Quantum Time-Reversal Symmetry

Shan Gao

Ce Wang

Time-reversal symmetry is of fundamental importance to physics. In the classical theory of time-reversal symmetry, the time-reversal symmetry of a quantum system is described by an anti-unitary operator, which is known as the time-reversal operator of the system. In this paper, we introduce and study a model of continuous-time quantum walk on a special locally infinite graph. After examining its spectral property, we investigate the time-reversal symmetry of the model. To our surprise, we find that its time-reversal symmetry can be described directly by a unitary operator, which contrasts sharply with that in the classical theory of time-reversal symmetry. Some other related results are also proven.

Ali Mostafazadeh

For an invertible (bounded) linear operator acting in a Hilbert space , we consider the consequences of the -symmetry of a non-hermitian Hamiltonian where is the time-reversal operator. If H is symmetric in the sense that , then -symmetry is equivalent to -weak-pseudo-hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non- -symmetric Hamiltonians that share the spectral properties of -symmetric Hamiltonians.

Jiří Pešek

Hsiang Shun Chou

Abstract Physicists believe that there is a corresponding conservation law for each principle of symmetry. In standard textbooks on quantum mechanics, however, it is asserted that the time-reversal symmetry does not lead to any conservation law. There seems to be a missing piece of the jigsaw puzzle of the connection between the principles of symmetry and the conservation laws. In this paper, we resolve this difficulty by considering the time-reversal invariance of the transition probability a basic physical requirement of the time-reversal symmetry. We show that this requirement, together with the time-reversal covariance of the Schrödinger equation, guarantees the hermiticity of the Hamiltonian which ensures the probability conservation. Thus we complete the missing piece by demonstrating that the time-reversal symmetry leads to the probability conservation. The present paper can help undergraduate students gain deeper insight into the connection between the principles of symmetry and the conservation laws in quantum mechanics.

Florian A. Scheck