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

The basic ingredients of the consistent histories' approach to quantum theory are a space $\UP$ of history propositions’ and a space $\text{D}$ 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 $\text{D}$, 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.