I am going through the derivation of the Ward identities in chapter 2 of Di Francesco, *Conformal Field Theory* and I am not sure how they go from equation 2.157:
∂∂xμ⟨jμa(x)Φ(x1)…Φ(xn)⟩=−in∑k=1δ(x−xi)⟨Φ(x1)…GaΦ(xi)…Φ(xn)⟩
to equation 2.161:
⟨Qa(t+)Φ(x1)Y⟩−⟨Qa(t−)Φ(x1)Y⟩=−i⟨GaΦ(x1)Y⟩
where
Qa=∫dd−1xj0μ(x)
Y=Φ(x2)…Φ(xn)
and t±=x01±ε. The authors say that 2.161 follows from integrating 2.157 in a "pill box", with time running from t− to t+ and x encompassing all space, expect small volumes centered about x2,…,xn.
It doesn't seem obvious to me that the left-hand side of 2.161 (involving the Q) follows from this procedure. In particular, I am not sure why we should ignore the surface term. I also wonder how explicitely the left-hand side of 2.161 would be changed if we included some other xi's in the integration volume.