A conformal field theory is a quantum field theory invariant under conformal transformations. Local operators are organized into conformal families labelled by scaling dimension and spin.
The scaling dimension of a local operator is its eigenvalue under dilations. Under , a scalar operator of dimension transforms by .
The conformal algebra in Euclidean dimensions is generated by translations , rotations , dilations , and special conformal transformations .
A conformal primary operator is annihilated at the origin by every special conformal generator. Acting with translation generators produces its conformal descendants.
A conformal descendant is obtained by applying one or more translation generators to a conformal primary operator. Its scaling dimension exceeds that of the primary by a nonnegative integer.
Positivity of descendant norms constrains the scaling dimension of a primary operator. For a nonidentity scalar primary in , unitarity requires .
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Conformal Field Theory (CFT) is a quantum field theory that is invariant under conformal transformations. These transformations include dilatations (scaling), translations, rotations, and special conformal transformations. The significance of CFTs lies in their mathematical properties and their applications in various areas of physics and mathematics, including statistical mechanics, string theory, and condensed matter physics.