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?
1 Answer
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.
