Symmetries of decoherence functionals

The basic ingredients of the “consistent histories” approach to quantum theory are a space ???????? of “history propositions” and a space ???? of “decoherence functionals.” In this article we consider such history quantum theories in the case where ???????? is given by the set of projectors ????(????) on some Hilbert space ????. Using an analog 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 standard quantum mechanics—when looked at from the perspective of the history program—some of these symmetries explicitly and relate them to some discussions that have appeared previously.