Quantum Fields, Stochastic PDE, and Reflection Positivity

We investigate stochastic quantization as a method to go from a classical PDE (with stochastic time $\lambda$) to a corresponding quantum theory in the limit $\lambda \to \mathrm{\infty }$. We test the method for a linear PDE satisfied by the free scalar field. We begin by giving some background about the importance of establishing the property of {\em reflection positivity} for the limit $\lambda \to \mathrm{\infty }$. We then prove that the measure determined through stochastic quantization of the free scalar field violates reflection positivity (with respect to reflection of the physical time) for every $\lambda <\mathrm{\infty }$. If a non-linear perturbation of the linear equation is continuous in the perturbation parameter, the same result holds for small perturbations. For this reason, one needs to find a modified procedure for stochastic quantization, in order to use that method to obtain a quantum theory.