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.

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

In ${\displaystyle d>2}$ dimensions conformal primary fields can be defined in two equivalent ways. ^[6]

First definition

Let ${\displaystyle {\hat {D}}}$ be the generator of dilations and let ${\displaystyle {\hat {K}}_{\mu }}$ be the generator of special conformal transformations. A conformal primary field ${\displaystyle {\hat {\phi }}_{\rho }^{M}(x)}$ , in the ${\displaystyle \rho }$ representation of the Lorentz group and with conformal dimension ${\displaystyle \Delta }$ satisfies the following conditions at ${\displaystyle x=0}$ :

1. ${\displaystyle \left[{\hat {D}},{\hat {\phi }}_{\rho }^{M}(0)\right]=-i\Delta {\hat {\phi }}_{\rho }^{M}(0)}$;
2. ${\displaystyle \left[{\hat {K}}_{\mu },{\hat {\phi }}_{\rho }^{M}(0)\right]=0}$.

Second definition

A conformal primary field ${\displaystyle {\hat {\phi }}_{\rho }^{M}(x)}$, in the ${\displaystyle \rho }$ representation of the Lorentz group and with conformal dimension ${\displaystyle \Delta }$, transforms under a conformal transformation ${\displaystyle \eta _{\mu \nu }\mapsto \Omega ^{2}(x)\eta _{\mu \nu }}$ as

${\displaystyle {\hat {\phi '}}_{\rho }^{M}(x')=\Omega ^{\Delta }(x){\mathcal {D}}{\left[R(x)\right]^{M}}_{N}{\hat {\phi }}_{\rho }^{N}(x)}$

where ${\displaystyle {R^{\mu }}_{\nu }(x)=\Omega ^{-1}(x){\frac {\partial x^{\mu }}{\partial x'^{\nu }}}}$ and ${\displaystyle {\mathcal {D}}{\left[R(x)\right]^{M}}_{N}}$ implements the action of ${\displaystyle R}$ in the ${\displaystyle SO(d-1,1)}$ representation of ${\displaystyle {\hat {\phi }}_{\rho }^{M}(x)}$.

Conformal field theory in D=2 dimensions

In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators ${\displaystyle L_{n},{\bar {L}}_{n},-\infty <n<\infty }$. Primaries are defined as the operators annihilated by all ${\displaystyle L_{n},{\bar {L}}_{n}}$ with n>0, which are the lowering generators. Descendants are obtained from the primaries by acting with ${\displaystyle L_{n},{\bar {L}}_{n}}$ with n<0.

The Virasoro algebra has a finite dimensional subalgebra generated by ${\displaystyle L_{n},{\bar {L}}_{n},-1\leq n\leq 1}$. Operators annihilated by ${\displaystyle L_{1},{\bar {L}}_{1}}$ 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[7]

In ${\displaystyle D\leq 6}$ 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 ${\displaystyle D>2}$ dimensions, superconformal primaries are annihilated by ${\displaystyle K_{\mu }}$ and by the fermionic generators ${\displaystyle S}$ (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 ${\displaystyle Q}$ on the superconformal primary. There exist also special chiral superconformal primary operators, which are primary operators annihilated by some combination of the supercharges.^[7]

In ${\displaystyle D=2}$ 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.^[8][9] 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^[10] and by Mack.^[3]

References

1. G Mack; Abdus Salam (1969). "Finite-component field representations of the conformal group". Annals of Physics. 53 (1): 174–202. Bibcode:1969AnPhy..53..174M. doi:10.1016/0003-4916(69)90278-4. ISSN 0003-4916.
2. Ferrara, Sergio; Raoul Gatto; A. F. Grillo (1973). Conformal Algebra in Space-Time and Operator Product Expansion. Springer-Verlag. ISBN 9783540062165.
3. G. Mack (1977). "All unitary ray representations of the conformal group SU(2, 2) with positive energy". Communications in Mathematical Physics. 55 (1): 1–28. doi:10.1007/bf01613145. S2CID 119941999. Retrieved 2013-12-05.
4. Polyakov, A. M. (1974). "Non-Hamiltonian approach to conformal quantum field theory". Soviet Journal of Experimental and Theoretical Physics. 39: 10. Bibcode:1974JETP...39...10P. ISSN 1063-7761.
5. Belavin, A.A.; A.M. Polyakov; A.B. Zamolodchikov (1984). "Infinite conformal symmetry in two-dimensional quantum field theory". Nuclear Physics B. 241 (2): 333–380. Bibcode:1984NuPhB.241..333B. doi:10.1016/0550-3213(84)90052-X. ISSN 0550-3213.
6. Campos Delgado, Ruben (2022). "On the equivalence of two definitions of conformal primary fields in d > 2 dimensions". Eur. Phys. J. Plus. 137 (9): 1038. arXiv:2112.01837. doi:10.1140/epjp/s13360-022-03228-y. S2CID 252258885.
7. Aharony, Ofer; Steven S. Gubser; Juan Maldacena; Hirosi Ooguri; Yaron Oz (2000). "Large N field theories, string theory and gravity". Physics Reports. 323 (3–4): 183–386. arXiv:hep-th/9905111. Bibcode:2000PhR...323..183A. doi:10.1016/S0370-1573(99)00083-6. ISSN 0370-1573. S2CID 119101855. Retrieved 2013-12-05.
8. Minwalla, Shiraz (1997). "Restrictions imposed by superconformal invariance on quantum field theories". Adv. Theor. Math. Phys. 2: 781–846. arXiv:hep-th/9712074. Retrieved 2013-12-05.
9. Grinstein, Benjamin; Kenneth Intriligator; Ira Z. Rothstein (2008). "Comments on unparticles". Physics Letters B. 662 (4): 367–374. arXiv:0801.1140. Bibcode:2008PhLB..662..367G. doi:10.1016/j.physletb.2008.03.020. ISSN 0370-2693. S2CID 5240874. Retrieved 2013-12-05.
10. Ferrara, S.; R. Gatto; A. Grillo (1974). "Positivity restriction on anomalous dimensions". Physical Review D. 9 (12): 3564–3565. Bibcode:1974PhRvD...9.3564F. doi:10.1103/PhysRevD.9.3564. ISSN 0556-2821. Retrieved 2013-12-05.