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  Conserved charge from Ward identity

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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:

$$\frac{\partial}{\partial x^{\mu}}\langle j^{\mu}_a(x)\Phi(x_1)\ldots\Phi(x_n)\rangle = -i\sum_{k = 1}^n\delta(x - x_i)\langle\Phi(x_1)\ldots G_a\Phi(x_i)\ldots\Phi(x_n)\rangle$$

to equation 2.161:

$$\langle Q_a(t_+)\Phi(x_1)Y\rangle - \langle Q_a(t_-)\Phi(x_1)Y\rangle = -i\langle G_a\Phi(x_1)Y\rangle$$

where

$$Q_a = \int\!\mathrm{d}^{d-1}{\bf x}\,j^0_{\mu}(x)$$
$$Y = \Phi(x_2)\ldots\Phi(x_n)$$

and $t_{\pm} = x_1^0 \pm \varepsilon$. The authors say that 2.161 follows from integrating 2.157 in a "pill box", with time running from $t_-$ to $t_+$ and ${\bf x}$ encompassing all space, expect small volumes centered about ${\bf x}_2, \ldots, {\bf x}_n$.

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 ${\bf x}_i$'s in the integration volume.

asked Jan 12 in Theoretical Physics by maharishi (25 points) [ no revision ]

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