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I've recently encountered a path integral of the form

$$\int \delta[a\phi+b\phi']\,L(\phi,\phi')\;\mathcal D\phi\mathcal D\phi'$$

(where $a$, $b$ are integers) and would like to eliminate one of the $\mathcal D\phi$ integrations.

Is this possible? A change of variables doesn't seem to work since e.g. for substitutions $u:=a\phi$ the path integral Jacobian isn't well-defined if $a\neq 1$.

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1 Answer 1

If e.g. $b=1$, it is obvious that the delta-functional – you should really call it $\Delta$ and not $\delta$ – "cancels" against the integral, giving $$\int L(\phi,-a\phi) {\mathcal D}\phi $$ where I just substituted the right value for $\phi'$. Similarly for $a=1$. Now, for general $a,b$, the Jacobian is just the simple power $b^{-N}$ where $N$ is the (naively infinite) dimension of the integral ${\mathcal D}\phi'$. In a sensible regulation, $N$ will really become the Euler characteristic $\chi$ of the manifold on which $\phi'$ is defined – the regulated "number of points" on a manifold. So the result is $$\int L(\phi,-a\phi/b) {\mathcal D}\phi \cdot b^{-\chi} $$ If there are many, $k$ fields per point, you need $k\chi$, not just $\chi$. In practice, the factor $b^{-\chi}$ is just a normalization constant that you need to renormalize to a right value anyway. It doesn't depend on any dynamical fields so you may pretty much ignore it.

You shouldn't be afraid of such simple manipulations.

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