Higher-spin theory
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 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 and not many examples have been worked out in detail except some specific toy models.
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.This gauge symmetry generalises that of massless spin-one and that of massless spin-two . Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above. For example, the equations are
where in the first bracket one needs terms more to make the expression symmetric and in the second bracket one needs permutations. The equations are gauge invariant provided the field is double-traceless and the gauge parameter is traceless.
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.
A theory for massive arbitrary higher-spin fields is proposed by C. Hagen and L. Singh. This massive theory is important because, according to various conjectures, 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. Interestingly, the Curtright field of such dual gravity model is of a mixed symmetry, hence the dual gravity theory can also be massive. Also the chiral and nonchiral actions can be obtained from the manifestly covariant Curtright action.
[|No-go theorems]
Possible interactions of massless higher spin particles with themselves and with low spin particles are 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 dateFlat 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 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, to 'too many' conservation laws that trivialise scattering so that.
Another well-known result is the Coleman–Mandula theorem. 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. 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 in the light-cone gauge.
Other no-go results include a direct analysis of possible interactions 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 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 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 as different from the AdS case.An AdS analog of the Coleman–Mandula theorem was obtained by Maldacena and Zhiboedov. 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. 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. 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 is plagued by non-localities, 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 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:where represents two helicity eigen-states of a massless spin- field in four dimensions. The action has two coupling constants: a dimensionless and a dimensionful which can be associated with the Planck length. Given three helicities fixed there is a unique cubic interaction, which in the spinor-helicity base can be represented as for positive. The main feature of chiral theory is the dependence of couplings on the helicities, which forces the sum to be positive. The theory is one-loop finite and its one-loop amplitudes are related to those of self-dual Yang-Mills theory. The theory can be thought of as a higher-spin extension of self-dual Yang–Mills theory. Chiral theory admits an extension to anti-de Sitter space, where it is a unique perturbatively local higher-spin theory with propagating massless higher-spin fields.
Conformal higher-spin gravity
Usual massless higher-spin symmetries generalise the action of the linearised diffeomorphisms from the metric tensor to higher-spin fields. In thecontext of gravity one may also be interested in conformal gravity that enlarges diffeomorphisms with Weyl transformations where is an arbitrary function. The simplest example of a conformal gravity is in four dimensions
One can try to generalise this idea to higher-spin fields by postulating the linearised gauge transformations of the form
where is a higher-spin generalisation of the Weyl symmetry. As different from massless higher-spin fields, conformal higher-spin
fields are much more tractable: they can propagate on nontrivial gravitational background and admit interactions in flat space. In particular, the action of conformal higher-spin
theories is known to some extent – it can be obtained as an effective action for a free conformal field theory coupled to the conformal higher-spin background.
Collective dipole
The idea is conceptually similar to the reconstruction approach just described, but performs a complete reconstruction in some sense. One begins with the free model partition function and performs a change of variables by passing from the scalar fields, to a new bi-local variable. In the limit of large this change of variables is well-defined, but has a nontrivial Jacobian. The same partition function can then be rewritten as a path integral over bi-local. It can also be shown that in the free approximation the bi-local variables describe free massless fields of all spins in anti-de Sitter space. Therefore, the action in term of the bi-local is a candidate for the action of a higher-spin theoryHolographic RG flow
The idea is that the equations of the exact renormalization group can be reinterpreted as equations of motions with the RG energy scale playing the role of the radial coordinate in anti-de Sitter space. This idea can be applied to the conjectural duals of higher-spin theories, for example, to the free model.Noether procedure
Noether procedure is a canonical perturbative method to introduce interactions. One begins with a sum of free actions and linearised gauge symmetries, which are given by Fronsdal Lagrangian and by the gauge transformations above. The idea is to add all possible corrections that are cubic in the fields and, at the same time, allow for field-dependent deformations of the gauge transformations. One then requires the full action to be gauge invariantand solves this constraint at the first nontrivial order in the weak-field expansion. Therefore, the first condition is. One has to mod out by the trivial solutions that result from nonlinear field redefinitions in the free action. The deformation procedure may not stop at this order and one may have to add quartic terms and further corrections to the gauge transformations that are quadratic in the fields and so on. The systematic approach is via BV-BRST techniques. Unfortunately, the Noether procedure approach has not given yet any complete example of a higher-spin theory, the difficulties being not only in the technicalities but also in the conceptual understanding of locality in higher-spin theories. Unless locality is imposed one can always find a solution to the Noether procedure or, the same time, by performing a suitable nonlocal redefinition one can remove any interaction. At present, it seems that higher-spin theories cannot be fully understood as field theories due to quite non-local interactions they have.