Complex classical fields: a framework for reflection positivity (Q2249441)

Constructing a rigorous mathematical foundation for quantum filed theory is a subtle question. The idea studied in this paper is a generalization of the Euclidean fields constructed by \textit{K. Osterwalder} and \textit{R. Schrader} [Commun. Math. Phys. 31, No. 2, 83--112 (1973; Zbl 0274.46047)]. In this model fields are linear transformations on a Hilbert space, and the fields are complex valued. The motivation for this model is given by Hamiltonians with complex-valued heat kernels, that arise, for example, in special relativity. One of the modifications that shows up is that charge conjugation is no longer just complex conjugation. The paper gives a careful analysis of a property called reflection positivity including a number of equivalent conditions together with related positivity conditions coming from spacial symmetry.