Primary field

In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. All other operators in a representation are called descendants; they can be obtained by acting on the primary with the raising generators.

Primary fields in a D-dimensional conformal field theory were introduced in 1969 by Mack and Salam where they were called interpolating fields. They were then studied by Ferrara, , and Grillo who called them irreducible conformal tensors, and by Mack who called them lowest weights. Polyakov used an equivalent definition as fields which cannot be represented as derivatives of other fields.

The modern terms primary fields and descendants were introduced by Belavin, Polyakov and Zamolodchikov in the context of two-dimensional conformal field theories. This terminology is now used both for D=2 and D>2.

The lowering generators of the conformal algebra in D>2 dimensions are the special conformal transformation generators ${\displaystyle K_{\mu }}$. Primary operators inserted at ${\displaystyle x=0}$ are annihilated by these generators: ${\displaystyle [K_{\mu },{\mathcal {O}}(0)]=0}$. The descendants are obtained by acting on the primaries with the translation generators ${\displaystyle P_{\mu }}$; these are just the derivatives of the primaries.

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