Higher Spin Field Equations Research Articles

Spinning particles and higher spin field equations

Relativistic particles with higher spin can be described in first quantization using actions with local supersymmetry on the worldline. First, we present a brief review of these actions and their use in first quantization. In a Dirac quantization scheme the field equations emerge as Dirac constraints on the Hilbert space, and we outline how they lead to the description of higher spin fields in terms of the more standard Fronsdal-Labastida equations. Then, we describe how these actions can be extended so that the propagating particle is allowed to take different values of the spin, i.e. carry a reducible representation of the Poincaré group. This way one may identify a four dimensional model that carries the same degrees of freedom of the minimal Vasiliev’s interacting higher spin field theory. Extensions to massive particles and to propagation on (A)dS spaces are also briefly commented upon.

Chiral higher spin theories and self-duality

We study recently proposed chiral higher spin theories — cubic theories of interacting massless higher spin fields in four-dimensional flat space. We show that they are naturally associated with gauge algebras, which manifest themselves in several related ways. Firstly, the chiral higher spin equations of motion can be reformulated as the self-dual Yang-Mills equations with the associated gauge algebras instead of the usual colour gauge algebra. We also demonstrate that the chiral higher spin field equations, similarly to the self-dual Yang-Mills equations, feature an infinite algebra of hidden symmetries, which ensures their integrability. Secondly, we show that off-shell amplitudes in chiral higher spin theories satisfy the generalised BCJ relations with the usual colour structure constants replaced by the structure constants of higher spin gauge algebras. We also propose generalised double copy procedures featuring higher spin theory amplitudes. Finally, using the light-cone deformation procedure we prove that the structure of the Lagrangian that leads to all these properties is universal and follows from Lorentz invariance.

Symmetries of higher-spin current interactions in four dimensions

Current interaction of massless fields in four dimensions is shown to break sp (8) symmetry of free massless equations of all spins down to the conformal symmetry su (2,2). This breaking is in agreement with the form of nonlinear higher-spin field equations.

Cauchy problem and Green's functions for first order differential operators and algebraic quantization

Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.

U(N|M) quantum mechanics on Kähler manifolds

We study the extended supersymmetric quantum mechanics, with supercharges transforming in the fundamental representation of U(N|M), as realized in certain one-dimensional nonlinear sigma models with Kähler manifolds as target space. We discuss the symmetry algebra characterizing these models and, using operatorial methods, compute the heat kernel in the limit of short propagation time. These models are relevant for studying the quantum properties of a certain class of higher spin field equations in first quantization.

Unconstrained higher spins of mixed symmetry II. Fermi fields

This paper is a sequel of [A. Campoleoni, D. Francia, J. Mourad, A. Sagnotti, Nucl. Phys. B 815 (2009) 289, arXiv:0810.4350 [hep-th]], and is also devoted to the local “metric-like” unconstrained Lagrangians and field equations for higher-spin fields of mixed symmetry in flat space. Here we complete the previous constrained on-shell formulation of Labastida for Fermi fields, deriving the corresponding constrained Lagrangians both via the Bianchi identities and via the requirement of self-adjointness. We also describe two types of unconstrained Lagrangian formulations: a “minimal” one, containing higher derivatives of the compensator fields, and another non-minimal one, containing only one-derivative terms. We identify classes of these systems that are invariant under Weyl-like symmetry transformations.

Spinning particles and higher spin fields on (A)dS backgrounds

Spinning particle models can be used to describe higher spin fields in first quantization. In this paper we discuss how spinning particles with gauged O(N) supersymmetries on the worldline can be consistently coupled to conformally flat spacetimes, both at the classical and at the quantum level. In particular, we consider canonical quantization on flat and on (A)dS backgrounds, and discuss in detail how the constraints due to the worldline gauge symmetries produce geometrical equations for higher spin fields, i.e. equations written in terms of generalized curvatures. On flat space the algebra of constraints is linear, and one can integrate part of the constraints by introducing gauge potentials. This way the equivalence of the geometrical formulation with the standard formulation in terms of gauge potentials is made manifest. On (A)dS backgrounds the algebra of constraints becomes quadratic, nevertheless one can use it to extend much of the previous analysis to this case. In particular, we derive general formulas for expressing the curvatures in terms of gauge potentials and discuss explicitly the cases of spin 2, 3 and 4.

A fiber approach to harmonic analysis of unfolded higher-spin field equations

In Vasiliev's unfolded formulation of higher-spin dynamics the standard fields are embedded on-shell into covariantly constant master fields valued in Lorentz-covariant slices of the star-product algebra of functions on the singleton phase space. Correspondingly, the harmonic expansion is taken over compact slices of that are unitarizable in a rescaled trace-norm rather than the standard Killing norm. Motivated by the higher-derivative nature of the theory, we examine indecomposable unitarizable Harish-Chandra modules consisting of standard massless particles plus linearized runaway solutions. This extension arises naturally in the above fiber approach upon realizing compact-weight states as non-polynomial analytic functions in .

On the uniqueness of minimal coupling in higher-spin gauge theory

We address the uniqueness of the minimal couplings between higher-spin fields and gravity. These couplings are cubic vertices built from gauge non-invariant connections that induce non-abelian deformations of the gauge algebra. We show that Fradkin-Vasiliev's cubic 2?s?s vertex, which contains up to 2s?2 derivatives dressed by a cosmological constant ?, has a limit where: (i) ????0; (ii) the spin-2 Weyl tensor scales non-uniformly with s; and (iii) all lower-derivative couplings are scaled away. For s = 3 the limit yields the unique non-abelian spin 2?3?3 vertex found recently by two of the authors, thereby proving the uniqueness of the corresponding FV vertex. We extend the analysis to s = 4 and a class of spin 1?s?s vertices. The non-universality of the flat limit high-lightens not only the problematic aspects of higher-spin interactions with ? = 0 but also the strongly coupled nature of the derivative expansion of the fully nonlinear higher-spin field equations with ??0, wherein the standard minimal couplings mediated via the Lorentz connection are subleading at energy scales (|?|)1/2??E??Mp. Finally, combining our results with those obtained by Metsaev, we give the complete list of all the manifestly covariant cubic couplings of the form 1?s?s? and 2?s?s?, in Minkowski background.

Real forms of complex higher spin field equations and new exact solutions

We formulate four-dimensional higher spin gauge theories in spacetimes with signature ( 4 − p , p ) and non-vanishing cosmological constant. Among them are chiral models in Euclidean ( 4 , 0 ) and Kleinian ( 2 , 2 ) signature involving half-flat gauge fields. Apart from the maximally symmetric solutions, including de Sitter spacetime, we find: (a) SO ( 4 − p , p ) invariant deformations, depending on one continuous and infinitely many discrete parameters, including a degenerate metric of rank one; (b) non-maximally symmetric solutions with vanishing Weyl tensors and higher spin gauge fields, that differ from the maximally symmetric solutions in the auxiliary field sector; and (c) solutions of the chiral models furnishing higher spin generalizations of type D gravitational instantons, with an infinite tower of Weyl tensors proportional to totally symmetric products of two principal spinors. These are apparently the first exact 4D solutions with non-vanishing massless higher spin fields.

The continuous spin limit of higher spin field equations

We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin field equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the spin to infinity with their product being fixed. The result allows to transform the Wigner equations to a gauge invariant Fronsdal-like form. We also give the generalisation of the Wigner equations to higher dimensions with fields belonging to arbitrary representations of the massless little group.

An abstract interface to higher spin gauge field theory

A comprehensive approach to the theory of higher spin gauge fields is proposed. By explicitly separating out details of implementation from general principles, it becomes possible to focus on the bare minimum of requirements that such a theory must satisfy. The abstraction is based on a survey of the progress that has been achieved since relativistic wave equations for higher spin fields were first considered in the 1930s. As a byproduct, a formalism is obtained that is abstract enough to describe a wide class of classical field theories. The formalism, viewed as syntax, can then be semantically mapped to a category of strongly homotopy Lie algebras, thus showing that the theory in some sense exists, at least as an abstract mathematical structure. Still, a concrete physicslike, implementation remains to be constructed. Lacking deep physical insight into the problem, an implementation in terms of generalized vertex operators is set up within which a brute force iterative determination of the first few orders in the interaction can be attempted.

Superfield Theories in Tensorial Superspaces and the Dynamics of Higher Spin Fields

We present the superfield generalization of free higher spin equations in tensorial superspaces and analyze tensorial supergravities with GL(n) and SL(n) holonomy as a possible framework for the construction of a non-linear higher spin field theory. Surprisingly enough, we find that the most general solution of the supergravity constraints is given by a class of superconformally flat and OSp(1|n)-related geometries. Because of the conformal symmetry of the supergravity constraints and of the higher spin field equations such geometries turn out to be trivial in the sense that they cannot generate a `minimal' coupling of higher spin fields to their potentials even in curved backgrounds with a non-zero cosmological constant. This suggests that the construction of interacting higher spin theories in this framework might require an extension of the tensorial superspace with additional coordinates such as twistor-like spinor variables which are used to construct the OSp(1|2n) invariant (`preonic') superparticle action in tensorial superspace.

Analysis of Higher Spin Field Equations in Four Dimensions

The minimal bosonic higher spin gauge theory in four dimensions contains massless particles of spin s=0,2,4,.. that arise in the symmetric product of two spin 0 singletons. It is based on an infinite dimensional extension of the AdS_4 algebra a la Vasiliev. We derive an expansion scheme in which the gravitational gauge fields are treated exactly and the gravitational curvatures and the higher spin gauge fields as weak perturbations. We also give the details of an explicit iteration procedure for obtaining the field equations to arbitrary order in curvatures. In particular, we highlight the structure of all the quadratic terms in the field equations.

Einstein's Equations and Associated Linear Systems

The relationship between Einstein's field equations and classical higher spin field equations is investigated using two-component spinor valued differential forms. Linear systems of equations associated to both the vacuum and coupled gravitational matter field equations are constructed. The latter equations are shown to be the integrability conditions of the linear systems.