# Schwinger Quantum action

Given the path integral

$$< -\infty | \infty > = N \int D[\phi ]e^{i \int dx (L+J \phi ) }$$

then , it would be true that (Schwinger)

$$\frac{\partial < -\infty | \infty >}{\partial J }= < -\infty |\delta \int L d^{4}x | \infty >$$

If so, could someone provide an exmple with the Kelin-gordon scalar field plus an interaction of the form $$\phi ^{4}$$

$$\int\!\mathcal{D}\phi\, \frac{\delta}{\delta \phi} \left( \dots \right) = 0$$
Essentially, one integrates by parts and hopes that the exponential damps out the boundary in function space, if such a boundary exists. Thus, the equations of motion for $$\phi$$ are obeyed inside the expectation value.
$$\int\!d^4x\, \frac{1}{2}\phi ( - \partial^2 - m^2 ) \phi + \frac{\lambda}{4!} \phi^4$$.