Can one define a path integral over differential forms, for instance $$\int [\mathcal{D}\phi]$$ where $\phi = X_{\mu}dx^{\mu}$ a one-form and $[\mathcal{D}\phi]$ is the path-integral measure? Does such a thing make sense?
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2Well, there's e.g. the path integral over the EM gauge field $A$ in pure QED. – Qmechanic Apr 25 '21 at 06:06
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@Qmechanic But the path integral is over $A_{\mu}$ and not $A_{\mu}dx^{\mu}$. – Dr. user44690 Apr 25 '21 at 06:18
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$A_{\mu}$ is (up to mathematical nuances) the coefficient of a one-form. – NDewolf Apr 25 '21 at 16:18
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@NDewolf I agree, but it is clearly not the same. I am trying to understand the path integral measure as an invariant measure in the space of differential forms. – Dr. user44690 Apr 26 '21 at 03:53