# Dependence of the Path Integral in the Function Space

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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 $H ^1 \Sigma \times H ^0 \Sigma$. However one can also prove that the problem is well-posed for initial data in $H^{3}\Sigma\times H^{3}\Sigma$. These two spaces define two different sets of functions and two different configuration spaces.

Does the path integral calculations give the same results for both cases?

edited Apr 17, 2014

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