Mathematical Foundations of Time Asymmetric Quantum Mechanics

1 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 Mathematical Foundations of Time Asymmetric Quantum Mechanics M Gadella Departamento de F´ısica Te´orica, At´omica y Optica. Facultad de Ciencias, Universidad de Valladolid, Paseo Bel´en 7. 47011 Valladolid, Spain E-mail: [email protected] Abstract. We review the mathematical tools that are suitable for a formulation of time asymmetry in quantum mechanics. In particular, Hardy functions on a half plane and rigged Hilbert spaces constructed with a subclass of Hardy functions. This time asymmetry often appears in quantum scattering and, in particular, in resonance scattering. We review the construction of Gamow vectors, often considered Gamow states for resonances. A brief summary of the fundamental ideas of time asymmetric quantum mechanics is presented in a last section. 1. Introduction In the framework of standard non-relativistic quantum mechanics, the time evolution governed by a self adjoint Hamiltonian His given by a group of unitary operators on a Hilbert space, depending on the parameter time t. In this case, tmay reach all possible real values, i.e., −∞ <t<∞. However, not all quantum dynamical processes are governed by a single Hamiltonian. Let us think in scattering, which intuitively and roughly speaking works as follows: Imagine a quantum particle. In the remote past, its quantum state is prepared as free. This means that its time evolution is given by some sort of free Hamiltonian H0. At some spatial localization, the particle enters into an interaction region, where is subject of some forces that we shall assume that come from the existence of a potential V. Then, inside the interaction region, time evolution of its state is determined by a Hamiltonian H=H0+V, which will transform this state. Eventually, the particle will abandon the interaction region and evolve again with H0to be detected in the far future. In this case, time evolution is given by a Hamiltonian pair {H0, H}. Although it is not necessary, we gain in intuition if we think on the interaction region as a bounded domain in space. A particularly interesting situation will occur when the particle spends in the interaction a time which is much larger than the time it would stay if the interaction would be switched off. In this case, we say that a metastable state or quantum resonance has been produced. Thus, we may think on resonances as the result of a capture of some particle by a center of forces and its release or emission. The whole process is called resonance scattering, while the process of emission is the decay. In atomic or nuclear physics, it is obvious the existence of quantum unstable states. A quantum particle (electron, alpha, etc.) is emitted spontaneously from an atom or nucleus and this is the observed situation. We may describe the situation in terms of resonance scattering, in 2 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 which we ignore the capture process and identify the unstable state with the quantum resonance. Here, we are solely interested in the process of decay. In fact, capture and decay are usually quite different process, which require different conditions. While the capture requires some conditions in the preparation of the incoming state, the emission or decay is spontaneous. Therefore, capture (or preparation of the metastable state) and decay are not mutually symmetric and they are not time reversal of each other. This type of asymmetry can also be observed in other quantum scattering processes. There are quantum scattering processes, which are very common but instead their time reversal processes are very improbable. A typical example has been discussed in [1]. All these situations give an idea of time asymmetry in quantum mechanics. Related notions are quantum irreversibility, which describes the non-invariance with respect to the time reversal operation and the existence of a time arrow, which distinguishes between past and future in a unique way [2]. Since we usually observe decay and not capture, we need a formalism that describes the decay of quantum unstable particles or resonances. A comment is in order here. In the past physicists have distinguished between resonances and decaying states [3]. Resonances were characterized by a bump in the cross section with width Γ, which was the measurable quantity. Then, a lifetime can be defined as τ:= ~/Γ. Decaying states were defined by its mean life. Measurements of mean life and cross section are independent, so that resonances and decaying states could not be fully identified until precision in both types of measurements were good enough to establish the accuracy of the formula τ=~/Γ. In the relativistic case this is not possible in general since the mean lives of decaying particles are times below the minimal time interval which is possible to measure. Yet, lifetimes of relativistic particles are often determined after the measurement of the cross section by the above formula τ:= ~/Γ. It is sometimes claimed that this relation can be fixed in the context of time asymmetric quantum mechanics [4]. Resonances may also be reasonably modeled by an interaction between a bound state and an external field, as is in the celebrated Friedrichs model [5–7]. Yet, the Friedrichs model may be described as a resonance scattering, in which scattering matrix, Møller wave operators, etc. are well defined [8]. Along this paper, we are going to review a mathematical formalism for the theory of quantum resonances, which are typical irreversible quantum processes, based in the use of Hardy functions on a half plane. Hardy functions are complex analytic functions, which properties we review in the sequel. Hardy functions seem to be the correct mathematical tool to describe irreversibility in Quantum mechanics. For other descriptions, which do not exclude Hardy functions, see [9,10]. 2. Rigged Hilbert spaces A rigged Hilbert space or Gelfand triplet is a tern of vector spaces [11–15] Φ⊂H⊂Φ×,(1) where: i) His an infinite dimensional (separable) Hilbert space. ii) Φ is a dense subspace of H. Dense means that for any vector ϕ∈ H, there is a sequence of vectors, {ϕn}in Φ such that ϕn7−→ ϕin terms of the norm topology on H. This means that any ϕ∈ H can be approached by vectors in Φ with arbitrary precision. In addition, Φ has its own topology, which is stronger than the topology Φ inherits from H. This means, in particular, that all convergent sequences in Φ are also convergent sequences in H, but the converse is not true. iii) In order to define Φ×, let us consider the set of mappings F: Φ 7−→ C, where Cis the field of complex numbers, such that: a) Fis antilinear on Φ, i.e., for any ϕ, ψ ∈Φ and any 3 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 pair α, β ∈C, we have that F(αϕ +βψ) = α∗F(ϕ)+β∗F(ψ), where the star denotes complex conjugation, and b) Fis continuous, which in particular means that if ϕn7−→ ϕin Φ, then, F(ϕn)7−→ F(ϕ)inC. These mappings form a linear space that we denote as Φ×.F(ϕ) is the complex number resulting of applying Fto the vector ϕ. For our purposes, we should use the Dirac notation: F(ϕ) = hϕ|Fi. Then, to any ϕ∈ H, we may associate a unique Fϕ∈Φ×, defined as hψ|Fϕi:= hψ|ϕi, for all ψ∈Φ. After identification of Fϕwith ϕ(that mathematicians often call an abus de langage, like that, in French), we conclude that H ⊂ Φ×. We do not want to enter in the discussion of the possible topologies on Φ×. Rigged Hilbert spaces have been used to (among other purposes): i) Give a rigorous setting to the Dirac formulation of Quantum Mechanics [12,14–16]. ii) Give a precise meaning to Gamow vectors or vector states for resonances [17–19]. iii) With the use of rigged Hilbert space of Hardy functions, provide a mathematical support for the time asymmetric quantum mechanics [20–23]. Let us briefly discuss some features related with rigged Hilbert spaces (RHS). Let Abe a linear operator on Hreduced by Φ. This means that for any ϕ∈Φ, we have that Aϕ ∈Φ, or AΦ⊂Φ, i.e., Φ is invariant under the action of A. Then, Amay be extended to a linear operator on Φ×by means of the duality formula: hAϕ|Fi=hϕ|AFi,∀ϕ∈Φ,∀F∈Φ×.(2) It has been proved that if Ais a self adjoint operator on an infinite dimensional (separable) Hilbert space H, there is always a dense subspace Φ, in the domain of A(the space of vectors in which Aacts) such that Φ reduces A[13]. This is one of the ingredients towards a rigorous Dirac formulation of quantum mechanics. From a RHS, we may construct infinitely many others which are, in some sense, equivalent to the original one. Let Ube an arbitrary unitary operator on H, or furthermore, Umight be a unitary operator between to Hilbert spaces Hand G, i.e., U:H 7−→ G. We know that unitary mappings transport topological properties from Hinto G. Moreover, there are some other important properties which are preserved relative to operators. In particular, if Ais self adjoint in H, its transformed by Uon G,UAU−1is self adjoint on G. Thus, if we have a RHS as in (1) and U:H 7−→ G is unitary, the triplet UΦ⊂ G ⊂ (UΦ)×(3) is a new RHS with the same properties than the original one. The topology on Φ is transported to UΦbyU, so that Φ and UΦ have the same topological properties [18]. 3. Hardy functions on a half plane Let us consider the open upper half plane, C+, of the complex plane, defined as the set of complex numbers with positive imaginary axis, i.e., C+:= {z∈C|Im z > 0}. By Cwe always mean the field of complex numbers. A Hardy function [18,19,24–26]f(z) on the upper half plane is a complex analytic function on C+such that sup α>0Z∞ −∞ |f(x+iα)|2dx < K < ∞.(4) This means that for any positive value of α, the function f(x+iα) is square integrable and that all integrals in (4) for all values of α > 0 are bounded by the same finite constant K. One 4 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 important consequence is that the function on the real line Rdefined as f(x) := limα→0f(x+iα) is also square integrable and Z∞ −∞ |f(x)|2dx < K. (5) It is customary to call f(x) the function of the boundary values of f(z). The function f(x) is uniquely determined almost elsewhere, which means that it may not be defined on a set of zero Lebesgue measure. The converse of this result is true: if we know that a square integrable function on the real line, f(x), gives the boundary values of a Hardy function on the upper half plane, f(z), then we can recover all values of f(z) for z∈C+. Due to a theorem by Titchmarsh [24,25], for all z∈C+, we have that f(z) = 1 2πi Z∞ −∞ f(x)dx x−z,(6) where f(x) is the boundary function of f(x). The proof of this result relies in the Cauchy theorem. Let us call H2 +the set of Hardy functions on the upper half plane. These functions have the following properties: i) We may identify any function f(z) in H2 +with its boundary function f(x). Since f(x) is square integrable, then f(x)∈L2(R), where L2(R) is the Hilbert space of all complex square integrable functions on the real line. Thus, H2 +⊂L2(R). ii) H2 +is a linear space, which is a subspace of L2(R). iii) According to the Titchmarsh theorem, we may recover the values of a function in H2 + from its boundary function, provided that we know that it is the boundary function of a Hardy function on the upper half plane. Then, how we can say that a square integrable function is the boundary function of one in H2 +? The answer was given by Paley and Wiener in a celebrated result [24–26]: The Fourier transform of a square integrable function f(x) on the real line is given by F(f) := b f(k) := 1 √2πZ∞ −∞ f(x)e−ikx dx (7) and it is also a square integrable function with the same norm. We say that f(x) is supported on an interval ∆ of the real line, either finite or infinite, if its is zero outside ∆. The Paley-Wiener theorem establishes that f(x)∈ H2 +if and only if f(x) is the Fourier transform of a function supported on the negative semi axis R−≡(−∞,0]. Thus, in order to recognize whether a function, f(x), is in H2 +, we take its inverse Fourier transform. If the function vanish outside R−, then it is in H2 +, otherwise it is not. iv) The Paley-Wiener theorem is a quite useful tool to construct all functions in H2 +. We only need to consider the space L2(R−) of all square integrable functions on R−and take their Fourier transforms. Then, we write that F[L2(R−)] = H2 +.(8) The Fourier transform is a unitary operation on a Hilbert space. Since L2(R−) is a subspace of L2(R), which is also a Hilbert space, then H2 +is a Hilbert subspace of L2(R). v) The values of a Hardy function on the upper half plane not only may be recovered by its boundary function on the real line, but also from its boundary values on the positive semi axis R+≡[0,∞) as shown by a result by van Winter [27]. This discussion requires the Mellin transform. Let f(x) be a square integrable function on R+. Its Mellin transform is given by M(f)(s) = fM(s) := 1 (2π)1/2Z∞ 0 f(x)xis−1/2dx , s ∈R.(9) 5 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 The Mellin transform of a square integrable function is also square integrable. The van Winter theorem states that a function f(x)∈L2(R+) is the boundary function on the positive semi axis R+of a Hardy function on the upper half plane if and only if its Mellin transform fM(s) satisfies Z∞ −∞ (1 + e2πs)|fM(s)|2ds<∞.(10) The values of f(x) for z=reiθ, 0 < r ≤ ∞ and 0 < θ ≤2πare given by: f(z) = 1 (2π)1/2Z∞ ∞ fM(s) (reiθ)−is−1/2ds . (11) Hardy functions in the lower half plane C−:= {z∈C|Im z < 0}are defined analogously. The space of Hardy functions in the lower half plane is denoted by H2 −. Functions in H2 −have similar properties than functions in H2 +with some minor differences. In particular, (6) needs of a minus sign right after the equal sign and in iii.) and iv.) we have to replace Fourier transforms supported in the negative semi axis by Fourier transforms of functions on the positive semi axis. There are some additional properties concerning functions in H2 ±: vi) The spaces H2 ±are subspaces of L2(R), which are also Hilbert spaces. Moreover, as a consequence of the Paley-Wiener theorem, each function in H2 ±is orthogonal to each function in H2 ∓: Z∞ −∞ f∗ ±(x)g∓(x)dx = 0 , f±(x)∈ H2 ±, g∓(x)∈ H2 ∓.(12) In addition: L2(R) = H2 +⊕H2 −,(13) where the sign ⊕means orthogonal direct sum. vii) The complex conjugate f∗(x) of a function f(x)∈ H2 ±is in H2 ∓,f∗(x)∈ H2 ∓. Furthermore, [f(z∗)]∗=f(z), where the star always denotes complex conjugation. viii.) Any function f(z)inH2 ±has the following asymptotic behavior for large values of |z|: |f(z)| ≈ |z|−1/2.(14) These are the most relevant properties of Hardy functions on a half plane. Next, we are going to refine the space of Hardy functions. 3.1. Smooth Hardy functions The Schwartz space Sis the set of all complex functions of a real variable, f(x) satisfying the following properties: i) Any function f(x)∈ S is continuous and has continuous derivatives of any order at all points. ii) Any function f(x)∈ S, as well as any of its derivatives, goes to zero at the infinity faster than the inverse of any polynomial. i.e., lim |x|→∞ xndmf(x) dxm= 0 , n, m = 0,1,2, . . . . (15) Functions in Sare called the Schwartz functions and have the following properties [28]: i) Sis a vector space over the field of complex numbers. 6 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 ii) Any function in Sis square integrable, so that S ⊂ L2(R). Furthermore, Sis dense in L2(R). iii) Scan be endowed with a metric (there exists a distance between vectors) topology, which is stronger than the norm topology inherited from L2(R). This means that the triplet of spaces S ⊂ L2(R)⊂ S×(16) is a RHS. The space S×is equivalent to the space of tempered distributions, with the only (very minor) difference that elements on S×are antilinear mappings on S, while tempered distributions are linear mappings on S. iv) Let [a, b] any interval in the real line R, either finite or infinite. Let S[a, b] the space of functions in Ssupported on [a, b]. One may prove that S[a, b]⊂L2[a, b]⊂S×[a, b],(17) where L2[a, b] is the Hilbert space of complex square integrable functions on the interval [a, b], is again a RHS. Recall that [a, b] could be either finite or infinite, as for instance [0,∞). In particular, S[a, b]is always dense in L2[a, b]. v) The Fourier transform of a Schwartz function is also a Schwartz function. Furthermore, the Fourier transform is an one to one onto mapping from Sonto itself that preserve the topological structre on S. Now, we are in the situation to construct the spaces of smooth Hardy functions. For that purpose, we shall use the Paley-Wiener theorem as an essential ingredient. Let us consider the Schwartz spaces S(R∓) and the space of Fourier transforms of functions in S(R∓), F[S(R∓)]. These spaces have the following properties: i) After the Paley-Wiener theorem, F[S(R∓)] ⊂ H2 ±. ii) The Fourier transform of a Schwartz function is also another Schwartz function and this operation is bijective (one to one and onto), so that F[S(R∓)] ≡ S ∩H2 ±.(18) iii) The spaces S∓are dense in L2(R∓). Since the Fourier transform is unitary, then, S∩H2 ± is dense in H2 ±with the norm topology. In addition due to the above comments on unitary mappings on RHS, the triplets: S ∩H2 ±⊂ H2 ±⊂(S ∩H2 ±)×(19) are well defined RHS. iv) Let us consider the space of restrictions to R+of functions in H±that we shall denote here as H2 ±R+. After the van Winter theorem, there exists one to one onto mappings (and therefore invertible) θ±: θ±:H2 ±7−→ H2 ±R+, θ±:S ∩H2 ±7−→ S ∩H2 ±R+.(20) Van Winter has also proved that H2 ±R+is dense in L2(R+). From here, we can also prove that S ∩H2 ±R+is also dense in L2(R+). The topology on H2 ±R+can be transported by θ±to S ∩H2 ±R+, so that the triplets S ∩H2 ±R+⊂L2(R+)⊂(S ∩H2 ±R+)×(21) are new RHS. It is important to point out that, although functions in S∩H2 ±R+can be uniquely extended to the negative semi axis R−(and to a half plane), as functions in S∩H2 ±R+, we are considering their values on the positive semi axis R+only. 7 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 4. Gamow states and their mathematical construction Once the metastable or decaying state or resonance (all these names represent the same object) has been prepared, it starts to decay. We may use an origin of times t= 0 at which the preparation of the decaying state is complete and starts to decay [3] (Although we shall stick to this notion of the origin of times, this determination is somehow ambiguous [29]). Then, assume that the state is represented at t= 0 by the state vector ψ. The non-decay probability at time t>0 is P(t) = |hψ|e−itHψi|2,(22) where His the interacting or total Hamiltonian, where {H0, H}is the Hamiltonian pair, responsible for the resonance scattering. Assume that ψis a normalized vector in a Hilbert space (in the subspace of scattering states) and that the Hamiltonian His semi bounded, i.e., that its spectrum has a lower bound. This is a condition that have most of known quantum Hamiltonians. Then, P(t) goes to zero as t7−→ ∞ [30]. The vector ψdenotes a decaying state if the non-decay probability is exponential function of the type e−αt,α > 0. This is because the observed decay rate is exponential [3]. However, under the above conditions imposed to ψand H,P(t) cannot be exponential. It could be approximately exponential for almost all times within the range of observation. However, it is far from being exponential for very short (Zeno era) and very long (Khalfin region) times [30]. Both deviations of the exponential regime have been reported to be found experimentally [31,32], but they are difficult to detect, the Zeno era because it is too short and the Khalfin region because it remains very little amount of undecayed material. Thus within a reasonable degree of accuracy, we may assume that resonances decay exponentially at all times. But then, this exponential decay cannot be produced by a normalizable state vector. The vector state that decays exponentially for all t>0 is called the (decaying) Gamow state and it cannot be normalized with the usual L2norm. It is well know that (non relativistic) resonances are very often associated to pairs of poles of the analytic continuation, S(p), to the complex plane of the scattering operator (or scattering matrix), in the momentum representation, located symmetrically with respect to the negative imaginary axis. If we shift to the energy representation, the lower half plane is transformed into the second sheet of the two sheeted Riemann surface corresponding to the transformation p=√2mE. Then, resonances are characterized by pairs of poles of the analytic continuation of S(E) to the second sheet, at values zR=ER−iΓ/2 and its complex conjugate, z∗ R. Here, ERis the resonant energy and Γ the width, which gives the mean life [3]. There are some other mathematical and physical definitions of resonances [3] that not always coincide [30]. The study of the relations of between these definitions needs to be completed. We are now in the position of defining the decaying Gamow vector,ψD, as an eigenvector of the total Hamiltonian Hwith the complex eigenvalue zR=ER−iΓ/2, i.e., HψD=zRψD and the growing Gamow vector,ψG, as an eigenvector of Hwith eigenvalue z∗ R=ER+iΓ/2, HψG=z∗ RψG. Gamow vectors ψDand ψGcannot be normalizable as they are eigenvectors of a self adjoint Hamiltonian with complex eigenvalues. This means that Gamow vectors do not belong to the Hilbert space in which His defined as a self adjoint operator. The advantage of this definition is that the decaying Gamow vector decays exponentially for all positive values of time (we should clarify this point later): e−itHψD=e−itERe−tΓ/2ψD.(23) The fact is that Gamow vectors belong to the antiduals of two, in principle different, RHS. Now, we proceed to explain the idea of their mathematical construction. In order to do it, we reduce the hypothesis to a minimum, further generalizations can be constructed without serious difficulties. Assume that H0and Hhave simple continuous spectrum 8 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 equal to R+= [0,∞), so that both have only scattering states. Assume that these operators are defined on certain Hilbert space H. Then, according to a spectral theorem [28] pp. 226227, there exists a unitary operator U:H 7−→ L2(R+) such that His transformed into the multiplication operator on L2(R+). To understand this, let us consider ψ(E)∈L2(R+) such that E ψ(E)∈L2(R+) (we use the argument Eto mean energy). On the space of functions with this property, let us define the multiplication operator Eas Eψ(E):=Eψ(E). Then, UHU−1=E. Note that Udiagonalizes Hby this operation. Concerning E, the following properties are relevant: i) The operator Eleaves the spaces S ∩H2 ±R+invariant, so that for any function ψ±(E)∈ S ∩H2 ±R+, we have that Eψ±(E) = Eψ±(E)∈ S ∩H2 ±R+. ii) The operator Ecan be extended to the dual (S∩H2 ±R+)×by means of the duality formula: If ψ±(E)∈ S ∩H2 ±R+and F±∈(S ∩H2 ±R+)×, then, hE ψ±(E)|F±i=hψ±(E)|E F±i,(24) so that Eacts on all vectors in (S ∩H2 ±R+)×. Next, we construct a new RHS as follows: Let Ube the unitary operator which diagonalizes Has above (UHU−1=E). If Φ±:= U−1[S ∩ H2 ±R+], we consider these new RHS given by Φ±⊂L2(R+)⊂Φ× ±. Then, UHΦ±=UHU−1[S ∩H2 ±R+] = E[S ∩H2 ±R+]⊂ S ∩H2 ±R+ =⇒HΦ±⊂U−1[S ∩H2 ±R+]=Φ±,(25) so that Φ±reduce H(HΦ±⊂Φ±). By the duality formula, hHϕ±|F±i=hϕ±|H F±i, valid for all ϕ∈Φ±and all F±∈Φ× ±, we extend Hinto the antiduals Φ× ±. Then, we may obtain some rigorous results that we list in the sequel. These results have been proved in some references like [18,19]: i) Assume that zR=ER−iΓ/2 is a resonance pole of the Smatrix associated to the Hamiltonian pair {H0, H}. Then, there exists a unique vector (which is a generalized function) δzR∈(S∩H2 +R+)×such that EδzR=zRδzR, i.e., δzRis an eigenvector of Ewith eigenvalue zR. This eigenvector lies in the dual (S∩H2 +R+)×and cannot be in L2(R+) due to the Hermiticity of E. Analogously, there exists δz∗ R∈(S ∩H2 −R+)×with Eδz∗ R=z∗ Rδz∗ R. The star always means complex conjugation. ii) Let us define the decaying Gamow vector as ψD:= U−1δzR∈Φ× +. Then, EδzR=zRδzR=⇒U−1EUU−1δzR=zRU−1δzR=⇒HψD=zRψD.(26) The latter identity makes sense in Φ× +. Analogously, we define the growing Gamow vector as ψG:= U−1δz∗ R∈Φ× −. Then, we have that HψG=z∗ RψG, equation valid in Φ× −. Thus Gamow vectors are defined as the eigenvectors of the total Hamiltonian Hwith eigenvalues given by resonance poles of the Smatrix in the energy representation. iii) Our structures have been constructed using Hardy spaces on a half plane. Hardy spaces on a half plane split the unitary group given by e−itE,t∈Rinto two semigroups. In fact [18,19]: If t≥0, eitEH2 +⊂ H2 +and also eitE[S∩H2 +R+]⊂ S∩H2 +R+. However, for any t0<0, there exists a function φ+(E)∈ S ∩H2 +R+, such that eit0Eφ+(E)/∈ S ∩H2 +R+. If t≤0, eitEH2 −⊂ H2 −and also eitE[S∩H2 −R+]⊂ S∩H2 −R+. However, for any t0>0, there exists a function φ−(E)∈ S ∩H2 −R+, such that eit0Eφ−(E)/∈ S ∩H2 −R+. 9 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012001 doi :10.1088/1742-6596/839/1/012001 vi) These ideas can be immediately carried to Φ±, since U−1eitEU=eitH. Thus: If t≥0, eitHΦ+⊂Φ+. Furthermore for any t0<0, there exists a vector ϕ+∈Φ+such that eit0Hϕ+/∈Φ+. If t≤0, eitHΦ−⊂Φ−. Furthermore for any t0>0, there exists a vector ϕ−∈Φ−such that eit0Hϕ−/∈Φ−. v) Let Φ ⊂H⊂Φ×be a RHS and Ua unitary operator on Hsuch that U†Φ⊂Φ, where U† is the adjoint of U. Then, Ucan be extended to an operator on Φ×by duality: If ϕ∈Φ and F∈Φ×are chosen arbitrarily, then, hU†ϕ|Fi=hϕ|UF i. This equation defines the action of U on Φ×in a unique way. In consequence: e−itHΦ× +⊂Φ× +,if and only if t>0 ; e−itHΦ× −⊂Φ× −,if and only if t<0.(27) Due to the properties of Hardy functions the evolution group e−itH has split into two semigroups, one for positive values of time and the other for negative values of time. On the other hand, we may reconstruct our spaces in such a way that relations (27) be valid for all values of time [33]. vi) Now, it comes the essential property of Gamow states, which has been proved in [18,19]: For t≥0, the time evolution of the decaying Gamow state is given by e−itH ψD=e−itERe−tΓ/2ψD.(28) This means that the decaying Gamow state has an exact exponential decay for t≥0. Due to the use of Hardy functions in the definition of our spaces, time evolution is not defined for ψD at times t<0. For t≤0, the time evolution of the growing Gamow state is given by e−itH ψG=e−itERe+tΓ/2ψG.(29) This means that the decaying Gamow state has an exact exponential grow for t≤0 or an exact exponential decay to the past. As for ψD, time evolution is not defined for ψGat times t>0. vii) The growing Gamow vector as well as all growing process that takes place in Φ× −for t<0 should not be confused with the process of capture, preparation or creation of the resonance. In fact, it is the time reversal of the decaying process: something that describes the same in the reverse direction of time. In particular, if Tis the time operator, TΦ× ±= Φ× ∓and TψD=ψG and also TψG=ψG. Nevertheless, another interpretation is possible. This interpretation is the founding stone of Time Asymmetric Quantum Mechanics. 5. Time Asymmetric Quantum mechanics As we have just pointed out in the previous section, we may reinterpret Φ× +in a different way than being the time reversal of Φ× −. This interpretation was proposed by A. Bohm and collaborators in a series of papers [4,20–23]. Thus, the notion of time asymmetric quantum mechanics (TAQM) comes from the idea according to which the processes described by vectors in the space Φ× −are not related with the time reversal of the decay as described by vectors in Φ× +. We have already mentioned in the Introduction that irreversibility in quantum mechanics include resonances but it goes beyond than resonances. In particular, it must consider many scattering situations in which the probability of the process in one direction of time is much smaller than the probability of its time reversal [1,2]. In order to introduce a formulation of TAQM, we need to add to the standard formulation of Quantum Mechanics a new axiom to the existing ones. This new axiom is called the Hardy space axiom. Since TAQM manifest itself on scattering processes, resonate or not, it is