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

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 Schr&#246;dinger 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 &#8220;unphysical&#8222;, 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>; 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>.

Abstract licence: CC BY

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