An Arrow of Time Operator for Standard Quantum Mechanics

TL;DR

This paper introduces a self-adjoint operator in quantum mechanics that unambiguously indicates the direction of time by monotonically decreasing expectation values, applicable to a broad class of quantum systems.

Contribution

It presents a novel construction of a time arrow operator in standard quantum mechanics, overcoming previous no-go theorems and applicable to systems with specific spectral properties.

Findings

Operator's expectation value decreases monotonically over time

Applicable to systems with finitely degenerate, absolutely continuous, semibounded spectra

Illustrated through scattering problem examples

Abstract

We introduce a self-adjoint operator that indicates the direction of time within the framework of standard quantum mechanics. That is, as a function of time its expectation value decreases monotonically for any initial state. This operator can be defined for any system governed by a Hamiltonian with a uniformly finitely degenerate, absolutely continuous and semibounded spectrum. We study some of the operator's properties and illustrate them for a large equivalence class of scattering problems. We also discuss some previous attempts to construct such an operator, and show that the no-go theorems developed in this context are not applicable to our construction.