Schwinger–Dyson equation

Remarks

The proof is integration by parts in the variable $\phi_i$. On almost every line parallel to the $i$-th axis, the function $g = F e^{-S}$ and its derivative $g' = (\partial_i F - F\,\partial_i S)e^{-S}$ are integrable. So $g$ has limits at $\pm\infty$, these limits are $0$, and $\int g' = 0$.

In field theory, $\phi_i$ are the values of a lattice field, and the formal continuum form reads $\langle \delta F/\delta\phi(x) \rangle = \langle F \, \delta S/\delta\phi(x) \rangle$. It expresses the invariance of the path integral under a shift of the integration variable.

Example: take the Euclidean quartic interaction, $S = \int \left(\tfrac12\abs{\nabla\phi}^2 + \tfrac12 m^2\phi^2 + \tfrac{\lambda}{4!}\phi^4\right)dx$, and $F = \phi(y)$. The equation becomes $$(-\Delta_x + m^2)\langle \phi(x)\phi(y) \rangle + \frac{\lambda}{3!}\langle \phi(x)^3\phi(y) \rangle = \delta(x - y).$$ For $\lambda = 0$ it says that the two-point function is the Green's function of $-\Delta + m^2$, the propagator.

Going deeper

For a student

The equations form an infinite system that links each correlation function to higher ones. They hold beyond perturbation theory and are used to derive Ward–Takahashi identities when $F$ is replaced by the variation under a symmetry. Truncating the system gives nonperturbative approximations, for example in studies of the quark propagator. In random matrix theory, the same integration by parts gives the loop equations.