  We explore a framework for complex classical fields, appropriate for
describing quantum field theories. Our fields are linear transformations on a
Hilbert space, so they are more general than random variables for a probability
measure. Our method generalizes Osterwalder and Schrader's construction of
Euclidean fields. We allow complex-valued classical fields in the case of
quantum field theories that describe neutral particles. From an analytic
point-of-view, the key to using our method is reflection positivity. We
investigate conditions on the Fourier representation of the fields to ensure
that reflection positivity holds. We also show how reflection positivity is
preserved by various space-time compactifications of Euclidean space.
