Stochastic fields from stochastic mechanics

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Abstract
Stochastic field theory for a real scalar field, considering both zero and positive temperatures, is developed from complements to Nelson’s stochastic mechanics. These complements include path integral formulas for the moments of the stochastic process, a functional differential equation for the generating functional, and a virial theorem. Using these and Yasue’s nonstandard analysis formulation of stochastic field theory, a rigorous meaning is given to the path integral formulas for the field moments and to the functional differential equation of the field’s generating functional.