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When I read about Altland and Simons “Condensed matter field theory”, I came across with the path integral (3.28).

$$\langle {q_f}|e^{-iHt/\hbar} |q_i\rangle = \det(\frac{i}{2\pi \hbar} \frac{\partial^2 S[q_{cl}]}{\partial q_i \partial q_f})^{\frac{1}{2}} e^{\frac{i}{\hbar}S[q_{cl}]}\tag{3.28}$$

Where the exponent of the determinant is $+1/2$. But another formula (3.25) says that:

$$\int Dx e^{-F[x]} \approx \sum_i e^{-F[x_i]} \det(\frac{A_i}{2\pi})^{\frac{-1}{2}} \tag{3.25}$$

Where the exponent of the determinant is $-1/2$.

Now I am just wondering why these two formula have these differences in the exponent in an explicit way.

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Eq. (3.25) is of course just the usual power $-1/2$ from a bosonic Gaussian integration. The power $+1/2$ of the van Vleck determinant in eq. (3.28) is more subtle. There is a proof of eq. (3.28) [in the context of 1D QM] in my Phys.SE answer here.

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The above answer is excellent. I just want to explain the difference in a heuristic way. The crucial point is that when we do path integral, we may assume different “measure” to the integral, so we will have an extra normalization factor in front of the integral. In this case, the normalization factor of (3.28) together with the ordinary Gaussian prefactor produces the overall prefactor with the opposite sign on the exponent through VVPM formula.

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