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since $\phi(x)$ is real, $\phi^*(q) = \phi(-q)$ implies Re$(\phi(q))$ is even and Im$(\phi(q))$ is odd, but nonetheless independent unlike the real and imaginary parts of $\phi(x)$. So in the new measure it must be understood that you are summing only over the even/odd functions. There for ...

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Yes, the path integral measure has units but they are mostly irrelevant because physically well-defined objects tend to be ratios of path integrals in which the basic portion cancels, along with its units. So the overall normalization factor in front of the path integral (including its units) drops out. But the path integral measure should be assigned the ...

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The following situation is not uncommon: classically a symmetry may be (spontaneously) broken, but, quantum mechanically, the symmetry is restored. Put differently, quantum fluctuations can, under certain, well understood conditions, destroy the classical asymmetry ("order"). The simplest example is probably the one-dimensional double well potential, ...

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I did not get my copy of Srednicki out but from what you have written... Srednicki is referencing the method of steepest descent. Although these notes look to be better than wikipedia. Another page that is directly applicable to the quantum field theory case is here. In short, exponential integrals may be estimated by the saddle points of the integrand. ...

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