Symmetries of Decoherence Functionals

TL;DR

This paper explores the symmetries of decoherence functionals within the consistent histories approach to quantum theory, establishing a symmetry group and explicitly calculating some symmetries for history quantum mechanics.

Contribution

It introduces the concept of symmetry groups of decoherence functionals and extends Wigner's theorem to history quantum theories.

Findings

Defined symmetry of a decoherence functional.

Proved that all symmetries form a group.

Explicitly calculated some symmetries in history quantum mechanics.

Abstract

The basic ingredients of the `consistent histories' approach to quantum theory are a space $\UP$ of `history propositions' and a space $\D$ of `decoherence functionals'. In this article we consider such history quantum theories in the case where $\UP$ is given by the set of projectors $\P(\V)$ on some Hilbert space $\V$. Using an analogue of Wigner's Theorem in the context of history quantum theories proven earlier, we develop the notion of a `symmetry of a decoherence functional' and prove that all such symmetries form a group which we call `the symmetry group of a decoherence functional'. We calculate---for the case of history quantum mechanics---some of these symmetries explicitly and relate them to some discussions that have appeared previously.