Hamiltonian constraints and unfree gauge symmetry
aa r X i v : . [ h e p - t h ] S e p HAMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY
V.A.ABAKUMOVA, S.L. LYAKHOVICH
Abstract.
We study Hamiltonian form of unfree gauge symmetry where the gauge parametershave to obey differential equations. We consider the general case such that the Dirac-Bergmannalgorithm does not necessarily terminate at secondary constraints, and tertiary and higher orderconstraints may arise. Given the involution relations for the first-class constraints of all gener-ations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form,including the differential equations constraining gauge parameters. All the field theories with un-free gauge symmetry share the common feature: they admit sort of “global constants of motion”such that do not depend on the local degrees of freedom. The simplest example is the cosmo-logical constant in the unimodular gravity. We consider these constants as modular parametersrather than conserved quantities. We provide a systematic way of identifying all the modularparameters. We demonstrate that the modular parameters contribute to the Hamiltonian con-straints, while they are not explicitly involved in the action. The Hamiltonian analysis of theunfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, includingexplicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRSTHamiltonian quantization method for the case of unfree gauge symmetry. The main distinctionis in the content of the non-minimal sector and gauge fixing procedure. The general formalismis exemplified by traceless tensor fields of irreducible spin s with the gauge symmetry parametersobeying transversality equations. Introduction
Gauge symmetry is usually understood as a set of the infinitesimal transformations of the fieldssuch that leaves the action intact, while the transformation parameters are the functions of space-time. Gauge symmetry is said unfree if the invariance of the action requires the gauge parametersto obey the PDE system. The general solution of the equations constraining gauge parametersmust involve arbitrary functions of all d space-time coordinates. If the solution includes arbitraryfunctions of d − free higher spin field theories, with gauge parameters constrained by transversality equations[3], [4]. The key distinction of UG from General Relativity (GR) with Λ-term is that Λ is aspecific constant fixed from the outset in the action of GR, while UG comprises dynamics withany cosmological constant. For discussion of the role of cosmological constant in UG and furtherreferences, we cite [5]. Also modifications of UG can be found in [6], [7], where Λ is defineddynamically, not as pre-fixed parameter in the action. All the field theories with unfree gaugesymmetry share the common feature: they admit the “global constants of motion” such that donot depend on the local degrees of freedom, with Λ of the UG being the simplest example. Thisgeneral fact is explained from various viewpoints in the recent articles [8], [9], [10]. As the specificvalues of these integration constants are defined by the field asymptotics, not the Cauchy data, weconsider them as modular parameters rather than conserved quantities. In the higher spin fieldanalogues of UG, for example, similar modular parameters exist, and their number grows withspin, although this fact has not previously been noticed.While the examples of unfree gauge symmetry have been known for a long time, the generaltheory of this class of gauge systems began to develop relatively recently. In the article [8], gen-eral structure is established for unfree gauge symmetry algebra in Lagrangian formalism, and themodification is proposed for the Faddeev-Popov (FP) method such that accounts for the con-straints imposed on gauge parameters. In the article [9], the BV-BRST field-antifield formalismis worked out for the systems with unfree gauge symmetry. In the article [10], general structuresare identified in the algebra of Hamiltonian constraints such that describe unfree gauge symmetry.Before this work, the equations constraining gauge parameters in Hamiltonian formalism havebeen unknown even in specific models. The article [10] assumes that the Dirac-Bergmann algo-rithm terminates at secondary constraints, no tertiary ones are allowed. In this work, we providethe Hamiltonian description of unfree gauge symmetry in the general case, with the sequence ofconstraints of any finite order. Besides the reason of generality, this is also motivated by specificmodels. While in UG the Dirac-Bergmann algorithm terminates at the stage of secondary con-straints, in the higher spin field theories with unfree gauge symmetry, the sequence of constraintsturns out linearly growing with spin, so the tertiary constraints arise for s = 3. The number ofmodular parameters is also growing with spin, and they all contribute to the constraints. The newphenomenon here is that the modular parameters, being connected to the non-trivial asymptotics Batalin-Vilkovisky–Becchi-Rouet-Stora-Tyutin.
AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 3 of the fields, can make the constraints explicitly depending on the space-time point x , even thoughthe original Lagrangian is x -independent. This phenomenon has previously unnoticed analoguein Lagrangian formalism.The main goal of this article is to work out Hamiltonian description of general unfree gaugesymmetry. Then, we also extend the BFV–BRST formalism to this class of theories, with mainmodifications related to the non-minimal sector of ghosts. The general formalism is exemplifiedby the massless spin- s theory where the irreducible representation is realized by traceless tensors[3]. To make the article self-contained we precede the Hamiltonian description of unfree gaugesymmetry with the corresponding Lagrangian formalism mostly providing the facts from [8], [9],with a more emphasis on modular parameters. We also provide a convenient formula for thedegree of freedom counting in Lagrangian formalism in the case of unfree gauge symmetry.2. Unfree gauge symmetry in Lagrangian formalism:completion functions, and modular parameters
Unfree gauge symmetry is a deviation from the usual assumptions implied by general theory ofgauge systems as it is formulated in the textbooks, see for example [11]. This deviation has animpact on basic statements of gauge theory. Notice the second Noether theorem, which connectsgauge symmetry of the action with Noether identities between Lagrangian equations. We canmention two assumptions implied by the theorem: (i) the gauge parameters are arbitrary functionsof x ; (ii) any on-shell vanishing local quantity reduces to a linear combination of the l.h.s. ofLagrangian equations and their derivatives. The first assumption is obviously invalid once thesymmetry is unfree. The second one is also inevitably violated for the case of unfree gaugesymmetry as it is explained in the articles [8], [9]. Let us rephrase the violation of the secondassumption: the local quantities τ a exist such that vanish on-shell, while they cannot be expandedin the l.h.s. of Lagrangian equations with local coefficients: ∃ τ a ( φ ) : τ a ≈ , τ a ( φ ) = K ia ( φ ) ∂ i S . (1)Here, we use the condensed notation. The condensed indices a, i include space-time point x anddiscrete labels. Summation over condensed indices includes integration over space-time, ∂ i S ( φ ) Batalin-Fradkin-Vilkovisky–Becchi-Rouet-Stora-Tyutin. By local quantity we mean the function of space-time coordinates, fields, and their derivatives of finite order.
V.A.ABAKUMOVA, S.L. LYAKHOVICH is a variational derivative of the action S ( φ ) by the field φ i , and the symbol ≈ means on-shellequality. So, violation of (ii) means that ideal I of on-shell vanishing local quantities is not spannedby the l.h.s. of Lagrangian equations ∂ i S = 0. The local quantities τ ∈ I (1) are called completionfunctions . The generating set of ideal I includes l.h.s. of Lagrangian equations and a number ofcompletion functions. In slightly different wording, any on-shell vanishing local quantity T ( φ ) isspanned off-shell by field equations and completion functions with the local expansion coefficients: T ( φ ) ≈ ⇔ T ( φ ) = T i ( φ ) ∂ i S ( φ ) + T a ( φ ) τ a ( φ ) . (2)The identities can exist between the Lagrangian equations and completion functions,Γ iα ( φ ) ∂ i S ( φ ) + Γ aα ( φ ) τ a ( φ ) ≡ , (3)where all the coefficients Γ( φ ) are local. These relations can be understood as modification ofthe usual Noether identities for the case when the theory admits completion functions. Upon notquite restrictive regularity assumptions (see in [8], [9]), the operators Γ aα ( φ ), being the coefficientsat completion functions, can admit at maximum a finite dimensional kernel:Γ aα ( φ ) u a = 0 ⇒ u a ∈ M = Ker Γ aα , dim M = n ∈ N . (4)The kernel M is understood as a moduli space of the field theory. Elements of M are parameterizedby finite number of constant parameters Λ. Being parameterized by constants, the elements of M can explicitly depend on the space-time point x . From the viewpoint of modified Noetheridentities (3), the completion functions τ a are defined modulo the kernel M (4). Specific elementof the kernel is defined by the asymptotics of the fields, as τ a should vanish on-shell everywhere,including boundary. From this perspective, the existence of completion functions (1) can beconsidered as a consequence of modified Noether identities (3) rather than a cause. Once thekernel of Γ aα is finite, the identities (3) mean that the local quantities τ a reduce on-shell to aspecific Λ-dependent function of x . This function can be subtracted from τ , so the completionfunctions vanish on-shell. On the other hand, Γ aα is a differential operator, and it does not haveinverse in the class of differential operators, as the kernel exists. Once Γ aα is not locally invertible,completion function τ a ( φ ), being on-shell vanishing local quantity, cannot be expressed from theidentities (3) as a linear combination of Lagrangian equations with local coefficients. In this sense,the identities (3) lead to existence of completion functions (1). AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 5
Be modified Noether identity (3) a consequence of existence of completion functions (1), or viceversa, anyway, it means that the action S ( φ ) enjoys unfree gauge symmetry. The unfree gaugetransformation is defined by the coefficients Γ iα of the identities (3), δ ε φ i = Γ iα ( φ ) ε α , (5)while the operators Γ aα define the equations constraining gauge parameters:Γ aα ( φ ) ε α = 0 . (6)Let us mention the terminology: operators Γ iα , being the coefficients at Lagrangian equations inmodified Noether identities (3), are understood as unfree gauge symmetry generators, while Γ aα ,being the coefficients at the completion functions in (3), are considered as operators of gaugeparameter constraints. Given the identities (3), the transformation (5) leaves the action intactoff-shell once the parameters obey conditions (6): δ ε S ( φ ) ≡ ∂ i S ( φ )Γ iα ε α ≡ − τ a Γ aα ε α = 0 . (7)In this way, we see that unfree gauge symmetry is a consequence of modified Noether identities(3). Proceeding from this observation, we can find the Hamiltinian counterpart of the unfree gaugesymmetry. It is sufficient to find the modified Noether identities (3) for the Hamiltonian equationswith constraints, and the equations for gauge parameters (6) are immediately identified. This isdone in the next section.Let us briefly explain the modification of the Faddeev-Popov (FP) ansatz needed to accountfor the unfree gauge symmetry. The modification is proposed in reference [8], where one can finda more detailed exposition of the method. In the section 4, we deduce this modified ansatz fromthe BFV-BRST formalism.The ghosts assigned to the unfree gauge transformations (5) are assumed to obey equationsΓ aα ( ϕ ) C α = 0 , gh ( C α ) = 1 , ǫ ( C α ) = 1 , (8)where Γ aα ( ϕ ) are the operators of gauge parameter constraints (6). Let us impose independentgauges χ I ( φ ). The index I is condensed, so it includes the space coordinates x µ . The dimension ofdigital part of the index should be equal to the number of unconstrained gauge parameters . Once In the next section, we explain the number of gauge conditions from the Hamiltonian perspective.
V.A.ABAKUMOVA, S.L. LYAKHOVICH we use independent gauge-fixing conditions, the number of unfree gauge parameters will exceedthe number of gauges, so FP matrix will be rectangular, δ ε χ I δε α = Γ iα ( φ ) ∂ i χ I ( φ ) . (9)Given the admissible gauge fixing conditions, the anti-ghosts¯ C I , gh (cid:0) ¯ C I (cid:1) = − , ǫ (cid:0) ¯ C I (cid:1) = 1 (10)are assigned to χ I ( φ ). The FP ansatz for path integral is adjusted to the case of unfree gaugesymmetry in the following way: Z = Z [ d Φ] exp n i ~ S F P ( ϕ ) o , Φ = { φ i , π I , C α , ¯ C I , ¯ C a } , (11)gh (cid:0) ¯ C a (cid:1) = − , ǫ (cid:0) ¯ C I (cid:1) = 1 ; gh ( π I ) = ǫ ( π I ) = 0 , (12)where the FP action reads S F P = S ( φ ) + π I χ I ( φ ) + ¯ C I Γ iα ( φ ) ∂ i χ I ( φ ) C α + ¯ C a Γ aα ( φ ) C α . (13)The Fourier multipliers ¯ C a to the ghost constraints Γ aα ( φ ) C α = 0 can be considered as anti-ghosts,on equal footing with the anti-ghosts ¯ C I assigned to the gauge-fixing conditions χ I ( φ ). In thesection 4, we shall see that these anti-ghosts naturally arise from the Hamiltonian BFV-BRSTformalism.Let us exemplify the above generalities about unfree gauge symmetry by the case of UG. Con-sider the unimodular metrics g µν ( x ) , det g = −
1, in d = 4. The usual explanations of gaugesymmetry in UG proceed from the idea that the symmetry is a diffeomorphism consistent withunimodularity condition. This imposes the transversality equation on the parameter. We goanother way, following the procedure above, and we shall see the same result.Lagrangian equations of UG read: δS [ g ] δg µν ≡ R µν − g µν R ≈ , S = Z d x R . (14)Taking divergence of the equations, and making use of Bianchi identity, we get ∇ ν δS [ g ] δg µν ≡ ∇ µ R ≈ . (15) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 7
Unlike GR, the divergence of the field equations does not identically vanish. Once ∂ µ R ≈
0, thescalar curvature is an on-shell constant, R ≈ Λ = const , where specific value of Λ is defined byasymptotics of g µν . So we have the modular parameter Λ, and completion function τ ≡ R − Λ ≈ τ cannot be represented as linear combination of equations (14) and their derivatives,so it is a completion function indeed. Then, we get modified Noether identities (3) for UG: ∇ ν δS [ g ] δg µν − ∇ µ τ ≡ . (16)This allows us to identify the unfree gauge symmetry transformations (5), and the gauge parameterconstraints (6): δ ε g µν = ∇ µ ε ν + ∇ ν ε µ , ∇ µ ε µ = 0 . (17)We can also mention one more example of completion function noticed in literature concerningMaxwell-like higher spin field theory [4]. In this theory, the double divergence of the tracefullsecond-rank tensor vanishes on-shell, ∂ µ ∂ ν ϕ µν ≈
0, while it does not reduce to the l.h.s. of thefield equations and their derivatives. This fact is emphasized in the article [12].In the end of this section, we provide, without proof, a receipt for covariant degree of freedom(DoF) counting in the theories with unfree gauge symmetry. In so doing, we assume that theLagrangian equations are involutive in the sense that they do not admit lower order differential consequences. The receipt can be deduced along the same lines as explained in the article [13] forthe gauge theories without constraints on gauge parameters.DoF number is calculated as follows: N DoF = n e o e − n s o s − n i o i + n c o c , (18)where n e , n s , n i . n c are the numbers, and o e , o s , o i , o c are the orders of Lagrangian equations ∂ i S = 0, gauge symmetry transformations δ ε ϕ i = Γ iα ε α , gauge identities Γ iα ∂ i S + Γ aα τ a = 0, andconstraints Γ aα ε α = 0, respectively. The order o e is defined by the highest order derivative inEoMs, o s is the order of gauge symmetry differential operator. The order of gauge identity, o i , isa sum of o s and o e , and o c is a sum of the order of constraint operator Γ aα and o s .Let us exemplify the DoF number count (18) by the case of UG in d = 4. We have nine equationsof the second order (14), n e = 9, o e = 2. There are four gauge symmetry transformations of thefirst order, and one first-order equation imposed on the gauge parameters (17), so n s = 4 , o s = V.A.ABAKUMOVA, S.L. LYAKHOVICH , n c = 1 , o c = 1 + 1 = 2. There exist four gauge identities (16), n i = 4, of the third order( o i = 1 + 2 = 3). So, according to (18), UG has four degrees of freedom by phase-space count,which corresponds to two “Lagrangian” DoF.3. Constrained Hamiltonian formalism:higher order constraints, modular parameters, and unfree gauge symmetry.
Any action functional can be brought to equivalent Hamiltonian form with primary constraints: S = Z dt (cid:0) p i ˙ q i − H T ( q, p, λ ) (cid:1) , H T ( q, p, λ ) = H ( q, p ) + λ α (1) T α ( q, p ) , (19)where q i , p i are canonical variables, and λ α are Lagrange multipliers. All these variables can beviewed as the fields φ = ( q, p, λ ), and then we can apply the general consideration of the previoussection to the action (19). As explained in the previous section, the unfree gauge symmetry (5),(6) is caused by modified Noether identities (3) which involve, besides the original Lagrangianequations ∂ i S ( φ ) = 0 and gauge generators Γ iα two more ingredients: completion functions τ a ( φ )and operators of gauge parameter constraints Γ aα . The key point in finding the unfree gaugesymmetry of any action functional is to find a modified Noether identities (3) involving the operatorΓ aα with a finite kernel (4). Once the identities are found, the coefficients at the equations definethe gauge generators, while the operators Γ aα give the equations imposed on the gauge parameters.Hamiltonian action (19), due to the canonical structure, is very convenient for algorithmicallydeducing modified Noether identities (3). The idea is quite simple: we apply the Dirac-Bergmannalgorithm of iterating constraints. We assume that no Lagrange multiplier is fixed, so all theconstraints are first-class. In the local field theory, the algorithm should terminate in a finitenumber of iterations. Termination of the algorithm is a (modified) Noether identity. Once themodified Noether identities (3) are established, one can find the gauge transformation for thefields φ = ( q, p, λ ) by identifying the coefficients at the corresponding equations, while the gaugeparameter constraints are defined by the coefficient at the completion functions in the identity. Asone can guess, the roles of completion functions are plaid in Hamiltonian formalism by secondaryconstraints of all generations. For the case when the sequence of constraints terminates at thesecondary constraints, without tertiary and higher order ones, this program has been alreadyimplemented in the article [10]. Here we consider the general case. When the secondary constraintslead to the higher order ones, and the involution coefficients include differential operators with AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 9 finite kernel, this can lead, in general, to explicit dependence of kernel elements on space-timecoordinates x . Through this mechanism, the explicit time dependence can arise in the higherorder constraints even if the original action is translation-invariant. The explicit x -dependence ofsecondary constraints is due to the field asymptotics which is defined by modular parameters.Let us consider iteration of secondary constraints to deduce Hamiltonian form of identity (3),and get in this way the unfree gauge symmetry (5), (6) for Hamiltonian action (19). EoM’s read: δSδp i ≡ ˙ q i − { q i , H T ( q, p, λ ) } = 0 ,δSδq i ≡ − ˙ p i + { p i , H T ( q, p, λ ) } = 0 ; (20) δSδλ α ≡ − (1) T α ( q, p ) = 0 . (21)Following the Dirac-Bergmann algorithm, we take time derivative of primary constraints (21) andcombine it with the evolutionary equations (20) to exclude the time derivatives. The result is atmost linear in λ . As the multipliers remain indefinite, all the coefficients at λ should be consideredas on-shell vanishing, so the derivative of the primary constraints reduces to the combination ofprimary and secondary constraints: ddt (1) T α ( q, p ) = { (1) T α ( q, p ) , H T ( q, p, λ ) } = (1) V β α ( q, p, λ ) (1) T β ( q, p ) + (1) Γ β α ( q, p, λ ) (2) T β ( q, p ) . (22)Unfree gauge symmetry corresponds to the case when the structure coefficient (1) Γ β α ( q, p, λ ) isa differential operator with finite kernel (4). This includes the case of zero kernel, while noinverse exists for (1) Γ in the class of differential operators. This has been first noticed in Ref. [10],though this article assumed no higher order constraints appear. Relation (22) defines secondaryconstraints (2) T modulo kernel of (1) Γ . The kernel is parameterized by finite set of constant modularparameters Λ. The elements of the kernel can be specific Λ-dependent functions of space-timepoint x . The latter fact means that (2) T can be explicitly time-dependent, (2) T β ( q, p, Λ , t ) = T β ( q, p ) + u β (Λ , t, q, p ) , (1) Γ β α u β (Λ , t, q, p ) = 0 . (23)Further examination of the stability of the secondary constraints has to account for the possibleexplicit time-dependence which can originate from the kernel of (1) Γ . The kernel depends, in itsown turn, on the asymptotics of the fields.
Consider now the sequence of n stability conditions of constraints labeled by index k, k =2 , . . . , n . The time derivatives of secondary constraints should vanish on-shell that leads to tertiaryconstraints, etc. Stability of the l -order constraints ( l ) T leads to ( l +1) T : ddt ( l ) T α l ( q, p ) = ∂∂t ( l ) T α l ( q, p ) + { ( l ) T α l ( q, p ) , H T ( q, p, λ ) } == l X m =1 ( l ) V β m α l ( q, p, λ ) ( m ) T β m ( q, p ) + ( l ) Γ β l +1 α l ( q, p, λ ) ( l +1) T β l +1 ( q, p ) , l = 2 , . . . , n − . (24)The coefficients ( l ) Γ at the constraints of next generation ( l +1) T are the differential operators with afinite kernel. Therefore, constraints of ( l + 1)-st generation are defined modulo the kernel elementsmuch like the secondary ones (23). In general, the kernel is different for different l ’s. The algorithmterminates when no further constraints appear: ddt ( n ) T α n ( q, p ) = ∂∂t ( n ) T α n ( q, p ) + { ( n ) T α n ( q, p ) , H T ( q, p, λ ) } = n X m =1 ( n ) V β m α n ( q, p, λ ) ( m ) T β m ( q, p ) . (25)Note, that constraints ( k ) T α k , k = 2 , . . . , n, contain modular parameters defined by asymptotics ofthe field and can be explicitly time-dependent. Once Γ’s are differential operators, the secondaryconstraints of all generations (22), (24) are not differential consequences of original variationalequations (20), (21), while they vanish on-shell, so they are completion functions (1).Notice that all the structure functions V, Γ in relations (22), (24), (25) are at most linear in λ ,so it is useful to introduce separate notation for the coefficients at λ ’s and λ -independent terms: ( r ) V β s α r ( q, p, λ ) = V β s α r ( q, p ) + U β s α r γ ( q, p ) λ γ , r, s = 1 , . . . , n ; (26) ( r ) Γ β r +1 α r ( q, p, λ ) = Γ β r +1 α r ( q, p ) + U β r +1 α r γ ( q, p ) λ γ , r = 1 , . . . , n , r + 1 ≤ n . (27)Once the secondary constraints ( k ) T , k = 2 , . . . , n of all generations play the role of completionfunctions (1), the relations of the Dirac-Bergmann algorithm (22), (24), (25) can be assembledinto the modified Noether identities (3): { (1) T α , q i } δSδq i + { (1) T α , p i } δSδp i + (cid:0) δ β α ddt − (1) V β α ( q, p, λ ) (cid:1) δSδλ β + (1) Γ β α ( q, p, λ ) (2) T β ≡ AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 11 { ( l ) T α l , q i } δSδq i + { ( l ) T α l , p i } δSδp i − ( l ) V β α l ( q, p, λ ) δSδλ β + l − X m =2 ( l ) V β m α l ( q, p, λ ) ( m ) T β m ++ (cid:0) − δ β l α l ddt + ( l ) V β l α l ( q, p, λ ) (cid:1) ( l ) T β l + ( l ) Γ β l +1 α l ( q, p, λ ) ( l +1) T β l +1 ≡ , l = 2 , . . . , n − { ( n ) T α n , q i } δSδq i + { ( n ) T α n , p i } δSδp i − ( n ) V β α n ( q, p, λ ) δSδλ β ++ n − X l =2 ( n ) V β l α n ( q, p, λ ) ( l ) T β l + (cid:0) − δ β n α n ddt + ( n ) V β n α n ( q, p, λ ) (cid:1) ( n ) T β n ≡ . (30)The coefficients at the variational equations in the identities (3) define unfree gauge variations (5)of corresponding variables, while the coefficients at completion functions define the constraintsimposed on the gauge parameters (6). Given the modified Noether identities in the Hamiltonianform (28), (29), (30), with q, p, λ being the fields, and the secondary constraints ( k ) T being thecompletion functions, we arrive at the Hamiltonian form of the unfree gauge symmetry: δ ε O ( q, p ) = n X r =1 { O , ( r ) T α r } ε α r ; (31) δ ε λ α = (cid:0) δ α β ddt + (1) V α β ( q, p, λ ) (cid:1) ε β + n X k =2 ( k ) V α β k ( q, p, λ ) ε β k , (32)while equations constraining gauge parameters (6) read (cid:0) δ α l β l ddt + ( l ) V α l β l ( q, p, λ ) (cid:1) ε β l + n X m = l +1 ( m ) V α l β m ( q, p, λ ) ε β m + ( l − Γ α l β l − ( q, p, λ ) ε β l − = 0 , (33)where l = 2 , . . . , n − (cid:0) δ α n β n ddt + ( n ) V α n β n ( q, p, λ ) (cid:1) ε β n + ( n − Γ α n β n − ( q, p, λ ) ε β n − = 0 . (34)As one can see, the gauge transformations are generated by the constraints of all generations (31),(32), while corresponding gauge parameters are bound by the differential equations (33), (34).One can verify by direct computation that transformations (31), (32) leave original Hamiltonianaction (19) intact. Given involution relations of Hamiltonian and constraints (22), (24), (25), thegauge variation (31), (32) of the action reads: δ ε S ≡ Z dt n n − X l =2 (cid:20)(cid:0) δ α l β l ddt + ( l ) V α l β l (cid:1) ε β l + n X m = l +1 ( m ) V α l β m ε β m + ( l − Γ α l β l − ε β l − (cid:21) ( l ) T α l + + (cid:20)(cid:0) δ α n β n ddt + ( n ) V α n β n (cid:1) ε β n + ( n − Γ α n β n − ε β n − (cid:21) ( n ) T α n − ddt n X r =1 ( r ) T α r ε α r ! o = 0 . (35)By virtue of equations (33), (34), imposed on the gauge parameters, the integrand reduces to thetotal derivative.Let us discuss the structure of equations (33), (34) constraining gauge parameters. To demon-strate key features of the equations, consider the toy model such that has only one constraint ofeach generation, so no indices α k are needed. The next simplification is that all the constraintscommute. So, the involution relations (22), (24), (25) get a simple form: { (1) T , H } = (1) Γ (2) T , { ( l ) T , H } = ( l ) Γ ( l +1) T , { ( n ) T , H } = 0 ; { ( r ) T , ( s ) T } = 0 , (36)where l = 2 , . . . , n − r, s = 1 , . . . , n . Given the involution relations, gauge transformations (31),(32) read: δ ǫ O = n X r =1 { O , ( r ) T } ε r , δ ǫ λ = ˙ ε . (37)The equations (33), (34) constraining gauge parameters ε r read:˙ ε r +1 + ( r ) Γ ε r = 0 , r = 1 , . . . , n − . (38)If operators ( r ) Γ , r = 1 , . . . , n −
1, were all invertible in the class of differential operators, one couldexpress all the gauge parameters ε r as the derivatives of the last one: ε r = (cid:18) ( r ) Γ (cid:19) − ddt (cid:18) ( r +1) Γ (cid:19) − . . . ddt (cid:18) ( n − Γ (cid:19) − ddt ε n . (39)Relation (39) is a general solution for equations (38). Given the solution, one can substitute allthe gauge parameters ε r , r = 1 , . . . , n −
1, in terms of the unique unconstrained parameter ε n , intothe gauge transformation (31), (32). In this way, we arrive at the gauge transformation withoutconstraints on gauge parameters but with higher derivatives of the unconstrained parameter. Themost general case of this type, when the higher order gauge transformation generators can beconstructed for the evolutionary equations with constraints, is considered in the article [14]. Theunfree gauge symmetry arises in the example above when at least one of operators Γ in involutionrelations (36) does not admit inverse in the class of differential operators. Notice the specialcase of this type, when operators are non-degenerate, i.e. ker Γ = 0, while no Γ − exist in theclass of differential operators. As the example, we can mention the unimodular gravity with AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 13 asymptotically flat metric. The role of Γ aα is plaid by partial derivative ∂ µ , whose kernel is aconstant. If the fields vanish at infinity, the kernel is zero, while no local inverse exists for theoperator. In this case, the higher order unconstrained symmetry can exist, though it is reducible.For the linear field theories, this class of gauge parameter constraints is described in reference[15] in Lagrangian formalism. The reducible unconstrained symmetry for this class of nonlineartheories will be considered elsewhere.Let us mention that the number of equations (33), (34) imposed on the gauge parameters equalsto the number of secondary constraints of all generations, while the number of gauge parametersis the number of constraints of all generations, including primary ones. All equations (33), (34)are independent, there are no identities among them, because every equation is resolved w.r.t. thederivative of a unique gauge parameter. Therefore, the number of independent gauge parametersequals to the number of primary constraints. If it was possible to locally express all the parametersin terms of independent ones and their derivatives, like in the example above, there would be m independent gauge transformations, where m is the number of primary constraints. On the otherhand, corresponding number of time derivatives of m independent gauge parameters ε n essentiallycontribute to the gauge transformations of dependent gauge parameters ε r , r = 1 , . . . , n − m independent parameters and their time derivatives would be involved inthe gauge transformation (31), where m is the total number of constraints. Hence, the on-shellgauge invariants should Poisson-commute on-shell with the constraints of all generations, even ifthe gauge symmetry is unfree. This would be true even if m independent higher order gaugetransformations cannot be explicitly extracted from m unfree first-order transformations in thelocal way. Here, we do not provide a more rigorous justification of this observation, limitingourselves to the explanations given above.Once the unfree gauge symmetry corresponds to the higher order symmetry with m indepen-dent parameters, it would be sufficient to impose m independent gauge-fixing conditions. Thisnumber of required gauge conditions remain the same, even if the independent gauge parameterscannot be explicitly found from equations (33), (34) in the local form. If the gauges are imposedonly on the phase-space variables, not Lagrange multipliers, then non-degeneracy condition of thegauges χ α reads: rank { χ α , T β } = m , (40) where T β stands for the complete set of all constraints, including primary, secondary, tertiary, etc., β = ( β , . . . , β n ). Once the number is different of the constraints and gauge-fixing conditions, thenon-minimal ghost sector have to be modified in the BFV-BRST formalism for the case of unfreegauge symmetry. This issue is considered in the next section.4. Hamiltonian BFV-BRST formalism for unfree gauge symmetry
Construction of the formalism begins with introducing the minimal sector of ghosts. Oncethe on-shell gauge invariants for the unfree gauge symmetry are defined by the requirement toPoisson-commute on-shell with the constraints of all generations, the minimal sector is introducedalong the same lines as for any first-class constrained system [11]. Every first-class constraint isassigned with canonical pair of ghosts with usual Grassmann parity and ghost number grading: ( r ) T α r → { C α r , ¯ P β r } = δ α r β r , gh ( C α r ) = − gh (cid:0) ¯ P α r (cid:1) = 1 ,ǫ ( C α r ) = ǫ (cid:0) ¯ P α r (cid:1) = 1 , r = 1 , . . . , n . (41)The Hamiltonian BFV-BRST generator of minimal sector begins with the constraints, Q min = n X r =1 C α r T α r + . . . , gh ( Q min ) = 1 , ǫ ( Q min ) = 1 , (42)where . . . stands for ¯ P -depending terms. These terms are iteratively defined by the equation { Q min , Q min } = 0 . (43)The ghost extension of the Hamiltonian begins with the original Hamiltonian H , H = H + . . . , gh ( H ) = 0 , ǫ ( H ) = 0 . (44)The specifics of the unfree gauge symmetry is that the completion functions (1), and hence thesecondary constraints may depend on the space-time coordinates, even if the original Lagrangianis x -independent. The x -dependence of the constraints is connected with the asymptotics ofthe fields. Once the constraints involve time, the BRST generator Q min can be explicitly time-dependent. The explicit time dependence of Q min results in appropriate modification [17] of theequation for H : ∂∂t Q min + { Q min , H} = 0 . (45) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 15
This equation defines the ¯ P -dependent terms in H . Equation (45) means that Hamiltonian H is not BRST-invariant. This is a natural consequence of the relations (24) which mean that theoriginal Hamiltonian is not invariant under the unfree gauge symmetry transformations (31), (33),(34). The Hamiltonian action (19), however, is gauge invariant, see (35). For a similar reason,the corresponding path integral is gauge-invariant in the BRST-BFV formalism, even though theHamiltonian H , being a solution of equation (45), is not a BRST invariant. This fact is provenfor general non-stationary constrained system in the reference [17].Consider now the non-minimal sector for the unfree gauge theory. Once the number of gaugefixing conditions coincides with the number of primary constraints, the same number of non-minimal sector ghosts is introduced, { P α , ¯ C β } = δ α β , gh ( P α ) = − gh (cid:0) ¯ C α (cid:1) = 1 , ε ( P α ) = ε (cid:0) ¯ C α (cid:1) = 1 . (46)The Lagrange multiplier canonical pairs are introduced for primary constraints (1) T α and gaugefixing conditions χ α : { λ α , π β } = δ α β , gh ( λ α ) = gh ( π α ) = 0 , ε ( λ α ) = ε ( π α ) = 1 . (47)Complete BRST generator extends the minimal sector one in the usual way, Q = Q min + π α P α . (48)Gauge-fixing conditions involve the time derivative of Lagrange multiplier and the function oforiginal phase-space variables, ˙ λ α − χ α ( q, p ) = 0 . (49)Given the gauge conditions, the gauge Fermion is introduced,Ψ = ¯ C α χ α + λ α ¯ P α , (50)and gauge-fixed Hamiltonian is defined by the usual rule, H Ψ = H + { Q , Ψ } . (51) This Hamiltonian provides conservation of the BRST generator Q much like H . The gauge-fixedBFV-BRST action reads: S ΨBRST = Z dt (cid:16) p i ˙ q i + π α ˙ λ α + n X r =1 ¯ P α r ˙ C α r + ¯ C α ˙ P α − H Ψ (cid:17) . (52)This action accounts for unfree gauge symmetry in two ways. First, the non-minimal sectoris asymmetric with the minimal one unlike the usual BFV formalism. Second, the secondaryconstraints, being a part of the BRST generator Q , may be explicitly time-dependent, even thoughthe original action does not involve time explicitly. Both of these features do not obstruct theusual reasoning that justifies Ψ-independence of the transition amplitude for this action, Z Ψ = Z [ D Φ] exp n i ~ S ΨBRST o , (53)where Φ = (cid:8) q i , p i , λ α , π α , C α , ¯ P α , C α , ¯ P α , . . . , C α n , ¯ P α n , P α , ¯ C α (cid:9) .Let us consider a theory (19) with constraints (22), (24), (25), with the involution relations { ( r ) T α r ( q, p ) , ( s ) T α s ( q, p ) } = U γ t α r α s ( q, p ) ( t ) T γ t ( q, p ) , r, s, t = 1 , . . . , n . (54)Assume that BRST generator Q min and Hamiltonian H are at most linear in the ghost momenta: Q min = n X r =1 C α r ( r ) T α r + 12 n X r,s,t =1 C β s C α r U γ t α r β s ¯ P γ t ; (55) H = H ( q, p ) + n X r =1 C α r (cid:16) n X s =1 ( r ) V β s α r ( q, p, λ ) ¯ P β s + ( r ) Γ β r − α r ( q, p, λ ) ¯ P β r − (cid:17) . (56)We also assume the following form of gauge-fixed Hamiltonian: H Ψ = H + { Q , Ψ } = H + λ α (1) T α + π α χ α + ¯ P α P α + n X r =1 ¯ C α { χ α , ( r ) T α r } C α r . (57)This is automatically true if the gauge conditions χ Poisson-commute to structure functions U inthe involution relations (54). Given the action, path integral (53) reads Z Ψ = Z [ D Φ] exp n i ~ Z dt h p i ˙ q i − H ( q, p ) − λ α (1) T α + π α ( ˙ λ α − χ α ) + − n X r =1 ¯ C α { χ α , ( r ) T α r } C α r + ¯ P α (cid:0) ˙ C α + n X s =1 ( s ) V α β s ( q, p, λ ) C β s (cid:1) + P α (cid:0) ¯ P α + ˙¯ C α (cid:1) + AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 17 + n X k =2 ¯ P α k (cid:16)(cid:0) δ α k β k ddt + ( k ) V α k β k (cid:1) C β k + n X m = k +1 ( m ) V α k β m C β m + ( k − Γ α l β k − C β k − (cid:17)io , (58)where Φ = (cid:8) q i , p i , λ α , π α , C α , ¯ P α , C α , ¯ P α , . . . , C α n , ¯ P α n , P α , ¯ C α (cid:9) . Integrating in path integral(58) over P α , ¯ P α , we arrive at the following answer for the transition amplitude Z Ψ = Z [ D Φ ′ ] exp n i ~ Z dt h p i ˙ q i − H ( q, p ) − λ α (1) T α + π α ( ˙ λ α − χ α ) + − n X r =1 ¯ C α { χ α , ( r ) T α r } C α r − ¯ C α (cid:0) ˙ C α + n X s =1 ( s ) V α β s ( q, p, λ ) C β s (cid:1) ++ n X k =2 ¯ P α k (cid:16)(cid:0) δ α k β k ddt + ( k ) V α k β k (cid:1) C β k + n X m = k +1 ( m ) V α k β m C β m + ( k − Γ α l β k − C β k − (cid:17)io , (59)where Φ ′ = (cid:8) q i , p i , λ α , π α , C α , C α , ¯ P α , . . . , C α n , ¯ P α n , ¯ C α (cid:9) .Let us discuss the path integral (59). The first line in (59) is the original action (19) and thegauge-fixing term. The second line is the FP term for the gauge transformations (31), (32). Thethird line has a natural interpretation from the viewpoint of the modified FP ansatz in Lagrangianformalism (11), (13). The ghost momenta ¯ P α k , k = 2 , . . . , n , can be viewed as Fourier multipliersat the constraints imposed on ghosts (cid:0) δ α k β k ddt + ( k ) V α k β k ( q, p, λ ) (cid:1) C β k + n X m = k +1 ( m ) V α k β m ( q, p, λ ) C β m + ( k − Γ α l β k − ( q, p, λ ) C β k − = 0 . (60)These ghost constraints mirror the equations imposed on gauge parameters in Hamiltonian for-malism (33), (34). So, equations (60) represent Hamiltonian form of the constraints (8) imposedon the ghosts in the case of unfree gauge symmetry. With this regard, the path integral (59) rep-resents the modified FP recipe (11), (13) for the Hamiltonian action (19), gauge symmetry (31),(32), and the constraints (33), (34) on the gauge parameters. So, proceeding from the amplitude(53) in the general Hamiltonian BFV-BRST formalism for unfree gauge symmetry, in the casewithout higher order ghost contributions (55), (57), we arrive at the modified FP path integral(11), (13). 5. Example: traceless massless spin s gauge fields Lagrangian, completion functions, and unfree gauge symmetry.
Let us consider atheory of traceless symmetric tensor field ϕ µ ...µ s , ϕ ννµ ...µ s = 0, in d -dimensional Minkowski space. The metric is chosen mostly negative, η µν = diag(1 , − , . . . , − L = ( − s (cid:16) ∂ ν ϕ µ ...µ s ∂ ν ϕ µ ...µ s − s ∂ ν ϕ νµ ...µ s ∂ ρ ϕ ρµ ...µ s (cid:17) ++ ( − s s ∂ ν (cid:0) ϕ νµ ...µ s ∂ ρ ϕ ρµ ...µ s (cid:1) . (61)The last term is a total divergence, so it does not contribute to the EoMs. We include it forconvenience when constructing the Hamiltonian formalism.The above Lagrangian describes irreducible massless spin- s representation of Poincar´e group.One of the advantages of this form of the irreducible higher spin theory, comparing to the FrondsdalLagrangian [18], is that it does not involve auxiliary fields. This Lagrangian can be viewed ashigher spin extension of linearized UG [1], [2]. In this section, we utilize this model for exemplifyingall the generalities about unfree gauge symmetry considered above in this article.The field equations for the Lagrangian (61) read: δSδϕ µ ...µ s ≡ − ( − s (cid:2) (cid:3) ϕ µ ...µ s − s∂ ( µ ∂ ν ϕ νµ ...µ s ) + s ( s − d + 2 s − η ( µ µ ∂ ν ∂ ρ ϕ νρµ ...µ s ) (cid:3) = 0 , (62)where round brackets ( µ . . . µ s ) mean symmetrization of all the included indices. Taking thedivergence of the l.h.s., we get the differential consequence, Cf. (15): ∂ µ δSδϕ µ ...µ s ≡ ( − s − d + 2 s − d + 2 s − (cid:0) ∂ ( µ τ µ ...µ s − ) − d + 2 s − η ( µ µ ∂ λ τ λ...µ s − ) (cid:1) ≈ , (63)where τ µ ...µ s − is a double divergence of the field, τ µ ...µ s − = ∂ ρ ∂ ν ϕ νρ...µ s − . (64)Relation (63) means that τ reduces on-shell to the element of the kernel of first-order differentialoperator. For s = 2, τ is a scalar, and relation (63) means just ∂ µ τ = 0, so τ is just on-shellconstant. In this case, the kernel is one-dimensional. For s ≥
3, relation (63) means τ µ ...µ s − ≈ Λ µ ...µ s − , (65)with Λ µ ...,µ s − being a solution of conformal Killing tensor equations, ∂ ( µ Λ µ ...µ s − ) − d + 2 s − η ( µ µ ∂ ν Λ ν...µ s − ) = 0 . (66) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 19
The space of conformal Killing tensors is finite dimensional, so τ is a completion function. SpecificΛ is defined by the asymptotic behavior of the fields. For example, if ϕ vanish at infinity, thenΛ = 0. In this most simple case, τ still remains a non-trivial completion function as it is a functionof field derivatives off-shell, not a fixed function of x . This linear function of ∂ ϕ vanishes on-shell, while it is not a linear combination of the Lagrangian equations (62). We detail the case ofnon-vanishing Λ below for s = 3.Once τ (64) is a completion function, relation (63) should be understood as modified Noetheridentity (3) because it binds Lagrangian equations with completion functions: ∂ µ δSδϕ µ ...µ s + ( − s d + 2 s − d + 2 s − (cid:0) ∂ ( µ τ µ ...µ s − ) − d + 2 s − η ( µ µ ∂ λ τ λ...µ s − ) (cid:1) ≡ . (67)Given the identities (3), it defines unfree gauge symmetry of the action: the coefficients at EoMsdefine the gauge generators (5), while the ones at completion functions define the equations (6)constraining the gauge parameters. In this way, the identities (67) define unfree gauge symmetry δ ε ϕ µ ...µ s = s∂ ( µ ε µ ...µ s ) , (68)where ε µ ...µ s − are traceless symmetric gauge parameters, ε ννµ ...µ s − = 0, subject to the transver-sality conditions ∂ ν ε νµ ...µ s − = 0 . (69)Transformations (68) and constraints (69) are noticed in the article [3] where the Lagrangian (61)is proposed. The completion functions (64), (65) are noticed here for the first time.5.2. Covariant degree of freedom count.
Let us now apply formula (18) to verify DoF numberof the spin- s theory (61) in explicitly covariant way. Given the EoMs (62), symmetry transforma-tions (68), gauge identities (67), and constraints on gauge parameters (69), we can compute all theingredients needed to count the DoF number by the recipe (18).The number of the second-order( o e = 2) Lagrangian equations (62) corresponds to the number of independent components oftraceless s -rank tensor, n e = d + s − s − d + s − s − = ( d + s − s !( d − (cid:0) d + d (2 s − − s − (cid:1) . (70)The number of first-order ( o s = 1) symmetry transformations (68) and third-order ( o i = 1 + 2 = 3) gauge identities (67) equals to the number of independent components of traceless ( s − n s = n i = d + s − s − − d + s − s − = ( d + s − s − d − (cid:0) d + d (2 s − − s − (cid:1) . (71)There exist second-order ( o c = 1 + 1 = 2) constraints on gauge parameters (69), whose numbercoincides with the number of independent components of traceless ( s − n c = d + s − s − − d + s − s − = ( d + s − s − d − (cid:0) d + d (2 s − − s − (cid:1) . (72)So, the expression (18) for DoF counting in case of a theory (61) reads: N DoF = h d + s − s − d + s − s − i · − h d + s − s − − d + s − s − i · −− h d + s − s − − d + s − s − i · h d + s − s − − d + s − s − i · . (73)For d = 4 this means N DoF (cid:12)(cid:12)(cid:12) d =4 = ( s + 1) · − s · − s · s − · . (74)Four DoF by the phase-space count corresponds to two “Lagrangian” modes, which is correctnumber for massless spin- s field in d = 4.5.3. Completion functions, asymptotics, moduli space for s = 3 . Let us elaborate on thecontribution of field asymptotics to the completion functions in the simplest higher spin case. For s = 3, Lagrangian (61) and field equations (62) read: L = − (cid:16) ∂ λ ϕ µνρ ∂ λ ϕ µνρ − ∂ µ ϕ µνρ ∂ λ ϕ λνρ (cid:17) − ∂ µ (cid:0) ϕ µνρ ∂ λ ϕ λνρ (cid:1) , ϕ ννµ = 0 ; (75) δSδϕ µνρ ≡ (cid:3) ϕ µνρ − ∂ ( µ ∂ λ ϕ λνρ ) + 6 d + 2 η ( µν ∂ λ ∂ σ ϕ σλρ ) = 0 . (76)Taking the divergence of the field equations, we get the differential consequence ∂ λ δSδϕ λµν ≡ dd + 2 (cid:16) ∂ ( µ ∂ ρ ∂ λ ϕ λρν ) − d η µ µ ∂ λ ∂ ρ ∂ ν ϕ νρλ (cid:17) ≈ . (77) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 21
Introduce the notation τ µ = ∂ ν ∂ λ ϕ µνλ . (78)Relation (77) means that τ µ must obey on-shell the equation for conformal Killing vector field, ∂ µ τ ν + ∂ ν τ µ − d η µν ∂ ρ τ ρ ≈ . (79)The general solution of the conformal Killing equation reads ∂ µ Λ ν + ∂ ν Λ µ − d η µν ∂ ρ Λ ρ = 0 ⇔ Λ µ = a µ + 2 η µν ω νρ x ρ + λx µ + b ν (cid:0) x µ x ν − η µν x ρ x ρ (cid:1) , (80)where a µ , λ, b µ , ω µν = − ω νµ are arbitrary (integration) constants, so there are ( d + 2)( d + 1)2constant parameters. Relation (79) means that τ µ reduces on-shell to Killing vector (80): ∂ ν ∂ λ ϕ µνλ ≈ Λ µ ( x ; λ, a, b, ω ) . (81)Let us shift the notation (78): τ µ = ∂ ν ∂ λ ϕ µνλ − Λ µ ( x ; λ, a, b, ω ). Then, τ µ vanishes on-shell, τ µ ≡ ∂ ν ∂ λ ϕ µνλ − Λ µ ( x ; λ, a, b, ω ) ≈ . (82)So, we have a function of the field derivatives such that vanishes on-shell, while it is not a linearcombination of the l.h.s. of Lagrangian equations (76) and their derivatives. This means that τ µ is a completion function, according to definition (1). Relation (82) can be considered as spin-3analogue of relation τ ≡ R − Λ ≈ d completion functions involving ( d + 1)( d + 2)2 constant parameters. Second, in the caseof UG, the completion function does not depend on space-time coordinates, while for s = 3 there isexplicit x -dependence. We see that the number of modular parameters does not directly correlateto the number of completion functions. Also, completion functions can be explicitly x -dependent,even if the Lagrangian is translation-invariant. Specific modular parameters λ, a, b, ω are definedby asymptotics of the fields. If the fields tend to zero at infinity, all the parameters vanish, whilethe equation τ µ = 0 will remain a non-trivial relation anyway.Notice that field equations (76) admit the solutions such that compatible with any modularparameters λ, a, b, ω in the completion function (82). Let κ (0) µνρ be a general solution vanishing atinfinity. It incudes the Cauchy data, corresponding to four local physical DoF in 4 d case. Double divergence of κ (0) µνρ inevitably vanishes. There is another solution, κ µνρ , with different asymptoticswhich includes the same number of local Cauchy data and arbitrary modular parameters: κ µνρ = κ (0) µνρ + A h a ( µ x ν x ρ ) − d + 2 η ( µν a ρ ) x λ x λ − d + 2 η ( µν x ρ ) a λ x λ i ++ B h η ( µα ω αβ x β x ν x ρ ) − d + 2 η ( µν η ρ ) α ω αβ x β x λ x λ i + Cλ h x ( µ x ν x ρ ) − d + 2 η ( µν x ρ ) x λ x λ i ++ Db λ x λ x ( µ x ν x ρ ) + Eb ( µ x ν x ρ ) x λ x λ + F η ( µν b ρ ) x λ x λ x σ x σ + Gη ( µν x ρ ) b λ x λ x σ x σ , (83)where A = 3( d + 4)( d − , B = 6( d + 4)( d + 1) , C = 1( d + 4)( d − ,D = 2( d + 7 d − d + 6)( d + 4)( d − , E = − d + 3 d − d + 6)( d + 4)( d − ,F = 3( d + 3 d − d + 6)( d + 4)( d + 2)( d − , G = − d ( d + 6)( d + 4)( d + 2)( d − . (84)For the solution κ , the double divergence of the field is a general conformal Killing vector (80): ∂ ν ∂ λ κ µνλ = Λ µ ( x ; λ, a, b, ω ) . (85)Once we have the completion function τ µ (82), relation (77) can be re-formulated as modifiedNoether identity (3): ∂ λ δSδϕ λµν − dd + 2 (cid:16) ∂ µ τ ν + ∂ ν τ µ − d η µν ∂ λ τ λ (cid:17) ≡ . (86)Given the modified Noether identity, the coefficient at the equations defines unfree gauge variationof the field, while the coefficient at completion function defines the equation constraining the gaugeparameters. In this way, we get unfree gauge symmetry of Lagrangian (75): δ ε ϕ µνλ = ∂ µ ε νλ + ∂ ν ε λµ + ∂ λ ε µν , (87) ∂ ν ε νµ = 0 , (88)where the gauge parameters are symmetric traceless tensors ε µν = ε νµ , ε νν = 0.5.4. Constrained Hamiltonian formalism for s = 3 case. Hamiltonian formalism for thetheory (61) is worked out in the article [3]. Our analysis extends the consideration of [3] in tworespects. First, the article [3] assumed that fields vanish at infinity. We admit non-trivial boundary
AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 23 conditions for the fields, and reveal contribution of the modular parameters to the Hamiltonianconstraints. Second, we demonstrate that involution relations of constraints and Hamiltoniandefine the unfree gauge symmetry.We begin constructing the Hamiltonian formalism with 1 + ( d −
1) decomposition of the fieldssuch that accounts for the traceless condition. The indices µ, ν, . . . = 0 , , . . . , d − i, j, . . . = 1 , . . . , d −
1. Metrics η ij = diag( − , . . . , − ϕ ij ≡ ˜ ϕ ij + 1 d − η ij ϕ , η ij ˜ ϕ ij = 0 , (89)and notice the consequences of symmetry and traceless properties of ϕ µνλ : ϕ = − ϕ ii ≡ − ϕ , ϕ i = − ϕ j ji ; (90)where ϕ jji = η jk ϕ kji . Given relations (89), (90), Lagrangian (75), being expressed in terms of thevariables ϕ ijk , e ϕ ij , ϕ , modulo total time derivative reads: L = −
12 ˙ ϕ ijk ˙ ϕ ijk + 32 ˙ ϕ ij j ˙ ϕ ikk + ˙ ϕ + 3 ˙ ϕ ijk ∂ k e ϕ ji − ϕ ijj ∂ k e ϕ ki − d − ϕ ij j ∂ i ϕ + 3 ˙ ϕ∂ i ϕ ijj −− (cid:16) ∂ i ϕ jkl ∂ i ϕ jkl + 3 ∂ i ϕ jkk ∂ i ϕ jll + 3 ∂ i e ϕ jk ∂ i e ϕ jk + d + d − d − ∂ i ϕ∂ i ϕ (cid:17) ++ 32 (cid:16) ∂ i ϕ ikl ∂ j ϕ jkl + ∂ i ϕ ikk ∂ j ϕ jll + 2 ∂ i e ϕ ik ∂ j e ϕ jk + 4 d − ∂ i e ϕ ij ∂ j ϕ (cid:17) . (91)The Lagrangian does not include ˙˜ ϕ ij . Making the Legendre transform w.r.t. ˙ ϕ ijk and ˙ ϕ , the actionis brought to the Hamiltonian form S H = Z d d x (cid:0) Π ijk ˙ ϕ ijk + Π ˙ ϕ − H − e ϕ ij e T ij (cid:1) , (92)where the Hamiltonian reads H = −
12 Π ijk Π ijk + 32 1 d Π ijj Π ikk + 14 Π + 3 d ( d −
1) Π ijj ∂ i ϕ −
32 Π ∂ i ϕ ijj ++ 12 (cid:16) ∂ i ϕ jkl ∂ i ϕ jkl + 3 ∂ i ϕ jkk ∂ i ϕ jll + d + 3 d ∂ i ϕ∂ i ϕ (cid:17) − (cid:16) ∂ i ϕ ikl ∂ j ϕ jkl − ∂ i ϕ ikk ∂ j ϕ jll (cid:17) , (93)and e T ij ≡ − (cid:16) ∂ k Π kij − d − η ij ∂ k Π kll (cid:17) = 0 , η ij e T ij ≡ , (94)are the primary constraints, with e ϕ ij being Lagrange multipliers. Let us examine stability of primary constraints (94):˙ e T ij = { e T ij , H } = − (cid:16) δ k ( i ∂ j ) − d − η ij ∂ k (cid:17) T ′ k = 0 . (95)where T ′ i = − (cid:0) ∂ i Π − ∂ i ∂ j ϕ jkk − ϕ ij j + 2 ∂ j ∂ k ϕ kji (cid:1) . (96)The coefficient at T ′ i in relation (95) is a linear differential operator with the finite kernel. Theequation for the null-vectors of the operator reads ∂ i Λ j + ∂ j Λ i − d − η ij ∂ k Λ k = 0 . (97)The equation above defines the conformal Killing vector field in ( d − x i , while the theory is Lorentz-invariant. Then, Lorentz boost will inevitably bringtime-dependence to any solution of (97). The time-dependence is fixed, as we shall see below, byfurther stability conditions. Stability condition (95) means that T ′ i (96) reduce to the solution ofequation (97), i.e. we arrive at secondary constraints T i ≡ T ′ i − Λ i ( x ) = 0 . (98)Given the secondary constraints, they have to conserve. The conservation condition reads:˙ T i = ∂ T i + { T i , H } = − ∂ Λ i ( x ) − (cid:16) δ ( ji ∂ k ) − d − η jk ∂ i (cid:17) e T jk + ∂ i T = 0 . (99)Relation (99) means we have tertiary constraint T ≡ T ′ + Λ ( x ) = 0 , (100)where T ′ = − (cid:16) d − ∂ i Π ij j + ∆ ϕ (cid:17) , (101)and Λ ( x ) is connected with Λ i ( x ) of (98) by the relation ∂ Λ i + ∂ i Λ = 0 . (102) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 25
Given tertiary constraint (100), it has to conserve,˙ T = ∂ T + { T , H } = ∂ Λ − d − ∂ i T i − d − ∂ i Λ i = 0 . (103)This relation does not result in any new constraint, while it is consistent if Λ ( x ) and Λ i ( x ) areconnected by one more relation ∂ Λ − d − ∂ i Λ i = 0 . (104)Relations (97), (102), (104) taken together are just 1 + ( d −
1) decomposition of conformal Killingequations (80) in d dimensions. So, Λ i ( x ), Λ ( x ) are the components of conformal Killing vector,Λ i = a i + 2( ω i x + ω ij x j ) + λx i + 2( b x + b j x j ) x i − b i ( x x + x j x j ) , (105)Λ = a + 2 w i x i + λx + b ( x x − x i x i ) + 2 b i x i x . (106)As soon as the Dirac-Bergrmann algorithm is completed, let us summarize its results. Completeset of constraints reads: e T ij = − (cid:16) ∂ k Π kij − d − η ij ∂ k Π kll (cid:17) ,T i = − (cid:0) ∂ i Π − ∂ i ∂ j ϕ jkk − ϕ ij j + 2 ∂ j ∂ k ϕ kji (cid:1) − Λ i ,T = − (cid:16) d − ∂ i Π ijj + ∆ ϕ (cid:17) + Λ , (107)where Λ i , Λ are defined by relations (105), (106). All the constraints Poisson-commute to eachother. There are non-trivial involution relations between the constraints and Hamiltonian: { e T ij , H } = − (cid:16) δ k ( i ∂ j ) − d − η ij ∂ k (cid:17) T k ,∂ T i + { T i , H } = − (cid:16) δ ( ji ∂ k ) − d − η jk ∂ i (cid:17) e T jk + ∂ i T ,∂ T + { T , H } = − d − ∂ i T i . (108)Once all the constraints are known, and structure coefficients of involution relations (22), (24),(25) are identified, they define the unfree gauge variations of the fields and Lagrange multipliers bythe general rules (31), (32). Also the structure coefficients define the equations (33), (34) imposedon the gauge parameters. Given the constraints (107) and involution relations (108), we applythe general rules, and arrive at unfree gauge symmetry transformations of the fields ϕ ijk , ϕ and Lagrange multipliers e ϕ ij : δ ε ϕ ijk = 3 ∂ ( i e ε jk ) + 3 d − η ( ij ∂ k ) ε , δ ε ϕ = 3 ∂ i ε i , (109) δ ε e ϕ ij = ˙ e ε ij + ∂ i ε j + ∂ j ε i − d − η ij ∂ k ε k . (110)Upon substitution of structure coefficients of involution relations (108) into general relations (33),(34), we get the constraints on gauge parameters for this model:˙ ε i + ∂ j e ε ji + 1 d − ∂ i ε = 0 , (111)˙ ε − ∂ i ε i = 0 . (112)This unfree gauge symmetry is parameterized by ( d − e ε ij , ε i , ε . Explicitly covariant un-free gauge symmetry (87), (88) of the original action (75) is parameterized by symmetric tracelesstensor ε µν . The gauge parameters e ε ij , ε i , ε of Hamiltonian form of the symmetry can be viewedas 1 + ( d −
1) decomposition of the d -tensor parameter ε µν : ε = − ε ii ≡ − ε , ε i ≡ ε i , ε ij ≡ e ε ij + 1 d − η ij ε . (113)As we see in this example, the Hamiltonian algorithm of section 3 allows one to systematicallyidentify all the unfree gauge symmetry transformations and modular parameters of the model,though the method is not explicitly covariant.5.5. BFV-BRST formalism for s = 3 case. In this subsection, we illustrate the general BFV-BRST formalism of section 4 by the spin-3 model (92).We begin with construction of the formalism by introducing the ghosts of the minimal sector.The ghost pairs are assigned to every constraint of the complete set (107): { e C ij , e ¯ P kl } = 12 (cid:0) δ ik δ jl + δ il δ jk (cid:1) − d − η ij η kl , { C i , ¯ P i } = δ ij , { C , ¯ P } = 1 , gh (cid:16) e C ij (cid:17) = − gh (cid:16) e ¯ P ij (cid:17) = gh (cid:0) C i (cid:1) = − gh (cid:0) ¯ P i (cid:1) = gh ( C ) = − gh (cid:0) ¯ P (cid:1) = 1 ,ǫ (cid:16) e C ij (cid:17) = ǫ (cid:16) e ¯ P ij (cid:17) = ǫ (cid:0) C i (cid:1) = ǫ (cid:0) ¯ P i (cid:1) = ǫ ( C ) = ǫ (cid:0) ¯ P (cid:1) = 1 . (114)Hamiltonian BRST generator reads: Q min = e C ij e T ij + C i T i + CT , { Q min , Q min } = 0 . (115) AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 27
Given the Hamiltonian H (93), and involution relations (108), the ghost-extended Hamiltonianreads: H = H − e C ij ∂ ( i ¯ P j ) + C i ∂ i ¯ P − C i ∂ j e ¯ P ji − d − C∂ i ¯ P i , ∂ Q min + { Q min , H} = 0 . (116)According to the prescriptions of section 4, the non-minimal sector ghosts are assigned only tothe primary constraints: { e P ij , e ¯ C kl } = 12 (cid:0) δ ik δ jl + δ il δ jk (cid:1) − d − η ij η kl , gh (cid:16) e P ij (cid:17) = − gh (cid:16) e ¯ C ij (cid:17) = 1 , ǫ (cid:16) e P ij (cid:17) = ǫ (cid:16) e ¯ C ij (cid:17) = 1 . (117)Also the momenta are introduced being canonically conjugate to the Lagrange multipliers, { e ϕ ij , e Π kl } = 12 (cid:0) δ ik δ jl + δ il δ jk (cid:1) − d − η ij η kl , gh (cid:0) e ϕ ij (cid:1) = − gh (cid:16) e Π ij (cid:17) = 0 , ǫ (cid:0) e ϕ ij (cid:1) = ǫ (cid:16) e Π ij (cid:17) = 0 . (118)Complete Hamiltonian BRST generator reads: Q = Q min + e Π ij e P ij . (119)Lorentz-like gauge conditions should be imposed, being explicitly resolved w.r.t. time derivativesof Lagrange multipliers. As explained in the section 4, the number of gauges should be the sameas the number of primary constraints. So, we choose the following gauges:˙ e ϕ ij − e χ ij = 0 , e χ ij ≡ − (cid:0) ∂ k ϕ kij − d − η ij ∂ k ϕ kll (cid:1) . (120)Given the gauges, the gauge Fermion reads:Ψ = e C ij e χ ij + e ϕ ij e ¯ P ij . (121)Following the general rule (51), the gauge-fixed Hamiltonian is constructed, H Ψ = H + { Q , Ψ } = H + e ϕ ij e T ij + e Π ij e χ ij + e ¯ P ij e P ij ++ e ¯ C ij { e χ ij , e T kl } e C kl + e ¯ C ij { e χ ij , T k } C k + e ¯ C ij { e χ ij , T } C . (122)
As a result, we arrive at the gauge-fixed BRST-invariant Hamiltonian action S ΨBRST = Z d d x h Π ijk ˙ ϕ ijk + Π ˙ ϕ + e Π ij ˙ e ϕ ij + e ¯ P ij ˙ e C ij + ¯ P i ˙ C i + ¯ P ˙ C + e ¯ C ij ˙ e P ij − H Ψ i . (123)Corresponding path integral reads: Z Ψ = Z [ D Φ] exp n i ~ S ΨBRST o , (124)where Φ = (cid:8) Π ijk , ϕ ijk , Π , ϕ, e Π ij , e ϕ ij , e ¯ P ij , e C ij , ¯ P i , C i , ¯ P , C, e ¯ C ij , e P ij (cid:9) . Integration over momenta e P ij , e ¯ P ij , Π ijk , Π, leads to the following result: Z = Z [ D Φ ′ ] exp n i ~ Z d d x h L + e Π ij ∂ µ ϕ µij + e ¯ C ij (cid:0) (cid:3) δ iµ δ jν + 2 ∂ i δ jµ ∂ ν (cid:1) C µν + ¯ P µ ∂ ν C νµ io , (125)where Φ ′ = (cid:8) ϕ µνρ , e Π ij , e ¯ C ij , C µν , ¯ P µ (cid:9) . Ghosts C, C i , e C ij can be viewed as 1 + ( d −
1) decompositionof ghost C µν , C νν = 0 (8), being d -dimensional symmetric traceless tensor, C = − C ii ≡ − C , C i ≡ C i , C ij ≡ e C ij + 1 d − η ij C . (126)Expression ∂ ν C νµ in the end of exponential of (125) can be viewed as a constraint imposed onthe ghosts which corresponds to the transversality condition imposed on gauge parameters. Theghost momenta ¯ P µ : ¯ P ≡ − ¯ P , assigned to the secondary and tertiary constraints, play the role ofFourier multipliers at the constraints imposed on ghosts for the original unfree gauge symmetry.With this regard, relation (125) is seen to reproduce the modified FP path integral (11), (13) forthe original action (75). 6. Conclusion
In this article we work out constrained Hamiltonian formalism corresponding to the unfree gaugesymmetry with gauge parameters constrained by differential equations. In the Hamiltonian form,the phenomenon of the unfree gauge symmetry has been clarified from viewpoint of involutionrelations between Hamiltonian and constraints. The key role is plaid by differential operatorsΓ, being the coefficients in the involution relations (22), (24) such that stand at the constraintsof the next generation in the stability conditions of the previous constraints. These structurecoefficients define the unfree gauge symmetry if they have a finite kernel. Even if Γ are non-degenerate (trivial kernel), but the inverse does not exist in the class of differential operators,
AMILTONIAN CONSTRAINTS AND UNFREE GAUGE SYMMETRY 29 we have unfree gauge symmetry. Given the structure coefficients of involution relations withthese properties, we arrive at the equations constraining the gauge parameters (33), (34). In thebest-known example of the unfree gauge symmetry, the unimodular gravity, the kernel of Γ isone-dimensional, and the corresponding modular parameter is the cosmological constant Λ. Themodular parameters are defined by the asymptotics of the fields. For example, even the free spin-2field theory [1], [2] with unfree gauge symmetry in Minkowski space (that corresponds to linearizedUG), admits solutions with non-vanishing Λ. These solutions correspond to non-vanishing fieldsat infinity. Analogous solutions with non-trivial modular parameters are noticed in section 5 forhigher spin fields with unfree gauge symmetry. The dynamics with non-trivial modular parametersare relevant upon inclusion of interactions as we expect. This issue will be addressed elsewhere.In section 4, we explain how the Hamiltonian BFV-BRST formalism is adjusted for the case ofunfree gauge symmetry. For the case when there are no higher-order ghost vertices, we deducefrom the phase-space path integral the modified FP quantization rules such that account for theunfree gauge symmetry by imposing corresponding constraints on the ghosts. In this way, we seethat the covariant quantization rules for the systems with unfree gauge symmetry are deducedfrom corresponding modification of Hamiltonian BFV-BRST quantization.
Acknowledgments.
The work is supported by the Foundation for the Advancement of Theoret-ical Physics and Mathematics “BASIS”.
References [1] E. Alvarez, D. Blas, J. Garriga and E. Verdaguer, “Transverse Fierz-Pauli symmetry”, Nucl. Phys. B (2006) 148 [hep-th/0606019].[2] D. Blas, “Gauge Symmetry and Consistent Spin-Two Theories”, J. Phys. A (2007) 6965 [hep-th/0701049].[3] E. D. Skvortsov and M. A. Vasiliev, “Transverse Invariant Higher Spin Fields”, Phys. Lett. B (2008) 301[hep-th/0701278 [hep-th]].[4] A. Campoleoni and D. Francia, “Maxwell-like Lagrangians for higher spins”, JHEP (2013) 168[arXiv:1206.5877 [hep-th]].[5] R. Percacci, “Unimodular quantum gravity and the cosmological constant”, Found. Phys. (2018) 1364-1379.[6] A. O. Barvinsky and A. Y. Kamenshchik, “Darkness without dark matter and energy – generalized unimodulargravity”, Phys. Lett. B (2017) 59 [arXiv:1705.09470 [gr-qc]].[7] A. O. Barvinsky, N. Kolganov, A. Kurov and D. Nesterov, “Dynamics of the generalized unimodular gravitytheory”, Phys. Rev. D (2019) 023542 [arXiv:1903.09897 [hep-th]]. [8] D. S. Kaparulin and S. L. Lyakhovich, “A note on unfree gauge symmetry”, Nucl. Phys. B (2019) 114735[arXiv:1904.04038 [hep-th]].[9] D. S. Kaparulin and S. L. Lyakhovich, “Unfree gauge symmetry in the BV formalism”, Eur. Phys. J. C no.8 (2019) 718 [arXiv:1907.03443 [hep-th]].[10] V. A. Abakumova, I. Y. Karataeva and S. L. Lyakhovich, “Unfree gauge symmetry in the Hamiltonian for-malism”, Phys. Lett. B (2020) 135208; [arXiv:1911.11548 [hep-th]].[11] M. Henneaux and C. Teitelboim, “Quantization of gauge systems”, Princeton, USA: Univ. Pr. (1992) 520 p.[12] D. Francia, G. L. Monaco and K. Mkrtchyan, “Cubic interactions of Maxwell-like higher spins”, JHEP (2017) 068 [arXiv:1611.00292 [hep-th]].[13] D. S. Kaparulin, S. L. Lyakhovich and A. A. Sharapov, “Consistent interactions and involution”, JHEP (2013) 097 [arXiv:1210.6821 [hep-th]].[14] S. L. Lyakhovich and A. A. Sharapov, “Normal forms and gauge symmetries of local dynamics,” J. Math.Phys. (2009) 083510 [arXiv:0812.4914 [math-ph]].[15] D. Francia, S. L. Lyakhovich and A. A. Sharapov, “On the gauge symmetries of Maxwell-like higher-spinLagrangians”, Nucl. Phys. B (2014) 248 [arXiv:1310.8589 [hep-th]].[16] S. L. Lyakhovich and A. A. Sharapov, “Normal forms and gauge symmetries of local dynamics”, J. Math.Phys. (2009) 083510 [arXiv:0812.4914 [math-ph]].[17] I. A. Batalin and S. L. Lyakhovich, “Generalized canonical quantization of nonstationary dynamical systemssubject to constraints”, Proceedings, 18th International Colloquium on Group Theoretical Methods in Physics(GROUP 18) Published in: In *Moscow 1990, Proceedings, Symmetries and algebraic structures in physics,pt. 1*, pp. 57-63.[18] C. Fronsdal, “Massless Fields with Integer Spin”, Phys. Rev. D (1978) 3624. Physics Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia.
E-mail address ::