Study of a self-adjoint operator indicating the direction of time within standard quantum mechanics

Abstract In [Y. Strauss, Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schrodinger evolution, J. Math. Phys. 51 (2010) 022104] a self-adjoint operator was introduced that has the property that it indicates the direction of time within the framework of standard quantum mechanics, in the sense that as a function of time its expectation value decreases monotonically for any initial state. In this paper we study some of this operatorʼs properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle.