For a conformal field $X$, Polchinski gives a relation between the time ordering $T$ (or equivalently the radial ordering ${\cal R}$) of a functional of identical fields and the normal ordering, which is $$ T(\mathscr{F}) ~=~ \exp\left(\frac{1}{2}\int d^2z_1d^2z_2 \Delta(z_1,z_2)\frac{\delta}{\delta X(z_1)}\frac{\delta}{\delta X(z_2)}\right) :\mathscr{F}: \tag{2.2.8} $$

Is there a similar expression for time order products of different fields, i.e. a fermion pair $\overline{\psi}(x)\psi(y)$?

  • 1
    $\begingroup$ Keep reading. In section 2.5 he introduces $bc$ CFT ($bc$ - anticommuting ghosts). Also, in volume 2 there is superconformal field theory with proper fermions. $\endgroup$
    – Kosm
    Jun 5 '17 at 14:01

Sure, the generalization to several fields $(\hat{A}_i)_{i\in I}$ is $$T(f(\hat{A})) ~=~ \exp\left(\frac{1}{2}\sum_{i,j\in I}\hat{C}_{ij}\frac{\partial}{\partial\hat{A}_j}\frac{\partial}{\partial\hat{A}_i} \right):f(\hat{A}):$$ where we have used DeWitt's condensed notation. We stress that for Grassmann-odd fields, it is important to pay attention to sign factors. See my Phys.SE answer here for notation & details.


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