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What should be the limits of integration for euclidean action $S(\phi)$ in 3d and 4d? This action is negatively exponentiated to calculate the decay rate. I suspect that it is variable limit problem.

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This doesn't make any sense. What limits and what action are you talking about? – Marek Jul 9 '11 at 17:38
This is just another Euclidean continuation confusion. The author is misinterpreting the decay of states in imaginary time with a decay rate of an unstable state. The actual decay rate is given by an instanton gas approximation, which works in imaginary time, but isn't an energy (it's the imaginary part of the energy of the state). – Ron Maimon Sep 3 '11 at 23:54

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The limits of integration are usually all space, when you want the vacuum correlation functions, or around a circle in time, if you want thermal correlation functions. A circle in time means a finite 0 to T time window, with periodic boundary conditions for the bosonic fields and anti-periodic boundary conditions for the fermions.

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