Expansion around a classical vacuum

[GargleBlast42]

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 $$\int D\phi e^{i S[\phi]}$$ 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 $$\lambda$$ in $$\phi^4$$ theory), but one doesn't assume $$\phi$$ 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.

 

[GargleBlast42]

I'm sorry for bumping this, but I would be really happy about any input.

 

[Demystifier]

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.

 

[genneth]

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.

 

[Avodyne]

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.