Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 301 / 3 / a / i

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- Solution i

Solution

The classical equations are

(□ + M^{2}) ϕ + λ \overset{ˉ}{ψ} ψ = 0, (i \neq \partial - m - λ ϕ) ψ = 0, i (\partial_{μ} \overset{ˉ}{ψ}) γ^{μ} + m \overset{ˉ}{ψ} + λ ϕ \overset{ˉ}{ψ} = 0.

For any time-ordered functional

O

, the three Schwinger-Dyson equations state that insertion of each left-hand side equals the corresponding contact terms obtained by

- i

times the functional derivative of

O

with respect to

p hi

,

\overset{ˉ}{ψ}

, or

ψ

, with the usual reversed order and Grassmann signs for fermionic derivatives.

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