. The change of representation does not change the physical situation whether it concerns free or interacting fields. There really is no “interaction picture’s assumption that there is a time at which the representation for the interaction is unitarily equivalent to the Fock representation for a free system” (Fraser, 2006, p. 54). 25 This ‘assumption’ has nothing to do with the representation being used. The ‘assumption’ is that with an adiabatic switching on of the interaction, the state vector for the interacting systems can be constructed from the state vector of free fields (Schweber, 1961, p. 320): aa0 -t t,tUlim ψϕ = ∞→ )( . This is what is supposed to be achieved in the adiabatic switching on of the potential that ‘connects’ a free field Hamiltonian with the interacting field Hamiltonian (Jauch & Rohrlich, 1976, p. 134; Schweber, 1961, p. 322): 0 t-ε 0 -t ε -t HV(t))e(Hlim (t)Hlim =+= ∞→∞→ , where ε is a parameter that is taken to zero in the end of the computations. The point is, as we have seen, that at infinite times before and after the adiabatic switching on/off of the interaction potential, the state vector in either the Heisenberg or the Dirac picture is assumed to be describing free fields. The question here is not thus the representation being used but whether it is possible to connect the interacting state to a free field state. It seems clear from the consequences of Haag’s theorem that the usual adiabatic switching on/off of the interaction will not do the trick. In this way we are in the situation of explaining how it is that with a mathematically incorrect procedure it is possible to develop applications from the theory that give so good results when compared with experiments. It is well known that in the applications of quantum electrodynamics there are problems with divergent integrals. These problems are circumvented in practice with renormalization techniques in which (basically) all the divergent integrals appearing in the series expansion of the S-matrix are related to corrections to the mass and charge of the fermions. Since the value of the mass and the charge are not defined by the theory but result from measurements, the terms in the series expansion that are divergent (but formally should be smaller and smaller) are taken to be part of the observed mass and charge. It might seem that by using renormalization techniques the consequences of Haag’s theorem might be evaded because “once infinite renormalization counter terms are introduced, the interaction picture is not mathematically well-defined” (Fraser, 2006, p. 2). From this it might seem that “renormalization addresses this problem not by refining the assumptions, but by rendering the canonical framework mathematically ill-defined” (Fraser, 2006, p. 90). But it would be rather strange, to say the least, that by considering 25 Earman and Fraser’s take it that in the infinite past and the infinite future “particles are assumed to be infinitely far apart and therefore not interacting” (Earman & Fraser, 2006, p. 321). In this case they consider that at t=±∞ “the representation is taken to be a Fock representation” (p. 321). And this is taken to be a Dirac picture (interaction representation) assumption: “In the interaction picture ... at t = ±∞ the representation is the Fock representation for the free field” (p. 321). At this point I thus disagree with Earman and Fraser’s presentation of the consequences of the Haag theorem. I simply do not think that the assumption that at infinite past we can take the state of a particular field to be a free state as described by a Fock space has to be seen as part of the assumptions of the Dirac picture. But this is a question of detail. The main aspect of their presentation is irrefutable: according to the Haag theorem the perturbative approach used in quantum electrodynamics to describe interactions is mathematically inconsistent. 104 an impoverished mathematical structure, suddenly, something physically equivalent to a unitary transformation connecting free and interacting field operators might emerge. In reality this argument does not provide any explanation for why the theory is effective, and this is a central question which has correctly been considered to be relevant: “There is, however, unfinished business in explaining why perturbation theory works as well as it does” (Earman & Fraser, 2006, p. 307). If we followed Fraser’s account we would be back where we started since, without taking into account the ill-defined mathematics of renormalization (and this is possible by considering only the lowest-order calculations), one could say that the adiabatic switching on/off of the interaction is also of doubtful mathematical rigor precisely due to the Haag theorem. This means that when imposing the adiabatic switching on/off we are developing an ill-defined approach. The primary question of the effectivity of the theory would remain, independently of the probable impossibility of taking into account the Haag theorem in this ill-defined mathematical context (now due to the adiabatic switching on/off of the interaction). Another aspect of not having a rigorous basis for a procedure to adiabatically construct the interacting states from the free states is the following. In the Fock space we have a direct connection of quanta to the normal modes of classical field configurations. It is from the Fourier splitting of a classical wave into positive and negative frequency normal modes, providing the basis for the configuration space, that the concept of quanta emerges (via a quantization procedure which results in associating quanta to each normal mode). In the case of interacting fields it is not possible to make this Fourier expansion (Fraser 2006, p. 136). This implies that it is not possible to use the Fock representation for a free field to represent the interacting Dirac and Maxwell fields (Earman & Fraser, 2006, p. 322). Following this line of reasoning, this means that the quanta concept is unavailable when we consider full interacting fields (Earman & Fraser, 2006, p. 330; Fraser, 2008, pp. 2-3). It could seem that the perturbative S-matrix approach could overcome this problem by providing a rigorous procedure to describe the interacting Dirac and Maxwell fields in the scattering problem by using only the Fock states representing free quanta from each field. According to the Haag theorem this is not possible. It seems then, according to Earman and Fraser, that we cannot use the concept of quanta when dealing with interacting fields. It does not seem that the effectiveness of the perturbative approach, in spite of Haag's theorem, is a consequence of the need for a mathematical ill-defined mass and charge renormalization. At least there is no argument that shows how the efficiency of this scheme might result from developing it from a mathematical structure that is illdefined due to renormalization. Moreover the renormalization technique is not the only element that makes the theory mathematically ill-defined. It seems that we cannot arrive at any solid conclusions by analysing from a mathematical perspective, what appears to be an ill-defined mathematical structure. My view is that the explanation for the good results of the perturbation theory approach to scattering (and bound state) problems in spite of the Haag theorem, will not be found in the mathematical structure of quantum electrodynamics. As will be seen below, the point about Haag’s theorem is that the question of its applicability, is not even addressed in the way the physical theory is really used: we have to consider the physical input assumptions of the theory together with its mathematical formulation and application, not the mathematical structure all by itself. As mentioned above, the theory is developed from a canonical quantization of two independent classical fields. The description of the interaction between the fields is given, like in the classical counterpart, by an interaction term. Formally we can adopt 105 whatever representation is mathematically possible. But since we are (apparently) dealing with an ill-defined mathematical structure one should not consider formal aspects of the theory, like the full formal S-matrix for a scattering process (which in the applications turns out to be an asymptotic series). We must consider the applications of the theory that provide results that we can check with experiments. In these circumstances the use of the interaction representation only attains a physical meaning at the level of applications in which we can consider different systems with an interaction that can be considered as a perturbation of their independent states. In this way the fields maintain their identity as separated physical systems even during the interactions. In this sense the use of the interaction representation is part of the applications. We simply use part of the Hamiltonian, which is possible since the theory was developed considering distinct parts in the Lagrangian, one corresponding to the free Dirac field, another to the free Maxwell field, and another to the interaction. This is the one pulled apart from the others in the interaction representation. The description of scattering is developed from the theory considering an initial state corresponding to a limited number of free particles (quanta), and with an adiabatic switching on of the interaction between the fields, a full interacting state ψ bis apparently obtained. The interacting state ψ a+ that corresponds to a well-defined number of quanta in the final state is defined in an equivalent way. The scattering amplitude S ab is given by (ψ b- , ψ a+ ) (Schweber, 1961, p. 323). The point is that we really do not work with these doubtful interacting states. What is going on is quite different. We are only considering a few terms of a perturbation expansion of the scattering matrix. When considering the applications we are taking advantage of the way the theory was developed. We always have clearly distinct fields. For the description of their interaction it is not necessary to have a description of both fields as a closed interacting system. On the contrary, as we will see next, if we try to make a full description of the interaction considering all the terms of the power series expansion of the S-matrix, it can “at best only be an asymptotic expansion” (Schweber, 1961, p. 644). One of the major achievements of Dyson in the development of quantum electrodynamics was showing that the perturbative expansion of the S-matrix is renormalized to all orders. As mentioned previously, quantum electrodynamics (QED) had tremendous problems of divergent integrals that made impossible but a few lower order calculations. This problem was circumvented by the procedure of mass and charge renormalization. Dyson showed, in a paper published in 1949, that the renormalization procedure could be applied to all orders of the perturbative expansion of the S-matrix (Schweber, 1994, pp. 527-544). Soon afterwards, in the summer of 1951, Dyson came out with a physical argument that strongly suggested that, after all, “all the power-series expansions currently in use in quantum electrodynamics are divergent after the renormalization of mass and charge” (Dyson, 1952b, p. 631). 26 According to Dyson, the series expansion of the S-matrix is divergent, and this has nothing to do with renormalization (Dyson 1952b). That is, even if there were no divergent integrals appearing in the terms of the S-matrix, the series would still be divergent. According to Dyson, if we try to make a full description of the interaction considering all the (infinite) terms of the power series expansion of the S- matrix, it can “only be an asymptotic series” (Schweber, 1994, p. 565). That is, according to Dyson’s physical argument, we can expect at some point that the term of 26 It is important to notice that Dyson’s is a heuristic physical argument not a rigorous mathematical derivation. In my view, this is not a deficiency of Dyson’s argumentation since we are considering a theory with an ill-defined mathematical structure. 106 order n+1 will not be smaller than the term of order n and the series starts to diverge: “the terms … will decrease to a minimum and then increase again without limit, the index of the minimum term being roughly of the order of magnitude 137” (Dyson, 1952b, p. 632). In Dyson’s view the “divergence will not prevent practical calculations being made with the series” (Dyson, 1952b, p. 632). But Dyson’s view was that in a certain way the theory only provided a basis for developing the S-matrix series expansion (Cushing, 1986, p. 122). In Dyson’s own words: “I had this rather positivistic view that all QED was the perturbative series. So if that failed you didn’t really have a theory” (quoted in Schweber, 1994, p. 565). Even if strict mathematical proof of the divergence of the S-matrix does not exist, further strong evidence in favor of Dyson’s claim has been given in the last decades (Aramaki 1989, 91-92; West, 2000, 180-181; Jentschura, 2004, pp. 86-112; Caliceti et al, 2007, pp. 5-6). I will now look in more details into Dyson’s argument and mention some of the contemporary results that reinforce his claims. During the summer of 1951 while visiting the ETH In Zurich Dyson found a heuristic argument that convinced him that the perturbation theory, used trough the S- matrix series expansion, diverges. According to Dyson All existing methods of handling problems in quantum electrodynamics give results in the form of powerseries in e 2 . The individual coefficients in these series are finite after mass and charge renormalization ... The purpose of this note is to present a simple argument which indicates that the power-series expansions obtained by integrating the equations of motion in quantum electrodynamics will be divergent after renormalization … The argument for divergence is as follows. According to Feynman, quantum electrodynamics is equivalent to a theory of the motion of charges acting on each other by a direct action at a distance, the interaction between two like charges being given by the formula e 2 δ + (s 122 ), (1) where e is the electron charge. The action-at-a-distance formulation is precisely equivalent to the usual formulation of the theory, in circumstances where all emitted radiation is ultimately absorbed. We shall suppose that conditions are such as to justify the use of the Feynman formulation of the theory. Then let F(e 2 ) =a 0 + a 2 e 2 + a 4 e 4 + ... (2) be a physical quantity which is calculated as a formal power series in e 2 by integrating the equations of motion of the theory over a finite or an infinite time. Suppose, if possible, that the series (2) converges for some positive value of e 2 ; this implies that F(e 2 ) is an analytic function of e at e = 0. Then for sufficiently small values of e, F(−e 2 ) will also be a well-behaved analytic function with a convergent power-series expansion. But for F(−e 2 ) we can also make a physical interpretation. Namely, F(−e 2 ) is the value that would be obtained for F in a fictitious world where the interaction between like charges is [−e 2 δ + (s 122 )] instead of (1). In the fictitious world, like charges attract each other. The potential between static charges, in the classical limit of large distances and large numbers of elementary charges, will be just the classical Coulomb potential with the sign reversed. But it is clear that in the fictitious world the vacuum state as ordinarily defined is not the state of lowest energy. By creating a large number N of electron-positron pairs, bringing the electrons together in one region of space and the positrons in another separate region, it is easy to construct a “pathological” state in which the negative potential energy of the Coulomb forces is much greater than the total rest energy and kinetic energy of the particles. This can be done without using particularly small regions or high charge densities, so that the validity of the classical Coulomb potential is not in doubt. Suppose that in the fictitious world the state of a system is known at a certain time to be an ordinary physical state with only a few particles present. There is a high potential barrier separating the physical state from the pathological states of equal energy: to overcome the barrier it is necessary to supply the rest-energy for the creation of many particles. Nevertheless, because of the quantummechanical tunnel effect, there will always be a finite probability that in any finite time-interval the system will find itself in a pathological state. Thus every physical state is unstable against the spontaneous creation of large numbers of particles. Further, a system once in a pathological state will not remain steady; there will be a rapid creation of more and more particles, an explosive disintegration of the 107 vacuum by spontaneous polarization. In these circumstances it is impossible that the integration of the equations of motion of the theory over any finite or infinite time interval, starting from a given state of the fictitious world, should lead to well-defined analytic functions. Therefore F(−e 2 ) cannot be analytic and the series (2) cannot be convergent. (Dyson, 1952b, pp. 631-632) Almost at the same time as Dyson set forward his argument, C. A. Hurst, working on the so-called φ 3 theory concluded that the perturbative approach was divergent. In particular He enumerated the number of graphs in field theories with the interaction Hamiltonian of general type and showed that the number of graphs with n vertices increases like n n/2 as n increases so that the perturbation expansion cannot converge unless the matrix elements decrease with correspondingly great rapidity as n increases. He studied the case of a three-scalar field interaction λφ 3 and obtained the conclusion, by evaluating a lower bound for the matrix elements, that in this case the perturbation expansion with respect to λ cannot converge. Hurst remarked that the excellent agreement of QED with experiment would indicate that the series is an asymptotic expansion about the singular point = 0, as Dyson conjectured. (Aramaki, 1989, pp. 91-92) Dyson’s and Hurst’s results stimulated W. Thirring to investigate the convergence in the λφ 3 theory, and he got the same conclusion regarding the series divergence (Aramaki, 1989, p. 92). Also the calculation of A. Petterman and A. Jaffe with particular models supported Dyson’s contention that the renormalized perturbation series diverges (Gill & Zachary, 2002, p. 29). More evidence for the perturbation series divergence was found in recent years. According to U. D. Jentschura A priori, it may seem rather unattractive to assume that the quantum electrodynamics perturbation series may be divergent even after the regularization and the renormalization. However, as shown by explicit nontrivial 30-loop calculations of renormalization group γ functions in a six-dimensional φ 3 theory, and in a Yukawa theory …, we believe that the ultimate divergence of the perturbative expansion can be regarded as a matter-of-fact, clearly demonstrated by explicit high-order calculations. (Jentschura, 2004, pp. 108-109) For Dyson his 1951 conclusion on the divergence of the S-matrix approach meant the end of his involvement with quantum electrodynamics. According to Dyson All my efforts up to that point had been directed toward building a complete convergent theory. Finding out that after all the series diverged convinced me that was as far as one could go … That was of course a terrible blow to all my hopes. It really meant that this whole program made no sense. (Quoted in Schweber, 1994, p. 565) This makes it even more imperative to justify the perturbative approach (which provides excellent agreement with experimental results), and to explain how to get rid of large-order terms of a divergent series that simply would make it impossible to use the theory. Just from a mathematical perspective the use of only a few terms of a divergent series is difficult to defend, but by considering the input physical assumptions of the theory the justification of throwing away large-order terms will become clear. In part it is related to the weakness of the interaction between the Maxwell and Dirac fields (Mandl & Shaw, 1984, p. 95). This by itself can explain why a few order terms in a perturbative approach can give so good results. But it does not by itself justify throwing away terms that should be smaller and smaller but which will ultimately result in a divergent series. 108 As mentioned, the scattering matrix amplitude S AB can be written as (ψ B– , ψ A+ ), where we have + −∞→ =, ΑΑ0 t ψ)φtU(0lim 0 and − ΒΒ +∞→ =, ψt)φU(0lim t . Here U is the time displacement unitary operator determined by solving the Schrödinger equation, ϕ A and ϕ B are the initial and final free states, and the in and out states ψ A+ and ψ B– are taken to be eigenstates of the full interacting Hamiltonian (Schweber, 1961, pp. 317-323). Formally these interacting states can be derived from the (complete) S-matrix. Since the S-matrix series expansion is divergent we can conclude that we cannot obtain these interacting states (Scharf, 1995, pp. 314-318), i.e. it is not possible to find solutions of the coupled non-linear Maxwell-Lorentz and Dirac equations as (infinite) power series expansions in the coefficient e (the electric charge). In other words, it is not possible to find a solution for a closed interacting system of an electromagnetic field and charged particles (Dirac field). Let us recall that the theory was structured around the idea that the interaction between radiation and matter is weak (due to the small coupling constant). We do not quantize the interacting system, but each field separately; i.e. when we speak of weak interaction this has two related aspects: one is that the interaction term in the Lagrangian (or Hamiltonian) is small; the other is that from this we are justified to consider the quantization of radiation and matter separately as the quantization of free fields, and to take the interaction term as a perturbative one. 27 As Dyson mentioned, quantum electrodynamics “is in its nature a perturbation theory starting from the non-interacting fields as … unperturbed systems” (Dyson, 1952a, p. 79). Since we are starting from the non-interacting fields, we need to use the adiabatic switching on/off trick to ‘connect’ the two quantized systems and so to be able to describe their interaction. Now, as we have just seen, if we try to describe within the theory the full weak interaction between radiation and matter (corresponding to determining the eigenstates of the full interacting Hamiltonian) we get into trouble (i.e. we obtain divergent results). We conclude then that, in practice (i.e. in the theory we really have and work with), the notion of weak interaction implies an intrinsic approximate approach, i.e. there is, in practice, an unbridgeable gap between the notion of weak interaction and the idea of a full (complete) description of the (weak) interaction (since a full description of the interaction would correspond to obtaining a complete expansion of the S-matrix, which is not possible). Thus the divergence of the S-matrix series expansion implies that we are unable to bridge the gap that exists between our starting physical assumption of two independent unperturbed systems and the (ideal) closed system of fully interacting radiation and matter; and there is a good reason for this. As Earman & Fraser showed from a formal consideration of fully interacting fields (corresponding to an exact solution that we cannot obtain in the case of quantum electrodynamics), we cannot describe them in terms of the Fock representation for free fields. This means that, formally, for a closed system of interacting fields we cannot use all the physical input of quantum electrodynamics associated with the notion of weak interaction of radiation and matter (in particular our starting physical assumption of two non-interacting fields); i.e. the formal considerations imply an, in principle, 27 It is important to remember that contrary to what formal presentations of the theory might lead us to think (e.g. Dyson, 1952a, pp. 58-59; Källén, 1972, pp. 75-80), we do not start with a Lagrangian for an interacting system of radiation and matter and then due to practical problems in solving a set of coupled non-linear equations we feel forced to resort to perturbative calculations. On the contrary the theory was pretty much developed along two separate lines – one of them the quantization of the free electromagnetic field (a subject not addressed here) the other the development of a relativistic equation for the electron –, from the start taking into account, implicitly, that we were dealing with two clearly distinct weakly interacting physical entities: (quantized) charged particles and (quantized) electromagnetic field. 109 incompatibility between the notion of weak interaction as implemented in the theory and the (formal) notion of a closed system of fully interacting fields. Also, as we have seen, considering the complete series expansion of the S-matrix would correspond to the description of a closed system of fully interacting fields. Here the formal results are valid, i.e. we would have two contradictory mathematical results. This is not the case since the series expansion of the S-matrix is divergent. We see then that we disregard the large-order terms not simply for pragmatic reasons but for physical reasons. Including these terms would correspond to an improper use of the mathematical structure of the theory beyond its physical content (in a tentative description of a closed system of radiation and matter): in quantum electrodynamics we have the concepts of radiation and matter, and of a weak interaction between them, not of fully interacting fields. 28 At this point I would like to establish a correspondence between getting meaningful mathematical results and the way we set up the physical concepts in the theory. In this case, (meaningful) approximate calculations of the weak interaction between different physical systems (radiation and matter). I think that the situation we are facing here can be illuminated by recalling some of Bohr’s views related to the Klein paradox and the problem with infinities in the theory (later addressed by renormalization). As we have seen, in the exchange of letters with Dirac in late 1929, Bohr addressed the negative energy problem of Dirac’s equation. Bohr called attention to the fact that it arises from not taking into account the elementary unit of electrical charge in the determination of the actual potential barrier, i.e. from not considering the physical concepts inscribed in the theory. According to Bohr, if calculations are made in which this fact and the limits in the determination of the electron’s position are not taken into account, we would be facing the actual limit of applying the idea of potentials in connection with possible experimental arrangements. In fact, due to the existence of an elementary unit of electrical charge we cannot build up a potential barrier of any height and steepness desired without facing a definite atomic problem. (Quoted in Moyer, 1981, pp. 1057) In a nutshell, to Bohr the paradox resulted from “an unlimited [mathematical] use of the concept of potentials in relativistic quantum mechanics” (quoted in Moyer, 1981, p. 1058; see also Darrigol, 1991, pp. 154-155). An analogous situation occurs with the (renormalizable) infinities in the theory. According to Alexander Rueger’s presentation of Bohr’s ideas, only for an [atomic] electron weakly interacting with the electromagnetic field could the radiation reaction, which would render the electron’s orbit unstable, be ignored; as Bohr stressed repeatedly, strong interactions would make the idea of approximately stationary states of the electron in the atom impossible. (Rueger, 1992, pp. 317-318) In these circumstances, Bohr recalls that “the whole attack on atomic problems … is an essentially approximate procedure, made possible only by the smallness of [the coupling constant]” (Bohr, 1932a, p. 378). That is, to Bohr the attempts to treat the radiation effects on rigorous lines by considering the atoms and the electromagnetic field as a closed quantum-mechanical system led to paradoxes arising from the appearance of an infinite energy of coupling between atoms and field. (Bohr, 1932b, p. 66) 28 With this justification for disregarding the large-order terms of the S-matrix series expansion, the excellent empirical results of the theory follow simply – in the perturbative approach – from the weakness of the interaction between the two separately quantized fields. 116 By the middle of 1929 Pauli (with whom J. R. Oppenheimer had began to collaborate) was seeking to improve the theory he had developed with Heisenberg. A three-man paper was being planned, where in particular Jordan’s criticism regarding gauge invariance would be addressed (Mehra & Rechenberg, 2000, p. 327). By July 1929 Heisenberg still felt that “the catastrophic self-interaction of the electron does not disturb me too much” (Mehra & Rechenberg, 2000, p. 328). Finally Heisenberg and Pauli published a second part of their quantum electrodynamics in 1930, and Oppenheimer published a separate note regarding specifically the self-energy problem. In their paper Heisenberg and Pauli obtained Oppenheimer ‘s result for the Coulomb self-energy of the electron. Now however they recognized that the infinite self-energy, “in many cases will make application of the theory impossible” (quoted Miller, 1994, p.34). Also, the fact that the self-energy problem could not simply be traced back to a similar situation occurring already in classical electrodynamics was soon revealed by Oppenheimer (1930b) who found out a new (infinite) contribution to the self-energy without any classical counterpart. Using Dirac’s second-order perturbation formula ∑ Ε− = jji 1 ji 1 fi fi E HH H , and adopting ψγψ=,−= ∫ r r r r r iej xdAjH tr1 for the perturbation term in the Hamiltonian H1, we can consider the particular case where i = j. This situation represents the perturbation to the energy of the electron’s state i arising from the self-interaction of the electron. Considering for simplicity a single free electron (with momentum p and energy E(p) = c(p2 + m2c2)1/2), the sequence of transitions i → j → i is e → e’ + photon → e. According to A. Pais the “virtual states [e’ + photon] correspond to all momentumconserving partitions of p between e’ and the photon. There are infinitely many such states” (Pais, 1986, p. 373). The self-energy of the electron is found to be )pW( r ~ ∫ dkk cE(p) ce 222 h h , in addition to smaller terms including the electrostatic (Coulomb) self-energy. What Oppenheimer obtained with this result was that while the classical self-energy diverges linearly (~ 1/r) as we take the electron radius to approach the point-like limit (r → 0), the quantum electrodynamical calculation predicted also a quadratic divergence of the term W(p). Also, W(p 1 ) – W(p 2 ) is not finite (as Heisenberg and Pauli initially expected) but also infinite, which means, “as Oppenheimer stressed, [that] self-energy effects causes infinite displacements of spectral lines” (Pais, 1986, p. 373). A disastrous result for quantum electrodynamics. 117 The situation of quantum electrodynamics during the thirties did not improve, on the contrary. As we have seen, to solve inconsistencies of his electron theory related to the existence of negative-energy solutions, Dirac proposed his hole theory. From this, a new infinity problem popped out. In Dirac’s hole theory we have an infinite sea filled with negative-energy electrons. This made Dirac consider that the electromagnetic field is generated by “the difference in the electric density from its value when the world is in its normal state (i.e. when every state of negative energy and none of positive energy is occupied)” (quoted in Pais, 1986, p. 378). That is, the Maxwell-Lorentz equation for the electric field is given, in Dirac’s hole theory, by Div E = – 4π (ρ – <ρ> vaccum ). Going back to chapter 3, we can recall that in his 1930 paper on the hole theory Dirac remarked that “in the general case of an arbitrary varying electromagnetic field we can make no hard-and-fast separation of the solutions of the wave equation into those referring to positive and those to negative kinetic energy” (Dirac, 1930, p. 361). The knowledge of this situation led Dirac to consider in more detail the effect of an ‘external’ electromagnetic field (that could simply result from the presence of a sole electron above the negative-energy sea) on the definition of the ‘normal’ state (vacuum state) of the negative-energy sea. According to Dirac when applied to space in which there is an electromagnetic field, … one must specify just which distribution of electrons is assumed to produce no field and one must also give some rule for subtracting this distribution from the actually occurring distribution in any particular problem. (Quoted Schweber, 1994, p. 114) In a letter to Bohr from September 10, 1933, Dirac summarized his findings: Peierls and I have been looking into the question of the change in the distribution of negative-energy electrons produced by a static electric field. We find that this changed distribution causes a partial neutralization of the charge producing the field. If it is assumed that the relativistic wave equation is exact, for all energies of the electron, then the neutralisation would be complete and electric charges would never be observable. A more reasonable assumption to make is that the relativistic wave equation fails for energies of the order 137mc 2 . If we neglect altogether the disturbance that the field produces in negative-energy electrons with energies less than –137mc 2 , then the neutralization of charge produced by the other negative-energy electrons is small and of order 1/137. We then have a picture in which all the charged particles of physics electrons, atomic nuclei, etc. have effective charges slightly less than their real charges, the ratio being about 136/137. The effective charges are what one measures in all low energy experiments, and the experimentally determined value for e must be the effective charge on an electron, the real value being slightly bigger. (Quoted in Schweber, 1994, p. 116) Dirac presented his results at the seventh Solvay Congress held in October 1933. Due to the fact that in relativistic mechanics the energy is given by W 2 = m 2 c 4 + c 2 p 2 , it can take positive and negative values. According to Dirac, “it has not been possible to develop a relativistic quantum theory of the electron in which the transitions from a positive to a negative value of the energy should be excluded” (Dirac, 1934a, p. 136). In particular transitions between the positive and negative energy states are “predicted in general for all processes putting into play exchanges of energy of the order mc 2 ” (p. 136). Dirac considered that it seems there are no reasons of principle against the applicability of the quantum mechanics to similar exchanges of energy. It is true that quantum mechanics does not seem applicable to phenomena which involve distances of the order of the classical radius e 2 /mc 2 , since the present theory can in no manner discuss the structure of the electron. But such distances, considered as electron wavelengths, correspond to energies of the order (ћc/e 2 )(mc 2 ), which are much greater than the changes in question. It seems that the most reasonable solution is to search for a physical meaning for the negative energy states. (pp. 136- 137) 118 Dirac went on to present again his hole theory which gave a physical interpretation for the negative-energy states: Let us admit that in the universe such as we know it, the states of negative energy are nearly all occupied by electrons, and that the distribution thus obtained is not accessible to our observation on account of its uniformity throughout space. Under these conditions every unoccupied negative energy state represents a break in that uniformity, and must reveal itself as a kind of hole. It is possible to admit that these holes constitute positrons. This hypothesis resolves the principal difficulties of the interpretation of the states of negative energy … the hole takes exactly the aspect of an ordinary particle, positively electrified. (p. 137) Dirac then addresses the problem facing his interpretation when an external field is present. As already mentioned when there is no external field it is simple to take into account the infinite negative-energy sea in the Maxwell-Lorentz equation by considering that “the distribution of electrons produces no field in which no state of positive energy is occupied. And that it is the deviations from that distribution which determine the fields” (p. 138). However as Dirac acknowledges, this hypothesis is completely satisfactory when it is a question of a region of space where there exists no field, and where the distinction between the positive energy states and those of negative energy is cleanly defined; but one must specify when it is a question of a region of space where the electromagnetic field is not zero in order to be able to lead to results free of all ambiguity. We must specify mathematically which distribution of electrons is supposed to produce no field, and also give a rule for subtracting that distribution from the one which exists effectively in each particular problem, in such a way as to obtain a finite difference that can figure into equation [div E = –4πρ], since, in general, the mathematical operation of subtraction between two infinities is ambiguous. (p. 138) Dirac set out to consider the case of a weak, time-independent electrostatic field using the Hartree-Fock approximation. Dirac defined the density matrix R as ( ) )(ψ)(ψ= ∑ 'q'q''q'Rq' r r r , where the ψ‘s are four-component wave functions that are solutions of Dirac’s equation for each individual electron, and the summation runs over all occupied negative energy states. The wave functions are determined in the Hartree-Fock approximation, where each electron is taken to move in an effective field that is the same for all electrons. The equation of motion for R is Η − = R HR R i & h , where H = cρ 1 (σ σσ σ, p) + ρ 3 mc 2 – eV is the Hamiltonian for an electron moving in the electric field E = –∇V; also due to the exclusion principle we must have R 2 = R. Dirac assumes that the distribution R 0 that produces no field is given by        −= W W 1 2 1 R 0 , where W = cρ1(σ σσ σ, p ) + ρ3mc2 is the kinetic energy of an electron. Dirac then looks for a “permanent state for which the equation of motion Η − = R HR R i & h reduces to HR – RH 119 = 0” (p. 139). In particular Dirac looks for a solution of the form R = R0 + R1 where R1 is a quantity of first order in V. Dirac considers that the quantity that interests us is the electric density corresponding to the distribution R 1 . In order to obtain it we must form the diagonal sum of R 1 , with respect to the spin variables, and then take the general diagonal element, multiplied by –e, of the resultant matrix with respect to the position variables x. (p. 140) Dirac denotes this quantity by D(R1). Dirac found that when doing the integration of the expression for D(R1), “the result contains an infinite logarithm” (p. 141). Dirac’s reaction was to use a cut off which rendered the result finite. But, according to Dirac there is a physical justification in the use of a cut off: We could believe, at first sight, that the presence of that infinity renders the theory unacceptable. However, we cannot assume that the theory applies when it is a question of energies greater than the order of 137mc 2 , and the most reasonable way to proceed seems to be to limit arbitrarily the domain of integration to a value of the momentum … corresponding to electron energies of the order indicated. (p. 141) This is an important point. As we have seen, Dirac had mentioned that quantum mechanics does not seem applicable to phenomena which involve distances of the order of the classical radius e 2 /mc 2 , since the present theory can in no manner discuss the structure of the electron … such distances, considered as electron wavelengths, correspond to energies of the order (ћc/e 2 )(mc 2 ) [@ 137mc 2 ]. (pp. 136-137) This goes along the lines of Bohr’s views on quantum electrodynamics. In Bohr’s terms an unbound limit of integration would mean to apply the theory not taking into account the physical assumptions used to set up the theory, since we would be disregarding that the theory treats the electron as a point-charge; in it we are always considering distances larger than the electron’s ‘diameter’ (Bohr, 1932b, pp. 63-64). In his Faraday lecture Bohr mentioned that The scope of the quantum mechanical symbolism is essentially confined, however, to problems where the intrinsic stability of the elementary electrical particles can be left out of consideration in a similar way as in the classical electron theory. In this connexion, it must not be forgotten that the existence of the electron even in classical theory imposes an essential limitation on the applicability of the mechanical and electromagnetic concepts. Indeed, the finite propagation of electromagnetic forces brings with it the existence of a fundamental length, the so-called “electron diameter” defining a lower limit for the extension of the region where the idealization according to which the electron is considered as a charged material point is justifiable. Not only would a concentration of the charge of the electron within a smaller space result in an essential modification of its mass, but we even meet here with a limitation of the unambiguous use of the idea of inertial mass. In fact, we lose any simple basis for a sharp separation between ponderomotoric forces and radiative reactions when we consider processes in which the electron undergoes a velocity change of the same order as the velocity of light within a length of path equal to the electron diameter. It is true that such considerations lose their significance to a large extent on account of the existence of the quantum of action which imposes an essential limit to the analysis of motion. The fertility of quantum mechanics as applied to the problem of atomic stability lies just in the fact that the linear dimensions of the regions ascribed to even the firmest electron-bindings outside the nucleus are still very large compared with the classical electron diameter. (Bohr, 1932a, pp. 377-378 [my emphases]) Returning to Dirac’s take on the logarithmically divergent integral, Dirac considered that if P is the value of the vector momentum … to which we limit the integration domain, the final result, obtained after a complicated integration, is: 120 ( ) ρ∇       π −ρ      − π −=− 2 2 22 1 mcc e 15 4 6 5 mc 2P log 3 2 c e x)D(Rxe h hh , where ρ is the electric density producing the potential V, so that ∇ 2 V = –4πρ, and where the terms containing the derivatives of ρ of order greater than second have been neglected. (Dirac, 1934a, p. 141) Now, Dirac’s view was that we could not apply the theory for energies greater than 137mc 2 . This means taking the cut off P to be of order 137mc. For this cut off value the first term in the expression for ( ) x)D(Rxe 1 − is equal to –(e 2 / ћc) ρ (Dirac took the second term not to be an important correction in the present conditions). Dirac’s interpretation of this result is as follows: As a result of the foregoing calculation, it would seem that the electric charges which one ordinarily observes on electrons and protons and the other particles of physics are not the actual charges which these particles carry (appearing in the fundamental equations) but are all slightly smaller, in a ratio of about 136/137. (Quoted in Schweber, 1994, pp. 115-116) Dirac tried to improve his approach by presenting a more systematic procedure, which, contrary to the previous case (Dirac 1934a), was relativistic and might be applied to the case of external time-dependent fields. In a letter to Bohr of November 10, 1933, Dirac mentioned his new approach: I have been working at the problem of the polarization of the distribution of negative-energy electrons, from a relativistic point of view. If I have not made a mistake, then there is just one relativistically invariant, gauge invariant treatment, which gets over all the difficulties connected with the infinites, to the accuracy with which the Hartree-Fock method applies … I have not yet seen whether this relativistic treatment leads to any kind of compensation of charge arising from the vacuum polarization. (Quoted in Schweber, 1994, p. 117) In his development of the density-matrix formalism Dirac again considers the density matrix R; and again it is considered that each electron moves in a definite electromagnetic field, which is the same for all electrons. This field will consist of a part coming from external causes and a part coming from the electron distribution itself, the precise way in which the latter part depends on the electron distribution being one of the problems we have to consider. (Dirac, 1934b, p. 146) Dirac’s objective was to find “some natural way of removing infinities from ( ) ∑ kkk xtRxt and ( ) ∑ α kkks xtRxt [which is the current density] so as to leave finite remainders” (p. 148). In the case of no external field, Dirac found that the singularities of ( ) 'k'k' 't''x'Rt'x' all occurred in the light cone (p. 151). In the case of an external field present, Dirac supposed “that the singularities are of the same form as in the case of no field, but have unknown coefficients” (p. 152). Dirac showed that the density matrix could be divided into two parts R = R a + R b , where R a contains all the singularities and “the electric and current densities corresponding to R b are those which are physically 121 present, arising from the distribution of electrons and positrons. (p. 156). Dirac’s idea was that with this division of R in two parts “we can remove the infinities” (p. 156). Since the singularities were located in the light cone, this means that when x = x’ – x’’ ≠ 0 R a is finite. According to Pais, “Dirac’s prescription for extracting finite results was: first subtract these singular terms, then let x → 0” (Pais, 1986, p. 382). Also, according to A. I. Miller, “the intent, of course, is to propose a counter-term – R a so that R – R a and, consequently, the measured charge densities are finite. This will be accomplished by Heisenberg” (Miller, 1994, p. 60). The uniqueness of Dirac’s subtraction method was immediately questioned (Miller, 1994, p. 60; Pais, 1986, p. 383). Heisenberg tried to improve Dirac’s method, and in the process, due to his use of a second quantized version of Dirac’s formalism in which electron and positrons were treated in a symmetrical way, came up with the existence of an “infinite self-energy of the light-quanta” (Heisenberg, 1934, p. 186). According to Heisenberg, when “compared to Dirac’s treatment, [his] paper emphasizes the significance of the conservation laws … and the necessity of formulating the basic equations of the theory in a manner extending beyond the Hartree approximation” (p. 169). In the first part of his paper Heisenberg used the density matrix formalism. Following Dirac’s approach Heisenberg considers that “one will have to subtract from the density matrix [R s ] another density matrix [S] which is determined uniquely by the external fields, in order to obtain the ‘real’ density matrix [r, i.e. a density matrix without singularities]” (p. 171). The problem with Dirac’s method is according to Heisenberg that it does not provide a unique specification for S and because of this for the equation of motion of the system (p. 172). Heisenberg’s idea is that “by taking into account the conservation laws of charge, energy, and momentum, the possibilities for S can be restricted insofar that a particular value can be distinguished as the simplest assumption” (p. 172). Heisenberg went on to calculate the vacuum polarization, obtaining a corrected second term for the induced charge density (Miller, 1994, p. 64). Like Dirac, Heisenberg took this term as having no physical significance. This is not the case. As shown in 1935 by E. Uehling, this term yields measurable effects. Uehling found for the hydrogen atom the level 2S to be displaced by ∆ν = –27 megacycles per second (Pais, 1986, p. 383). In the second part of his paper Heisenberg extended the formalism, treating the Dirac wave function as a quantized Dirac field and also treating the electromagnetic field as a quantized field. In this part Heisenberg gave a symmetrical treatment to electrons and positrons being both treated on equal footing as quanta of the quantized Dirac field (Heisenberg, 1934, p. 183). Adopting Hamiltonian methods, Heisenberg noted that the subtraction of infinites could be done order by order in perturbation theory, noticing nevertheless the presence of self-energy-like terms that the subtraction procedure could not remove (Schweber, 1994, p. 118). According to Heisenberg “the perturbation method can be continued in principle, unless an infinite self-energy causes the method to diverge” (Heisenberg, 1934, p. 184). This formalism enabled Heisenberg to treat, in particular, the creation and annihilation of electron-positron pairs. Heisenberg found that the process by which a photon creates an electron-positron pair, which subsequently annihilates creating a photon, gives rise to a logarithmically divergent term. Heisenberg interpreted this term (in analogy to the electron’s self-energy) as the (infinite) self-energy of the light quanta: “we shall treat the matter density associated with a light quantum and in particular the self-energy of the light quantum derived on the basis of this matter density ... The energy of this matter field becomes infinite, in precise analogy to the infinite self-energy of the electrons” (p. 185). 122 Regarding the self-energy problems there was not much improvement during the thirties. The only solid result was V. F. Weisskopf’s demonstration that the electron self-energy is ‘only’ logarithmically divergent (Pais, 1986, p. 385). Going back to Oppenheimer’s calculation of the electron’s self-energy (Oppenheimer, 1930b), Oppenheimer found using single-particle theory (i.e. Dirac’s equation as a one-electron equation) a new quantum-mechanical term in the electron’s self-energy that diverges quadratically. Weisskopf completed in early 1934 a second-order calculation of the electron’s self-energy in hole theory (Weisskopf, 1934, p. 158). Weisskopf divided the electron’s electromagnetic field in two parts: a rotation-free part and a divergence-free part. According to Weisskopf the electrostatic self-energy is given by ∫ Φρ=Ε rd' 2 1 S r, and the electrodynamic self-energy is given by ∫ )Α(−=Ε rdi 2 1 trtr D r r r where i tr is the divergent-free part of the current density. Expanding the self-energy operators by powers of the electric charge, in the second-order calculation only terms proportional to e 2 are kept. The electrostatic self-energy calculated in the Dirac singleelectron theory diverges linearly. Weisskopf made his calculation in hole theory, i.e. considering a multi-electron system where all the states with negative energy are occupied. According to Weisskopf, “to calculate the self-energy of a multi-electron system, it is advantageous to use the method of quantized waves, in which the charge and current densities act as operators on the eigenfunctions, whose variables are the occupation numbers N k (p) of the stationary states p k , k=1, …, 4 of the free electron” (p. 160). He then calculated the diagonal element of the self-energy operator E S for a particular occupation of states. In this way, in the expression for the electrostatic selfenergy there is a summation over all the negative-energy states plus the occupied positive-energy state: the electron whose self-energy we want to calculate. Weisskopf found that “the electrostatic self-energy diverges logarithmically in the ‘hole’ theory” (p. 163). In this work Weisskopf made a mistake in the calculation of the electrodynamic part of the self-energy, obtaining initially a quadratic divergence (like in Oppenheimer’s calculation). Soon after W. Furry called Weisskopf’s attention to his error. In a correction to his first paper, also published in 1934, Weisskopf presented the corrected result. He found like in the case of the electrostatic part of the self-energy that in Dirac’s hole theory E D also had a logarithmic divergence (Miller, 1994, p. 61). 31 Writing to Weisskopf in February 1935, Heisenberg called attention to a shortcoming in Weisskopf’s calculation. Heisenberg repeated Weisskopf’s calculation and found that ∫ )+( + =Ε k dk pcm2 pcmh e 222 222 2 S 31 In 1939 Weisskopf showed that the self-energy of the electron was logarithmically divergent to every approximation in an expansion of the self-energy in powers of the fine structure constant α (Weisskopf, 1939). 123 ∫ )/−( + =Ε k dk p34cm pcmh mc 222 222 D . Heisenberg found these results suspicious because one must expect on relativistic grounds that ∫ +=Ε+Ε k dk pcm h e constant 222 2 DS . (Quoted in Schweber, 1994, p. 125) According to Schweber, “the lack of proper relativistic covariance was to plague all self-energy calculations in the prewar period” (Schweber, 1994, p. 125). 3 The submergence of infinites in quantum electrodynamics Things changed drastically in 1947, with W. Lamb’s experimental results on the shift of the 2 2 S 1/2 state relative to the 2 2 P 1/2 states in the hydrogen atom. H. A. Bethe coming from a conference in Shelter Island, held on 2-4 June 1947, did some calculations on a train going from New York to Schenectady. In the conference W. Lamb presented his recent results on the level shift. Bethe made a nonrelativistic calculation of the Lamb shift, taking into account the suggestion by J. Schwinger, Weisskopf, and Oppenheimer that the self-energy of the electron was responsible for the shift in the energy levels, and Kramers’s idea of mass renormalization (Pais, 1986, pp. 455-456; Schweber, 1994, p. 228-231). According to Bethe: “Kramers suggested that what one really ought to do was to renormalize the mass of the electron, taking into account its interaction with its own electromagnetic field. Then only those parts of the self-energy which are not contained in the mass of the particle would be observable” (quoted in Mehra & Rechenberg, 2001, p. 1039). Bethe calculated the self-energy W of a bound electron and subtracted to it the self-energy W 0 of a free electron (with the same average kinetic energy). This corresponds to Kramers’s idea of mass renormalization. The difference W–W 0 was according to Bethe “the true shift of the levels due to interactions” (quoted in Schweber, 1994, p. 231). This expression is logarithmically divergent. Bethe considered that there should be (in the relativistic theory) a natural cut-off at energies around mc 2 (which is not the case). By taking into account this ad hoc cut-off, Bethe was able to obtain a result in good agreement with the observed value. As seen above, the idea of renormalization, in the case of the electron’s charge, was basically present in Dirac’s report to the Solvay conference of 1933 (Dirac, 1934a). His ideas are stated more clearly in the letter to N. Bohr written after the preparation of the report: We then have a picture in which all the charged particles of physics, electrons, atomic nuclei, etc. have effective charges slightly less than their real charges, the ratio being about 136/137. The effective charges are what one measures in all low energy experiments, and the experimentally determined value of e must be the effective charge of an electron, the real value being slightly bigger. (Quoted in Schweber, 1994, p. 116) 32 32 Another statement of the idea of charge renormalization due to the effect of the vacuum polarization can be seen in Weisskopf’s paper on vacuum polarization from 1936. According to Weisskopf “the polarizability could in no way be observed, but would only multiply all charges and field strengths by a constant factor” (Weisskopf , 1936, p. 208). 124 A similar approach regarding the electron self-energy started to emerge in the end of the thirties, in Dirac’s own work, and in H. A. Kramers’s investigation of the renormalization of the electromagnetic mass at the classical level (as a first step for dealing with the problem at the quantum level). Kramers’s intention was to sidestep the problem by obtaining a consistent model for a finite size electron – that avoided the classical self-energy divergence –, considering from the start the experimental mass of the electron (that contained the mechanical mass and the electromagnetic mass). In this way Kramers “tried to present the theory in such a fashion that the questions of the structure and the finite extension of the particles are not explicitly involved and that the quantity that is introduced as the ‘particle mass’ is from the very beginning the experimental mass” (Kramers, 1938, p. 254). But mass renormalization was only put to use in quantum electrodynamics in 1947, in the quantum-mechanical (non-relativistic) train-ride calculation of Bethe. A few months after the conference, Schwinger worked on a non-covariant relativistic calculation of the Lamb shift using the mass and charge renormalization recipe, and obtained finite results to order e 2 /ħc. Knowing of G. Breit’s suggestion that the electron might have an intrinsic magnetic moment different from the one predicted by the Dirac equation – that explained the discrepancy with the experimental results regarding the hyperfine structure of the hydrogen atom –, Schwinger calculated the socalled anomalous magnetic moment for an electron in an externally applied homogeneous magnetic field, which accounted for the previous hyperfine discrepancies between theory and experiment (Mehra & Rechenberg, 2001, p. 1045). Schwinger published his results in a short note from late December 1947. This note did not include the precise results of his Lamb shift calculations due to discrepancies between the Coulomb (Lamb shift) and the magnetic field (anomalous magnetic moment) calculations, because there existed a difference in the result for the magnetic moment in the case of the electron in an atom and a free electron (Schweber, 1994, p. 319; Mehra, 1994, pp. 238-239). In this brief note Schwinger presented his view on the need for renormalization in quantum electrodynamics. According to Schwinger Attempts to evaluate radiative corrections to electron phenomena have heretofore been beset by divergent difficulties, attributable to self-energy and vacuum polarization effects. Electrodynamics unquestionably requires revision at ultra-relativistic energies, but is presumably accurate at moderate relativistic energies. It would be desirable, therefore, to isolate those aspects of the current theory that essentially involve high energies, and are subject to modification by a more satisfactory theory, from aspects that involve only moderate energies and are thus relatively trustworthy. This goal has been achieved by transforming the Hamiltonian of current hole theory electrodynamics to exhibit explicitly the logarithmically divergent self-energy of a free electron, which arises from the virtual emission and absorption of light quanta. The electromagnetic self-energy of a free electron can be ascribed to an electromagnetic mass, which must be added to the mechanical mass of the electron. Indeed, the only meaningful statements of the theory involve this combination of masses, which is the experimental mass of a free electron. It might appear, from this point of view, that the divergence of the electromagnetic mass is unobjectionable, since the individual contributions to the experimental mass are unobservable. However, the transformation of the Hamiltonian is based on the assumption of a weak interaction between matter and radiation, which requires that the electromagnetic mass be a small correction (~(e 2 /ħc)m 0 ) to the mechanical mass m 0 …It is important to notice that the inclusion of the electromagnetic mass with the mechanical mass does not avoid all divergences; the polarization of the vacuum produces a logarithmically divergent term proportional to the interaction energy of the electron in an external field. However, it has long been recognized that such a term is equivalent to altering the value of the electron charge by a constant factor, only the final value being properly identified with the experimental charge. Thus the interaction between matter and radiation produces a renormalization of the electron charge and mass, all divergences being contained in the renormalization factors. (Schwinger, 1948a, p. 416) 125 The calculations mentioned in the note were done using non-covariant hole-theoretic methods. This is the motive for the discrepancy in the calculations. Afterwards, Schwinger developed a covariant formulation of the Heisenberg and Pauli quantum electrodynamics. Nevertheless, “there were a great many ambiguities in the procedure” (Schweber, 1994, p. 333) of identifying the divergent contributions. Also the calculation method was terribly complicated and threatened to become insurmountable in higherorder calculations. A key aspect of Schwinger’s covariant formulation was the deduction of an equation (later known as Tomonaga-Schwinger equation) that was a manifestly Lorentz covariant generalization of the Schrödinger equation. This was a functional derivative equation that describes the state function ψ as a functional ψ[ σ ] of a general three-dimensional surface σ in space-time: [ ] [ ] σψ= )(δσ σ δψ t)H(x, x ci h . According to J. A. Wheeler’s notes on Schwinger’s presentation of his covariant formulation at the Pocono conference (held from March 30 to April 2, 1948), “these equations contain nothing more than Heisenberg-Pauli formalism and would not be required if one knew how to carry out Heisenberg-Pauli calculations consistently” (quoted in Schweber, 1994, p. 324). Schwinger applied his formalism in the determination of the radiative corrections to the motion of an electron in an external electromagnetic field, i.e. the calculation of the anomalous magnetic moment and the Lamb shift. At the Ann Arbor summer school (from July 19 to August 7, 1948), Schwinger gave a more detailed presentation of his formalism including an improved treatment of vacuum polarization. The value for the Lamb shift given by Schwinger was c1040 2 1 5 1 8 3 ln2 E mc ln n3 8 2 3 43 Μ=        +−+− ∆ Ζ π α , which includes the 1/5 Uehling term, and where the 1/2-term is the magnetic moment effect (which were not included in the expression Schwinger presented at the Pocono conference). Schwinger’s lectures were based on his paper ‘quantum electrodynamics I’ (Schwinger 1948b) and a first draft of a sequel where Schwinger applied the formalism to determine the vacuum polarization and the self-energy (Schwinger 1948c). In the introduction to the first paper Schwinger gave further insights on his views regarding quantum electrodynamics. According to Schwinger The unqualified success of quantum electrodynamics in applications involving the lowest order of perturbation theory indicates its essential validity for moderately relativistic particle energies. The objectionable aspects of quantum electrodynamics are encountered in virtual processes involving particles with ultra-relativistic energies. The two basic phenomena of this type are the polarization of the vacuum and the self-energy of the electron. The phrase “polarization of the vacuum” describes the modification of the properties of an electromagnetic field produced by its interaction with the charge fluctuations of the vacuum. In the language of perturbation theory, the phenomenon considered is the generation of charge and current in the vacuum through the virtual creation and annihilation of electron-positron pairs by the electromagnetic field. If the electromagnetic field is that of a light quantum, the vacuum polarization effects are equivalent to ascribing a proper mass to the photon. Previous calculations have yielded non-vanishing, divergent 132 650b0a56 1 560b0a 2(1) d2)d (6,1)K (5,Kt6)r (4,5)K (3,Kie 2) 1, 4; (3,K ττ)(δ×−= − ∫∫ where dτ = d3xdt. In the case of a delayed interaction the story goes as follows. Since “the Coulomb potential does not act instantaneously, but is delayed by a time r56, taking the speed of light as unity. This suggests simply replacing r56−1δ(t56) in [the expression for K(1)(3, 4;1, 2)] by something like r56−1δ(t56 − r56) to represent the delay in the effect of b on a.”(p. 772). According to Feynman things are not that easy because “when this interaction is represented by photons they must be of only positive energy, while the Fourier transform of δ(t56 − r56) contains frequencies of both signs” (p. 773). Because of this, Feynman uses instead the expression 1 xi 0 ix)(x)dex) − ω− ∞ + π+(δ=π/ω=(δ ∫ . This is not the whole story. After taking into account the contribution due to the case when t5 < t6 (which corresponds to a emitting the quantum that b receives), generalizing to an interaction described also by the vector potential, and adapting the formalism to the case of electrons described by the Dirac equation, Feynman arrives at the expression (1 − αa · αb)δ+(s562) = βaβbγaµγbµδ+( s562). In this way, in the case of electrons interacting through an electromagnetic field, the amplitude is given by 65ba 2 56baba 2(1) d2)d (6,1)K (5,Ks6) (4,5)K (3,Kie 2) 1, 4; (3,K ττ)(δ×γγ−= ++µµ++ ∫∫ . Feynman calls it his fundamental equation for quantum electrodynamics. According to Feynman it “describes the effect of exchange of one quantum (therefore first order in e2) between two electrons. It will serve as a prototype enabling us to write down the corresponding quantities involving the exchange of two or more quanta between two electrons or the interaction of an electron with itself” (p. 772). Feynman then gives a description of the meaning of the equation presenting a graphical support to his interpretation: a Feynman diagram. It goes as follows: It can be understood (see Fig. 1) as saying that the amplitude for “a” to go from 1 to 3 and “b” to go from 2 to 4 is altered to first order because they can exchange a quantum. Thus, “a” can go to 5 (amplitude (K + (5, 1)) emit a quantum (longitudinal, transverse, or scalar γ aµ ) and then proceed to 3 (K + (3, 5)). Meantime “b” goes to 6 (K + (6, 2)), absorbs the quantum (γ bµ ) and proceeds to 4 (K + (4, 6)). The quantum meanwhile proceeds from 5 to 6, which it does with amplitude δ + (s 562 ). We must sum over all the possible quantum polarizations it and positions and times of emission 5, and of absorption 6. Actually if t 5 > t 6 it would be better to say that “a” absorbs and “b” emits but no attention need be paid to these matters, as all such alternatives are automatically contained in [the fundamental equation]. (pp. 772-773) Feynman first applied his technique to the case of the electron’s self-energy. Since the calculations turn out to be easier in the momentum-energy space, Feynman presented rules to calculate the amplitude working with momentum and energy variables. Feynman then shows how his regularization scheme works. In this paper Feynman also addresses the problem of vacuum polarization making use of the gauge invariant regularization procedure developed by Pauli and Villars. 133 Figure 1: The fundamental interaction (described by Feynman’s fundamental equation). Exchange of one quantum between two electrons. To see regularization at work I will look into the problems of the electron’s selfenergy and photon’s self-energy. In the case of the electron’s self-energy, Feynman’s ‘fundamental formula’ reduces to )(δττΚγΚγΚ−=Κ ++µ+µ+ ∫ ∫ 2 4343 2(1) sd1)d,3(3),4(4),2(ie1),2( . According to Feynman, this first order correction to the amplitude K+(2, 1), “arises because the electron instead of going from 1 directly to 2, may go (Fig. 2) first to 3, (K+(3, 1)), emit a quantum (γµ), proceed to 4, (K+(4, 3)), absorb it (γµ), and finally arrive at 2 (K+(2, 4)). The quantum must go from 3 to 4 δ+(s432))” (p. 773). Feynman shows how this expression is related with the self-energy of an electron, which turn out to be 4 2 4343 2 ds)xexp(ipu)3),4(u(e τ)(δ⋅γΚγ=∆Ε +µ+µ ∫ . Figure 2: Interaction of an electron with itself As mentioned, for easiness in the calculations Feynman works in the momentumenergy space. In this case the self-energy is the matrix element between u and u (taken from the plane wave solution for a free Dirac electron: u exp(−px)) of the matrix 134 kdkm)k(pi)e 4212 ∫ − µ − µ γ−−γπ/( . According to Feynman: The equation can be understood by imagining (Fig. 3) that the electron of momentum p emits (γ µ ) a quantum of momentum k, and makes its way now with momentum p − k to the next event (factor (p − k− m) −1 ) which is to absorb the quantum (another γ µ ). The amplitude of propagation of quanta is k −2 . (There is a factor e 2 /πi for each virtual quantum). One integrates over all quanta. The reason an electron of momentum p propagates as 1/(p − m) is that this operator is the reciprocal of the Dirac equation operator, and we are simply solving this equation. Likewise light goes as 1/k 2 , for this is the reciprocal D’Alembertian operator of the wave equation of light. The first γ µ represents the current which generates the vector potential, while the second is the velocity operator by which this potential is multiplied in the Dirac equation when an external field acts on an electron. (p. 775) Figure 3: Interaction of an electron with itself. Momentum-energy space. Up to this point, things are moving smoothly. However if we calculate the integral to obtain the self-energy it turns out, as already mentioned, to be infinite. Using contemporary notation, the problems in the calculation of the electron self-energy are all concentrated in this apparently harmless integral (Mandl & Shaw, 1984, p. 187): iεmk)(p 4mk2p2 iεk 1 kd )2π( e (p)ie 2 0 2 0 2 4 4 2 0 2 0 +−− − / − / + = ∑ ∫ . As it stands this integral is divergent. From Lorentz invariance Σ(p) can be put in the form (p)m)p(m)Bp(A(p) c ∑ − / +− / += ∑ , where m is the electron’s (experimental) mass, and, in particular, A = Σ(p) when γµpµ = m (Mandl & Shaw, 1984, p. 189). This term provides a correction δm = –e02A of electromagnetic origin to the bare mass m0 of the electron, which can be seen as resulting from the interaction of the electron with its own field. In this way, at the level of quantum electrodynamical applications, the electron’s mass that is experimentally measured corresponds to a renormalized mass where the electron’s self-energy is taken into account. For k Ø ∞ the previous integral is logarithmically divergent (this is the famous ultra-violet divergence). A way out is to make a “change in the fundamental laws” 135 (Feynman, 1961, p. 137): the photon propagator 1/k 2 is multiplied by a relativistically invariant convergence factor, assumed by Feynman to be c(k 2 ) = –λ 2 /(k 2 – λ 2 ). This change has to be seen as a formal calculational device, a mathematical trick to get rid of the logarithmic divergence in the integral. If we try to see it as a new theory distinct from the one derived from classical electrodynamics we obtain a non-hermitian interaction Lagrangian that implies that probability is not conserved. Also from a physical point of view the use of this convergence factor is equivalent to considering “an additional interaction of the electron-positron field with a vector field whose quanta have mass λ and whose propagators are –(k 2 – λ 2 ) –1 ” (Schweber, 1961, p. 519). With this prescription it is possible to calculate the integral. With this regularization procedure we have m Λ ln 8 π 3m A 2 −= , where Λ is a cut-off parameter (Mandl & Shaw, 1984, 191). It turns out that the only contribution from the self-energy which is not renormalized is the finite integral Σ c (p). From this term the radiative correction (due to the electron self-energy) to the lowestorder calculations is obtained. If we stopped here, we would have an experimentally measurable radiative correction dependent on an arbitrary cut-off parameter Λ. 33 To get things right, after the renormalization we have to make the cut-off parameter go to infinity, so that the radiative correction term “remains well-defined and finite in this limit and independent of the details of the regularization procedure” (Mandl & Shaw, 1984, p. 191). This method to overturn the problem of infinites in quantum electrodynamics was summarized by Feynman in one of his quantum electrodynamical rules: “(1) Put in an arbitrary cutoff factor c(k 2 ) = – [λ 2 /(k 2 – λ 2 )] for each propagator 1/k 2 . (2) Express everything in m exp = m – δm. (3) Take the limit as λ Ø ∞ and keep m exp fixed.” (Feynman, 1962, p. 143). 4 Different views on renormalization It is usually held that the divergence problem in quantum electrodynamics is due to a failure of the theory at ultra-relativistic energies, that is, to the fact that there is no upper bound to the energy of the virtual quanta that are exchanged during interactions. This might lead to the idea that the cut-off parameter serves like a “boundary line separating the knowable region from the unknowable” (Cao & Schweber, 1993, p. 52). But since there is no indication on where to put this cut-off, it seems that “we cover our ignorance by calculating only quantities which are independent of the exact value of the cut-off” (Teller, 1988, p. 87). This procedure results in a change from the approximative regularized version of the theory to a recovered quantum electrodynamics with renormalized mass and charge. This means changing “the status of the cutoff from a tentative, and tantalizing, threshold energy to a purely formalistic device” (Cao & Schweber, 1993, p. 53). Even if Feynman was trying to achieve a consistent regularized theory, and published his method as a provisional one while searching for a “correct 33 This problem does not arise with δm because we consider it to be ‘absorbed’ in the experimental measurable mass m exp = m 0 + δm, which is seen as an amalgamation of the bare mass and the electromagnetic mass, and whose magnitude – as a phenomenological parameter – is determined not by the theory but from experiments. 136 form of f + [the function that substitutes the delta function appearing in the interaction term] which will guarantee energy conservation” (Feynman, 1949b, p. 778), it seems that it ended up being what Bethe had envisaged from the beginning: a mathematical calculational device designed to overcome the divergence problems in some integrals. Also, as previously mentioned, from a physical point of view the regularized theory is completely different from quantum electrodynamics. It does not have a divergence problem because of the presence of an auxiliary vector field. This field can be seen as a formal mathematical device if, and only if, after the renormalization we recover a cutoff independent theory. If we tried to maintain the regularized theory, so that we did not have to deal with the problem of infinites in the calculations, we would be working not with quantum electrodynamics but with another (inconsistent) field theory. We see in the case of Feynman (like previously with Schwinger) a lack of understanding of the possible implications at a conceptual level of the regularization scheme and renormalization. As we have seen Feynman as looking for a consistent modification of quantum electrodynamics. Contrary to Dirac (with Bohr), Feynman does not relate the regularization to structural aspects of the theory (i.e. to the adoption in Dirac’s equation of a point-like electron) that might imply (even if in an inconsistent way) an upper bond to the possible energy exchanges. Contrary to a common view I think that the divergence of the integrals and the use of the cut-off trick do not reveal where the theory stops being good and a ‘true’ theory should come into play. It reveals structural problems in the construction of the theory, that are impossible to ignore when we have some integrals, that should (from a physical point of view) provide small radiative corrections to lower-order calculations, and end up blowing apart in a proliferation of infinites. In quantum electrodynamics the description of the interaction between ‘particles’ (like photons and electrons) as quanta of the Maxwell and the Dirac fields is given by the perturbative expansion of the scattering matrix that describes the interaction (and it is in the terms of the S-matrix that the divergent integrals appear). Considering the second-order term of the scattering matrix S 2 (x 1 , x 2 ) in configuration space, the mathematical expression of the terms related with the divergent part of the electron’s self-energy are dependent on δ(x 1 – x 2 ). This means that “all the divergences in S 2 (x 1 , x 2 ) come from terms proportional to δ(x 1 – x 2 ) and to its derivatives which differ from zero only in the infinitesimal neighbourhood of the point x 1 = x 2 ” (Bogoliubov & Shirkov, 1959, p. 299). The divergence problem does not arise solely from the fact that there is in the theory no upper bound to the energy of the virtual quanta. It results from the ‘coincidence’ in the theory of the inexistence of a natural limit to the energy of virtual quanta and from the local character of the interaction between the fields in quantum electrodynamics. One other aspect of the mass renormalization procedure is that when we go beyond the lowest-order approximations, and the mass renormalization is needed, there is a mismatch between the conceptual basis of the theory and its description of matter: conceptually quantum electrodynamics is developed from the idea of two independently quantized fields – one of them describing matter – that interact. The electrons (and positrons) are described in quantum electrodynamics as quanta of the quantized Dirac field, having a (bare) mass associated with them. At the same time the higher-order calculations require considering the mass of the electron as having a non detachable contribution from the Maxwell field. The observed or experimental mass of the electron results in the applications of the theory from a simultaneous contribution from the bare mass of quanta of the Dirac field and the self-energy of these quanta due to the interaction of the Dirac and Maxwell fields. 137 That the divergence in the calculation of the electron’s self-energy reveals structural problems related to distinct aspects of the theory does not mean that the theory does not have other less noticed limitations, also related to the divergence of integrals. This can be seen on a closer look at the second-order calculation of the photon self-energy. Again we have a divergent integral and again a regularization scheme is used and a renormalization is made. The (second-order) photon self-energy leads to a modification of the photon propagator: D µν ’(k)= g µν D(k)+ D(k)Π µν (k)D(k), where Π µν (k) is a quadratically divergent integral and D(k) is the bare photon propagator. Considering the requirement of Lorentz and gauge invariance, the second-order tensor Π µν (k) must have the form ( g µν k 2 – k µ k ν )Π(k 2 ) (Jauch & Rohrlich, 1976, p. 189). As it stands, for k 2 = 0 we have Π µµ (0) ≠ 0. This would mean that the propagator we obtain taking into account this second-order correction is not the propagator for a zero mass photon but the propagator for a massive neutral vector boson (Sakurai, 1967, p. 275). To recover our photon we must recall that Π µν (k) must be gauge invariant. Imposing this condition, we must have Π µν (k)k ν = 0. From this we obtain the ambiguous result that the quadratically divergent integral pd )m(p 2mp iπ 1 π 2α (0)Π 4 222 22 2 µ µ ∫ + + −= , must be identically zero (Jauch & Rohrlich, 1976, p. 190). The only way to circumvent this situation is to consider that “the integral is, strictly speaking, meaningless, since it is divergent” (Schweber, 1961, p. 552). The pragmatic view is that we need a ‘functioning’ theory that is gauge invariant and provides a zero mass for the photon in the lower terms of the perturbation expansion of the S-matrix that are used in practice. This, when evaluating Π µν (k), can be done by taking into account the divergent integral Π µµ (0) and subtracting it from Π(k 2 ), which leads to a logarithmically divergent integral. Using a gauge invariant regularization scheme we have Π(k 2 )= C + k 2 Π f (k 2 ), where Π f (k 2 ) is a finite correction term that as Λ Ø ∞ “tends to a well-defined finite limit which is independent of the detailed form of the regularization procedure” (Mandl & Shaw, 1984, p. 187), and C is logarithmically divergent as Λ Ø ∞ (Sakurai, 1967, p. 277). With this procedure we obtain a regularized photon propagator that includes second-order photon self-energy effects. The next step is to incorporate the regularized constant C in a parameter of the theory whose value is experimentally determined, so that we can take the cut-off limit to infinity and recover quantum electrodynamics from the regularized ‘theory’. Considering, for example, the second-order correction to the electron-electron or Møller scattering 34 due to the self-energy of the photon, the change in the Møller scattering amplitude amounts to → 2 2µ ' 21µ ' 1 2 ik )uγu)(uγu(e 34 I will look into some general aspects of the Møller scattering in the next chapter. [Document text truncated for crawler view.]