In the path integral approach one defines in some heuristic way the functional path integral \begin{equation} Z=\int{\cal{D}}\phi e^{iS(\phi)} \end{equation} and the one claims that one must integrate over all paths.
I understand that the domain of the integral is the configuration space of the theory.
My question is:
How does the integral depend on our initial choice of configuration space?
EDIT:
For example, in a globally hyperbolic spacetime with compact initial Cauchy surface $\Sigma$ one can have well-posed problems for the scalar field, $\phi$ with initial data in the Sobolev Spaces $H^{1}(\Sigma)\times H^{0}(\Sigma)$. However one can also prove that the problem is well-posed for initial data in $H^{k}(\Sigma)\times H^{k-1}(\Sigma)$.
These two well-possessedness results gives two different configuration spaces $H^{1}$ in the first case and $H^{k}$ in the second.
How does the path integral change in this case?