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| 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.
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| ==History of the concept==
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| Primary fields in a ''D''-dimensional conformal field theory were introduced in 1969 by Mack and Salam<ref>{{Cite journal
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| | doi = 10.1016/0003-4916(69)90278-4
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| | issn = 00034916
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| | volume = 53
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| | issue = 1
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| | pages = 174-202
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| | author = G Mack
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| | coauthors = Abdus Salam
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| | title = Finite-component field representations of the conformal group
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| | journal = Annals of Physics
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| | date=1969
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| | accessdate = 2013-12-05
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| | url = http://adsabs.harvard.edu/abs/1969AnPhy..53..174M
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| }}</ref> where they were called ''interpolating fields''. They were then studied by Ferrara, Gatto, and Grillo<ref>{{Cite book
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| | publisher = Springer-Verlag
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| | isbn = 9783540062165
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| | last = Ferrara
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| | first = Sergio
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| | coauthors = Raoul Gatto, A. F. Grillo
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| | title = Conformal Algebra in Space-Time and Operator Product Expansion
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| | date = 1973
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| }}</ref> who called them ''irreducible conformal tensors'', and by Mack<ref name=Mack>{{Cite journal
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| | volume = 55
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| | issue = 1
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| | pages = 1-28
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| | author = G. Mack
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| | title = All unitary ray representations of the conformal group SU(2, 2) with positive energy
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| | journal = Communications in Mathematical Physics (1965-1997)
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| | accessdate = 2013-12-05
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| | date = 1977
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| | url = http://projecteuclid.org/euclid.cmp/1103900926
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| }}</ref> who called them ''lowest weights''. Polyakov<ref>{{Cite journal
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| | issn = 1063-7761
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| | volume = 39
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| | pages = 10
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| | last = Polyakov
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| | first = A. M.
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| | title = Non-Hamiltonian approach to conformal quantum field theory
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| | journal = Soviet Journal of Experimental and Theoretical Physics
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| | accessdate = 2013-12-05
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| | date = 1974
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| | url = http://adsabs.harvard.edu/abs/1974JETP...39...10P
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| }}</ref> used an equivalent definition as fields which cannot be represented as derivatives of other fields.
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| The modern terms ''primary fields'' and ''descendants'' were introduced by Belavin, Polyakov and Zamolodchikov<ref>{{Cite journal
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| | doi = 10.1016/0550-3213(84)90052-X
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| | issn = 05503213
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| | volume = 241
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| | issue = 2
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| | pages = 333-380
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| | last = Belavin
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| | first = A.A.
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| | coauthors = A.M. Polyakov, A.B. Zamolodchikov
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| | title = Infinite conformal symmetry in two-dimensional quantum field theory
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| | journal = Nuclear Physics B
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| | date=1984
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| | accessdate = 2013-12-05
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| | url = http://adsabs.harvard.edu/abs/1984NuPhB.241..333B
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| }}</ref> in the context of two-dimensional conformal field theories. This terminology is now used both for ''D''=2 and ''D''>2.
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| ==Conformal field theory in ''D''>2 spacetime dimensions==
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| The lowering generators of the [[conformal algebra]] in ''D''>2 dimensions are the [[special conformal transformation]] generators <math>K_\mu</math>. Primary operators inserted at <math>x=0</math> are annihilated by these generators: <math>[K_\mu, \mathcal{O}(0)]=0</math>. The descendants are obtained by acting on the primaries with the translation generators <math>P_\mu</math>; these are just the derivatives of the primaries.
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| ==Conformal field theory in ''D''=2 dimensions==
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| In two dimensions, conformal field theories are invariant under an infinite dimensional [[Virasoro algebra]] with generators <math>L_n, \bar{L}_n, -\infty<n<\infty</math>. Primaries are defined as the operators annihilated by all <math>L_n, \bar{L}_n</math> with ''n''>0, which are the lowering generators. Descendants are obtained from the primaries by acting with <math>L_n, \bar{L}_n</math> with ''n''<0.
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| The Virasoro algebra has a finite dimensional subalgebra generated by <math>L_n, \bar{L}_n, -1\le n\le 1</math>. Operators annihilated by <math>L_1, \bar{L}_1</math> 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.
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| Quasi-primary fields in two-dimensional conformal field theory are the direct analogues of the primary fields in the ''D''>2 dimensional case.
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| ==Superconformal field theory<ref name=MAGOO>{{Cite journal
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| | doi = 10.1016/S0370-1573(99)00083-6
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| | issn = 03701573
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| | volume = 323
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| | issue = 3-4
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| | pages = 183-386
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| | last = Aharony
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| | first = Ofer
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| | coauthors = Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz
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| | title = Large N field theories, string theory and gravity
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| | journal = Physics Reports
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| | accessdate = 2013-12-05
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| | date = 2000
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| | url = http://inspirehep.net/record/499969?ln=en
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| }}</ref>==
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| In <math>D\le 6</math> 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.
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| In ''D''>2 dimensions, superconformal primaries are annihilated by <math>K_\mu</math> 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.<ref name=MAGOO/>
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| In ''D''=2 dimensions, superconformal field theories are invariant under [[super Virasoro algebra|super Virasoro algebras]], which include infinitely many fermionic operators. Superconformal primaries are annihilated by all lowering operators, bosonic and fermionic.
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| ==Unitarity bounds==
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| In unitary (super)conformal field theories, dimensions of primary operators satisfy lower bounds called the unitarity bounds.<ref>{{Cite journal
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| | volume = 2
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| | pages = 781-846
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| | last = Minwalla
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| | first = Shiraz
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| | title = Restrictions imposed by superconformal invariance on quantum field theories
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| | journal = Adv.Theor.Math.Phys.
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| | accessdate = 2013-12-05
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| | date = 1997
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| | url = http://inspirehep.net/record/452061?ln=en
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| }}</ref><ref>{{Cite journal
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| | doi = 10.1016/j.physletb.2008.03.020
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| | issn = 03702693
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| | volume = 662
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| | issue = 4
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| | pages = 367-374
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| | last = Grinstein
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| | first = Benjamin
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| | coauthors = Kenneth Intriligator, Ira Z. Rothstein
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| | title = Comments on unparticles
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| | journal = Physics Letters B
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| | accessdate = 2013-12-05
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| | date = 2008
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| | url = http://inspirehep.net/record/776996?ln=en
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| }}</ref> 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<ref>{{Cite journal
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| | doi = 10.1103/PhysRevD.9.3564
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| | issn = 0556-2821
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| | volume = 9
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| | issue = 12
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| | pages = 3564-3565
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| | last = Ferrara
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| | first = S.
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| | coauthors = R. Gatto, A. Grillo
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| | title = Positivity restriction on anomalous dimensions
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| | journal = Physical Review D
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| | accessdate = 2013-12-05
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| | date = 1974
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| | url = http://inspirehep.net/record/89113?ln=en
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| }}</ref> and by Mack <ref name=Mack/>
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Primary Field}}
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| [[Category:Conformal field theory]]
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