  General history quantum theories are quantum theories without a globally
defined notion of time. Decoherence functionals represent the states in the
history approach and are defined as certain bivariate complex-valued
functionals on the space of all histories. However, in practical situations --
for instance in the history formulation of standard quantum mechanics -- there
often is a global time direction and the homogeneous decoherence functionals
are specified by their values on the subspace of homogeneous histories.
  In this work we study the analytic properties of (i) the standard decoherence
functional in the history version of standard quantum mechanics and (ii)
homogeneous decoherence functionals in general history theories. We restrict
ourselves to the situation where the space of histories is given by the lattice
of projections on some Hilbert space H. Among other things we prove the
non-existence of a finitely valued extension for the standard decoherence
functional to the space of all histories, derive a representation for the
standard decoherence functional as an unbounded quadratic form with a natural
representation on a Hilbert space and prove the existence of an
Isham-Linden-Schreckenberg (ILS) type representation for the standard
decoherence functional.
