Operator Product Expansion (OPE) in Conformal Field Theory

Question

We denote local operators of a conformal field theory (CFT) as $\mathcal{O}_i$ where $i$ runs over the set of all operators. Formally, the operator product expansion (OPE) is given by,

$$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega})$$

where we have left the $\langle...\rangle$ implicit. My question is primarily, in practice, how does one determine the operator product expansion for an explicit case? Is the OPE similar to a Laurent expansion? In which case, can one determine the residues from the OPE? I would appreciate an explicit non-trivial example.

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I'm following String Theory and M-Theory by Becker, Becker and Schwarz, and the OPE is dealt with somewhat rapidly.

This post imported from StackExchange Physics at 2014-05-04 11:18 (UCT), posted by SE-user user45666

Answer 1

In the limit of the positions of two operator insertions approaching one another, the product of those two operators can be approximated as a series of local operators. This can be done to arbitrary accuracy and within conformal field theory, the expression is exact.

The series described above will in general have singular behaviour. At first, this might sound bad, but this is actually the key point of the whole formalism. In fact, all the physical information we are interested in is encoded in the singularities. As a consequence, regular terms are usually not even written down.

An explicit example would be the OPE of the conformal stress energy tensor with itself, given by

$$T(z)T(z')=\frac{\partial T(z')}{z-z'}+\frac{2T(z')}{(z-z')^2}+\frac{c}{2(z-z')^4}+\dots,$$

where $c$ is the central charge and dots denote regular terms. Furthermore, this expression is to be understood as an operator expression within a vacuum expectation value. This is, for convenience, often omitted in the literature.

The terms I have written down above are singular in the limit $z\rightarrow z'$ and correspond to the residues of the OPE. As you have correctly suggested, the latter is formally equivalent to a Laurent series.

But why is the OPE of interest, other than for providing a convenient way of writing products of operators? It turns out that it encodes the transformation behaviour of an operator under conformal transformations. This can be shown by working out the respective Ward identities. Furthermore, the OPE contains information about the mode algebra of the operators, i.e. the Virasoro algebra.

As proof of the above statements would exceed the boundaries of my answer, I refer at this point to the literature. I can recommend both the book by Polchinski and two sets of lecture notes freely available on the internet by Maximilian Kreuzer and David Tong.

This post imported from StackExchange Physics at 2014-05-04 11:18 (UCT), posted by SE-user Frederic Brünner

Answer 2

A hint about OPEs: They exist for CFTs in any dimension. They play nicely with the holomorphic phenomena in 2d, but they don't require it. They encode information about the singularities which occur when you multiply operator-valued distributions.

Explicit examples in higher dimension are computationally challenging.

This post imported from StackExchange Physics at 2014-05-04 11:18 (UCT), posted by SE-user user1504