Symmetry and History Quantum Theory: An analogue of Wigner's Theorem

TL;DR

This paper extends Wigner's theorem to history quantum theories, characterizing physical symmetries in terms of projectors and decoherence functionals, and explores their implications for the equivalence of different quantum theories.

Contribution

It introduces a notion of physical symmetry for history quantum theories and provides an exhaustive characterization using an analogue of Wigner's theorem.

Findings

Defined physical symmetries of history quantum theories (PSHQT)

Proved an analogue of Wigner's theorem for these theories

Showed how symmetries relate different history quantum theories

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 $\V$. We define the notion of a `physical symmetry of a history quantum theory' (PSHQT) and specify such objects exhaustively with the aid of an analogue of Wigner's theorem. In order to prove this theorem we investigate the structure of $\D$, define the notion of an `elementary decoherence functional' and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT `physically equivalent' and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.