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

TL;DR

This paper investigates a self-adjoint operator in quantum mechanics that signals the direction of time by its expectation value decreasing monotonically, analyzing its spectral properties and applying it to a free particle example.

Contribution

It derives the spectrum and eigenstates of the operator indicating the time direction, expanding understanding of its mathematical properties within quantum mechanics.

Findings

Operator's spectrum and eigenstates are explicitly derived.

The operator's behavior is exemplified in the free particle case.

Monotonic expectation value decreases demonstrate time direction indication.

Abstract

In [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 operator's properties. In particular, we derive its spectrum and generalized eigenstates, and treat the example of the free particle.