# Integrated Ward Identity

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Suppose you have the following ward identity: $$\int_{M} d^4x\ \epsilon(x)\ \partial_{\mu} \langle j_{\mu}(x)O(y)\rangle = - \ \langle\delta O(y)\rangle$$ where $\delta O(y)$ can be written in the following way to get the usual local form of the ward identity: $$\delta O(y) = \int_M d^4x \ \delta(x-y)\ \epsilon(x)\ \delta O(y)$$

Now suppose we choose $\epsilon$ (which is arbitrary) such that it's costant in M. Is it possible to use Stokes theorem and get :  $$\int_{\partial M} d\sigma_\mu\ \langle j_{\mu}(x)O(y)\rangle = - \ \langle\delta O(y)\rangle$$ or are we neglecting contact terms given by the $T$ product by doing so? Moreover is it possible to do the spectral decomposition of $\langle j_{\mu}(x)O(y)\rangle$ even if it's integrated?

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