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Relative Information and Quantum Free Particle exp(ipx) Part 2 Francesco R. Ruggeri Hanwell, N.B. Oct. 29, 2025 In Part 1, we noted that (1) suggests that quantum mechanics is based on relative information. (1) provides the example of a person Robert who may be at home or at the office. A direct measurement of his location would settle the matter, but if this is not possible, (1) suggests there is relative information, namely a second person, Mary, who knows Robert’s whereabouts. Thus, one may obtain the value of Robert’s location without measuring it. We suggested in Part 1, that this scenario fits the momentum variable. A direct measurement involves an impulse hit (if one does not know rest mass mo). In classical physics, if a direct measurement of impulse cannot be done, one does not know the momentum. Quantum mechanics of a free particle, however, involves exp(ipx) which allows for 2-slit interference. Thus, one may find p from interference pattern results without measuring p using an impulse hit. In Part 1, we suggested that the relationship between x and p in exp(ipx) arises from introducing probability into Newtonian 2-body elastic scattering. We suggested that for an initial (e1,e2) energy and (p1,p2) momentum vectors, any (ei,ej), (pi,pj) which conserve energy and momentum have the same product probability. This suggests exp(ip) as a probability for p as there is no real value weight. P, however, depends on the direction of the x-axis and so to have p,x to the right and -p, x to the left yield the same probability, one may introduce exp(ipx). Generalizing, one may create the Lorentz invariant exp(-iEt+ipx). Here, we wish to provide a second argument for p being linked with x. Again, there is an interaction and inherent probability, as in the above case, and one must again make the assumption of equal probabilities for (ei,ej), (pi,pj) pairs. We use the example of 1-D reflection-refraction at an n1-n2 index of refraction junction. We argue that conservation of probability: 1= P(reflect) + P(refract) may be written in terms of dynamic variables as: AAp – p BB = p2 CC, where AA, BB and CC are fluxes of the incident, reflected and refracted photons. Physically, this is a momentum balance equation which would hold on average or in a steady stream case. In other words, it represents an equilibrium scenario and this usually requires many particles, not one. The problem is that the probabilities to reflect and refract hold for a single photon. How can a single photon possibly sense an equilibrium scenario which contains the incident, reflected and refracted photon probabilities? In a previous note, we argued that this is possible if there is an inherent uncertainty in x associated with the incident, reflected and refracted photons. This then represents a probability in x linked to a free particle and so should be represented by a unit modulus complex number exp(i f(p,x)). If p→-p and x→ -x, one should have the same result. Furthermore, if one assumes that P(p1)P(p2) = P(p1+p2) as in the elastic Newtonian scattering problem, one once again arrives at exp(ipx). Thus, a relative variable, namely x, appears, but in this case it exists to allow for a single particle to create an equilibrium scenario which involves all possible outcomes. This same physics seems to occur in the 2-slit scenario as well. This, then, seems to be the basis of free particle quantum physics, the idea of a relative variable which allows one to sample different outcomes.
Relative Variable In Part 1, we noted that (1) suggests that quantum mechanics is a theory involving relative information. In particular, a person, Robert, may either be at home or at the office. It seems that one should perform a direct measurement of his location. In (1), however, it is suggested that one may need to use relative information, i.e. a second person Mary who knows Robert’s whereabouts. This bypasses performing a direct measurement of his location. In Part 1, we suggested that this fits the case of a quantum free particle wavefunction exp(Ipx). If one does not know rest mass mo, one would need a direct impulse hit measurement to find p. The relative information variable x, however, allows one to measure a wavelength (through a 2-slit interference experiment) instead. No impulse measurement is needed. This, however, involves introducing an uncertainty in x linked to p which manifests itself physically, as in the 2-slit case. The big question is: Why does x appear with p in exp(ipx)? In Part 1, we argued that it occurs in the following manner. One may suggest that probability exists in Newtonian 2-body elastic scattering. For an initial (e1,e2) energy and (p1,p2) momentum set, one may postulate that any (ei,ej) (pi,pj) set which conserves energy and momentum has the same product probability as the initial case. For momentum, this suggests exp(ip) as probability, but p is based on the direction of the x axis. Reversing the direction leads to exp(-ip) which should not be. One might suggest using exp(i |p|), but this cannot be because P(p)P(-p) should represent 0 momentum, not exp(i 2 |p|). Thus, we suggested introducing the relative information variable x, i.e. exp(ipx) which we generalized to exp(-iEt+ipx) to be Lorentz invariant ((1)) Here, we suggest a different example which shows the appearance of a relative information variable for a different reason, but keeps the same (ei,ej) (pi,pj) assumption. One Dimensional Reflection-Refraction Consider one dimensional reflection-refraction at an n1-n2 junction (index of refraction). The probability equation is: 1 = P(reflect) + P(refract) ((2)) For a photon with E=pc and E, the same for the incident, reflected and refracted photons, one may write ((2)) in terms of dynamical variables: AAp -BB p = CC p2 ((3)) Here AA, BB and CC are the fluxes of the incident, reflected and refracted photons. In previous notes, we have argued that physically ((3)) is a pressure balance equation. Pressure balance, however, only occurs in equilibrium and in classical physics, and requires many particles. One might argue that a steady stream physical scenario may be created to exemplify
((3)), but the same probabilities must hold for a single photon approaching the n1-n2 junction. How can a single photon possibly know about all outcomes and equilibrium? In a previous note, we argued that a way for this to happen is to introduce a spatial region of uncertainty associated with each p. In such a case, what looks like an interaction at an n1-n2 point really occurs in a delta x which allows the single particle to sample the incident, reflected and refracted cases and create an equilibrium as a single particle using single particle probabilities. Then, one outcome is physically chosen, but based on the equilibrium result. This can only happen if there exists a relative information variable associated with p, namely x here. The fundamental reason for this variable in interactions is to allow one to create an equilibrium of probability. This same reasoning holds for two slit interference as well. Given the uncertainty in x suggested above, one considers the two possibilities of the particle going through one slit or the other. To do so, however, one must find an explicit expression for the probability linked with p and x. Given that one is dealing with a free particle (in both cases), the probability cannot have a real value weight. Thus, we suggest: exp(i f(p,x)) ((1)) Next, exp(i f(p,x) exp(i f(-p,x) = exp(i function2(x)) ((2)) This suggests that f(p,x) must be odd in p. If we use the same assumption as in the Newtonian elastic scattering case, then: exp(i f(p1,x)) exp(i f(p2,x)) = exp( i f(p1+p2,x)) ((3)) A simple solution is exp(ipx). ((4)) Thus, one may arrive at the quantum result ((4)) by considering a physical example, n1-n2 reflection-refraction, together with the probability rule of elastic Newtonian scattering (all (ei,ej) (pi,pj) sets which conserve energy and momentum have the same probability) to arrive at a quantum probability ((4)). The key idea of using the reflection-refraction case (which also appears in the 2-slit case) is that a single particle has a probability linked to the relative information x which means that there is an uncertain spatial region which allows a single particle to create a probabilistic equilibrium of all possible outcomes before choosing one. Thus, this seems to be analogous to a many-body classical equilibrium situation as argued above, but is based on a single particle. We suggest that this is why many papers in the literature try to use fluid dynamics equations to derive quantum mechanics as these equations are inherently based on conservation, but we nevertheless reject that approach as the equilibrium exists at a single particle level. Conclusion In conclusion, (1) suggests that quantum mechanics is based on relative information. In particular, (1) argues that one may find Robert’s location either through a direct measurement of
his location or from information from a second person, Mary. In Part 1, we suggested that this scenario fits exp(ipx), with x being the relative information variable. One does not have to measure p directly through an impulse hit measurement, but may use 2-slit interference based on x. In Part 1, we tried to derive exp(ipx) based on elastic Newtonian scattering, arguing that for an initial (e1,e2) (p1,p2) any (ei,ej) (pi,pj) which conserve energy and momentum, have the same probability. Using ideas of reversing the x axis and Lorentz invariance, we arrived at exp(-iEt+ipx). Here we suggest that the notion of a relative information variable to p may be seen from the point of view of a single particle creating an equilibrium scenario using probability. We still retain, however, the equal probability of (ei,ej) (pi,pj) pairs. We consider the case of 1-D reflection-refraction at an n1-n2 junction. The probability balance equation may be written in terms of dynamic variables and becomes a pressure balance equation involving the flux of the incident, reflected and refracted photons. This is fine for a steady state picture, but the same probabilities must hold for a single photon. How can a single photon sense an equilibrium? We suggested in a previous note that this requires an uncertainty in position associated with each momentum. Thus, a relative information variable is introduced. We make arguments above, including using the (ei,ej) (pi,pj) equal probability assumption to arrive at exp(ipx). Thus, the relative information variable is directly linked with an uncertainty in this variable which allows a single particle to sample various outcomes, i.e. a single particle may establish an equilibrium situation as if many particles were present. Then, one outcome is chosen physically, but it matches the overall balance result. The probability used to sample the various outcomes is exp(ipx) because it is consistent with (ei, ej) (pi,pj) pairs having the same probability if they conserve momentum and energy. References Di Biagio, A. and Rovelli, C. Relative Information, Relative Facts (2025 https://www.semanticscholar.org/reader/c0ccc246970146c31bd6a60cee5cc53f7f6117e3
