# Ward Identities in Conformal Theory

For a 2D free boson model, $$S=\frac{1}{2} g \int d^2 x \; \partial_\mu \varphi\;\partial^\mu \varphi$$ The energy-momentum tensor should be $$T_{\mu \nu}=g\left(\partial_\mu \varphi \partial_\nu \varphi-\frac{1}{2} \eta_{\mu \nu} \partial_\rho \varphi \partial^\rho \varphi\right)$$ This equation should hold at operator level, which means that $$T^\mu_\mu = 0$$ at operator level.

But this would lead to a contradiction as the Ward identity states that $$\left\langle T_\mu^\mu(x) X\right\rangle=-\sum_{i=1}^n \delta\left(x-x_i\right) \Delta_i\langle X\rangle$$ According to the argument above, the LHS of the equation would be fixed to zero, which is clearly in contradiction with the RHS. What is the problem of my above argument?

• The problem is that the matrix element is divergent and so has to be renormalized. Consequently the actual opertor in the Ward identity differes from the operator in your second line. Commented Jun 13 at 18:01
• Another way to say this is that $g^{\mu\nu}\langle T_{\mu\nu}\rangle_{\rm reg} \ne \langle g^{\mu\nu} T_{\mu\nu}\rangle_{\rm reg}$. Commented Jun 13 at 18:16
• I'm still wondering how to explicitly calculate the correlation functions with the trace of energy-momentum. Is there any way to do that? Commented Jun 14 at 2:29
• This is done in any CFT book surely? Commented Jun 14 at 11:36
• No, this is omitted in Di Francesco's Conformal Field Theory. The trace of energy-momentum tensor is just an intermediate step to obtain the Ward identites in holomorphic coordinates, and i can't find any proof using direct path integrals or operator algebra. Commented Jun 14 at 11:46