For a physical theory to make sensible predictions in the sense of having a stationary phase approximation of the action to expand about, there must be a lowest energy state called the vacuum.
The energy of the vacuum is completely arbitrary since a central scalar constant may be added to the Hamiltonian to globally shift the phase without changing the observable dynamics, and so the vacuum energy may take negative values so long as it is bounded below.
This requirement is for instance what prompted the Dirac sea interpretation to address the Dirac equation's prediction of negative energy solutions, precisely because they generate an algebra of creation operators that can lower the energy ad infinitum.
To rectify this situation, the Witt algebra is centrally extended to provide a richer variety of Hilbert space modules to choose from, including the so-called positive energy representations, while leaving intact almost all of the Lie bracket relations between operators.
This shows that any two-dimensional conformal field theory is also a quantum integrable system.
[5] The space of states, also called the spectrum, of a CFT, is a representation of the product of the two Virasoro algebras.
, A CFT is called rational if its space of states decomposes into finitely many irreducible representations of the product of the two Virasoro algebras.
[6] A CFT is called diagonal if its space of states is a direct sum of representations of the type
The CFT is called unitary if the space of states has a positive definite Hermitian form such that
While unitarity is necessary for a CFT to be a proper quantum system with a probabilistic interpretation, many interesting CFTs are nevertheless non-unitary, including minimal models and Liouville theory for most allowed values of the central charge.
The torus partition function coincides with the character of the spectrum, considered as a representation of the symmetry algebra.
In a two-dimensional conformal field theory, properties are called chiral if they follow from the action of one of the two Virasoro algebras.
If the space of states can be decomposed into factorized representations of the product of the two Virasoro algebras, then all consequences of conformal symmetry are chiral.
on its position is assumed to be determined by It follows that the OPE defines a locally holomorphic field
-point function of primary fields yields From the last two equations, it is possible to deduce local Ward identities that express
In two dimensions, this method leads to exact solutions of certain CFTs, and to classifications of rational theories.
According to the left and right global Ward identities, three-point functions of such fields are of the type where the
For the three-point function to be single-valued, the left- and right-conformal dimensions of primary fields must obey This condition is satisfied by bosonic (
Three-point structure constants also appear in OPEs, The contributions of descendant fields, denoted by the dots, are completely determined by conformal symmetry.
Four-point conformal blocks are complicated functions that can be efficiently computed using Alexei Zamolodchikov's recursion relations.
If one of the four fields is degenerate, then the corresponding conformal blocks obey BPZ equations.
When a correlation function can be written in terms of conformal blocks in several different ways, the equality of the resulting expressions provides constraints on the space of states and on three-point structure constants.
The equality of the three resulting expressions is called crossing symmetry of the four-point function, and is equivalent to the associativity of the OPE.
The consistency of a CFT on all Riemann surfaces also requires modular invariance of the torus one-point function.
A minimal model is a CFT whose spectrum is built from finitely many irreducible representations of the Virasoro algebra.
These degenerate representations are labelled by pairs of integers that form the Kac table, For example, the A-series minimal model with
is compact, then this CFT is rational, its central charge takes discrete values, and its spectrum is known.
Logarithmic conformal field theories are two-dimensional CFTs such that the action of the Virasoro algebra generator
This contrasts with the power-like dependence of the two- and three-point functions that are associated to lowest weight representations.
include:[13] The known torus partition function[14] suggests that the model is non-rational with a discrete spectrum.