aa r X i v : . [ h e p - t h ] A p r Prepared for submission to JHEP
BRST Theory for Continuous Spin
Anders K. H. Bengtsson a, a School of Engineering, University of Bor˚as, All´egatan 1, SE-50190 Bor˚as, Sweden.
E-mail: [email protected]
Abstract:
Some puzzling aspects of higher spin field theory in Minkowski space-time,such as the tracelessness constraints and the search for an underlying physical principle,are discussed. A connecting idea might be provided by the recently much researchedcontinuous spin representations of the Poincar´e group. The Wigner equations, treated asfirst class constraints, yields to a four-constraint BRST formulation. The resulting fieldtheory, generalizing free higher spin field theory, is one among a set of higher spin theoriesthat can be related to previous work on unconstrained formulations. In particular, it isconjectured that the unconstrained higher spin theory of Francia and Sagnotti is a limit ofa continuous spin theory. Furthermore, a simple analysis of the constraint structure revealsa hint of a physical rationale behind the trace constraints.
Keywords:
Higher spin field theory, Continuous spin representations, BRST methods,Higher spin gravity. Work supported by the Research and Education Board at the University of Bor˚as. ontents – 1 –
Introduction
A most mysterious, and awkward, aspect of higher spin gauge field theory is the doubletracelessness and tracelessness constraints for fields ϕ µ ...µ s gauge parameters θ µ ...µ s − respectively ϕ αβαβµ ...µ s = 0 for s ≥ ,θ ααµ ...µ s − = 0 for s ≥ . Their presence, and consequently their need to be treated, is a nuisance. Superficially,and technically, it is since long well understood why they appear. There is a mismatchbetween the number of physical degrees of freedom of the higher spin massless states,and the number of components of the corresponding tensor gauge fields. This mismatchcannot be accounted for by ordinary gauge transformations only. Instead, these extra traceand double trace constraints are imposed on the gauge parameters and the gauge fieldsrespectively. This is all well known since the canonical papers [1, 2] on higher spin gaugefields.In the original BRST treatment of higher spin gauge field theory the trace constraintswere treated as second class and imposed as equations on the states [3, 4]. Alternativeformulations were proposed about ten years ago, employing non-local actions [5, 6] or extracompensator fields [7–10] . These latter approaches essentially amount to introducing newdegrees of freedom in order that the concomitant extra gauge invariance alone can fix themismatch in numbers of degrees of freedom. Still, one cannot escape the impression thatsomething is missing in the basic understanding.There is the related problem of understanding the underlying physical principle, ifany such exists, behind higher spin gauge fields. In [12], mechanical models were brieflydiscussed for higher spin gauge fields as a step towards basing a physical picture of theinteractions on such a model. Although looking promising at the outset, such an approach isfraught with problems that presumably go back to the free theory itself and its constraintstructure. In [13] and [14] a tentative role for the tracelessness constraint (or a similarcondition) in relation to the cubic vertex were noted.The recent interest in the continuous spin representations of the Poincar´e group [15–17] now seems to offer a hope for finding the missing idea that might connect these looseends. Indeed, the four Wigner equations [18] for the wave function, can be interpreted asfirst class constraints of a mechanical model. The resulting BRST theory is the subject ofthe present paper. Sections 2 through 4 sets up the background for the BRST treatmentof section 5. The tentative underlying physics is discussed in section 6. Some concludingremarks are in section 7 and some technical details about the ghost complexes are relegatedto section 8. For a more extensive set or references as well as a thorough discussion, see [11]. – 2 –
Constraint structure
In a BRST approach to free higher spin gauge fields, the gauge transformations are gen-erated by the first class constraints of an underlying mechanics model, while the traceconditions can be imposed through second class constraints. This works well for the freetheory and reproduces the Fronsdal theory. Let us work in four dimensions (with a − + ++metric) and briefly iterate the steps.We start with a classical or first-quantized two-particle relativistic mechanical systemwith center of motion ( x µ , p ν ) and relative ( ξ µ , π ν ) coordinates and momenta, but wedon’t specify any action, instead working directly from the constraints. For the relativecoordinates we also use oscillators ( α µ , α † ν ), or holomorphic coordinates classically, definedin terms of the relative coordinates and momenta as α µ = 1 √ ξ µ + iπ µ ) , α † µ = 1 √ ξ µ − iπ µ ) . (2.1)We take ξ µ and π ν to be dimensionless. Classically we have { x µ , p ν } = η µν , { ξ µ , π ν } = η µν , (2.2)and quantum mechanically[ x µ , p ν ] = iη µν , [ ξ µ , π ν ] = iη µν , [ α µ , α † ν ] = η µν . (2.3)Excluding explicit occurrence of the center of motion coordinate x µ there are six bilinearscalars in terms of these variables G = − p , G + = α · p, G − = α † · p, (2.4) T = 12 α · α, T † = 12 α † · α † , N = 12 ( α · α † + α † · α ) = α † · α + 2 . (2.5)From this set we can chose various linear combinations as first and second class constraintsby (weakly) equating to zero. Once such a choice is made, ghost coordinates and momentacan be introduced corresponding to the first class set and the BRST operator Q constructed.Then a free field theory can be set up using BRST-BV techniques. The standard choiceis simply taking the set { G = 0 , G + = 0 , G − = 0 } as first class. Then the tracelessnessconstraints (on fields and parameters) are given by T | state i = 0 with the T operatoraugmented with a ghost contribution.One obvious problem with this approach is that it is very formal, inspired as it is (andinitially was) by string field theory [19–21]. Where, is the physics? For strings, there isvibration dynamics that motivates the introduction of oscillators. For higher spins, suchdynamics is not at all obvious. There are various possible limits of strings (such as the muchresearched zero-tension limit , the straight line (or rigid) string [22] and the discrete string[23]) that furnish constraints that are essentially built from different linear combinationsof the bilinear terms (2.4) and (2.5). But the physical intuition is weak. A nice picture This limit has been studied by many authors throughout the history of string theory. – 3 –s a one-dimensional ”spring model” of two relativistic particles bound by a harmonicpotential. Such vibrations could correspond to a constraint π + ξ = 0. However, thisconstraint, corresponding to the oscillator Hamiltonian, would fix the excitation level andreduce the field theory to a single spin theory, whereas we want to accommodate all spins.Clearly, almost any sensible two-particle action with symmetries will produce constraintsthat are various linear combinations of the set (2.4) and (2.5). For a thorough discussion,see reference [24]. Now for some input from continuous spin theory. In the Bargmann-Wigner paper [18] theequations are given as p ψ = 0 , (3.1) ξ ψ = − lψ, (3.2) p · ξψ = 0 , (3.3) p · ∂/∂ξψ = − i Ξ ψ. (3.4)where ψ ( p, ξ ) is a wave function, p the momentum operator and ξ an internal four-vector oflength l ( l = 1 in [18]). These equations look conspicuously much like constraint equationsfor higher spin gauge fields. To make the connection more precise, let us instead follow[31, 32]. These authors consider the equations p | phys i = 0 , (3.5)( w − µ ) | phys i = 0 . (3.6)to be satisfied by physical states | phys i and where w µ is the Pauli-Lubanski vector w µ = 12 ǫ µνρσ m νρ p σ . (3.7)They find the most general solution in terms of four constraints. We will not need the mostgeneral case, so let us just pick the set (using slightly different conventions) G = − p , (3.8) G = p · π, (3.9) G = p · ξ − µω , (3.10) G = − π + ω . (3.11)These are essentially the same equations as considered in [33]. Classically, we can think ofthe equations as set of four first class constraints G i ≈
0. The non-zero Poisson bracketsare { G , G } = − G , { G , G } = − G . (3.12) The theory goes back to Wigner [25]. A set of classic references are [26–29]. The recent literatureinclude [30–33]. – 4 –ith µ zero and neglecting G these are precisely the first class constraints that uponBRST-quantization yield a free field theory containing all integer spin and where the fieldequations are precisely those of Fronsdal after imposing the trace constraint. As willargued in the next section, the continuous spin constraint G can serve a role similar tothe tracelessness constraint. Before using the input from continuous spin representations, let us first see what can bedone with the constraint structure from a purely formal point of view. The idea is totreat the trace constraint as first class. Counting phase space degrees of freedom (d.o.f)as ( , , cm . + (4 , rel . that is 16 d.o.f. in all. The threefirst class constraints { G = 0 , G + = 0 , G − = 0 } remove 3 · { T = 0 , T † = 0 } remove 2 d.o.f. We are left with 8 d.o.f. which in phasespace distribute as (3 , cm . + (1 , rel . . This precisely corresponds to a tower of masslessfields with helicities ± λ . There is a simple way to treat a pair of second class constraintsas first class: take one of them as a first class constraint and treat the remaining one as agauge condition. Two requirements must be met: the new constraint algebra must be firstclass and the new constraint must be Hermitean (real).Then it is clear that we cannot take T = 0 as first class since it is not Hermitean. Herewe need not run through the full analysis, suffice it to say that any linear combination ofthe set { T, T † , N } that is Hermitean would do. We will study three cases in this paper,see equation (4.10) below.Let us now be specific and pick G from equation (3.11) as the fourth first classconstraint. Using (2.1) we find G = − π + ω = T + T † − N + ω . (4.1)We keep G as it is, but linearly recombine G and G into G + = p · α − µ √ ω , (4.2) G − = p · α † − µ √ ω , (4.3)so that G + + G − = √ G , G + − G − = i √ G . (4.4)The non-zero commutators of the resulting first class algebra is[ G + , G − ] = − G , (4.5)[ G + , G ] = G − − G + , (4.6)[ G − , G ] = G − − G + . (4.7)In the following, µ/ √ ω will be denoted by β .– 5 – slight generalization It is interesting to see whether the G constraint can be generalized since this is really wherewe depart from standard BRST higher spin. Instead of G we may attempt a constraint G with the following non-zero commutators with the set { G , G + , G − } [ G + , G ] = σ − G − + σ + G + , (4.8)[ G − , G ] = − σ + G − − σ − G + . (4.9)Taking G = N corresponds to σ + = 1 and σ − = 0 but then we must have β = 0. Likewisetaking G = T + T † corresponds to σ + = 0 and σ − = 1 and again we must have β = 0.The case G = G corresponds to σ − = − σ + = 1 and then we can have β = 0. This is thecontinuous spin case. The different cases can be captured by writing G = σ − (cid:0) T + T † (cid:1) + σ + N + ω (4.10)where from now on we use the notation G for this generalized case. Taking σ + = σ − = 1would correspond to taking G = ξ + ω , but since π and ξ can be interchanged throughcanonical transformations [32], nothing new is gained by this.For clarity, the interesting cases are summarized in the Table 1: Higher spin with”trace constraint” T + T † = 0 (abbreviated HS T ), Higher spin with ”number constraint” N = 0 (abbreviated HS N ) and Continuous spin (abbreviated CS).Type of theory σ + σ − CommentHS T β = 0 but allows ω = 0HS N β = 0 but allows ω = 0CS − β = 0, ω = 0 (but allows β = 0, ω = 0) Table 1 . Theories considered.
A four-constraint theory of the type HS N has been studied previously in [34]. The free field theory of massless higher spin fields has been extensively studied by manyauthors during the last decade (for review and references, see [11]). Much effort has beenspent in trying to circumvent the tracelessness constraints, one interesting approach beingthe unconstrained formulation using compensator fields originally proposed in [5, 6]. Insection 5.6 we will see how that formulation can be understood in the present framework.
Without G and with β = 0, we would have the higher spin BRST-operator Q hs Q hs = c G + c + G + + c − G − + 2 c + c − b . (5.1)– 6 –or ghost and vacuum conventions, see section 8. The fields and gauge parameters areexpanded as | Φ i = ( A + F c + b − + Hb − c ) | + i , (5.2) | Θ i = θb − | + i . (5.3)Here A contains the symmetric integer spin tensors, H are auxiliary fields and F containsthe traces of the fields in A upon imposing the trace constraint. All fields are expandedover the oscillator basis. For the complete field and anti-field BV-complex, the reader isreferred to [14]. Field equations and gauge transformations are given by Q hs | Φ i = 0 , (5.4) δ | Φ i = Q hs | Θ i . (5.5)The content of the field equations can be extracted as (cid:0) G A + G − H (cid:1) |−i = 0 , (5.6) (cid:0) G F + G + H (cid:1) c + b − |−i = 0 , (5.7) (cid:0) G + A − G − F − H (cid:1) c + | + i = 0 . (5.8)These equations are sometimes called the ”triplet” equations [6, 7]. The last equationcan be solved algebraically for the auxiliary field H . Then applying the trace constraint (cid:0) T + b + c − (cid:1) | Φ i = 0, traces of the A fields are related to the F fields through A = T F . Thestandard Fronsdal equations result for the integer spin field components in A . Indeed, ifall the fields are expanded over the oscillator basis, generically as φ = φ + φ µ α † µ + φ µν α † µ α † ν + · · · , (5.9)and the shorthand notation A ( s ) for a gauge field A µ µ ...µ s with s symmetrized indices isused, equations (5.6) and (5.8) yield p A ( s ) − p (1 p · A s ) + p (1 p F s − = 0 , (5.10)with p = − i∂ . The trace constraint finally gives F ( s − = A ′ ( s ) (where the prime denotes atrace), thus recovering the Fronsdal equations. However, the field equation (5.10) for thedoublet ( A, F ) is gauge invariant even without imposing the trace constraint, and thus isunconstrained.It can be noted that this is all true even if β = 0 in the constraints G + and G − . Anon-zero β would give level, i.e. spin, mixing, a phenomena that will also result in thecontinuous spin case. With this background, let us return to the four-constraint theory. Since the fourth con-straint is Hermitean, the fourth ghost pair ( c , b ) must also be Hermitean. Then thecontinuous spin BRST-operator Q cs is Q cs = Q hs + c G − c ( c + + c − )( b + − b − ) . (5.11)– 7 –he slightly more general form is Q gen = Q hs + c G + σ + c ( b − c − − b + c + ) + σ − c ( b + c − − b − c + ) . (5.12)The interpretation of the extra terms is clear. When σ + = 1 and σ − = 0, the extra termis a ghost number operator. On the other hand, when σ + = 0 and σ − = 1, the extra termis a ghost trace operator.The presence of a second Hermitean ghost pair leads to a second vacuum degeneracyabove the degeneracy between | + i and |−i . The vacuum structure becomes somewhatmore complicated. Using the notation | ± ± i so that the first entry refers to the 0 -ghostsand the second to the 3 -ghosts, the vacuum complex is given in Table 2.0 0 ↑ b ↑ b ←− b | + + i −→ c ←− b | − + i −→ c c ↓ ↑ b c ↓ ↑ b ←− b | + −i −→ c ←− b | − −i −→ c c ↓ c ↓ Table 2 . Doubly degenerate vacuum complex.
This complex is built from the generic principle0 ←− b | + i −→ c ←− b |−i −→ c . (5.13)As for Grassmann parities, it is consistent to chose | + + i and | − −i to have parity 0 and | + −i and | − + i to have parity 1. The full ghost complex is given in section 8 in Table 4.The presence of the two Hermitean ghost pairs makes it difficult, perhaps impossible, toset up a BRST Lagrangian of the type h Φ | Q | Φ i (or alternatively a BV master action S ) inany simple way [35]. Therefore we will recourse to field equations for the time being . Thephysical fields reside in the (mechanical) ghost number − − | Φ i = ( A + F c + b − + Hb − c + Bb − c ) | + + i , (5.14) | Θ i = θb − | + + i , (5.15)where B is the new field corresponding to the trace constraint. This question is further discussed in section 5.6. – 8 –he field equations following from Q cs | Φ i = 0 now become (cid:0) G A + G − H (cid:1) | − + i = 0 , (5.16) (cid:0) G F + G + H (cid:1) c + b − | − + i = 0 , (5.17) (cid:0) G + A − G − F − H (cid:1) c + | + + i = 0 , (5.18) (cid:0) G B − G H (cid:1) b − | − −i = 0 , (5.19) (cid:0) G + B + G F + A − F (cid:1) c + b − | + −i = 0 , (5.20) (cid:0) G − B + G A + A − F (cid:1) | + −i = 0 . (5.21)Here, the first three equations are the same as in higher spin theory. The last two equationscan also be written more generally, corresponding to (4.8) and (4.9), as (cid:0) G + B + G F + σ − A + σ + F (cid:1) c + b − | + −i = 0 , (5.22) (cid:0) G − B + G A − σ + A − σ − F (cid:1) | + −i = 0 . (5.23)Working with these equations, we can compare different constraint structures. Since the starting point for this investigation was the puzzle of the tracelessness con-straints, let us begin with how they come about in the present theory by analyzing thefield equations. From now on the ghosts and the vacua are dropped from the notation butall equations should be thought of as acting on a vacuum | vac i for the bosonic oscillators.Using the field equation (5.23), we can solve for the field F in the form σ − F = (cid:0) G − σ + (cid:1) A + G − B, (5.24)and since G computes the trace, among other things, this equation will allow us to derivea variant of the higher spin equations. Writing G = T + ˜ G where ˜ G = T † − N + ω ,equation (5.24) allows us to write the higher-spin similar field equation as p A − G − G + A + 1 σ − G − G − T A = − σ − G − G − (cid:0) T † − N + ω − σ + (cid:1) A − σ − G − G − G − B. (5.25)The left hand side corresponds to Fronsdal’s equations, although due to the special formof the CS constraints G − = p · α † − β and G + = p · α − β there are extra contributions, oforder β and β mixing fields from various excitation levels. For instance, the field equationfor the spin 0 field A (0) will contain terms ∼ βp · A (1) and ∼ β A ′ (2) . This was found inthe recent paper [17]. The phenomena repeats itself on the spin 1 level where there occurderivatives of the spin 2 field and the trace of the spin 3 field. A theory with four constraints was studied in the paper [36]. These authors treat the second classconstraints in a different way. – 9 –he right hand side of (5.25) provides source terms. It is zero for spin 1. For the spin2 field A , it provides a source term of the form − ω − σ + σ − p µ p ν A . (5.26)This term is not present in the HS T theory (see Table 1). The last term on the right handside starts to contribute source terms for the spin 3 field A (3) where it produces a termof the form p µ p µ p µ B . The term G − G − G − B corresponds to the compensator field of[7, 37] first appearing at the spin 3 level. For more comments on this, see section 5.6.However, equation (5.22) have a very similar structure and it allows us to write σ − A = − (cid:0) G + σ + (cid:1) F − G + B. (5.27)The structure is thus quite complex and working spin level by spin level soon becomesconfusing. In order to analyze the full content of the equations let us arrange them in threegroups. Dynamical equations G A + G − H = 0 , (5.28) G F + G + H = 0 , (5.29) G B − G H = 0 , (5.30) Auxiliary equation H = 12 (cid:0) G + A − G − F (cid:1) , (5.31) Trace equations σ − F = (cid:0) G − σ + (cid:1) A + G − B, (5.32) σ − A = − (cid:0) G + σ + (cid:1) F − G + B. (5.33)In principle, we do not expect to have six independent field equations for four differentfields, so there must be redundancies in these equations. Since the trace equations arefirst order in derivatives they should really be counted as one equation. This leaves oneequation to many. In section 5.5 we will see that equation (5.30) can either be seen as atrivial identity or as a consequence of the trace equations. The effective number of fieldequations are therefore just four. With the field and gauge parameter expanded over the ghost complex as in (5.14) and(5.15) the gauge transformations become δA = G − θ, (5.34) δF = G + θ, (5.35) δH = − G θ, (5.36) δB = − G θ. (5.37)– 10 –he invariance of the field equations (5.28) to (5.33) are direct consequences of the gaugeconstraint algebra. The structure of the field equations suggest two formal ways of reducing them.
Solving for A and F The two fields A and F are coupled through the auxiliary field H . Inserting H from (5.31)into (5.28) and (5.29) and collecting the fields into a vector, we get G " AF + 12 " G − G + − G − G − G + G + − G + G − AF = 0 . (5.38)Then the trace equations (5.32) and (5.33) can be written in matrix form as T " AF = − " G − G + B, (5.39)where the matrix T is T = " G − σ + − σ − σ − G + σ + . (5.40)It is formally invertible with inverse T − = 1 σ − − σ + G " G + σ + σ − − σ − G − σ + . (5.41)This means that the doublet of fields ( A, F ) is expressible in terms of the B field as " AF = − T − " G − G + B. (5.42)The inverse T − is well defined in all cases. For instance in the HS T case we have T − T = (1 − G + G − · · · ) " G − G . (5.43)There is no issue of convergence since these operators act on fields B | vac i and for any finiteexcitation level (spin level) only a finite number of terms contribute. Let us however focuson the CS case. Then we get T − = 1 G " G − − G + 1 . (5.44)At least as long as ω = 0 we can expand 1 /G in a formal power series.– 11 –ombining field equations (5.30) and (5.31), the equation for B can be written as G B − G h G + , − G − i " AF = 0 . (5.45)Then using (5.42) we get G B + 12 G h G + , − G − i T − " G − G + B = 0 . (5.46)Thus it seems that we have reduced the content of the field equations to the equation(5.46) for the independent field B , the dependent field doublet ( A, F ) being determined byequation (5.39). However, equation (5.46) is void of content. Using the gauge algebra, itcan be shown that for CS 12 G h G + , − G − i T − " G − − G + = − G , (5.47)as an identity. Solving for B Alternatively, we can view the trace equation (5.39) as expressing G − B and G + B in termsof A and F . Then multiplying the G − B equation by G + and the G + B equation by G − and subtracting gives (cid:0) G + G − − G − G + (cid:1) B = G (cid:0) G − F − G + A (cid:1) , (5.48)which is precisely the field equation (5.30) with H substituted for through (5.31). Thisshows that the set of field equations are compatible , but it also shows that the field B should properly be regarded as a redundant field, although it seems to have a dynamicalfield equation. Its field equation is however not independent, but follows from the traceequations for the doublet ( A, F ) and the equation for H .It is indeed possible to gauge B to anything (but not zero). Let us see how this worksout on the component level. Expand the B field as in (5.9) and the gauge parameter as θ = θ + θ µ α † µ + θ µν α † µ α † ν + · · · , (5.49)where the indexing θ s indicates to which primary gauge field A ( s ) the parameter belongs.Then we get for the first few levels δB = − θ ′ + (2 − ω ) θ ,δB µ = − θ ′ + (3 − ω ) θ µ ,δB µν = − θ ′ µν + (4 − ω ) θ µν − η µν θ . Which they must be, derived as they are from a nilpotent BRST-operator based on a closed first classalgebra. – 12 –hat is important here is that the scalar component B of B can be gauged to anythingwithout using up any of the freedom of the spin 1 gauge parameter θ . What is used isthe trace part of the spin 3 gauge parameter θ , i.e. what is otherwise set to zero in aconstrained formulation. A similar argument holds for B µ and so on. We can also make contact with the work of Francia and Sagnotti on unconstrained formu-lations of the higher spin field equations. For that purpose, consider the generalized casewith G = σ − (cid:0) T + T † (cid:1) + σ + N + ω . In their paper [37] they quote the ”local non-Lagrangiancompensator equations” F µ ...µ s = 3 ∂ µ ∂ µ ∂ µ α µ ...µ s + . . . , (5.50) ϕ ρσρσµ ...µ s = 4 ∂ · α µ ...µ s + ( ∂ µ α ρρµ ...µ s + . . . ) , (5.51)where F is a field equation for the higher spin field ϕ and α is the compensator field.As we saw in equation (5.25), a compensator-like term G − G − G − B is produced whenthe field F is substituted for in the field equation for the field A . Our equation (5.25)would be a generalization of (5.50). To find the generalization of (5.51), consider the traceequations again. Substituting F from (5.32) into (5.33) we get σ − A = − ( G + σ + ) (cid:0) ( G − σ + ) A + G − B (cid:1) − σ − G + B = − ( G − σ ) A − ( G + σ + ) G − B − σ − G + B. Using the constraint algebra we get( σ − − σ ) A = − G A − G − G B − σ + G − B − σ − G + B, and precisely in the CS case when σ − − σ = 0 we get G A = − G − G B + 2 G − B − G + B. (5.52)This equation generalizes equation (5.51). As already remarked, it seems not possible to construct an action of the h Φ | Q | Φ i form forthis type of four-constraint theory. One natural attempt would be to try h Φ | ( c ± c ) Q | Φ i as this has the correct ghost number. It does not produce the correct field equations.On the level of components this shows up as a (most likely unavoidable) mixing of thedynamic equations and the trace equations. It does however suggest a possible actionwith Lagrange multiplier fields | Λ i| Λ i = ( e A + e F c + b − + e Hb − c + e Bb − c (cid:1) | + + i , (5.53) The question needs more thought though. – 13 –here we denote the components of the Lagrange multiplier fields by tildes. We take asour tentative action A = − h Φ | c Q cs | Φ i + ǫ h Φ | ( c Q cs − Q cs c ) | Λ i , (5.54)with ǫ parameterizing the weighting of the second term in relation to the first. The formof the second term is needed for reality since c and Q does not anti-commute. This formalso ensures gauge invariance (the field Λ does not vary under gauge transformations).Varying with respect to | Φ i then produces c Q cs | Φ i − ǫ ( c Q cs − c + c − ) | Λ i = 0 . (5.55)where the term with c + c − is picked up when c is anti-commuted through Q cs . Due to theghost-vacuum structure, the equation breaks up into four components δ/δA ⇒ G A + G − H = ǫ (cid:0) ( G − σ + ) e A − σ − e F + G − e B (cid:1) , (5.56) δ/δF ⇒ G F + G + H = ǫ (cid:0) ( G + σ + ) e F + σ − e A + G + e B (cid:1) , (5.57) δ/δH ⇒ G + A − G − F − H = − ǫ e B, (5.58) δ/δB ⇒ G + e A − G − e F − e H = 0 . (5.59)where it is indicated from which variations of components the equations follow if the com-ponent action is worked out from (5.54). Then varying with respect to | Λ i produces( c Q cs − c + c − ) | Φ i = 0 . (5.60)Working out the components of this equation yield δ/δ e A ⇒ ( G − σ + ) A + G − B − σ − F = 0 , (5.61) δ/δ e F ⇒ ( G + σ + ) F + G + B + σ − A = 0 , (5.62) δ/δ e H ⇒ B = 0 , (5.63) δ/δ e B ⇒ G + A − G − F − H = 0 . (5.64)The system is clearly over-constrained in that equations (5.63) and (5.64) together with(5.58) implies B = 0 and H = ǫ e B . This defect can be remedied by choosing a Lagrangemultiplier | Λ i with e H = e B = 0, effectively removing the last two field equations.Then defining the Fronsdal matrix operator F = " G G + 12 " G − G + − G − G − G + G + − G + G − , (5.65)the field equations (5.56)-(5.58) can be reduced to F " AF = ǫ T " e A e F (5.66) Spacetime integrations are implicit in all formulas involving actions. – 14 –f the remaining field equations, (5.61) and (5.63) can be written (see (5.39)) T " AF = − " G − G + B. (5.67)It is easy check that the operator matrices F and T commute. Thus applying T onboth sides of equation (5.66) yields FT " AF = ǫ T " e A e F (5.68)and using (5.67) we get − F " G − G + B = ǫ T " e A e F (5.69)However, since the left hand side of this equation is identically zero, we get T " e A e F = 0 (5.70)On-shell we can chose for a solution e A = e F = 0 or at least T " e A e F = 0 (5.71)thus reproducing the contiuous spin field equations of section 5.3. There is a nice physical rationale for the constraint structure. Think of a mechanical systemconsisting of two point particles with coordinates t µ and b µ and corresponding canonicalmomenta u µ , d µ . These phase space variables can be called ”end-point” variables. Thecenter of motion x µ and relative coordinates ξ µ are defined by x µ = 12 ( t µ + b µ ) , ξ µ = 12 ( t µ − b µ ) . (6.1)The canonical conjugate momenta p µ , π µ are p µ = u µ + d µ , π µ = u µ − d µ . (6.2)Consider the two end-points to be moving with the velocity of light so that the relevantconstraints are u ≈ d ≈ . (6.3)A further natural constraint is to require the center of motion momentum p to have aconstant Lorentz-scalar product with the relative coordinate ξ so that ξ · p ≈ β. (6.4)– 15 –he last constraint is requiring the Lorentz-scalar product of the end-point momenta u and d to be constant u · d ≈ ω . (6.5)This last constraint corresponds to G . It is easy to see that these constraints can belinearly recombined into the constraints used in this paper. However, a little bit more can be said, in that the fourth constraint cannot be avoided,it must be included. This is because the three constraints in equations (6.3) and (6.4) donot form a first class algebra by themselves. One way of seeing this is to linearly recombinethe constraints in (6.3) into2( u + d ) = p + π ≈ , u − d = π · p ≈ . (6.6)Then, working with Poisson brackets we get { p + π , ξ · p } = − π · p , { π · p, ξ · p } = − p , (6.7)which is not first class since we do not have p ≈
0. However, requiring u · d = ( p − π ) ≈ p constraint. Again we have an algebra with four first classconstraints.This leads to a puzzle regarding the conventional three-constraint higher spin theory.What does it correspond to in terms of end-point coordinates and momenta? Transcribingthe set { p , α · p , α † · p } first to { p , ξ · p , π · p } and then to end-point variables, we getthe constraints ( t − b ) · ( u + d ) ≈ , u + u · d ≈ , d + u · d ≈ . (6.8)The interpretation is that in conventional higher spin theory, the end-points of the under-lying mechanical model do not move with the velocity of light. To recover that, the traceconstraint u · d must again included. What’s even more puzzling, the operator u · d plays astill not fully understood role in the interacting theory. It looks like it provides the internaldynamics of the mechanical system. Already the triplet BRST-theory of free higher spin gauge fields is free from tracelessnessconstraints, and thus ”unconstrained”. But it is so at the price of having at least a doubletof fields (
A, F ) where F propagates lower spin unphysical components. The fields A and F are independent but coupled through the field equations. The objectives of unconstrainedformulations are to minimize the number of extra fields needed. The fields F are substitutedfor by trace-like equations at the price of introducing Lagrange multiplier fields into theaction. It seems from the present work that such formulations can be understood, in aBRST framework, as arising from various four-constraint theories. Somewhat surprisingly If one does not want to consider the CS case, the right hand sides of the equations (6.4) and (6.5) canbe set to zero. – 16 –t seems that a four-constraint theory modeling the continuous spin representations doesprecisely this, at least in the special case with β = ω = 0 where there is no level mixing.But clearly, more work is needed to sort out the details.Furthermore, as argued above, there seems to be an underlying physical rationalefor the trace constraints. When expressed at the level of Poincar´e covariant fields, it ismanifested not through conventional gauge invariance, but instead through these awkwardtracelessness constraints. But rather than being just an inconvenience, this very fact mayhint at a dynamical principle behind the interactions, based not on the harmonic oscillatorequation π + ξ = 0, but on the u · d equation. The ghost structure is well known but let us fix notation. Corresponding to the repa-rameterization constraint G we have the pair of Hermitean ghosts ( c , b ) with anti-commutator { c , b } = 1. Their presence leads to a degenerate vacuum with |−i = c | + i and | + i = b |−i . On the other hand, the gauge constraints G + and G − go with theconjugate ghost pairs ( c + , b + ) and ( c − , b − ). Their essential properties are summarized in( c − ) † = c + , ( b − ) † = b + and the anti-commutators { c + , b + } = 1, { c − , b − } = 1.The full ghost complex that results is given in table 3, where gh m denotes the (me-chanical) ghost number. A | + i vacuum is given ghost number − / c -ghosts and b -ghosts have ghost numbers 1 and − b − and c + operators are creatorswhile b + and c − are annihilators.gh m ( · ) 3/2 1/2 -1/2 -3/2 c | + i | + i c + c | + i c + | + i b − c | + i b − | + i c + b − c | + i c + b − | + i Table 3 . Higher spin ghost complex
The continuous spin ghost complex with the doubly degenerate vacuum structure canthen be constructed as in table 4.gh m ( · ) 2 1 0 -1 -2 | − −i | + −i , | − + i | + + i c + | − −i c + | − + i b − | − −i b − | − + i b − | + + i c + | + −i c + b − | − + i b − | + −i c + b − | − −i c + b − | + −i c + b − | + + i c + | + + i Table 4 . Continuous spin ghost complex
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