Hi everyone, I have a severe confusion about the notions of "expanding the theory around a classical vacuum" and "considering small fluctuations around a classical vacuum" which I find in QFT textbooks. My problem is: in the path integral [tex]\int D\phi e^{i S[\phi]}[/tex] one doesn't integrate only over field configurations close to the vacuum, but over all field configurations. And when one is considering a perturbative expansion, this expansion is in the coupling constant (like [tex]\lambda[/tex] in [tex]\phi^4 [/tex] theory), but one doesn't assume [tex]\phi[/tex] to be small, or am I wrong? So the questions would be: Why does one require the field configurations to be small fluctuations around a classical vacuum? And what would happen if I was expanding the theory about a field configuration that is not a classical vacuum (except that the mass could be possibly negative)? The first question is more important for me. I would be very grateful for any clarification.
You are right that one integrates over all values of fields, not only the small ones. The assertion that field is small means something else. It refers to a physical value of field, such as the boundary value appearing in the definition of the path integral. In particular, if you calculate the vacuum-to-vacuum transition, then the boundary values of the field are zero, which, of course, are small.
Do you have any idea how to actually compute these integrals? If not, I'm afraid that the answer won't make sense --- the entire apparatus is rather formal, which is to say, it is a series of methods to circumvent the problem that evaluating these integrals exactly is impossible.
There is a strong analogy with evaluating an ordinary integral of this type by the method of stationary phase. One first finds the point(s) of stationary phase, and then approximates the integral as a gaussian (which equates to treating the fluctuations as "small" in some formal sense) around each such point. Corrections to the gaussian correspond to doing perturbation theory in QFT.