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Physical Review A

Microphysical irreversibility and the time-reversal operation

Arno Böhm

Published Mar 1, 1995• 23 citations
Publisher

Abstract

The division of a scattering experiment into the preparation of a state and the registration of an observable in this state defines an obvious but commonly ignored quantum-mechanical arrow of time. This leads to irreversibility on the microphysical level. The time evolution semigroup for exponentially decaying Gamow states is an example of this microphysical irreversibility. A time-reversal operator exists which is compatible with the quantum-mechanical arrow of time.

DOI
10.1103/physreva.51.1758

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The division of a scattering experiment into the preparation of a state and the registration of an observable in this state defines an obvious but commonly ignored quantum-mechanical arrow of time. This leads to irreversibility on the microphysical level. The time evolution semigroup for exponentially decaying Gamow states is an example of this microphysical irreversibility. A time-reversal operator exists which is compatible with the quantum-mechanical arrow of time.
The exponential decay (or growth) of resonances provides an arrow of time which is described as the semigroup time evolution of Gamow vector in a new formulation of quantum mechanics. Another direction of time follows from the fact that a state must first be prepared before observables can be measured in it. Applied to scattering experiments, this produces another quantum mechanical arrow of time. The mathematical statements of these two arrows of time are shown to be equivalent. If the semigroup arrow is interpreted as microphysical irreversibility and if the arrow of time from the prepared in-state to its effect on the detector of a scattering experiment is interpreted as causality, then the equivalence of their mathematical statements implies that causality and irreversibility are interrelated.
Quantum mechanics in the Rigged Hilbert Space formulation describes quasistationary phenomena mathematically rigorously in terms of Gamow vectors. We show that these vectors exhibit microphysical irreversibility, related to an intrinsic quantum mechanical arrow of time, which states that preparation of a state has to precede the registration of an observable in this state. Moreover, the Rigged Hilbert Space formalism allows the derivation of an exact golden rule describing the transition of a pure Gamow state into a mixture of interaction-free decay products.
The time reversal and irreversibility in conventional quantum mechanics are compared with those of the rigged Hilbert space quantum mechanics. We discuss the time evolution of Gamow and Gamow-Jordan vectors and show that the rigged Hilbert space case admits a new kind of irreversibility which does not appear in the conventional case. The origin of this irreversibility can be traced back to different initial-boundary conditions for the states and observables. It is shown that this irreversibility does not contradict the experimentally tested consequences of the time-reversal invariance of the conventional case but instead we have to introduce a new time reversal operator.
The connection between the micro and macro-arrow of time is discussed in the frame of the stochastic quantum hydrodynamic analogy. The presence of fluctuations that in the case on non-linear interactions leads to the breaking of the quantum mechanics on large scale and then to the macroscopic irreversibility and arrow of time, gives also rise to the time reversal breaking in the micro-scale quantum stochastic evolution (micro-arrow of time). The quantum irreversibility and the theoretical tools to calculate the time reversal asymmetry in the elementary particle decays are briefly discussed.
The connection between the micro and macro-arrow of time is discussed in the frame of the stochastic quantum hydrodynamic analogy (SQHA). The presence of fluctuations that in the case on non-linear interactions leads to the breaking of the quantum mechanics on large scale and then to the macroscopic irreversibility with a defined arrow of time, gives also rise to the time reversal breaking in the micro-scale quantum evolution (micro-arrow of time). The quantum irreversibility with time reversal asymmetry is briefly discussed.