Symmetries of Decoherence Functionals

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 $\V$. 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.