Quantum Fields, Stochastic PDE, and Reflection Positivity

We investigate stochastic quantization as a method to go from a classical PDE (with stochastic time $λ$) to a corresponding quantum theory in the limit $λ\to\infty$. 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 $λ\to\infty$. 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 $λ<\infty$. 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.

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