Delta functional in path integrals

Question

In a few articles dealing with path integral quantization I came across some calculations where apparently identities of the form \begin{equation} \int (\mathcal{D}\Phi)\, \delta(-\partial_0\Phi+j)\,\,g(\Phi)=\det(\partial_0)^{-1}g(\frac{1}{\partial_0}j) \end{equation} have been used. I know that the quantities appearing on the right hand side are quite delicate to define and their evaluation very much depends on the functional space in question. Moreover, is there any "standard reference" (also accessible for physicists, if possible) that deals with the proper treatment of such expressions?

Answer 1

Are you asking for rigor$^1$ for a path integral? Heuristically, it is just a substitution

$$\tag{1}\Phi~\longrightarrow~ \Psi~=~ \partial_0 \Phi. $$

The path integral measure then changes as

$$\tag{2}{\cal D}\Phi~\longrightarrow~ {\cal D}\Psi~=~~\det(\partial_0)~{\cal D}\Phi, $$

so that the path integral becomes

$$ \int\! {\cal D}\Phi~ \delta(j-\partial_0\Phi)~g(\Phi) ~=~\det(\partial_0)^{-1}\int\!{\cal D}\Psi~\delta(j-\Psi)~g(\partial^{-1}_0\Psi) $$$$ \tag{3}~=~\det(\partial_0)^{-1}~g(\partial^{-1}_0j).$$

Of course in the specific physical application, one should check that the substitution (1) is an invertible map when pertinent boundary conditions are imposed. Another issue is whether there should be an absolute value of the determinant in eq. (2).

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$^1$ Rigor in QFT is e.g. discussed in this Phys.SE post and links therein.