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Firstly I am not sure as to whether I am using the word "condensate" in the right context. In QFT contexts I think I see it getting used to mean the space-time independent solution which would solve the Euler-Lagrange equations of the action that would sit in the exponent in the path-integral - which in general might be different from the classical action. I would like to know why are these kinds of solutions so important - because this is picking out some special configurations among the entire space of classical solutions which would in general include non-trivially dynamical solutions.
Now when one is doing a "small" fluctuation about the condensate and integrating out degrees of freedom to get an effective action for one of the fluctuation variables then there are two issues which confuse me -
In multi component fields (like say complex ones which can be thought of as the modulus and the phase) what drives the choice as to which fluctuation is to be integrated out? (..in the complex case I guess in general people talk to the effective action for the phase fluctuation..)
What is most confusing to me is to understand how to determine whether the space-time derivatives of the fluctuations are big or small. If one is doing the calculation to say second order then does one keep the products and squares of the derivatives of the fluctuation at the same level of perturbation as the squares and products of the fluctuation themselves? I can't see a natural scale for the derivatives of the fluctuations to which I can compare the derivatives to decide whether they are large or small.
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