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.