Section 4
Operator product expansion

This section is technically the most demanding of the module, and some of the constructions and proofs will have to remain on a sketchy level.
I begin by defining fields on a fairly general level, allowing me to give an interpretation of the standard generators of the Virasoro algebra, the free fermion algebra, and the free boson algebra, which we have been discussing in Sections 1-3, as Fourier modes of formal Laurent series with values in a space of linear operators. I discuss the state-field correspondence and show how it is implemented in CFTs through the definition of so-called n-point functions together with a property called reflection positivity which ensures that the 2-point functions encode the scalar product on the pre-Hilbert space.
The properties of n-point functions lead to the definition of the operator product expansion (OPE) and its representations on n-point functions. I explain how the OPE translates into the commutator and anticommutator relations of the Fourier modes in terms of contour integral methods, as it is common in the physics literature.
Finally I explain the implementation of conformal invariance into the picture through conformal representations of the OPE. I discuss the constraints that conformal invariance can pose on the form of the OPE and in particular the so-called primary and quasi-primary fields and their most important properties in terms of the representation theory of the Virasoro algebra.