Next Article in Journal
Influence of Chronic Obstructive Pulmonary Disease and Moderate-To-Severe Sleep Apnoea in Overnight Cardiac Autonomic Modulation: Time, Frequency and Non-Linear Analyses
Previous Article in Journal
Bayesian Network Modelling of ATC Complexity Metrics for Future SESAR Demand and Capacity Balance Solutions
Previous Article in Special Issue
Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles
Article Menu
Issue 4 (April) cover image

Export Article

Open AccessArticle

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

Institute of Industrial Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8574, Japan
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;
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)
PDF [1036 KB, uploaded 17 April 2019]


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

Figure 1

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).

Share & Cite This Article

MDPI and ACS Style

Hatano, N.; Ordonez, G. Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems. Entropy 2019, 21, 380.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top