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Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems

1
Institute of Industrial Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8574, Japan
2
Department of Physics and Astronomy, Butler University, 4600 Sunset Avenue, Indianapolis, IN 46208, USA
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(4), 380; https://doi.org/10.3390/e21040380
Received: 24 December 2018 / Revised: 11 March 2019 / Accepted: 3 April 2019 / Published: 8 April 2019
(This article belongs to the Special Issue Coherence in Open Quantum Systems)
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Abstract

It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called “unphysical”, they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 . View Full-Text
Keywords: open quantum system; time-reversal symmetry; arrow of time; resonant state open quantum system; time-reversal symmetry; arrow of time; resonant state
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Hatano, N.; Ordonez, G. Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems. Entropy 2019, 21, 380.

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