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arXiv:quant-ph/9902085 · Physical Review A

Time-asymmetric quantum physics

Arno Böhm

Published Aug 1, 1999• 57 citations
arXivHTMLPDF

Abstract

A quantum theory that applies to the (closed) universe needs to be time asymmetric because of the cosmological arrow of time. The preparation $\ensuremath{\Rightarrow}$ registration arrow of time (a state must be prepared before an observable can be detected in it) of the quantum mechanics of measured systems provides a phenomenological reason for an asymmetric semigroup time evolution. The standard theory in the Hilbert space (HS) is inadequate for either since the mathematics of the HS allows only reversible unitary group evolution and time symmetric boundary conditions. The mathematical theory that describes time-asymmetric quantum physics in addition to providing the mathematics for the Dirac kets is the rigged Hilbert space (RHS) theory. It uses a pair of RHS's of Hardy class with complementary analyticity property, one for the prepared states (``in states'') and the other for the registered observables (``out states''). The RHS's contain Gamow kets which have all the properties needed to represent decaying states and resonances. Gamow kets have only asymmetric time evolution. The neutral kaon system is used to show that quasistationary microphysical systems can be experimentally isolated if their time of preparation can be accurately identified. The theoretical predictions for a Gamow ket have the same features as the observed decay probabilities, including the time ordering. This time ordering is the same as the time ordering in the probabilities of histories for the quantum universe. The fundamental quantum mechanical arrow of time represented by the semigroup in the RHS is therefore the same as the cosmological arrow of time, assuming that the universe can be considered a closed quantum system.

DOI
10.1103/physreva.60.861

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A quantum theory that applies to the (closed) universe needs to be time asymmetric because of the cosmological arrow of time. The preparation \ensuremath⇒\ensuremath{\Rightarrow}\ensuremath⇒ registration arrow of time (a state must be prepared before an observable can be detected in it) of the quantum mechanics of measured systems provides a phenomenological reason for an asymmetric semigroup time evolution. The standard theory in the Hilbert space (HS) is inadequate for either since the mathematics of the HS allows only reversible unitary group evolution and time symmetric boundary conditions. The mathematical theory that describes time-asymmetric quantum physics in addition to providing the mathematics for the Dirac kets is the rigged Hilbert space (RHS) theory. It uses a pair of RHS's of Hardy class with complementary analyticity property, one for the prepared states (in states'') and the other for the registered observables (out states''). The RHS's contain Gamow kets which have all the properties needed to represent decaying states and resonances. Gamow kets have only asymmetric time evolution. The neutral kaon system is used to show that quasistationary microphysical systems can be experimentally isolated if their time of preparation can be accurately identified. The theoretical predictions for a Gamow ket have the same features as the observed decay probabilities, including the time ordering. This time ordering is the same as the time ordering in the probabilities of histories for the quantum universe. The fundamental quantum mechanical arrow of time represented by the semigroup in the RHS is therefore the same as the cosmological arrow of time, assuming that the universe can be considered a closed quantum system.
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.
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 .
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. 1 Introduction—Time Asymmetry Time asymmetry, irreversibility, time reversal non-invariance are different concepts and they are (probably) not (all) related to each other [1], [2], [3]. These concepts are usually called arrows-of-time. Time asymmetry comes from time asymmetric boundary conditions of time symmetric equations; its most prominent consequence is causality. The radiation arrow of time is its example from classical physics. Irreversibility
The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theory allows for the in-states ϕ+ϕ^+ϕ+ and the out-states (observables) ψ−ψ^-ψ− of the S-matrix elements (ψ−,ϕ+)=(ψout,Sϕin)(ψ^-,ϕ^+)=(ψ^{out},S ϕ^{in})(ψ−,ϕ+)=(ψout,Sϕin) any elements of the HS $\H$, {ψ−}={ϕ+}(=)˝\{ψ^-\}=\{ϕ^+\}(=\H){ψ−}={ϕ+}(=)˝​, the RHS theory chooses the boundary conditions~: ϕ+∈Φ−⊂⊂˝Φ−×ϕ^+\inΦ_-\subset\H\subsetΦ_-^\timesϕ+∈Φ−​⊂⊂˝​Φ−×​, ψ−∈Φ+⊂⊂˝Φ+×ψ^-\inΦ_+\subset \H\subset Φ_+^\timesψ−∈Φ+​⊂⊂˝​Φ+×​, where Φ−Φ_-Φ−​ (Φ+Φ_+Φ+​) are Hardy class spaces associated to the lower (upper) half-plane of the second sheet of the analytically continued S-matrix. This can be phenomenologically justified by causality. The two RHS's for states ϕ+ϕ^+ϕ+ and observables ψ−ψ^-ψ− provide new vectors which are not in $\H$, e.g. the Dirac-Lippmann-Schwinger kets ∣E±∈Φ∓×|E^{\pm}\inΦ_{\mp}^{\times}∣E±∈Φ∓×​ (solutions of the Lippmann-Schwinger equation with ±iε\pm iε±iε respectively) and the Gamow vectors ∣ER−iΓ/2±∈Φ∓×|E_R-iΓ/2^\pm\inΦ_{\mp}^\times∣ER​−iΓ/2±∈Φ∓×​. The Gamow vectors ∣ER−iΓ/2−|E_R-iΓ/2^-∣ER​−iΓ/2− have all the properties that one heuristically needs for quasistable states. In addition, they give rise to asymmetric time evolution expressing irreversibility on the microphysical level.