Paths in the path integral

Question

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?

Answer 1

You do integrate over all paths in configuration space, but beware : differentiable paths contribute to a measure of 0 in the integral. The real contribution comes from fractal paths of dimension 2 (cf "The Dimension of a Quantum-Mechanical Path" by Abbott and Wise).

This "spreading" of the path is the equivalent in path integrals of the Heisenberg uncertainty principle, something of the form

$\langle m \frac{x_{k+1} - x_k}{\varepsilon} x_k \rangle - \langle x_k m \frac{x_{k} - x_{k-1}}{\varepsilon} \rangle = \frac{\hbar}{i} \langle 1 \rangle$

(cf Feynman and Hibbs)

the angle brackets indicating a path integration of some functional with some action. It means that there is no real speed, but only an average one, since the paths are all non-differentiable at every points. The speed has a standard deviation linked to the standard deviation of the position of your measurement (this is also expressed in the informal relation you sometimes see in path integral book : $dx^2 \propto dt$)

In phase space, things get a bit more complicated, and only discontinuous paths in phase space contribute ("Feynman Path Integrals in a Phase Space" by Berezin).

The same logic applies for fields, and indeed you have to be careful to not integrate over the same field configuration twice if it's a gauge field.