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Expansion around a classical vacuum

  1. May 15, 2011 #1
    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.
  2. jcsd
  3. May 17, 2011 #2
    I'm sorry for bumping this, but I would be really happy about any input.
  4. May 18, 2011 #3


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    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.
    Last edited: May 18, 2011
  5. May 18, 2011 #4
    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.
  6. May 18, 2011 #5


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    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.
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