Higher-spin theory or higher-spin gravity is a common name for field theories that contain massless fields of spin greater than two. Usually, the spectrum of such theories contains the graviton as a massless spin-two field, which explains the second name. Massless fields are gauge fields and the theories should be (almost) completely fixed by these higher-spin symmetries. Higher-spin theories are supposed to be consistent quantum theories and, for this reason, to give examples of quantum gravity. Most of the interest in the topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher-spin theories to weakly coupled conformal field theories. Only certain parts of these theories are known at present (in particular, standard action principles are not known) and not many examples have been worked out in detail except some specific toy models (such as the higher-spin extension of pure Chern–Simons, Jackiw–Teitelboim, selfdual (chiral) and Weyl gravity theories).
Free higher-spin fields Systematic study of massless arbitrary spin fields was initiated by Christian Fronsdal. A free spin-s field can be represented by a tensor gauge field.
δ Φ μ 1 μ 2 . . . μ s = ∂ μ 1 ξ μ 2 . . . μ s + permutations {\displaystyle \delta \Phi _{\mu _{1}\mu _{2}...\mu _{s}}=\partial _{\mu _{1}}\xi _{\mu _{2}...\mu _{s}}+{\text{permutations}}}
This (linearised) gauge symmetry generalises that of massless spin-one (photon) δ A μ = ∂ μ ξ {\displaystyle \delta A_{\mu }=\partial _{\mu }\xi } and that of massless spin-two (graviton) δ h μ ν = ∂ μ ξ ν + ∂ ν ξ μ {\displaystyle \delta h_{\mu \nu }=\partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }} . Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above. For example, the equations are
◻ Φ μ 1 μ 2 . . . μ s − ( ∂ μ 1 ∂ ν Φ ν μ 2 . . . μ s + permutations ) + 1 2 ( ∂ μ 1 ∂ μ 2 Φ ν
ν μ 3 . . . μ s + permutations ) = 0 {\displaystyle \square \Phi _{\mu _{1}\mu _{2}...\mu _{s}}-\left(\partial _{\mu _{1}}\partial ^{\nu }\Phi _{\nu \mu _{2}...\mu _{s}}+{\text{ permutations}}\right)+{\frac {1}{2}}\left(\partial _{\mu _{1}}\partial _{\mu _{2}}\Phi ^{\nu }{}_{\nu \mu _{3}...\mu _{s}}+{\text{permutations}}\right)=0}
where in the first bracket one needs s − 1 {\displaystyle s-1} terms more to make the expression symmetric and in the second bracket one needs s ( s − 1 ) / 2 − 1 {\displaystyle s(s-1)/2-1} permutations. The equations are gauge invariant provided the field is double-traceless Φ ν
ν
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