Conserved charge from Ward identity

Question

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.

Answer 1

They volume integrate the Ward identity between two hypersurfaces of constant $t$ ($t_+$ and $t_-$). It is important that this volume only contains the $\pmb x_1$ insertion, because it is the only one that is going to contribute on the right (because the delta functions vanish except the one that falls in the integration volume)

So the right-hand side is clear, the only contribution in the sum comes when $k=1$.

The left-hand side, being a volume integral of a divergence, gives you the surface flux integral, through Stokes' theorem. It is actually what it is written there, the $t_+$ face minus the $t_-$ face.

If there where more $x_i$ insertions inside the volume, the LHS would be the same, but the RHS would contain a sum of each term inside.

This is the way of translating into path integral language the action of a conserved charge (through the commutator) to a given operator for any quantisation of the theory. Moreover, you can deform the contour so that it is a sphere surrounding the $x_1$ insertion, so that action of charges are just given by spheres in the path integral formulation.