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Journal of Mathematical Physics

Stochastic symmetry breaking of time reversal invariance

Ronald F. Fox

Published Sep 1, 1973• 5 citations
Publisher

Abstract

An equation for the time evolution of the density matrix for a nonrelativistic quantum mechanical system is presented. It involves a Hamiltonian which contains a stochastic contribution. Before stochastic averaging is performed, it is shown that the density matrix equation is time reversal invariant, whereas after stochastic averaging is performed, it is shown that the averaged density matrix equation is not time reversal invariant.

DOI
10.1063/1.1666461

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An equation for the time evolution of the density matrix for a nonrelativistic quantum mechanical system is presented. It involves a Hamiltonian which contains a stochastic contribution. Before stochastic averaging is performed, it is shown that the density matrix equation is time reversal invariant, whereas after stochastic averaging is performed, it is shown that the averaged density matrix equation is not time reversal invariant.
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Time‐reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle, and not all statistical theories conserve this particular symmetry—most notably, hydrodynamic equations and kinetic equations such as the Boltzmann equation. Here, we consider quantum kinetic generalizations of the Boltzmann equation using the method of reduced density operators, leading to the quantum generalization of the Bogolyubov‐Born‐Green‐Kirkwood‐Yvon (BBGKY) hierarchy. We demonstrate that all commonly used approximations, including Vlasov; Hartree‐Fock; and the non‐Markovian generalizations of the Landau, T‐matrix, and Lenard‐Balescu equations, are originally time‐reversal invariant, and we formulate a general criterion for time reversibility of approximations to the quantum BBGKY hierarchy. Finally, we illustrate, through the example of the Born approximation, how irreversibility is introduced into quantum kinetic theory via the Markov limit, making the connection with the standard Boltzmann equation. This paper is a complement to paper I (Scharnke et al., J. Math. Phys., 2017, 58, 061903), where the time‐reversal invariance of quantum kinetic equations was analysed in the frame of the independent non‐equilibrium Green functions formalism.
It is commonly thought that a state-dependent quantity, after being averaged over a classical ensemble of random Hamiltonians, will always become independent of the state. We point out that this is in general incorrect: if the ensemble of Hamiltonians is time reversal invariant, and the quantity involves the state in higher than bilinear order, then we show that the quantity is only a constant over the orbits of the invariance group on the Hilbert space. Examples include fidelity and decoherence in appropriate models.
Time reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle and not all statistical theories conserve this particular symmetry, most notably hydrodynamic equations and kinetic equations such as the Boltzmann equation. In this article, it is shown analytically that quantum kinetic generalizations of the Boltzmann equation that are derived using the nonequilibrium Green functions formalism as well as all approximations that stem from Φ-derivable self-energies are time reversal invariant.