Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 304 / 2 / b

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- Solution b

Solution

The analogous Grassmann Gaussian integral, normalized by

Z_{0} [0, 0] = 1

, gives

Z_{0} [\overset{η}{ˉ}, η] = exp [- \int d^{4} x, d^{4} y \overset{η}{ˉ} (x) S_{F} (x - y) η (y)],

with the free Dirac propagator

S_{F} (x - y) = \int \frac{d ^{4} p}{( 2 π ) ^{4}} \frac{i ( \neq p + m ) e ^{- i p \cdot (x - y)}}{p ^{2} - m ^{2} + i ϵ} .

Indeed

(i \neq \partial_{x} - m) S_{F} (x - y) = i δ^{(4)} (x - y)

.

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