# path integrals: how/why can the phase be identified with the action?

In Peskin & Schroeder, chapter 9 introduces the functional methods.

The idea, to recall, is simply to sum over all the possible paths:

$U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . \text{phase}} = \int Dx(t) e^{i . \text{phase}}$

Then, it happens that the phase could be identified with the action. But I have not understood how this could be done ... Could anyone clarify that point?

In the classical (small $\hbar$) limit you can find the dominant contribution to the path integral by the method of stationary phase. Because of that it is natural to identify the phase with something that is extremised in classical dynamics, the obvious candidate being the action. –  Michael Brown Jan 19 '13 at 10:14