The representation theory of decoherence functionals in history quantum theories

In the first part of this paper the general perspective of history quantum theories is reviewed. History quantum theories provide a conceptual and mat hematical framework for formulating quantum theories without a globally def ined Hamiltonian time evolution and for introducing the concept of space-ti me event into quantum theory. On a mathematical level a history quantum the ory is characterized by the space of histories, which represent the space-t ime events, and by the space of decoherence functionals, which represent th e quantum mechanical states in the history approach. The second part of thi s paper is devoted to the study of the structure of the space of decoherenc e functionals for some physically reasonable spaces of histories in some de tail. The temporal reformulation of standard Hamiltonian quantum theories s uggests to consider the case that the space of histories is given by (i) th e lattice of projection operators on some Hilbert space or, slightly more g enerally, (ii) the set of projection operators in some von Neumann algebra. In the case (i) the conditions are identified under which decoherence func tionals can be represented by, respectively, trace class operators, bounded operators, or families of trace class operators on the tensor product of t he underlying Hilbert space by itself. Moreover, we discuss the naturally a rising representations of decoherence functionals as sesquilinear forms. Th e paper ends with a discussion of the consequences of the results for the g eneral axiomatic framework of history theories.