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Imaginary-Time Path Integrals from Klein-Gordon Equation

The ordinary Feynman-Kac-Itô formula gives a path integral representation of the Schrödinger semi-groups. We discuss here an analogous probabilistic expression for the (positive energy) solutions of the imaginary-time Klein-Gordon equation in a static external electromagnetic field. When the external field is not purely magnetic, our result is different from the path integral associated to the semi-group t ∊ [0, + ∞) → exp[−t(H−mc2)/ℏ] of a relativistic Schrödinger operator H = {c2[−iℏ∇−(e/c)A]2 + m2c4}1/2 + V.