Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / d

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- Solution d

Solution

The quadratic momentum-space operator is inverted by the Proca propagator. With the Feynman propagator boundary prescription,

Δ_{μν} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{- i}{p ^{2} - m ^{2} + i ϵ} (η_{μν} - \frac{p _{μ} p _{ν}}{m ^{2}}) e^{- i p \cdot (x - y)} .

Multiplication by the Fourier-space Proca operator gives

i δ_{μ}^{ν}

, with the overall sign determined by the convention for the Green function.

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