Editing Feynman diagram (section)

=== Schwinger representation<!--'Schwinger representation' redirects here--> ===
The Euclidean scalar propagator has a suggestive representation:

:<math> \frac{1}{p^2+m^2} = \int_0^\infty e^{-\tau\left(p^2 + m^2\right)}\, d\tau </math>

The meaning of this identity (which is an elementary integration) is made clearer by Fourier transforming to real space.

:<math> \Delta(x) = \int_0^\infty d\tau e^{-m^2\tau} \frac{1}{ ({4\pi\tau})^{d/2}}e^\frac{-x^2}{ 4\tau}</math>

The contribution at any one value of {{mvar|τ}} to the propagator is a Gaussian of width {{math|{{sqrt|''τ''}}}}. The total propagation function from 0 to {{mvar|x}} is a weighted sum over all proper times {{mvar|τ}} of a normalized Gaussian, the probability of ending up at {{mvar|x}} after a random walk of time {{mvar|τ}}.

The path-integral representation for the propagator is then:

:<math> \Delta(x) = \int_0^\infty d\tau \int DX\, e^{- \int\limits_0^{\tau} \left(\frac{\dot{x}^2}{2} + m^2\right) d\tau'} </math>

which is a path-integral rewrite of the '''Schwinger representation'''<!--boldface per WP:R#PLA-->.

The Schwinger representation is both useful for making manifest the particle aspect of the propagator, and for symmetrizing denominators of loop diagrams.