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where <math> \hat A </math> is a differential operator with <math> \varphi </math> and {{mvar|J}} functions of [[spacetime]], and <math> D\varphi </math> indicates integration over all possible paths. In analogy with the matrix version of this integral the solution is
:<math>\int \exp\left[ \int d^4x \left (-\frac 1 2 \varphi \hat A \varphi +J\varphi \right) \right ] D\varphi \; \propto \; \exp \left( {1\over 2} \int d^4x \; d^4y J(x) D( x - y) J(y) \right)</math>
where
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