Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 3 / c

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- Solution c

Solution

Fourier transformation sends

\partial_{μ}

to

i q_{μ}

. Write the connected current three-point function as the two full Dirac propagators joined to the amputated vertex:

⟨ j^{μ} ψ \overset{ˉ}{ψ} ⟩_{m co nn} = G (p_{1}) V_{3}^{μ} (q, p_{1}, p_{2}) G (p_{2}), q = p_{1} - p_{2} .

The two contact terms remove one propagator at a time. Multiplying the transformed identity by

G^{- 1} (p_{1})

on the left and

G^{- 1} (p_{2})

on the right yields

i q_{μ} V_{3}^{μ} (q, p_{1}, p_{2}) = i e [G^{- 1} (p_{1}) - G^{- 1} (p_{2})], p_{2} = p_{1} - q .

This is the momentum-space Ward-Takahashi identity for the full vertex and full propagator.

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