The classical equations are
$$
(\Box+M^2)\phi+\lambda\bar\psi\psi=0,
\qquad
(i\not\partial-m-\lambda\phi)\psi=0,
\qquad
i(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi+\lambda\phi\bar\psi=0.
$$
For any time-ordered functional $\mathcal O$, 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 $\mathcal O$ with respect to $phi$, $\bar\psi$, or $\psi$, with the usual reversed order and Grassmann signs for fermionic derivatives.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-301/3/a/i/solution