# How to derive modes of energy-momentum tensor in string theory?

It was never clear to me how to actually obtain the modes of an energy-momentum tensor in string theory.

Let's consider the energy-momentum tensor $$T_{\psi}(z) = \frac{1}{2}: (\partial_z \psi^{\mu}(z)) \psi_\mu (z): = \sum_{m \in z} L_m z^{-m-2}.$$

How do I derive that the modes are given by $$L_{m \neq 0} = \frac{1}{2} \sum_{r \in z} \psi_{m-r}^\mu \psi_r ^\nu \eta_{\mu \nu}.$$

First I was thinking about just computing the OPE $$\frac{1}{2}: (\partial_z \psi^{\mu}(z)) \psi_\mu (z):$$ but this feels weird since both fields are a function of $$z$$.

• Plug in the mode expansion for $\psi$. Mar 7 at 21:02