# Why is there an difference between the exponent of the determinant of these two path integral?

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.

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.