Higher Spin Theory - Part I

Higher Spin Theory - Part I

Rakibur Rahman 
Physique Théorique et Mathématique & International Solvay Institues
Université Libre de Bruxelles, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium
E-mail: rakibur.rahman@ulb.ac.be
Speaker.
Abstract

These notes comprise a part of the introductory lectures on Higher Spin Theory presented in the Eighth Modave Summer School in Mathematical Physics. We construct free higher-spin theories and turn on interactions to find that inconsistencies show up in general. Interacting massless fields in flat space are in tension with gauge invariance and this leads to various no-go theorems. While massive fields exhibit superluminal propagation, appropriate non-minimal terms may cure such pathologies as they do in String Theorya fact that we demonstrate. Given that any interacting massive higher-spin particle is described by an effective field theory, we compute a model independent upper bound on the ultraviolet cutoff in the case of electromagnetic coupling in flat space and discuss its implications. Finally, we consider various possibilities of evading the no-go theorems for massless fields, among which Vasiliev’s higher-spin gauge theory is one. We employ the BRST-antifield method for a simple but non-trivial gauge system in flat space to find a non-abelian cubic coupling and to explore its higher-order consistency.

Higher Spin Theory - Part I

 

Rakibur Rahmanthanks: Speaker.

Physique Théorique et Mathématique & International Solvay Institues

Université Libre de Bruxelles, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium

E-mail: rakibur.rahman@ulb.ac.be

\abstract@cs

Eighth Modave Summer School in Mathematical Physics, 26th august - 1st september 2012 Modave, Belgium

1 Motivations & Outline

In Quantum Field Theory fundamental particles carry irreducible unitary representations of the Poincaré group, and therefore can have arbitrary (integer or half-integer) values of the spin, at least in principle. The motivations for studying higher-spin (HS) particles are manifold.

  1. While free HS fields are fine, severe problems show up as soon as interactions are turned on. For massless particles, there exist powerful no-go theorems [1, 2, 3, 4, 5] that forbid them from interacting in flat space with electromagnetism (EM) or gravity when their spin exceeds certain value. One would like to evade these theorems in order write down interacting theories for massless HS fields.

  2. Massive HS particles do exist in the form of hadronic resonances, e.g., (1670), (1690), (2040) etc. These particles are composite, and their interactions are described by complicated form factors. However, when the exchanged momenta are small compared to their masses, one should be able to describe their dynamics by consistent local actions.

  3. Massive HS excitations show up in (open) string spectra. In fact, they play a crucial role in that some of the spectacular features of String Theory, e.g., planar duality, modular invariance and open-closed duality, rest heavily on their presence.

  4. It is conjectured that String Theory describes a broken phase of a HS gauge theory [6, 7, 8, 9]. The underlying HS theory may provide a better understanding of what String Theory is.

  5. Vasiliev’s HS gauge theory in  [8, 10] is conjectured to be holographically dual to Vector Models [11], and the first non-trivial checks of the duality appear in [12]. This duality may be regarded as the simplest non-trivial example of AdS/CFT correspondence, and may help us understand some aspects of gauge/gravity dualities in general.

In what follows we will present an introduction to HS fields and their interactions. In Section 2, we start with construction of free HS theories for massive and massless fields. In Section 3, we discuss the difficulties that one faces when interactions are turned on. In particular, Section 3.1 presents the various no-go theorems [1, 2, 3, 4, 5] for interacting HS massless fields in flat space, all of which essentially derive from the fact that interactions are in tension with gauge invariance, while Section 3.2 shows that massive HS fields exhibit superluminal propagation in external backgroundsthe notorious Velo-Zwanziger problem [13]. Appropriate non-minimal terms may cure the Velo-Zwanziger acausality, and in Section 4 we show how String Theory provides remedy to this pathology. In Section 5, we consider the fact that interacting massive HS particles are described by effective field theories, compute a model independent upper bound on the ultraviolet cutoff for their electromagnetic coupling in flat space, and discuss the implications of this result. Finally, in Section 6 we study interactions of HS gauge fields by considering the different possibilities of evading the aforementioned no-go theorems, e.g., adding a mass term, working in or in AdS space, or considering higher-derivative interactions. We work out a simple example of a non-abelain cubic coupling in flat space by using the BRST-antifield formalism and explore its higher-order consistency.

2 Construction of Free Higher Spin Theories

The task of constructing a theory of HS fields dates back to 1936, when Dirac tried to generalize his celebrated spin- equation [14]. A systematic study of HS particles, though, was initiated by Fierz and Pauli in 1939 [15]. Their approach was field theoretic that focused on the physical requirements of Lorentz invariance and positivity of energy. The works of Wigner on unitary representations of the Poincaré group [16] and of Bargmann and Wigner on relativistic wave equations [17], however, made it clear that one could replace positivity of energy by the requirement that a one-particle state carries an irreducible unitary representation of the Poincaré group.

2.1 Massive Fields

The two Casimir invariants of the Poincaré group are

(2.0)

where is the Pauli-Lubanski pseudovector. They define respectively mass and spinthe two basic quantum numbersa field may possess. Let us consider the first Wigner class, which corresponds to a physical massive particle of mass and spin . In this case, we have and . The transformation properties under the Lorentz group is uniquely determined for a bosonic field by the usual choice of the representation . The field is then a totally symmetric rank- tensor, , which is traceless:

(2.0)

The Casimir then demands that the Klein-Gordon equation be satisfied:

(2.0)

The representation is however reducible under the rotation subgroup, because it contains all spin values from down to . The Casimir requires that all lower spin values be eliminated; this is done by imposing the divergence/transversality condition:

(2.0)

The latter is a necessary condition in order for the total energy to be positive definite [15].

How many degrees of freedom (DoF) does this field propagate in space-time dimensions? Note that the number of independent components of symmetric rank- tensor is given by

(2.0)

The tracelessness condition is a symmetric rank- tensor that removes of these components. Similarly, the divergence constraint (2.1) should eliminate many, but its trace part has already been incorporated in the tracelessness of the field itself, so that the actual number is less by . Then the total number of propagating DoFs are

(2.0)

In particular when , this number reduces to as expected.

A fermionic field of spin , on the other hand, is represented by a totally symmetric tensor-spinor of rank , , which is -traceless:

(2.0)

It transforms as the representation of the Lorentz group , and satisfies

(2.0)
(2.0)

The counting of DoFs is analogous to that for bosons. In dimensions, a symmetric rank- tensor-spinor has independent components, where . The -trace and divergence constraints, being symmetric rank- tensor-spinors, each eliminates of them. But there is an over-counting of constraints, since only the traceless part of the divergence condition (2.0) should be counted. The number of dynamical DoFsfields and conjugate momentais therefore

(2.0)

In in particular, this is as expected.

Fierz and Pauli noted in [15] that attempts to introduce electromagnetic interaction at the level of the equations of motion (EoM) and constraints lead to algebraic inconsistencies for spin greater than 1. If one modifies the Eqs. (2.1)-(2.1) and (2.0)-(2.0) by directly replacing ordinary derivatives with covariant ones, they are no longer mutually compatible111We will work out an example of this inconsistency in Section 3.. To avoid such difficulties it was suggested that all equations involving derivatives be obtainable from a Lagrangian. This is possible only at the cost of introducing lower-spin auxiliary fields, which must vanish when interactions are absent.

Fronsdal [18] and Chang [19] spelled out a procedure for introducing these auxiliary fields to construct HS Lagrangians. Singh and Hagen [20] achieved the feat in 1974 by writing down an explicit form of the Lagrangian for a free massive field of arbitrary spin. The Singh-Hagen Lagrangian for an integer spin is written in terms of a set of symmetric traceless tensor fields of rank . For a half-integer spin , the Lagrangian incorporates symmetric -traceless tensor-spinors: one of rank , another of rank , and doublets of rank . When the Eqs. (2.1)-(2.1) and (2.0)-(2.0) are satisfied, all the lower spin fields are forced to vanish.

To understand the salient features of the Singh-Hagen construction, we consider the massive spin-2 field, which is described by a symmetric traceless rank-2 tensor . Let us try to incorporate the Klein-Gordon equation, , and the transversality condition, , into a Lagrangian as

(2.0)

where the constant is to be determined. While varying the above action one must keep in mind that is symmetric traceless. One therefore obtains the following EoMs:

(2.0)

where is the space-time dimensionality. The divergence of Eq. (2.0) gives

(2.0)

The transversality condition can be recovered by setting as well as requiring . The latter condition, however, does not follow from the EoMs. This problem can be taken care of by introducing an auxiliary spin-0 field , so that the condition follows from the Lagrangian. Let us add the following terms to the Lagrangian (2.0):

(2.0)

where are constants. The double divergence of the EoMs now gives

(2.0)

The EoM for the auxiliary scalar , on the other hand, reduces to

(2.0)

Eqs. (2.0)-(2.0) comprise a linear homogeneous system in the variables and . The condition and the vanishing of the auxiliary field must follow if the associated determinant does not vanish. The latter is given by

(2.0)

Note that becomes algebraic, not containing , proportional to and hence non-zero if

(2.0)

Thus one constructs the following Lagrangian for a massive spin-2 field

(2.0)

One can now check that this Lagrangian yields and , so that the Klein-Gordon equation and the transversality condition follow from it.

One can carry out this procedure for higher spins to find that the following pattern emerges: For an integer spin , one must successively obtain , for . At each value of one needs to introduce an auxiliary symmetric traceless rank- tensor field. Similarly for a half-integer spin , the transversality condition, , can be obtained by introducing a symmetric -traceless tensor-spinor of rank , , provided that one also satisfies the conditions: and . These can be achieved by successively obtaining and , for . Then for each one must introduce two symmetric -traceless rank-() tensor-spinors. The explicit form of the Lagrangian for an arbitrary spin is rather complicated and is given in Ref. [20].

2.2 Massless Fields

In 1978, Fronsdal and Fang [21] considered the massless limit of the Singh-Hagen Lagrangian to find considerable simplification of the theory. For the bosonic case, all the auxiliary fields decouple in this limit, except for the one with the highest rank . Furthermore, the two symmetric traceless rank- and rank- tensors can be combined into a single symmetric tensor, , which is double traceless: . The Lagrangian reduces to [21]

(2.0)

where prime denotes trace: , and denotes divergence: . The Lagrangian (2.0) has acquired a gauge invariance with a symmetric traceless gauge parameter :

(2.0)

Let us illustrate the Fronsdal-Fang formulation [21] by considering again the spin-2 case. In the massless limit, the Singh-Hagen spin-2 Lagrangian (2.0) reduces to

(2.0)

We combine and into a single field :

(2.0)

The tracelessness of then gives

(2.0)

This reduces the Lagrangian (2.0) to

(2.0)

Here the new field is symmetric but not traceless. The Lagrangian (2.0) is nothing but the linearized Einstein-Hilbert action with identified as the metric perturbation around Minkowski background. This describes a massless spin-2 particle, and the corresponding gauge symmetry is just the infinitesimal version of coordinate transformations:

(2.0)

Gauge invariance and the trace conditions on the field and the gauge parameter play a crucial role in the Fronsdal-Fang construction in that they ensure absence of ghosts, and give the correct number of propagating DoFs. To do the DoF count, we write down the EoMs that follow from the Lagrangian (2.0):

(2.0)

where is the Fronsdal tensor defined as

(2.0)

Another equivalent form of the EoMs is the so-called Fronsdal form:

(2.0)

This contains precisely the correct number of conditions to determine the components of the symmetric double-traceless field because is vanishing. A symmetric rank- tensor with vanishing double trace has independent components in dimensions. Now that Eq. (2.0) enjoys the gauge invariance (2.0), the symmetric traceless rank- gauge parameter enables us to remove components by imposing an appropriate covariant gauge condition, e.g.,222This is the arbitrary-spin generalization of the Lorenz gauge for and the de Donder gauge for .

(2.0)

where is traceless because . This reduces Eq. (2.0) to

(2.0)

Thus indeed describes a massless field. Eq. (2.0) however does not completely fix the gauge since gauge parameters satisfying are still allowed. Therefore, we can gauge away another set of components. Thus the total number of propagating DoFs turns out to be [22]:

(2.0)

In particular, this leaves us exactly with 2 DoF for all in .

For the fermionic case too we can take the massless limit of the Singh-Hagen Lagrangian. For spin , one is left in this case only with three symmetric -traceless tensor-spinors of rank and respectively; the other auxiliary fields decouple completely. These three can be combined into a single symmetric rank- tensor-spinor, , which is triple -traceless: . This leads to the following Lagrangian [21]:

(2.0)

The resulting EoMs read

(2.0)

where the fermionic Fronsdal tensor is given by

(2.0)

One can also rewrite the EoMs in the Fronsdal form:

(2.0)

This equation contains the appropriate number of independent components because is triple -traceless if is. The Lagrangian (2.0) and the EoMs (2.0)-(2.0) enjoy a gauge symmetry with a symmetric -traceless tensor-spinor parameter :

(2.0)

To get the DoF count let us note that a symmetric triple -traceless rank- tensor-spinor has components in dimensions. However, the -traceless part of is actually non-dynamical as Eq. (2.0) shows. Therefore, we have constraints. Now, the symmetric -traceless gauge parameter enables us to choose, for example, the gauge333 Here the gauge condition does not involve derivatives, so that it cannot convert constraints into evolution equations. This is in contrast with the bosonic case where the gauge condition (2.0) renders the non-dynamical componentsthe traceless part of dynamical as seen from Eq. (2.0).

(2.0)

where is -traceless because . This allows further gauge transformations for which the gauge parameter satisfies

(2.0)

Note that the left-hand side of Eq. (2.0) is -traceless because is. The gauge condition (2.0) and the residual gauge fixing eliminate components each. In this way, one finds that the number of dynamical DoFsfields and conjugate momentais [22]:

(2.0)

In in particular, this is 4 as expected. We parenthetically remark that the gauge fixing (2.0) does not reduce the EoMs (2.0) to the Dirac equation. However, one can use part of the residual gauge invariance, which satisfies the condition (2.0), to set . Then the field equations become , which indeed describe a massless fermion.

Apart from their gauge theoretic and geometric aspects444One can construct HS analogs of the spin-2 Christoffel connection [22]a hierarchy of generalized Christoffel symbols linear in derivatives of the field with simple gauge transformation properties. The wave equations are also very simple in terms of these objects. The trace conditions on the HS field and the gauge parameter can also be easily understood in this approach. See also Ref. [23] for geometric formulations of HS gauge fields., massless HS theories are interesting since they give rise to the massive theories through proper Kaluza-Klein (KK) reductions [24]. This constructionmuch simpler than the original Singh-Hagen onejustifies the choice of auxiliary fields in the latter. For the bosonic case, for example, a massless spin- Lagrangian in dimensions KK reduces in dimensions to a theory containing symmetric tensor fields of spin through . The ()-dimensional gauge invariance gives rise to the Stückelberg symmetry in dimensions. The tracelessness of the higher dimensional gauge parameter has non-trivial consequences; one can gauge away some, but not all, of the auxiliary (Stückelberg) fields. The gauge-fixed Lagrangian is then equivalent to the Singh-Hagen one up to field redefinitions [24].

3 Turning on Interactions

As we already mentioned, turning on interactions for HS fields at the level of EoMs and constraints, by replacing ordinary derivatives with covariant ones, results in inconsistencies. Let us take, for example, the naïve covariant version of Eqs. (2.1)-(2.1) for massive spin 2 coupled to EM

(3.0)

where . Now the Klein-Gordon equation and the transversality condition give

(3.0)

The non-commutativity of covariant derivatives then results in unwarranted constraints. For example, for a constant EM field strength , one obtains

(3.0)

This constraint does not exist when the interaction is turned off, so that the system of equations (3.0) does not describe the same number of DoFs as the free theory. Such difficulties can be avoided, as Fierz and Pauli suggested [15], by taking recourse to the Lagrangian formulation. However, as we will see in Sections 3.1 and 3.2, the Lagrangian approach is not free of difficulties either; inconsistencies show up for both massless and massive HS fields when interactions are present.

3.1 No-Go Theorems for Massless Fields

No massless particle with has been observed in Nature. Neither is there any known string compactification that gives a Minkowski space with massless particles of spin larger than two. Indeed, interactions of HS gauge fields in flat space are severely constrained by powerful no-go theorems [1, 2, 3, 4, 5]. Below we will closely follow Ref. [5] to give an overview of some of the most important no-go theorems.

3.1.1 Weinberg (1964)

One can show, in a purely -matrix-theoretic approach, that there are obstructions to consistent long-range interactions mediated by massless bosons with  [1]. Let us consider an -matrix element with external particles of momenta , and a massless spin- particle of momentum . The matrix element factorizes in the soft limit as:

(3.0)

where is the polarization tensor of the spin- particle; it is transverse and traceless:

(3.0)

The polarization tensor is redundant in that it has more components than the physical polarizations of the massless particle. This redundancy is eliminated by demanding that the matrix element vanish for spurious polarizations, i.e., for

(3.0)

where is also transverse and traceless. One finds from Eq. (3.0) that the spurious polarizations decouple, for any generic momenta , only if

(3.0)

For , this is possible when . This is nothing but the conservation of charge. For , Eq. (3.0) reduces to , which can be satisfied for generic momenta if (i) , and (ii) . The first condition imposes energy-momentum conservation, while the second one requires all particles to interact with the same strength with the massless spin 2 (graviton). The latter is simply the equivalence principle. For , Eq. (3.0) has no solution for generic momenta.

This argument does not rule out massless bosons with spin larger than 2, but say that they cannot give rise to long-range interactions. Similar arguments were used in Ref. [2] for half-integer spins to forbid interacting massless fermions with . These theorems leave open the possibility that massless HS particles may very well interact but can mediate only short-range interactions.

3.1.2 Aragone-Deser (1979)

In order to completely rule out massless HS particles, one should consider a truly universal interaction which no particle can avoid. Gravitational coupling is one such interaction. In fact, Eq. (3.0) shows that the graviton interacts universally with all matter in the soft limit .

Aragone and Deser [3] studied the gravitational coupling of a spin- gauge field to find that such a theory is fraught with grave inconsistencies: the unphysical gauge modes decouple only in the free theory. The system of a spin- fielddescribed by the tensor-spinor minimally coupled to gravity, is governed by the action:

(3.0)

The redundancy in the field should be eliminated, as in the free case, by the gauge invariance

(3.0)

Under this gauge transformation, however, the action (3.0) transforms as

(3.0)

i.e., the action is invariant only in flat space where the Riemann tensor vanishes. In other words, the gauge modes do not decouple for an interacting theory. It is easy to see that local non-minimal terms, regular in the neighborhood of flat space, do not come to the rescue [3].

The Aragone-Deser no-go theorem is based on Lagrangian formalism, and therefore has a very important implicit assumption: locality. Some non-locality in the Lagrangian appearing, for example, in the guise of a form factor for gravitational couplings may remove the difficulties. It is also possible to have a non-minimal description involving a larger gauge invariance than (3.0). All these issues can be addressed in the -matrix language by considering, for example, scattering of massless HS particles off soft gravitons.

3.1.3 Weinberg-Witten (1980) & Its Generalization

Weinberg and Witten took the -matrix approach to prove the following important theorem [4]: “A theory that allows the construction of a conserved Lorentz covariant energy-momentum tensor for which is the energy-momentum four-vector cannot contain massless particles of spin .” They considered in 4D a particular matrix element: the elastic scatt