Primary field explained

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.

History of the concept

Primary fields in a D-dimensional conformal field theory were introduced in 1969 by Mack and Salam^[1] where they were called interpolating fields. They were then studied by Ferrara, Gatto, and Grillo^[2] who called them irreducible conformal tensors, and by Mack^[3] who called them lowest weights. Polyakov^[4] 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^[5] in the context of two-dimensional conformal field theory. This terminology is now used both for D=2 and D>2.

Conformal field theory in D>2 spacetime dimensions

The lowering generators of the conformal algebra in D>2 dimensions are the special conformal transformation generators

K _{\mu} . Primary operators inserted at x =0 are annihilated by these generators: [K _{\mu,} l{O}(0)]=0 . The descendants are obtained by acting on the primaries with the translation generators P _{\mu} ; these are just the derivatives of the primaries.

Conformal field theory in D2 dimensions

In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators

L _{n,} \bar{L} _{n,} -inf ty<n< inf ty . Primaries are defined as the operators annihilated by all L _{n,} \bar{L} _{n} with n>0, which are the lowering generators. Descendants are obtained from the primaries by acting with L _{n,} \bar{L} _{n} with n<0.

The Virasoro algebra has a finite dimensional subalgebra generated by

L _{n,} \bar{L} _{n,} -1 \le n\le 1 . Operators annihilated by L _{1,} \bar{L} _{1} are called quasi-primaries. Each primary field is a quasi-primary, but the converse is not true; in fact each primary has infinitely many quasi-primary descendants. Quasi-primary fields in two-dimensional conformal field theory are the direct analogues of the primary fields in the D>2 dimensional case.

Superconformal field theory^[6]

In

D\le 6 dimensions, conformal algebra allows graded extensions containing fermionic generators. Quantum field theories invariant with respect to such extended algebras are called superconformal. In superconformal field theories, one considers superconformal primary operators.

In

D> 2 dimensions, superconformal primaries are annihilated by K _{\mu} and by the fermionic generators S (one for each supersymmetry generator). Generally, each superconformal primary representations will include several primaries of the conformal algebra, which arise by acting with the supercharges Q on the superconformal primary. There exist also special chiral superconformal primary operators, which are primary operators annihilated by some combination of the supercharges.^[6]

In

D =2 dimensions, superconformal field theories are invariant under super Virasoro algebras, which include infinitely many fermionic operators. Superconformal primaries are annihilated by all lowering operators, bosonic and fermionic.

Unitarity bounds

In unitary (super)conformal field theories, dimensions of primary operators satisfy lower bounds called the unitarity bounds.^{[7] [8]} Roughly, these bounds say that the dimension of an operator must be not smaller than the dimension of a similar operator in free field theory. In four-dimensional conformal field theory, the unitarity bounds were first derived by Ferrara, Gatto and Grillo^[9] and by Mack.^[3]

Notes and References

1. 10.1016/0003-4916(69)90278-4. 0003-4916. 53. 1. 174–202. G Mack. Abdus Salam. Finite-component field representations of the conformal group. Annals of Physics. 1969. 1969AnPhy..53..174M .
2. Book: Ferrara , Sergio . Springer-Verlag. 9783540062165. Raoul Gatto . A. F. Grillo . Conformal Algebra in Space-Time and Operator Product Expansion. 1973.
3. 55. 1. 1–28. G. Mack. All unitary ray representations of the conformal group SU(2, 2) with positive energy. Communications in Mathematical Physics. 2013-12-05. 1977. 10.1007/bf01613145. 119941999.
4. 1063-7761. 39. 10. Polyakov. A. M.. Non-Hamiltonian approach to conformal quantum field theory. Soviet Journal of Experimental and Theoretical Physics. 1974. 1974JETP...39...10P .
5. 10.1016/0550-3213(84)90052-X. 0550-3213. 241. 2. 333–380. Belavin. A.A. . A.M. Polyakov . A.B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Physics B. 1984. 1984NuPhB.241..333B .
6. 10.1016/S0370-1573(99)00083-6. 0370-1573. 323. 3–4. 183–386. Aharony. Ofer. Steven S. Gubser . Juan Maldacena . Hirosi Ooguri . Yaron Oz. Large N field theories, string theory and gravity. Physics Reports. 2013-12-05. 2000. hep-th/9905111. 2000PhR...323..183A . 119101855.
7. 2. 781–846. Minwalla. Shiraz. Restrictions imposed by superconformal invariance on quantum field theories. Adv. Theor. Math. Phys.. 2013-12-05. 1997. hep-th/9712074.
8. 10.1016/j.physletb.2008.03.020. 0370-2693. 662. 4. 367–374. Grinstein. Benjamin. Kenneth Intriligator . Ira Z. Rothstein . Comments on unparticles. Physics Letters B. 2013-12-05. 2008. 0801.1140 . 2008PhLB..662..367G . 5240874.
9. 10.1103/PhysRevD.9.3564. 0556-2821. 9. 12. 3564–3565. Ferrara. S.. R. Gatto . A. Grillo . Positivity restriction on anomalous dimensions. Physical Review D. 2013-12-05. 1974. 1974PhRvD...9.3564F .