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I'm learning Feynman's path-integral from Tuckerman's statistical mechanics textbook. I got stuck a bit at chapter 12.6.1 Path-integral molecular dynamics, I wish I can get some hints here! enter image description here

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The part that confuses me is from (12.6.1) to (12.6.2), I can clearly see how the 2nd and 3rd term in the exponential function were obtained, but $\frac{P_{k}^2}{2m'}$ I must assume it comes from by recasting the prefactor, $(\frac{mP}{2\pi \beta \hslash^2})^{P/2}$, as a set of Gaussian integrals, but Im not certain how it is derived, if someone can interpret more detail about the derivation.

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Yes this is just the identity $$\int dp_1\ldots dp_n \exp\left(-\frac{\alpha}{2}\sum_{i=1}^np_i^2\right) = \left(\frac{2\pi}{\alpha} \right)^{n/2}$$ which comes from factoring the exponential and doing $n$ independent Gaussian integrals.

It looks like they have some factors of $2\pi$ out front wrong (but those don't matter) and they're also playing around with redefining constants in terms of $n$ (which they call $P$) during the same step.