Physical symmetries of quantum histories

Gell-Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals play a key role. Physical symmetrie s of quantum history systems are shown, under mild physical assumptions, to correspond to isometric Jordan *-automorphisms of the underlying von Neuma nn algebra. Unitary representations of groups of symmetries of quadratic fo rms over operator algebras are discussed and related to symmetries of decoh erence functionals.