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Importance of euclidean field configurations of finite action

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In Coleman's Aspects of symmetry he writes about significance of euclidean field configurations of finite action:

The naive anser, sometimes given in the literature, is that configurations of infinite euclidean action are unimportant in the functional integral, since, for such configurations, $e^{-\frac{S}{\hbar}}$ is zero. This is wrong. In fact, it is configurations of finite action are unimportant; to be precise, they form a set of measure zero in function space. This has nothing to do with divergences in quantum field theory; it is true even for the ordinary harmonic oscillator. The only reason we are interested in doing semiclassical approximations, and a configuration of infinite action does indeed give zero if it is used as the center point of a Gaussian integral.
 

I don't understand this. Precisely, I don't understand when semiclassical approximation enters the game. We use it when impose the Wick rotation, so I don't understand why the semiclassical approximation of the amplitude is different from evaluation of the euclidean path integral by using saddle point approximation.

asked Mar 4, 2016 in Theoretical Physics by NAME_XXX (1,010 points) [ revision history ]
edited Mar 4, 2016 by NAME_XXX

What precisely is your question? The nonanalytic contributions are proportional to $e^{-s/\hbar}$ and arise from tunneling. They are usually obtained in a semiclassical approximation via classical instantons in imaginary time.

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