Derivation of Propagators

Proof:
Since ,
So,

This is evaluated as a contour integral» using the residue theorem», and noting that the integral on C₂ vanishes in the lower half plane if x⁰ > 0, and in the upper half plane if x⁰ < 0.

The Photon Propagator

Using the above lemma,
where if and if is the momentum from j to i. Provided that the integrals converge at , the step functions can be summed to unity in the integrand.

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The Dirac Propagator

For Dirac particles, the creation and annihilation operators obey
and
Hence, using the above lemma,
where if and if is the momentum from j to i (in the direction of the arrow). Provided that the integrals converge at , the step functions can be summed to unity in the integrand.
For a Dirac particle,p⁰ > 0 , and we can simplify the denominator by shifting the pole under the limit, replacing 2ip⁰ + ε² with iε. Thus, the Dirac propagator arrowed from j to i is

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