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arXiv:0803.3233 · Journal of Physics A Mathematical and Theoretical

Quantal time asymmetry: mathematical foundation and physical interpretation

Arno Böhm, Peter William Bryant, Yoshihiro Sato

Published Jul 15, 2008• 6 citations
arXivHTMLPDF

Abstract

For a quantum theory that includes exponentially decaying states and Breit–Wigner resonances, which are related to each other by the lifetime–width relation , where τ is the lifetime of the decaying state and Γ is the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz–Rigged Hilbert Space of the Dirac formalism. One has to distinguish between prepared states, using a space , and detected observables, using a space , where −(+) refers to analyticity of the energy wavefunction in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: a state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime–width relation is that Φ + and Φ − are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.

DOI
10.1088/1751-8113/41/30/304019

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For a quantum theory that includes exponentially decaying states and Breit–Wigner resonances, which are related to each other by the lifetime–width relation , where τ is the lifetime of the decaying state and Γ is the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz–Rigged Hilbert Space of the Dirac formalism. One has to distinguish between prepared states, using a space , and detected observables, using a space , where −(+) refers to analyticity of the energy wavefunction in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: a state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime–width relation is that Φ + and Φ − are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.
The states (Schrödinger picture) and observables (Heisenberg picture) in the standard quantum theory evolve symmetrically in time, given by the unitary group with time extending over − ∞ < t < + ∞ . This time evolution is a mathematical consequence of the Hilbert space boundary condition for the dynamical differential equations. However, this unitary group evolution violates causality. Moreover, it does not solve an old puzzle of Wigner: How does one describe excited states of atoms which decay exponentially, and how is their lifetime τ related to the Lorentzian width Γ? These question can be answered if one replaces the Hilbert space boundary condition by new, Hardy space boundary conditions. These Hardy space boundary conditions allow for a distinction between states (prepared by a preparation apparatus) and observables (detected by a registration apparatus). The new Hardy space quantum theory is time asymmetric, i.e, the time evolution is given by the semigroup with t 0 ≤ t < + ∞ , which predicts a finite "beginning of time" t 0 , where t 0 is the ensemble of time at which each individual system has been prepared. The Hardy space axiom also leads to the new prediction: the width Γ and the lifetime τ are exactly related by τ = ℏ/Γ .
The exact relation τ = ℏ/Γ between the width Γ of a resonance and the lifetime τ for the decay of this resonance could not be obtained in standard quantum theory based on the Hilbert space or Schwartz space axiom in non-relativistic physics as well as in the relativistic regime. In order to obtain the exact relation, one has to modify the Hilbert space axiom or the Schwartz space axiom and choose new boundary conditions based on the Hardy space axioms in which the space of the states and the space of the observables are described by two different Hardy spaces. As consequences of the new Hardy space axioms, one obtains, instead of the symmetric time evolution for the states and the observables, asymmetrical time evolutions for the states and observables which are described by two semi-groups. A relativistic resonance obeying the exponential time evolution can be described by a relativistic Gamow vector, which is defined as superposition of the exact out-plane wave states with a Breit-Wigner energy distribution of the width Γ.
The exact relation τ = ~/Γ between the width Γ of a resonance and the lifetime\nτ for a decay of this resonance could not be obtained in the conventional quantum\ntheory based on the Hilbert space as well as on the Schwartz space in which the spaces\nof the states and of the observables are described by the same space. Furthermore,\nboth dynamical evolution of the states (in Schrodinger picture) and observables (in\nHeisenberg picture) are symmetrically in time, given by an unitary group with time\nextending over −∞ < t < +∞. This time symmetric evolution is a mathematical\nconsequence of Von Neumann theorem for the dynamical differential equations,\nSchrodinger equation for the state or Heisenberg equation for the observable, under\neither the Hilbert space or Schwartz space boundary condition. However, this unitary\ngroup evolution violates causality. In order to get a quantum theory in which the\nexact relation and causality are obtained, one has to replace the Hilbert space or the\nSchwartz space boundary condition by a new boundary condition based on the Hardy\nspace axioms in which the space of the states and the space of the observables are\ndescribed by two distinguishable Hardy spaces. As a consequence of the new Hardy\nspace axiom, one obtains, instead of the time symmetric evolution for the states and\nthe observables, time asymmetrical evolutions for the states and observables which\nare described by two semi-groups. The time asymmetrical evolution predicts an finite\nbeginning of time t0(= 0) for quantum system which can be experimental observed\nby directly measuring the lifetime of the decaying state as the beginning of time of\nthe ensemble in scattering experiments. A resonance obeying the exponential time\nevolution can then be described by a Gamow vector, which is defined as superposition\nof the exact out-plane wave states with exact Breit-Wigner energy distribution.
This article is a review of time asymmetric quantum theory and its consequences applied to the resonance and decay phenomena. We first give some phenomenological results about resonances and decaying states to support the popular idea that resonances characterized by a width \Gamma and decaying states characterized by a lifetime \tau are different appearances of the same physical entity. Based on Weisskopf-Wigner (WW) methods, one obtains approximately \frac{\hbar}{\tau}\approx \Gamma. However, using standard axioms of quantum physics it is not possible to establish a rigorous theory to which the various WW methods can be considered as approximations. In standard quantum theory, the set of states and the set of observables are mathematically identified and described by the same Hilbert space H. Modifying this Hilbert space axiom to a Hardy space axiom one distinguishes the prepared (in) states and detected (out) observables. This leads to semi-group time evolution and to beginnings of time for individual microsystems. As a consequence of this time asymmetric theory one derives \frac{\hbar}{\tau} = \Gamma as an exact relation, and this unifies resonances and decaying states. Finally, we show that this unification can also be extended to the relativistic regime.
Abstract A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in‐states of scattering theory) and the space of observables (out‐“states” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law.
The meaning of time asymmetry in quantum physics is discussed.On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group.Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution.Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)).If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ = Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t 0 ≤ t < ∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system.The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry.From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.