# Ward identities without time-ordering

In the book Conformal Quantum Field Theory in D-Dimensions, they state on pg. 181 the following two identities in relation to Ward identities of an Abelian internal symmetry (so the infinitesimal variation is given by $$\delta \varphi(x) = \varphi(x)$$): and The first states that the conservation of current only holds in correlation functions up to contact terms - the usual statement of the Ward identity. The second, if I'm understanding correctly, seems to state that the current is conserved exactly in the absence of time-ordering. I can believe that the time-ordering would become important as two points in the correlation function approach each other, but I do not know how to derive the second equation. Where does this come from, and is it specific to Abelian internal symmetries?

That might look surprising, but it's correct. It is not hard to believe that the $$\delta$$ functions may come from the derivatives of the step functions $$\Theta$$ in the definition of the time ordering. However, we can understand this result even better.
The contributions that violate conservation of $$j_\mu$$ happen when two points overlap. This is why they are called contact terms. We can try and formulate both Wightman and time ordered correlators in a unified way and see where these contact terms might appear.
This unified method is the $$i\varepsilon$$ prescription. Recall that operators are always time ordered in imaginary time. If we want $$\phi_1(t_1)$$ and $$\phi_2(t_2)$$ to be time ordered we need to make $$t_1 - t_2 \to (t_1-t_2)(1+i\varepsilon)\,.$$ so that the order depends on the sign of $$t_1-t_2$$. Whereas for Wightman functions we want the same order to hold at all times, so we set $$t_1 - t_2 \to t_1-t_2 +i\varepsilon\,.$$ This makes a huge difference. Now there is no contact term anymore because if $$t_1 \to t_2$$ their time difference is $$i\varepsilon$$. True, $$\varepsilon \to 0$$, but as a limit. Which means that the $$\delta$$ functions always evaluate to zero.