STOCHASTIC PDE , REFLECTION POSITIVITY AND QUANTUM FIELDS By Arthur

We outline some known relations between classical random fields and quantum fields. In the scalar case, the existence of a quantum field is equivalent to the existence of a Euclidean-invariant, reflectionpositive (RP) measure on the Schwartz space tempered distributions. Martin Hairer recently investigated random fields in a series of interesting papers, by studying non-linear stochastic partial differential equations, with a white noise driving term. To understand such stochastic quantization, we consider a linear example. We ask: does the measure on the solution induced by the stochastic driving term yield a quantum field? The RP property yields a general method to implement quantization. We show that the RP property fails for finite stochastic parameter λ, although RP does hold in the limiting case λ =∞.