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. It is important to note that 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,^[1][2] Jackiw–Teitelboim,^[3] selfdual (chiral)^[4][5] and Weyl gravity theories^[6][7]).

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.^[8]

${\displaystyle \delta \Phi _{\mu _{1}\mu _{2}...\mu _{s}}=\partial _{\mu _{1}}\xi _{\mu _{2}...\mu _{s}}+{\text{permutations}}}$

This (linearised) gauge symmetry generalises that of massless spin-one (photon) ${\displaystyle \delta A_{\mu }=\partial _{\mu }\xi }$ and that of massless spin-two (graviton) ${\displaystyle \delta h_{\mu \nu }=\partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }}$. Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above. For example, the equations are

${\displaystyle \square \Phi _{\mu _{1}\mu _{2}...\mu _{s}}-\left(\partial _{\mu _{1}}\partial ^{\nu }\Phi _{\nu \mu _{2}...\mu _{s}}+{\text{ permutations}}\right)+{\frac {1}{2}}\left(\partial _{\mu _{1}}\partial _{\mu _{2}}\Phi ^{\nu }{}_{\nu \mu _{3}...\mu _{s}}+{\text{permutations}}\right)=0}$

where in the first bracket one needs ${\displaystyle s-1}$ terms more to make the expression symmetric and in the second bracket one needs ${\displaystyle s(s-1)/2-1}$ permutations. The equations are gauge invariant provided the field is double-traceless ${\displaystyle \Phi ^{\nu }{}_{\nu }{}^{\lambda }{}_{\lambda \mu _{5}...\mu _{s}}=0}$ and the gauge parameter is traceless ${\displaystyle \xi ^{\nu }{}_{\nu \mu _{3}...\mu _{s-1}}=0}$.

Essentially, the higher spin problem can be stated as a problem to find a nontrivial interacting theory with at least one massless higher-spin field (higher in this context usually means greater than two).

A theory for massive arbitrary higher-spin fields is proposed by C. Hagen and L. Singh.^[9][10] This massive theory is important because, according to various conjectures,^[11][12][13] spontaneously broken gauges of higher-spins may contain an infinite tower of massive higher-spin particles on the top of the massless modes of lower spins s ≤ 2 like graviton similarly as in string theories.

The linearized version of the higher-spin supergravity gives rise to dual graviton field in first order form.^[14] Interestingly, the Curtright field of such dual gravity model is of a mixed symmetry,^[15] hence the dual gravity theory can also be massive.^[16] Also the chiral and nonchiral actions can be obtained from the manifestly covariant Curtright action.^[17][18]

No-go theorems

Possible interactions of massless higher spin particles with themselves and with low spin particles are (over)constrained by the basic principles of quantum field theory like Lorentz invariance. Many results in the form of no-go theorems have been obtained up to date^[19]

Flat space

Most of the no-go theorems constrain interactions in the flat space.

One of the most well-known is the Weinberg low energy theorem^[20] that explains why there are no macroscopic fields corresponding to particles of spin 3 or higher. The Weinberg theorem can be interpreted in the following way: Lorentz invariance of the S-matrix is equivalent, for massless particles, to decoupling of longitudinal states. The latter is equivalent to gauge invariance under the linearised gauge symmetries above. These symmetries lead, for ${\displaystyle s>2}$, to 'too many' conservation laws that trivialise scattering so that ${\displaystyle S=1}$.

Another well-known result is the Coleman–Mandula theorem.^[21] that, under certain assumptions, states that any symmetry group of S-matrix is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group. This means that there cannot be any symmetry generators transforming as tensors of the Lorentz group – S-matrix cannot have symmetries that would be associated with higher spin charges.

Massless higher spin particles also cannot consistently couple to nontrivial gravitational backgrounds.^[22] An attempt to simply replace partial derivatives with the covariant ones turns out to be inconsistent with gauge invariance. Nevertheless, a consistent gravitational coupling does exist^[23] in the light-cone gauge (to the lowest order).

Other no-go results include a direct analysis of possible interactions^[24][25] and show, for example, that the gauge symmetries cannot be deformed in a consistent way so that they form an algebra.

Anti-de Sitter space

In anti-de Sitter space some of the flat space no-go results are still valid and some get slightly modified. In particular, it was shown by Fradkin and Vasiliev^[26] that one can consistently couple massless higher-spin fields to gravity at the first non-trivial order. The same result in flat space was obtained^[23] by Bengtsson, Bengtsson and Linden in the light-cone gauge the same year. The difference between the flat space result and the AdS one is that the gravitational coupling of massless higher-spin fields cannot be written in the manifestly covariant form in flat space^[27] as different from the AdS case.

An AdS analog of the Coleman–Mandula theorem was obtained by Maldacena and Zhiboedov.^[28] AdS/CFT correspondence replaces the flat space S-matrix with the holographic correlation functions. It then can be shown that the asymptotic higher-spin symmetry in anti-de Sitter space implies that the holographic correlation functions are those of the singlet sector a free vector model conformal field theory (see also higher-spin AdS/CFT correspondence below). Let us stress that all n-point correlation functions are not vanishing so this statement is not exactly the analogue of the triviality of the S-matrix. An important difference from the flat space results, e.g. Coleman–Mandula and Weinberg theorems, is that one can break higher-spin symmetry in a controllable way, which is called slightly broken higher-spin symmetry.^[29] In the latter case the holographic S-matrix corresponds to highly nontrivial Chern–Simons matter theories rather than to a free CFT.

As in the flat space case, other no-go results include a direct analysis of possible interactions. Starting from the quartic order a generic higher-spin gravity (defined to be the dual of the free vector model, see also higher-spin AdS/CFT correspondence below) is plagued by non-localities,^[30][31] which is the same problem as in flat space.

Various approaches to higher-spin theories

The existence of many higher-spin theories is well-justified on the basis of AdS/correspondence, but none of these hypothetical theories is known in full detail. Most of the common approaches to the higher-spin problem are described below.

Chiral higher-spin gravity

Generic theories with massless higher-spin fields are obstructed by non-localities, see No-go theorems. Chiral higher-spin gravity^[4][5] is a unique higher-spin theory with propagating massless fields that is not plagued by non-localities. It is the smallest nontrivial extension of the graviton with massless higher-spin fields in four dimensions. It has a simple action in the light-cone gauge:

${\displaystyle {\mathcal {S}}=\int \mathrm {d} ^{4}x\left}$

where ${\displaystyle \Phi _{\lambda }(x)}$ represents two helicity eigen-states ${\displaystyle \lambda =\pm s}$ of a massless spin-${\displaystyle s}$ field in four dimensions (for low spins one finds $$Zdroj:https://en.wikipedia.org?pojem=Higher-spin_theory
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