Editing Conformal field theory (section)

=== Behaviour under conformal transformations ===

Any conformal transformation <math>x\to f(x)</math> acts linearly on fields {{tmath|1= O(x) \to \pi_f(O)(x) }}, such that <math>f\to \pi_f</math> is a representation of the conformal group, and correlation functions are invariant:
: <math>
\left\langle\pi_f(O_1)(x_1)\cdots \pi_f(O_n)(x_n) \right\rangle = \left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle.
</math>
'''Primary fields''' are fields that transform into themselves via {{tmath|1= \pi_f }}. The behaviour of a primary field is characterized by a number <math>\Delta</math> called its '''conformal dimension''', and a representation <math>\rho</math> of the rotation or Lorentz group. For a primary field, we then have 
: <math> 
\pi_f(O)(x) = \Omega(x')^{-\Delta} \rho(R(x')) O(x'), \quad \text{where}\ x'=f^{-1}(x).
</math>
Here <math>\Omega(x)</math> and <math>R(x)</math> are the scale factor and rotation that are associated to the conformal transformation {{tmath|1= f }}. The representation <math>\rho</math> is trivial in the case of scalar fields, which transform as {{tmath|1= \pi_f(O)(x) = \Omega(x')^{-\Delta} O(x')
}}. For vector fields, the representation <math>\rho</math> is the fundamental representation, and we would have {{tmath|1= \pi_f(O_\mu)(x) = \Omega(x')^{-\Delta} R_\mu^\nu(x') O_\nu(x') }}.

A primary field that is characterized by the conformal dimension <math>\Delta</math> and representation <math>\rho</math> behaves as a highest-weight vector in an [[induced representation]] of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension <math> \Delta</math> characterizes a representation of the subgroup of dilations. In two dimensions, the fact that this induced representation is a [[Verma module]] appears throughout the literature. For higher-dimensional CFTs (in which the maximally compact subalgebra is larger than the [[Cartan subalgebra]]), it has recently been appreciated that this representation is a parabolic or [[generalized Verma module]].<ref name="pty16"/>

Derivatives (of any order) of primary fields are called '''descendant fields'''. Their behaviour under conformal transformations is more complicated. For example, if <math>O</math> is a primary field, then <math>\pi_f(\partial_\mu O)(x) = \partial_\mu\left(\pi_f(O)(x)\right)</math> is a linear combination of <math> \partial_\mu O</math> and {{tmath|1= O }}. Correlation functions of descendant fields can be deduced from correlation functions of primary fields. However, even in the common case where all fields are either primaries or descendants thereof, descendant fields play an important role, because conformal blocks and operator product expansions involve sums over all descendant fields.

The collection of all primary fields {{tmath|1= O_p }}, characterized by their scaling dimensions <math>\Delta_p</math> and the representations {{tmath|1= \rho_p }}, is called the '''spectrum''' of the theory.