Editing Chern–Simons theory (section)

==Quantization==

To [[canonical quantization|canonically quantize]] Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a [[Hilbert space]]. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a [[Cauchy surface]], in fact, a state can be defined on any surface.

Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. [[Edward Witten|Witten]] has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of [[Virasoro conformal block#Larger symmetry algebras|conformal block]]s of the G WZW model at level k.

For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable [[group representation|representation]]s of the [[affine Lie algebra]] corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.