What is the point of the Pauli-Van Vleck-Morette determinant?

Question

I wish to get a better feel for the Heat kernel ansatz below

$$\hat{K}(s \mid x, y)=\frac{\Delta^{1 / 2}(x, y)}{(4 \pi s)^{d / 2}} g^{1 / 2}(y) e^{-\sigma(x, y) / 2 s-s m^2} \sum_{n=0}^{\infty} s^n \hat{a}_n(x, y).$$

Can anyone explain to me why do we need the Pauli-Van Vleck-Morette determinant $\Delta(x, y)$ in the ansatz above?

Edit: Thanks to an answer here I understand the point of this determinant but I still don't understand the requirement that the heat kernel ansatz have it?

Answer 1

Briefly speaking, the heat kernel has an interpretation as an Euclidean path integral. Semiclassically, the functional determinant from integrating over quantum fluctuations produces the van Vleck determinant, cf. e.g. my Phys.SE answer here.