aa r X i v : . [ h e p - t h ] F e b GENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS
S.L. LYAKHOVICH
Abstract.
A systematic procedure is proposed for inclusion of Stueckelberg fields. The proce-dure begins with the involutive closure when the original Lagrangian equations are complementedby all the lower order consequences. The involutive closure can be viewed as Lagrangian analogueof complementing constrained Hamiltonian system with secondary constraints. The involutivelyclosed form of the field equations allows for explicitly covariant degree of freedom number count,which is stable with respect to deformations. If the original Lagrangian equations are not involu-tive, the involutive closure will be a non-Lagrangian system. The Stueckelberg fields are assignedto all the consequences included into the involutive closure of the Lagrangian system. The iterativeprocedure is proposed for constructing the gauge invariant action functional involving Stueckelbergfields such that Lagrangian equations are equivalent to the involutive closure of the original theory.The generators of the Stueckelberg gauge symmetry begin with the operators generating the clo-sure of original Lagrangian system. These operators are not assumed to be a generators of gaugesymmetry of any part of the original action, nor are they supposed to form an on shell integrabledistribution. With the most general closure generators, the consistent Stueckelberg gauge invari-ant theory is iteratively constructed, without obstructions at any stage. The Batalin-Vilkoviskyform of inclusion the Stueckelberg fields is worked out and existence theorem for the Stueckelbergaction is proven. Introduction
In 1938, Stueckelberg proposed [1] to reformulate the Proca action for the massive vector fieldin a gauge invariant way by introducing the scalar field. Since then, the general idea attributedto Stueckelberg has been widely used to equivalently reformulate the original non-gauge theoryin a gauge invariant way by introducing some extra fields. Historical review of ideas about, andapplications of the Stueckelberg method can be found in the article [2].The most often used scheme of introducing the Stueckelberg fields follows the pattern of theoriginal work [1] implying that Lagrangian includes a gauge invariant part, and the non-gaugeinvariant part. In the case of massive vector field, the gauge invariant part is the Maxwell La-grangian, while the non-invariant part is the massive term. Then, the finite gauge transformation defining the symmetry of the invariant part is made of the fields in the entire Lagrangian. Thismakes the Lagrangian depending on the gauge parameters. After that, the gauge parameters aretreated as the Stueckelberg fields. Under this scheme, the Stueckelberg gauge transformations ofthe original fields are the same as in the theory without non-invariant part of the Lagrangian. Inthis sense, the broken gauge invariance of the original fields is restored. Once the Stueckelbergfields are included as the parameters of gauge transformations of the invariant part of the action,their own gauge symmetry is defined as the composition of original gauge transformations. In thisway, the gauge transformation of Stueckelberg fields compensates the change of the non-invariantpart of the Lagrangian caused by the transformation of original fields. The equivalence to the orig-inal theory is established by imposing the gauge conditions fixing the Stueckelberg fields to zero.In this gauge, the Stueckelberg theory reduces to the original one. This pattern of inclusion theStueckelberg fields, sometimes referred to as the “Stueckelberg trick”, is summarized and studiedin very general form in the article [3] where one can also find a review of the vast contemporaryliterature on this topic.The “Stueckelberg trick” works well in many models, but it does not seem consistent as ageneral method as it is more art than science. The division of the Lagrangian into invariant andnon-invariant parts is rather arbitrary, as is the predefined choice of gauge transformations. Forexample, given any transformations of the fields that form the Lie group, any invariant of the groupcan be added to and subtracted from any Lagrangian. Thus, one can get a division into invariantand non-invariant parts with respect to any transformation. Thereafter, the above pattern can beapplied, resulting in a model with almost any gauge symmetry that is not necessarily relevant tothe original dynamics. Even the number of gauge parameters can be any within this approach. Inthe case of the Proca action, one could shift the original vector field not by a gradient of a scalar,but – for example – by a divergence of anti-symmetric second rank tensor. In this case, another partof the Lagrangian – the square of divergence of original vector field – is considered as an invariant,and the rest as a non-invariant part. The method works, and results in the gauge invariant theoryof massive spin one with reducible gauge symmetry parameterized by antisymmetric tensor.Another commonly used scheme of inclusion of Stueckelberg fields is the method of conversionof the Hamiltonian second class constrained systems into the first class ones. The first version ofthe method [4], [5], [6] implied to extend the phase space of the original system by new canonicalvariables whose number coincides with the number of second class constraints. After that, the
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 3 constraints and Hamiltonian are continued into the extended space to convert the system intothe first class. Once the gauge conditions are imposed killing the conversion variables, the systemreduces to the original second class system. The conversion variables can be viewed as Hamiltonianversion of Stueckelberg fields. The original proposal for conversion implied to linearly include theconversion variables into the effective first class constraints. In general, the gauge symmetry, beinggenerated by the effective first class constraints, is non-abelian. Later on, the abelian conversionscheme has been proposed [7], [8], [9]. Under this scheme, the original second class constraints areextended by a power series in the conversion variables to become abelian first class constraints.Any original phase space function, including Hamiltonian, is extended by the conversion variablesto Poisson commute with the effective abelian first class constraints. The existence theorem of theabelian conversion is proven by the homological perturbation theory (HPT) tools in the article[9]. In the article [10], the conversion method is extended to the general second class systems onsymplectic manifolds, with the constraints not necessarily being scalar functions, but sections ofa bundle over the symplectic base. In this setup, the conversion is proven to exist, though notnecessarily abelian. This conversion scheme allows one to extend Fedosov deformation quantisationto the second class constrained systems.Notice important distinctions of the Hamiltonian conversion method from the described above“Stueckelberg trick” which is widely applied in Lagrangian formalism. The starting point of theHamiltonian conversion is a complete system of the second class constraints, including primaryand secondary ones. The Hamiltonian equations are first order, while primary constraints are zeroorder. All these equations are variational. The secondary constraints are zero order differentialconsequences of the variational equations, and they are not variational by themselves. It is thecompletion of the original equations by the lower order consequences which allows one to explic-itly count the degree of freedom (DoF) number. The same applies not only to the constrainedHamiltonian equations, but to any system of field equations. The completion of the system bythe lower order consequences is known as the invloutive closure. Given the involutive closure ofthe equations, one can count degree of freedom number in a covariant manner, not appealing tothe 1 + ( d −
1) decomposition. Simple and explicitly covariant DoF number counting recipe isworked out in the article [11] for any involutively closed system of field equations, not necessarilyLagrangian. For the involutive closure of Lagrangian system, the general recipe (see relation (8) ofthe article [11]) can be further simplified. The covariant DoF count is explained in the Appendix
S.L. LYAKHOVICH of this article. The Hamiltonian scheme of including Stueckelberg fields proceeds from involutiveclosure of variational equations, in particular the number of conversion variables coincides withthe number of constraints. The Lagrangian pattern of the Stueckelberg trick does not account forthe structure of involutive closure, even the number of the Stueckelberg fields is unrelated to thenumber of lower order consequences of the Lagrangian equations. Notice one more essential dis-tinction between the pattern of Lagrangian “Stueckelberg trick” and the Hamiltonian conversionprocedure. The first one proceeds from certain integrable distribution on the space of fields, whichis considered as gauge symmetry of “invariant part” of the action. The second class constraintsare unrelated, in general, to any integrable distribution, and the Hamiltonian conversion methodsdo not employ any predefined transformations of the original fields.One of the motivations for introducing the Stueckelberg fields is the idea to provide consistentinclusion of interactions by controlling compatibility of Stueckelberg gauge symmetry when thefree theory is deformed. This idea works well in various examples, see [3] and references therein.However, it does not seem a consistent general scheme, as it controls just algebraic consistency ofthe Stueckelberg symmetry, not the number of propagating DoF’s, while the artificial symmetry isnot necessarily reasonably related to the structure of the dynamics. In the article [11], the methodis proposed to consistently include interactions proceeding from the involutive closure of fieldequations and without introducing Stueckelberg fields. Proceeding from the free field equationsbrought to the involutive form, the method allows one to iteratively find all the consistent verticesin the equations. Even though the original non-involutive equations are Lagrangian, the involutiveclosure is not Lagrangian system. Therefore, not all the vertices are necessarily Lagrangian. Forapplications of this scheme of inclusion of interactions, see, for example [11], [12], [13], [14], [15].While non-Lagrangian vertices may have their own advantages, in particular they can be stablein higher derivative field theories [16], [14], it seems interesting to have a method of identifying allthe consistent Lagrangian vertices. The way to construct all the consistent Lagrangian vertices isbriefly noticed in the next section as a side remark.The main subject of this article is to work out a method of inclusion the Stueckelberg fieldswhich proceeds from the involutive closure of Lagrangian equations. In this sense, the methodcan be considered as the Lagrangian counterpart of the conversion method for the Hamiltoniansecond class constrained systems.
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 5
Notation.
DeWitt’s condensed notation [17] is adopted, where the indices cover both space time-points x and a set of numerical labels. As a rule, all the indices are understood as condensed, theexceptions are clear from the context. The derivatives ∂ i are variational w.r.t. the fields fields φ i .Summation over condensed indices includes integration over x . Sign ≈ is the on shell equality.2. Involutive closure of the Lagrangian equations and the Stueckelberg fields
In condensed notation, given the action S ( φ ), Lagrangian equations read ∂ i S ( φ ) = 0 . (1)Suppose the action admits no gauge symmetry. This means that matrix of second derivatives ofthe action does not have on-shell kernel. Inclusion of Stueckelberg fields in the case with gaugesymmetry can be considered along the similar lines, though it is slightly more complex. It will beaddressed elsewhere.Let us complement equations (1) with the consequences τ α ( φ ) = − Γ iα ( φ ) ∂ i S ( φ ) , (2)such that the system ∂ i S ( φ ) = 0 , τ α = 0 (3)is involutive. Here, the involution means that the system does not admit any lower order con-sequence which is not already included. For a review of the involution concept in the partialdifferential equation (PDE) theory, and various applications, one can consult the book [18]. Ifthe original Lagrangian equations are not involutive, by adding consequences, it will be broughtto the involution. This can be considered as Lagrangian analogue of Dirac-Bergmann algorithmof constrained Hamiltonian formalism with the consequences τ a being analogues of the secondclass constraints. The difference is that the order of derivatives is considered w.r.t. any space-time coordinate. Much like the Hamiltonian conversion, the Lagrangian procedure of inclusionStueckelberg fields begins with involutive closure of equations of motion. S.L. LYAKHOVICH
In principle, one can include the consequences of the higher order than the original Lagrangianequations. The only requirement is that the system of Lagrangian equations and their conse-quences (3) are involutive — i.e. any lower order consequence is already contained among theseequations . Example of this sort is considered in the end of this section.The consequences τ α are supposed independent. This also means that the generators Γ iα , beingthe local differential operators, are also independent. The specific conditions of independence areexplained in the next section. The over-complete set of generators Γ would lead to the reducibleStueckelberg gauge symmetry. This case is not considered in the article, though it can be ofinterest for some models.The involutive system (3) admits gauge identitiesΓ iα ( φ ) ∂ i S ( φ ) + τ α ( φ ) ≡ τ α (2). The involutivelyclosed system (3) is non-Lagrangian as such though it is equivalent to the original Lagrangiansystem (1). In non-Lagrangian systems, the second Noether theorem does not apply, so the gaugeidentities are not necessarily related to a gauge symmetry. Since the original Lagrangian equations(1) do not have gauge symmetry, then their involutive closure (3) will not be gauge invariant either,because they define the same mass shell.The involutive closure (3) of the Lagrangian system is characterised by three types of numberswhich determine the DoF number: (i) the orders of original Lagrangian equations (1); (ii) theorders of the consequences τ a included into involutive closure (3)of the system; (iii) the ordersof differential operators Γ iα generating consequences of Lagrangian equations included into theinvolutive closure. The DoF number is a certain linear combination of these three types of integers.The DoF counting is detailed in the Appendix, see relation (65) and corresponding explanations.Let us make a side remark on consistent inclusion of interactions, proceeding from the involutiveclosure (3) of the Lagrangian system. Given the free Lagrangian without gauge symmetry, theproblem is to find all the vertices such that the DoF number remains unchanged upon inclusion This also has a counterpart in the constrained Hamiltonian formalism. Given the Lagrangian with the first orderderivatives, one can construct the Hamiltonian formalism as if the acceleration were included. Then, the phasespace would include auxiliary coordinates, absorbing velocities, and also extra momenta. These extra variables aresuppressed by the second class constraints. In principle, these constraints can be converted into the first class, byusual conversion procedure. This conversion procedure, at Lagrangian level, would correspond to the involutivelyclosed system constructed by inclusion of the higher order consequences of original Lagrangian equations.
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 7 of interactions. The procedure is quite simple. At first, one has to bring the system of thefree Lagrangian equations into the involutive form, by complementing them with all the lowerorder consequences. Given the involtutive closure, the DoF number is fixed. The second step toconsistent inclusion of interactions is to simultaneously deform the free action S and the generatorsΓ iα of the consequences, S ( φ ) = 12 M ij φ i φ j S int = X k =0 ( k ) S ( φ ) , ( k ) S = 1 k + 2 M i ...i k +2 φ i . . . φ i ; (5)Γ iα G iα ( φ ) = X k =0 ( k ) G iα , ( k ) G iα = G iαj ...j k φ j . . . φ j k , (6)where the vertices M i ...i k +2 , G iαj ...j k are field-independent poly-differential operators. The firstoperator M ij is the same as in the free theory, and (0) G iα coincides with the generator of theconsequences included into the involutive closure at the free level. Given the vertices in the action(5) and the generators of consequences (6), one gets the deformation of the consequences includedinto the involutive closure (3) τ α = − Γ iα ∂ i S
7→ − G iα ∂ i S int = T α = X k =0 ( k ) T α , ( k ) T α = T αi ...i k +1 φ i . . . φ i k +1 ; (7) T αi ...i k +1 = k +1 X l =0 G iα ( j ...j l M j l +1 ...j k +1 ) i . (8)Here, the round brackets mean symmetrization of corresponding indices. Upon inclusion of inter-action, the deformed action (5) and consequences (7) by construction obey the gauge identity: G iα ∂ i S int + T α ≡ . (9)One can see once again that from the perspective of algebraic consistency, any interaction isadmissible for the field equations without gauge symmetry. The consistency of interactions isprovided not just by algebraic consistency but also the DoF number should remain unchangedupon deformation. This condition can be easily controlled making use of the involtive form offield equations. The vertices (5), (6) will be consistent if the following two conditions are met: (i)the system ∂ i S int = 0 , T α = 0 (10) S.L. LYAKHOVICH remains involutive; (ii) the DoF number (65) for equations (10) remains the same as it is for thefree system. Upon inclusion of interaction, the orders of Lagrangian equations, generators, andconsequences, being ingredients of the DoF number count, can increase, or remain unchanged.If they do not increase, the DoF number obviously remains unchanged. These three orders canincrease, however, in the correlated way without changing the DoF number, once relation (65) stillholds true. Therefore, the interaction can be consistent, in principle, even if the higher derivativesare involved.Below in this section, the iterative procedure is described for inclusion of Stueckelberg fields.Under this procedure, the Stueckelberg field ξ α is assigned to every consequence τ α ( φ ) includedinto the involutive closure (3) of original Lagrangian equations. The Stueckelberg action is soughtfor as a power series in the fields ξ α : S St ( φ, ξ ) = ∞ X k =0 S k , S k ( φ, ξ ) = W α ...α k ( φ ) ξ α · · · ξ α k , k > , (11)where S ( φ ) is the original action S ( φ ), and the first expansion coefficient W α is defined by τ α (2): W α ( φ ) = ∂ S St ( φ, ξ ) ∂ξ α (cid:12)(cid:12) ξ = 0 = τ α . (12)Hence, at ξ = 0, the field equations for the Stueckelberg action reproduce the involutive closure(3) of the original Lagrangian equations.The equivalence of the Stueckelberg theory to the original one is provided by the gauge symmetryof the action (11) such that the fields ξ α can be gauged out, with ξ α = 0 being admissible gaugefixing condition. This means, the number of gauge parameters should coincide with the numberof consequences τ α (2) included into involutive closure of the original Lagrangian system. Thegauge transformations are iteratively sought for order by order of the Stueckelberg fields δ ǫ φ i = R iα ( φ, ξ ) ǫ α , δ ǫ ξ γ = R γα ( φ, ξ ) ǫ α , R iα ( φ, ξ ) = X k =0 ( k ) R iα , R γα ( φ, ξ ) = X k =0 ( k ) R γα , (13) ( k ) R iα ( φ, ξ ) = R iαβ ...β k ( φ ) ξ β . . . ξ β k , ( k ) R γα ( φ, ξ ) = R γαβ ...β k ( φ ) ξ β . . . ξ β k . (14) ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 9
The Stueckelberg action (11) is supposed to be invariant with respect to the gauge transformations(13). The gauge symmetry (13) of the action is equivalent to the Noether identities δ ǫ S St = 0 , ∀ ǫ α ⇔ R iα ∂ i S St + R γα ∂ S St ∂ξ γ ≡ . (15)These identities can be expanded in Stueckelberg fields, (cid:18) R γα ( φ, ξ ) ∂∂ξ γ + R iα ( φ, ξ ) ∂∂φ i (cid:19) S St ≡ X k =0 k X m =0 (cid:18) ( k − m ) R γα ∂ S ( m +1) ∂ξ γ + ( k − m ) R iα ∂ S ( m ) ∂φ i (cid:19) ≡ . (16)Once the Noether identities are valid for every order in ξ , each term in the sum over k shouldvanish separately. In this way, the requirement of the gauge symmetry results in the sequence ofrelations k X m =0 (cid:18) ( k − m ) R γα ∂ S ( m +1) ∂ξ γ + ( k − m ) R iα ∂ S ( m ) ∂φ i (cid:19) ≡ , k = 0 , , , . . . . (17)For k = 0, given the boundary condition (12), the above relations reduce to the identities betweenconsequences τ and Lagrangian equations, (0) R γα τ γ + (0) R iα ∂ i S ≡ . (18)Any identity between τ α and ∂ i S reduces to the linear combination of the identities in the closureof the original system (4). This means, the identity (18) reads (0) R γα (cid:0) τ γ + Γ iγ ∂ i S (cid:1) ≡ , (19)where (0) R γα can be any non-degenerate matrix. Below, we stick to the simplest choice (0) R γα = δ γα . (20)This choice does not restrict the generality for two reasons. First, as demonstrated below, itadmits consistent inclusion of Stueckelberg fields for any action and any generating set of theconsequences included into the involutive closure of original Lagrangian system (3). Second, anyother choice of non-degenerate (0) R γα can be absorbed by the change of the gauge parameters ǫ α . Accurate formulation of completeness of the identities (4) is provided in the next section, see in particular relations(26), (27). Here we proceed from the intuitive understanding that the consequences τ , being defined as independent linear combinations of the equations by relations (2), cannot admit any other dependency with ∂ i S besides the onefollowing from the definition. Given relation (19), the choice (20) defines zero order of the Stueckelber gauge symmetry (13)for the original fields: (0) R iα = Γ iα . (21)Given zero order of the expansion for the gauge transformations (13) in ξ , and zero and first orderin the Stuekelberg action (11), S St ( φ, ξ ) = S ( φ ) + τ α ( φ ) ξ α + . . . , δ ǫ φ i = Γ iα ( φ ) ǫ α + . . . , δ ǫ ξ α = ǫ α + . . . , (22)all the higher orders are iteratively defined by relations (17), both for the action, and for thegauge transformations. The procedure of resolving relation (17) with certain k for S ( k +1) and ( k ) R α is inductive. Relations (18), (20), (21) solve equations (17) for k = 0. Substituting this solutioninto ((17) with k = 1, we get the equation for S (2) and (1) R α . This equation is labeled by index α ,and it is linear in ξ β . Once the relation has to be met for any ξ β , the equation is a square matrix.The symmetric part of the matrix defines the structure coefficient W αβ of S (2) , while the anti-symmetric part defines the structure coefficients of (1) R α . The solution involves certain ambiguityrelated to the fact that gauge generators are defined modulo on-shell vanishing contributions, andup to a linear combination. It is the ambiguity which is common for any gauge theory. Given ( l ) R α , S ( l +1) , l = 0 , , . . . , k , they all are substituted into equation (17) of the order k + 1 that defines ( k +1) R α and S ( k +2) . This iterative procedure is unobstructed at any stage. This is seen from thealgebraic consequences of the gauge identity studied in the next section. The formal proof ofconsistency of the procedure for inclusion the Stueckelberg feilds is provided in Section 4 by theHPT tools.3. Gauge algebra of the involutive closure of Lagrangian system
In this article, the class of field theories is considered such that the action does not admitgauge symmetry , while the Lagrangian equations are not involutive. The involutive closure (3) isnon-Lagrangian as such, though it is equivalent to the original Lagrangian system (1). Since theoriginal equations (1) are complemented by their consequences τ α (2), the extended system (3)admits gauge identities (4). These identities are unrelated to any gauge symmetry. This is possiblebecause the involutively closed system (3) is non-Lagrangian, so the second Noether theorem does Inclusion of Stueckelberg fields in the models enjoying another gauge symmetry can be considered along the samelines, though the procedure would need some adjustments.
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 11 not apply. The procedure of inclusion Stueckelberg fields described in the previous section isintended to convert the identities (4) into Noetherian ones by introducing the new fields ξ andgauge symmetry such that the generators of the identities are converted into gauge generators(22).The gauge identities (4) turn out having a sequence of consequences of their own. The gaugeidentities and their consequences are understood as the gauge algebra of the involutive closure(3). It is the algebra which is behind existence of the solution to the conversion equations (17) inevery order. This algebra is considered in this section.Let us begin with the remarks concerning equivalence relations for the generators of conse-quences Γ iα (2). First, notice that the original action S ( φ ) is assumed having no gauge symmetry.Hence, if an identity occurs between Lagrangian equations, the identity generator is trivial, κ i ∂ i S ≡ ⇔ ∃ E ij = − E ji : κ i = E ij ∂ j S . (23)Since the consequences τ α (2) are supposed independent, any identity between them is trivial inthe similar sense κ α τ α ≡ ⇔ ∃ E αβ = − E βα : κ α = E αβ τ β . (24)The generators of the of invloutive closure Γ iα (2) are considered equivalent if they result in thesame consequences τ α . Hence, the difference between equivalent generators Γ iα and Γ ′ iα is a trivialgenerator of the identity between the original equations:Γ iα ( φ ) ∂ i S ( φ ) − Γ ′ iα ( φ ) ∂ i S ( φ ) ≡ ⇔ Γ iα − Γ ′ iα = E ijα ∂ j S, E ijα = − E jiα . (25)Once the consequences τ α (2) are independent, any set of identities (we label the set elements bythe index A ) among the involutive equations (3) is spanned by the identities (4):Λ iA ∂ i S + Λ αA τ α ≡ ⇔ ∃ U αA : Λ iA ∂ i S + Λ αA τ α ≡ U αA (cid:0) Γ i ∂ i S + τ α (cid:1) . (26)The expansion coefficients U aA define the generators of the identities Λ A modulo natural ambiguityΛ iA = U αA Γ iα + E ijA ∂ j S + E iαA τ α , Λ αA = U αA − E iαA ∂ i S + E αβA τ β , E ijA = − E jiA , E αβA = − E βαA , (27) with E A being arbitrary. All the relations above are valid, in principle, for any regular system offield equations admitting irreducible generating set for gauge identities. These relations do notimply that the equations (3) follow from Lagrangian equations (1).Now, let us exploit the fact that the original field equations are Lagrangian to deduce theconsequences of the gauge identities (4). Consider the action of variational vector field Γ α = Γ jα ∂ j onto the consequence τ β (2). On shell, this amounts to the second variation of the action S alongthe field Γ α . Since the variational derivatives commute, the second variations of the action alongvariational vector fields commute on shellΓ iα ( φ ) ∂ i τ β ( φ ) ≈ Γ iα Γ jβ ∂ ij S = Γ iβ Γ jα ∂ ij S . (28)Once the matrix Γ iα ( φ ) ∂ i τ β is symmetric on shell, off shell the symmetry can be broken by thecontributions proportional to τ α and ∂ i S :Γ iα ( φ ) ∂ i τ β = W αβ + R iαβ ∂ i S + R γαβ τ γ , W αβ = W βα . (29)The matrix W αβ is off shell symmetric. On shell, W αβ ≈ Γ iα Γ jβ ∂ ij S . The structure coefficients R γαβ , R iαβ do not have certain symmetry w.r.t. the lower labels. Consider antisymmetric part ofrelations (29), and use the definition of τ α (2) (cid:16) Γ i [ α ( φ ) ∂ i Γ jβ ] − R j [ αβ ] (cid:17) ∂ j S − R γ [ αβ ] τ γ ≡ . (30)The relation above is the identity between the original Lagrangian equations ∂ i S and their conse-quences τ α . Any identity of this type reduces to the basic identity (4) according to relation (26).The coefficients in this identity are connected with each other by relation (27). Applying (27) tothe specific identity (30) we arrive at the following relations defining the set of structure functions R iαβ , R γαβ involved in (30) in terms of a single independent structure coefficient U γαβ and arbitrarystructure functions E :Γ jα ∂ j Γ iβ − Γ jβ ∂ j Γ iα − U γαβ Γ iγ − R iαβ + R iβα − E jiαβ ∂ j S − E iγαβ τ γ = 0; (31) U µαβ − R µαβ + R µβα + E jµαβ ∂ j S − E µναβ τ ν = 0 . (32)Let us briefly comment on relations (31), (32). The first of them demonstrates that the gener-ators of the consequences Γ iα do not necessarily form an on-shell integrable distribution, unlike ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 13 the generators of gauge symmetry. These generators commute on shell to their linear combina-tions with the structure coefficients U γαβ , while R iαβ describes deviation from integrability of thedistribution. Both U γαβ and R iαβ are defined modulo natural off-shell ambiguity, given the originalgenerators Γ iα . Relations (32) identify the anti-symmetric part of structure function R γαβ involvedin the relation (28) as the involution coefficient U γαβ in commutation relations of generators (31).Symmetric parts of R γαβ , R iαβ can be absorbed by on-shell vanishing part of symmetric structurefunction W αβ defined by relation (29).Notice that relations (29), (31), (32) are the immediate consequences of the identity (4). Theyfollow from the fact that the set of identities (4) is complete and irreducible. In their turn, theidentities (4) follow from the definition of the functions τ as independent linear combinations (2)of the l.h.s. of Lagrangian equations (1). Once the original equations are consistent, all theirconsequences, including relations (29), (31), (32) cannot be inconsistent.Once the structure coefficients R γαβ , R iαβ , W αβ are found from relations (29), (31), (32) modulonatural ambiguity, they define the first order of the Stueckelberg gauge symmetry generators andthe second order of Stueckelberg action, (1) R iα = R iαβ ξ β , (1) R γα = R γαβ ξ β , S (2) = 12 W αβ ξ α ξ β . (33)Notice that if the generators Γ iα of consequences included into involutive closure of Lagrangianequations form integrable distribution, (1) R iα will vanish, as R iαβ = 0. In this case, the Stueckelbergsymmetry does not mix up the original fields with the Stueckelberg ones, much like it happensin the “Stueckelberg trick”. If the distribution generated by Γ is not integrable, i.e. R iαβ = 0,the deviation from integrability is included into the generator of Stueckelberg symmetry at thefirst order in ξ , compensating non-commutativity of zero order term. As demonstrated in thenext section, the iterative procedure consistently continues in the higher orders, and it inevitablyresults in the gauge transformations with on-shell integrable distribution.Concluding this section, let us mention that relations (29), (31), (32) can have further conse-quences involving higher structure functions. These higher structures contribute to the higherorders in the Stueckelberg action and gauge generators. All these higher relations should be alsoconsistent as the original Lagrangian equations are supposed having no contradictions, and henceany inconsistency is impossible in their consequences. BV master equation for the Stueckelberg gauge symmetry
In this section, the BV formalism is rearranged to serve as a tool for consistent inclusion ofStueckelberg fields.If the Stueckelberg action S St ( φ, ξ ) (11) and the corresponding gauge generators R iα ( φ, ξ ) , R βα ( φ, ξ )(13), (15) would have been known from the outset, it could be considered as the usual Lagrangiangauge theory without any distinction between Stueckelberg fields ξ α and original fields φ i . Then,the master action can be constructed for the gauge system along the usual lines of the BV for-malism [19], [20]. The ghosts C α are assigned to all the gauge parameters ǫ α , and the anti-fieldsare introduced for all fields, including ghosts. The usual Grassmann parity and ghost numbergradings are imposed on the fields and antifields : ǫ ( φ i ) = ǫ ( ξ α ) = 0 , ǫ ( C α ) = 1 , gh ( φ i ) = gh ( ξ α ) = 0 , gh ( C α ) = 1; ǫ ( φ ∗ i ) = ǫ ( ξ ∗ α ) = 1 , ǫ ( C ∗ α ) = 0 , gh ( φ ∗ i ) = gh ( ξ ∗ α ) = − , gh ( C ∗ α ) = − . (34)The BV action is sought for as a solution to the master equation( S BV , S BV ) = 0 , gh ( S BV ) = 0 , ǫ ( S ) = 0 , (35)where ( · , · ) is the anti-bracket( A, B ) = ∂ R A∂ϕ I ∂ L B∂ϕ ∗ I − ∂ R A∂ϕ ∗ I ∂ L B∂ϕ I , ϕ I = ( φ i , ξ α , C α ) , ϕ ∗ I = ( φ ∗ i , ξ ∗ α , C ∗ α ) . (36)To solve the master equation, the usual setup of the BV formalism for irreducible gauge systemswould imply imposing the boundary condition on the action, S BV ( ϕ, ϕ ∗ ) = S St ( φ, ξ ) + C α (cid:0) R iα ( φ, ξ ) φ ∗ i + R γα ( φ, ξ ) ξ ∗ γ (cid:1) + . . . . (37)Here, the first term is the gauge invariant action, and the second one includes generators of thegauge symmetry of the action multiplied by corresponding ghosts C α and anti-fields φ ∗ i , ξ ∗ γ . Thisterm has anti-ghost degree 1. The dots stand for the terms of higher anti-ghost degrees. Theanti-ghost degree is imposed in usual way [21]: agh ( φ ∗ i ) = agh ( ξ α ) = 1 , agh ( C ∗ α ) = 2; agh ( φ i ) = agh ( ξ α ) = agh ( C α ) = 0 . (38) For simplicity, we consider the case with even gauge symmetries and even original fields. Adjustments are madeto the odd case by inserting known sign factors.
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 15
Given the boundary condition (37), the higher terms can be iteratively found from the masterequation (35) by expansion with respect to the anti-ghost degree. The unique existence of thesolution can be proven by the usual HPT tools [20], [21].From the perspective of including Stueckelberg fields in the BV formalism, the boundary condi-tion (37) is unsuitable, because neither the Stueckelberg action S St is known from the outset, norare the gauge generators R βα , R iα . The action is known up to the first order in Stueckelberg fields(12), while the gauge generators are known only at ξ α = 0, see (20), (21). Hence, the known partof the boundary condition (37) reads S BV ( ϕ, ϕ ∗ ) = S ( φ ) − τ α ( φ ) ξ α + C α Γ iα ( φ ) φ ∗ i + C α ξ ∗ α + . . . , (39)where the dots stand for the higher orders of anti-fields and Stueckelberg fields. So, to find thesolution for the master equation (35) of the Stueckelberg theory one has to proceed from theboundary condition (39) iterating the solution order by order w.r.t. anti-fields and Stueckelbergfields. This means, another resolution degree has to be imposed instead of the anti-ghost number(38) such that would be nonzero for Stueckelberg fields. The boundary condition (39) should be,at maximum, of the first order in the resolution degree, so ξ has to be assigned the weight 1. So,we impose the following resolution degree: deg ( ξ α ) = deg ( ξ ∗ α ) = deg ( φ ∗ i ) = 1 , deg ( C ∗ α ) = 2 , deg ( C α ) = deg (Φ i ) = 0 . (40)The solution to the master equation (35) is sough for as the expansion of the action S BV ( ϕ, ϕ ∗ )w.r.t. the resolution degree, S BV ( ϕ, ϕ ∗ ) = X k =0 ( k ) S , deg ( k ) S = k . (41)Once the solution is found in all the orders resolution degree, the complete Stueckelberg action isextracted as zero order w.r.t. to the anti-ghost number (i.e. with switched off anti-fields), whilethe Stueckelberg gauge generators are defined by the first order of S BV w.r.t. the anti-ghost degree(i.e. as the coefficients at ξ ∗ γ and φ ∗ i ).Consider the master action up to the next order of the resolution degree after the boundarycondition (39), S BV ( ϕ, ϕ ∗ ) = S ( φ ) − τ α ξ α + C α (cid:0) Γ iα ( φ ) φ ∗ i + ξ ∗ α (cid:1) + + 12 W αβ ξ α ξ β + C α (cid:0) R γαβ ξ β ξ ∗ γ + R iαβ ξ β φ ∗ i (cid:1) ++ 12 C β C α (cid:0) U γαβ C ∗ γ + φ ∗ j φ ∗ i E ijαβ + ξ ∗ µ φ ∗ i E µiαβ + ξ ∗ µ ξ ∗ ν E µναβ (cid:1) + . . . , (42)where all the structure coefficients are supposed to be functions of the original fields. The firstline in this expression is the boundary condition (39) defined by the original action, and by thegenerators Γ iα of the consequences τ α included in the involutive closure of the system. The nextlines include the most general expression with the resolution degree 2, and ghost number 0. Thestructure coefficient involved in (2) S are defined by the master equation. Let us expand the l.h.s.of the master equation w.r.t. the resolution degree up to the first order. Notice that ( k ) S , k > S BV , S BV ) = 2(Γ iα ∂ i S + τ α ) C α ≡ , (43)( S BV , S BV ) = 2 ξ γ (Γ iα ∂ i τ γ − R iαγ ∂ i S − R βαγ τ β − W γα ) C α −− C α C β (cid:0) φ ∗ i (Γ jα ∂ j Γ iβ − Γ jβ ∂ j Γ iα − U γαβ Γ iγ − R iαβ + R iβα − E jiαβ ∂ j S − E iγαβ τ γ ) −− ξ ∗ µ ( U µαβ − R µαβ + R µβα + E jµαβ ∂ j S − E µναβ τ ν ) (cid:1) = 0 . (44)Relation (43) is valid, given the gauge identity (4). The first order of the master equation (44)holds by virtue of identities (29), (31), (32) upon identification of the structure coefficients in theexpansion (42) with corresponding structure functions in the mentioned identities.As we have seen, the solution to (35) exists up to the second order w.r.t. resolution degree (40).Let us consider the general order k . Substitute the expansion (41) into the master equation andtake k -th order. It has the following structure( S BV , S BV ) k = δ ( k +1) S + B k ( S, (1) S , . . . , ( k ) S ) , (45)where B k involves only ( l ) S ) , l ≤ k , and the operator δ reads: δO = − ∂ R O∂φ ∗ i ∂ i S − ∂ R O∂ξ ∗ α τ α + ∂ R O∂C ∗ α (cid:0) φ ∗ i Γ iα + ξ ∗ α (cid:1) + ∂ R O∂ξ α C α . (46)By virtue of identity (4), the operator δ squares to zero, δ O = ∂ R O∂C ∗ α (cid:0) Γ iα ∂ i S + τ α (cid:1) ≡ , (47) ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 17 so it is a differential. Obviously, δ decreases the resolution degree by one, deg ( δ ) = − . (48)Notice that δ is acyclic in the strictly positive resolution degree because the identities (3) areindependent, i.e. δX = 0 , deg ( X ) > ⇔ ∃ Y : X = δY . (49)By Jacobi identity ( S, ( S, S )) ≡ , ∀ S . Expanding the identity w.r.t. the resolution degree, onecan see that B k of relation (45) is δ -closed, δB k = 0 , k > . (50)Then, because of (49), B k is δ -exact, ∃ Y k +1 : B k = δY k +1 , deg ( Y k +1 ) = k + 1 . (51)Substituting (51) into (45) we arrive at the relation δ (cid:18) ( k +1) S + Y k +1 (cid:19) = 0 . (52)This provides solution for ( k +1) S ( k +1) S = − Y k +1 + δZ k +2 , deg ( Z k +2 ) = k + 2 . (53)The solution is unique modulo natural δ -exact ambiguity.In this way, one can iteratively find the master action of the Stueckelberg theory, given theoriginal action, generators Γ iα of consequences τ α (2) included into the involutive closure (3)of Lagrangian system. The solution is unobstructed at any order. Once the master action isfound, its zero order w.r.t. the anti-ghost number defines complete Stueckelberg action, while theStueckelberg gauge generators are defined by the first order of agh , in accordance with (37).In the end of this section, notice some similarity between the BV formalism for the Stueckelbergembedding described above, and the BV formalism for the field theories with unfree gauge sym-metry [22]. In the latter case, the BV formalism [22] also involves the compensatory fields ξ with By unfree gauge symmetry, we mean the case when the gauge parameters are constrained by differential equations.The most known example of the unfree gauge symmetry is the unimodular gravity where the gauge parameters areconstrained by transversality equation. ghost number 0 and resolution degree 1. These fields compensate, in a sense, the constraints im-posed on the gauge parameters making the effective gauge symmetry parameters unconstrained.These variables can be viewed, in a broad sense, as Stueckelberg fields. In the case of unfreegauge symmetry, however, there is no pairing between the gauge symmetry parameters and thecompensator fields, unlike the case of Stueckelberg symmetry considered in this article.5.
Concluding remarks
Let us make concluding remarks and discuss further perspectives. At first, let us sum up theproposed scheme of including Stueckelberg fields.Proposed inclusion of Stueckelberg fields proceeds from involutive closure of Lagrangian sys-tem (3) which includes the consequences (2) of the original Lagrangian equations. The involutivesystem (3) does not admit any lower order consequence, while the original Lagrangian equationsare not necessarily involutive. The Hamiltonian analogue of the involutive closure is the com-pletion of the original equations by secondary constraints, being zero order consequences of theoriginal primary constraints and the first order Hamiltonian equations. The involutive closure ofLagrangian equations allows explicitly covariant degree of freedom count, see relations (61), (65)in the Appendix. The explicit control of the DoF number in the involutive closure allows one toconsistently include interactions in the covariant form for the second class systems, see (5), (6),(7). This method has been first proposed in the article [11] (for applications of the method inspecific models see, for example, [12], [13], [14], [15]). The original proposal of [11] allows oneto find all the consistent vertices in the field equations, including non-Lagrangian one, withoutdistinctions between variational and non-variational interactions. In this articles we added a sideremark which allows one to identify all the consistent Lagrangian interactions. In this article,however, the involutive closure is considered not for its own sake, but as a starting point forinclusion of Stueckelberg fields.The proposed procedure implies to introduce the Stueckelberg field ξ α for every consequence(2) added to the Lagrangian equations to form the involutive closure (3). The Stueckelberg action(11) and gauge symmetry generators (13) are sought for as the power series in Stueckelberg fieldsproceeding from the requirement of gauge symmetry. The boundary conditions (22) for the actionand gauge generators are defined by the original action and generators Γ iα of consequences τ α included into the involutive closure (3). Given the boundary conditions, relations (17), being the ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 19 expansion coefficients of the Noether identities for Stueckelberg gauge symmetry, allow one toiteratively find order by order all the expansion coefficients of the action and gauge generators.This iterative procedure of inclusion Stueckelberg fields is unobstructed at every stage. Uponinclusion of Stueckelberg fields, the theory has the same DoF number (see in the Appendix),while the gauge fixing ξ = 0 is admissible. This means the equivalence of the Stueckelberg theoryto the original one. It is worse to mention that existence of the Stueckelberg embedding of theoriginal system does not impose any restriction on the generators Γ iα (2) besides independence ofconsequences τ α included into involutive system (3. In particular, the generators of consequencesΓ, being the leading terms in Stueckelberg gauge symmetry transformations of original fields(22), are not required to form the integrable distribution, even on shell. This contrasts to thelogic of the most common form of ”Stueckelberg trick” which implies that the Stueckelberg gaugetransformations of the original fields begin with the gauge generators of gauge symmetry of apart of original action. Being the gauge generators, they inevitably commute to each other, atleast on shell. As is seen from this article, the integrability of the distribution generated by Γ’s isunnecessary for existence of consistent Stueckelberg embedding. The algebraic structure behindthe consistency of the proposed Stueckleberg embedding is described in Section 3. This algebrafollows from the gauge identities (4) of the involutive closure (3) much like the gauge algebra ofthe gauge invariant system follows from the Noether identities. In the case of Noether identities ofgauge invariant system, the structure relations of the gauge algebra are deduced proceeding fromthe three factors: (i) dependence of the equations, (ii) independence of the generators of identities,and (iii) the gauge symmetry of the equations. It is the third factor which does not apply tothe gauge identity (4), while the first two ones do. Therefore, similar structure relations followfrom the first two factors, while the integrability of the gauge distribution, being a consequenceof the third factor is not required. The structure relations of the gauge algebra of the closureof Lagrangian system, being described in Section 3, are equivalent to the equations of Section 2defining the Stueckelberg action and Stueckelberg gauge gauge generators. Once the gauge algebrais consistent (as it is a consequence of the non-contradictory identities (4)), this explains why theStueckelberg embedding is unobstructed at all the stages.In this article, also BV formalism is proposed to perform the Stueckelberg embedding proceedingfrom the involutive closure of Lagrangian system (3). There are three key distinctions from theconventional BV-formalism [19], [20]. The first difference is that the boundary conditions (39) imply to specify, besides the action, also the generators of consequences Γ iα and the consequences τ α included into the involutive closure of Lagrangian equations (3). The second difference is theset of variables. Once the original action is non-gauge invariant, no ghost would be introducedunder the usual BV procedure [19]. For inclusion of the Stueckelberg embedding, the ghostsare assigned to every generator of consequence Γ iα , and the Stueckelberg field ξ α is introducedfor every consequence τ α (34). The anti-fields are introduced for all the fields, including ghostsand Stueckelberg fields. The third distinction is that the solution of the master equation isiterated w.r.t. a different resolution degree (40) which counts the orders both anti-fields andStueckelberg fields. The solution of master equation exists at every order of the iterative procedure,no obstruction can arise. Once the master action is found, the anti-field independent part gives theStueckelberg action, while the the first order in the anti-fields gives the generators of Stueckelberggauges symmetry. This gives another way to construct the Stueckelberg embedding of the originaltheory.Concerning further perspectives, two directions can be mentioned for developing the proposedmethod. The first direction is related to the possibilities of applying this method to specific modelsof field theory. From this perspective the potential advantage of the method is that it allows oneto control the degree of freedom number in an explicitly covariant fashion at every stage, beit inclusion of interactions, on Stueckelberg embedding. Among the models of current interestvarious generalizations of gravity can be mentioned where the original Lagrangian equations arenot involutive, and hence the second class constraints arise. On the other hand, in these models donot offer any obvious hint for the “Stueckelberg trick” which would correspond to the conversionof the second class constraints at Hamiltonian level. In particular, this concerns generalizationsof unimodular gravity [23], [24] and “new massive gravity” in 3 d [25], [26], [27]. The proposedmethod seems an appropriate tool to study dynamics in the models of this type. These models,being complex by themselves, are not suitable, however, as touchstones for the first article no thegeneral method, so the applications will be studied elsewhere.Another aspect of possible future developments concerns generalizations of the method as such.With this regard, the inclusion of the case of gauge invariant actions whose Lagrangian equationsadmit the lower order consequences. The extension of the procedure for inclusion Stueckelbergfields seems straightforward. One more issue concerns the case where the reducible set of conse-quences is included into the involutive closure (3) of Lagrangian system. Notice that the involutive ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 21 closure is not unique, as the higher order consequences can be added without breaking involu-tivity, and also the generating set of consequences can be chosen over-complete. Therefore, thesame action can admit different involutively closed extensions of Lagrangian equations. The setof Stueckelberg fields is defined by specific involutive closure of Lagrangian equations. That iswhy, the multiple choice is possible for the set of Stueckelberg fields for the same Lagrangian,including, in principle, reducible sets of gauge generators. One more potential aspect of interest inthis story is related to gauge fixing. Hypothetically, there can be the case when the gauge fixingis admissible which kills the original fields while the physical degrees of freedom are described bythe Stueckelberg fields. This would allow to construct dual formulations of the same dynamicsconnecting them by different schemes of inclusion Stueckelberg fields and different gauge fixing.Finally, let us mention the issue of locality of the Stueckelberg embedding procedure proposedin this article. The existence of the embedding is proven in Section 4 in terms of condensednotation that could potentially hide the obstructions related to locality. There is, however, anon-rigorous reason to believe that the locality problem should not arise. Once the action andthe generators of consequences (2) are of the finite order, the structure functions of the algebra(see Section 3), being consequences of the identities (4), should involve the finite number of thederivatives. The theory involve a natural invertible operator W αβ (29) which defines the kineticterm for Stueckelberg fields. The inverse can be non-local, but the embedding procedure does notneed to invert W . Acknowledgements.
The author thanks A. Sharapov for valuable discussions of this work andcomments on the manuscript. The part of the work concerning inclusion of Stueckelberg fieldsis supported by the Foundation for the Advancement of Theoretical Physics and Mathematics“BASIS”. The BV-formalism for inclusion of Stueckelberg fields is worked out with the supportof the Ministry of Science and Higher Education of the Russian Federation, Project No. 0721-2020-0033.
AppendixDegree of freedom count in involutive closure of Lagrangian systems
In this Appendix, the relations are provided for DoF number count in the involutively closedsystems. First these relations have been obtained in the article [11] for general involutive fieldequations , not necessarily being the involutive closure of any Lagrangian system. Here, theserelations are slightly re-arranged to be more convenient for making simplifications related to thesystems being the involutive closure of Lagrangian equations (3).In the Appendix, all the indices are by default understood as numerical labels, not condensedones. Exceptions are specially reported in each case.The degree of freedom number is counted in terms of orders of equations, gauge identities andgauge symmetries. Let us explain the definitions of these orders.Consider a system of field equations E I ( φ, ∂φ, . . . , ∂ n I φ ) = 0 , (54)where n I is the maximal order of partial derivatives involved in the equation with the label I .The number n I is considered as the order of equation n I . The partial derivatives by all space-timecoordinates are treated on the equal footing, and the order n I accounts for the mixed derivatives.For example, the order of the equation ∂ φ∂x∂t = 0 is 2.A system of equations is considered involutive if it does not admit such consequences of a lowerorder that are not yet included in the original system. If the system is not involutive, it canbe always complemented by the lower order consequences, to make it involutive. In principle,higher order consequences can be also added, to make the system involutive. Completion of thesystem by the consequences is understood as involutive closure of the system, if no new lowerorder consequences exist. The involutive closure has the same solutions as the original system. Inthis sense, the involutive closure is equivalent to the original system.Let the equations (54) admit gauge identities L IA E I ≡ , (55) Certain regularity assumptions are made for deducing these relations, see in [11]. Here, the regularity issues arenot addressed.
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 23 where L IA are the differential operators with the field depending coefficients, L IA = l IA X k =0 L IAµ ...µ k ( φ, ∂φ, ∂ φ . . . ) ∂ µ . . . ∂ µ k l IA ∈ N . (56)Operator L IA is called the generator of gauge identity, and l IA is the order of the operator. Theorder i A of the gauge identity (55) with the label A is defined by the following rule: i A = max { I } (cid:0) l IA + n I (cid:1) , (57)i.e. it is the maximal aggregate order of the identity generator and the equation it acts on.Suppose the involutive system (54) admits gauge symmetry transformations, δ ǫ φ i = R iα ǫ α , δ ǫ E I ≈ , ∀ ǫ α , (58)where the gauge generators R iα are the differential operators with the field-depending coefficients, R iα = r iα X k =0 R iµ ...µ k α ( φ, ∂φ, . . . ) ∂ µ · · · ∂ µ k . (59)The gauge variation (58), by definition, leaves the mass shell invariant.The order r α of the gauge transformation with specific parameter ǫ α is defined as the maximalnumber of derivatives acting on the parameter in the transformation of any field, r α = max { i } (cid:0) r iα (cid:1) . (60)Given the involutive equations with the complete set of gauge identities, and gauge symmetries,the DoF number is counted by the rule N DoF = X I n I − X A i A − X α r α (61)So, the DoF is computed as follows: the total order of the identities and the total order of gaugesymmetries are subtracted from the total order of the equations. Notice two interesting features ofthis counting recipe. First, it does not involve the number of fields. Second, zero orders relations(of any sort, be it equation, identity, or gauge symmetry) do not contribute to the DoF number.This relation for N DoF has been first deduced in the article [11] in a slightly different form. The recipe (61), being equivalent to the original one, is more convenient for simplifications accountingfor specifics of involutive closure of Lagrangian systems.Let us apply the DoF number counting recipe (61) first to the involutive closure of Lagrangiansystem (3). Denote as n α the order of consequence τ α (2), and let g iα be the order of the differentialoperator Γ iα generating the consequence, and n i is the order of Lagrangian equation (1), Byconstruction (2), the order of consequence τ α cannot exceed the maximal aggregate order of thegenerator of consequence and original Lagrangian equations (1) n α ≤ max { i } (cid:0) g iα + n i (cid:1) . (62)If only lower-order consequences are included, then this is a strict inequality. With the conse-quences whose order is higher than the original equations, the equality is possible. The maximumof the l.h.s. inequality (62) is reached at certain i , which is denoted ¯ i ,max { i } (cid:0) g iα + n i (cid:1) = g ¯ iα + n ¯ i . (63)The order of the corresponding operator Γ ¯ iα is unique for given α . This order is denoted just g α .In the identity (4), the coefficient at τ α has zero order. Given the inequality (62), the order ofidentity (4), being defined by the rule (57), reads i α = g α + n ¯ i . (64)As a result, for the DoF number of any Lagrangian system without gauge symmetry can becounted making use of the orders related to its involutive closure (3): N DoF = X i n i + X α ( n α − i α ) = X i =¯ i n i + X α ( n α − g α ) . (65)The above formula allows one to count DoF number of any Lagrangian system in explicitly co-variant way. In this form, it works for the case without gauge symmetry (second class constrainedsystems, from Hamiltonian perspective). To account for the gauge symmetry, one has to sub-tract the total order of gauge symmetry generators (see (61)) in the irreducible case. Furtheradjustments can be made for the reducible gauge symmetry. If the Lagrangian system had gauge symmetry, one should additionally subtract the total order of gauge symmetrytransformations, see (59), (60)
ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 25
Now, let us discuss the DoF number upon inclusion Stueckelberg fields. Consider the Stueckel-berg action and gauge symmetry transformations, S St = S ( φ ) + τ α ξ α + 12 W αβ ξ α ξ β + · · · , δ ǫ ξ α = ǫ α + · · · , δ ǫ φ i = Γ iα ǫ α + · · · , (66)where · · · stand for the higher orders in ξ . From the perspective of the DoF number counting,there are two main distinctions between the involutive closure (3) of the original Lagrangiansystem, and the Stueckelberg theory (66). First, the consequences τ α are replaced by Lagrangianequations for ξ α τ α ( φ ) ∂ S St ∂ξ α = τ α + W αβ ξ β + · · · , (67)where the indices are condensed. Denote ¯ n α the order of the equation for Stueckelberg field ξ α Notice that the operator W αβ results from the action of the operator Γ iα on τ β . This means thatthe total order of the equations ∂ S St ∂ξ α exceeds the total order of the equations τ β by the total orderof operators Γ, i.e. X α ¯ n α = X α ( n α + g α ) . (68)Hence, inclusion of Stueckelberg fields increases the positive contribution to the DoF number (65)by P α g α . The second relevant distinction of the Stueckelberg theory (66) from the involutiveclosure of original Lagrangian equations (3) is the gauge symmetry. The order of the gaugetransformation is g α , at least in the first approximation in Stueckelberg fields. Hence, the negativepart of the DoF count changes to the same number P α g α . These two changes cancel each other,so the DoF number remains unchanged.Below, we illustrate involutive closure, inclusion of Stueckelberg fields and DoF counting by twosimple examples: mechanical toy model, and Proca Lagrangian. Example 1. Mechanical toy model.
As a toy example, consider L = ˙ x . The Lagrangianequation, ¨ x = 0, is involutive, as it does not admit any lower order differential consequence.However, one could add a higher differential consequence, being just a derivative of the equation.This extension is also involutive system, as no lower order consequences exist. The system (3) inthis case reads, δSδx = − ¨ x = 0 , τ = ... x = 0 , (69) Here, the labels are understood as condensed indices. and the generator of the consequence Γ is just ddt . The identity (4) between the equations (3) inthis case reads τ + Γ δSδx ≡ ... x − ... x ≡ x , i.e. n i = 2. There is one consequence of the third order τ = ... x ,i.e. n α = 3. There is one generator of consequence Γ = ddt , it is of the first order, i.e. g = 1.Substituting these numbers into (65), one gets N DoF = 2 + 3 − (2 + 1) = 2, as it should be. Now,given the involutive system of Lagrangian equation and its consequence (69), let us include theStueckelberg fields following the procedure of Section 2. The operator W (29) in this case is just d dt , so the Stueckelberg Lagrangian (modulo total derivative) and gauge symmetry transformations(66) read L St = 12 (cid:16) ˙ x + ¨ ξ (cid:17) , δ ǫ ξ = ǫ , δ ǫ x = − ˙ ǫ . (71)The degree of freedom count adds one to the positive part of the sum (65) as the third orderconsequence ... x = 0 is replaced by the fourth order equation for ξ . Simultaneously, 1 is added tothe negative contribution, because the first order gauge symmetry is switched on. Obviously theNoether identity for Lagrangian (71) at ξ = 0 reproduces the identity (70) of the system (69).The equivalence of the Stueckelberg Lagrangian to the original one is obvious, as the gauge fixingcondition ξ = 0 is admissible, and the Stueckelberg equations reduce to (69) in this gauge.Let us comment on this elementary example from the perspective of conversion method for theHamiltonian second class constrained systems. The Lagrangian L = ˙ x could be considered asincluding accelerations. Then, the Ostrogradski method should be applied. The velocity wouldbecome an auxiliary canonical coordinate, whose conjugate momentum should vanish due to theprimary constraint. Conservation of the primary constraint results in the secondary constraintwhich connects the canonical momentum of the original coordinate with the velocity. The pairof primary and secondary constraints is of the second class. If they are converted into the firstclass, we arrive at gauge invariant Hamiltonian action. All the momenta can be excluded by theinverse Legendre transformation in this action, and we arrive at Lagrangian (71). This exampledemonstrates that the covariant procedure of inclusion Stueckelberg fields proposed in Section 2directly corresponds to the Hamiltonian conversion of the second class systems. It also illustrates ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 27 the fact that the method works well proceeding from any extension of Lagrangian system by theconsequences, including the higher order ones, if the starting point is an involutively closed system.
Example 2. Proca action.
Let us exemplify DoF count recipe (65) by the Proca equations formassive vector field, δS P roca δA µ ≡ (cid:0) δ µν ( (cid:3) − m ) − ∂ µ ∂ ν (cid:1) A ν = 0 . (72)The Proca equations is a system are of four equations of the second order, so n i = 2 , i = 1 , , , τ = − ∂ µ δS P roca δA µ ≡ m ∂ µ A µ . (73)So, we have one first order consequence τ , to be added for the sake of involutive closure. Thismeans, n α = 1. The generator Γ µ = ∂ µ of the single consequence is the operator of divergence.It has the order one, g = 1. No other equations and consequences are included in the involutiveclosure (72), (73) of Proca system. The gauge identity (4) for the Proca system reads, ∂ µ δS P roca δA µ + τ ≡ . (74)The identity is of the order 3, as the first order operator ∂ acts on the second order Proca equations.Now, let us calculate the DoF number applying relation (65). The total order of the equations inthe involutive closure of the Proca system is 9: there are 4 second order Proca equations (72), plusone first order consequence, P i n i + P α n α = 4 × P α i α = 3. Substituting these numbers into the formula (65), we get 9 − d = 4.Given the involutive closure of the Proca equations, let us introduce the Stueckelberg fieldfollowing the prescription of Section 2. The matrix W (29) in this case is the d’Alembertian, andno higher order corrections appear. So we arrive to the standard Stueckelberg equations, gaugesymmetries, and Noether identities δ S δξ = m ( ∂ µ A µ − (cid:3) ξ ) = 0 , δ S δA µ = ( δ µν (cid:3) − ∂ ν ∂ µ ) A ν + m ( A µ − ∂ µ ξ ) = 0; (75) δ ǫ ξ = ǫ , δ ǫ A µ = ∂ µ ǫ ; δ S δξ − ∂ µ δ S δA µ ≡ . (76)Let us apply to this system the DoF number count recipe (61). There are five second orderequations (75), so the total order of the equations is ten. There is one first order gauge symmetry,and one third order gauge identity. So, the DoF number is 10 − − ENERAL METHOD FOR INCLUDING STUECKELBERG FIELDS 29
References [1] St¨uckelberg, Ernst C.G. (1938). ”Die Wechselwirkungskr¨afte in der Elektrodynamik und in der Feldtheorieder Kr¨afte”. Helvetica Physica Acta (in German). : 225[2] H. Ruegg and M. Ruiz-Altaba, “The Stueckelberg field,” Int. J. Mod. Phys. A (2004), 3265-3348doi:10.1142/S0217751X04019755 [arXiv:hep-th/0304245 [hep-th]].[3] N. Boulanger, C. Deffayet, S. Garcia-Saenz and L. Traina, “Consistent deformations of free massive fieldtheories in the Stueckelberg formulation,” JHEP , 021 (2018) [arXiv:1806.04695 [hep-th]].[4] L. D. Faddeev and S. L. Shatashvili, “Realization of the Schwinger Term in the Gauss Law and the Possibilityof Correct Quantization of a Theory with Anomalies,” Phys. Lett. B (1986), 225-228[5] I. A. Batalin and E. S. Fradkin, “Operator Quantization of Dynamical Systems With Irreducible First andSecond Class Constraints,” Phys. Lett. B (1986), 157-162 [erratum: Phys. Lett. B (1990), 528][6] I. A. Batalin and E. S. Fradkin, “Operatorial Quantization of Dynamical Systems Subject to Second ClassConstraints,” Nucl. Phys. B (1987), 514-528[7] E. S. Egorian and R. P. Manvelyan, “BRST QUANTIZATION OF HAMILTONIAN SYSTEMS WITH SEC-OND CLASS CONSTRAINTS,” YERPHI-1056-19-88 (1988).[8] E. S. Egorian and R. P. Manvelyan, “Quantization of dynamical systems with first and second class con-straints,” Theor. Math. Phys. (1993), 173-181[9] I. A. Batalin and I. V. Tyutin, “Existence theorem for the effective gauge algebra in the generalized canonicalformalism with Abelian conversion of second class constraints,” Int. J. Mod. Phys. A (1991) 3255.[10] I. Batalin, M. Grigoriev and S. Lyakhovich, “Non-Abelian conversion and quantization of non-scalar second-class constraints,” J. Math. Phys. (2005) 072301; [hep-th/0501097].[11] D. S. Kaparulin, S. L. Lyakhovich and A. A. Sharapov, “Consistent interactions and involution,” JHEP (2013) 097 [arXiv:1210.6821 [hep-th]].[12] I. Cortese, R. Rahman and M. Sivakumar, “Consistent Non-Minimal Couplings of Massive Higher-Spin Par-ticles,” Nucl. Phys. B (2014), 143-161 doi:10.1016/j.nuclphysb.2013.12.005 [arXiv:1307.7710 [hep-th]].[13] M. Kulaxizi and R. Rahman, “Higher-Spin Modes in a Domain-Wall Universe,” JHEP (2014), 193doi:10.1007/JHEP10(2014)193 [arXiv:1409.1942 [hep-th]][14] V. A. Abakumova, D. S. Kaparulin and S. L. Lyakhovich, “Stable interactions in higher derivative field theoriesof derived type,” Phys. Rev. D (2019) no.4, 045020 doi:10.1103/PhysRevD.99.045020 [arXiv:1811.10019[hep-th]].[15] R. Rahman, “The Involutive System of Higher-Spin Equations,” Nucl. Phys. B (2021), 115325;[arXiv:2004.13041 [hep-th]].[16] D. S. Kaparulin, S. L. Lyakhovich and A. A. Sharapov, “Classical and quantum stability of higher-derivativedynamics,” Eur. Phys. J. C (2014) no.10, 3072, [arXiv:1407.8481 [hep-th]]. [17] B. S. DeWitt, “Dynamical theory of groups and fields,” Conf. Proc. C (1964) 630701 [Les Houches Lect.Notes (1964) 585].[18] W.M. Seiler, “Involution: The Formal Theory of Differential Equations and its Applications in ComputerAlgebra”, Springer-Verlag, Berlin-Heidelberg, 2010.[19] I. A. Batalin and G. A. Vilkovisky, “Gauge Algebra and Quantization,” Phys. Lett. (1981) 27.[20] I. A. Batalin and G. A. Vilkovisky, “Existence Theorem for Gauge Algebra” J.Math.Phys. (1985) 172-184.[21] M. Henneaux and C. Teitelboim, “Quantization of gauge systems,” Princeton, USA: Univ. Pr. (1992) 520 p.[22] D. S. Kaparulin and S. L. Lyakhovich, “Unfree gauge symmetry in the BV formalism,” Eur. Phys. J. C (2019) no.8, 718; [arXiv:1907.03443 [hep-th]][23] A. O. Barvinsky and A. Y. Kamenshchik, “Darkness without dark matter and energy – generalized unimodulargravity,” Phys. Lett. B (2017), 59-63; [arXiv:1705.09470 [gr-qc]].[24] A. O. Barvinsky, N. Kolganov, A. Kurov and D. Nesterov, “Dynamics of the generalized unimodular gravitytheory,” Phys. Rev. D (2019) no.2, 023542; [arXiv:1903.09897 [hep-th]].[25] E. A. Bergshoeff, O. Hohm and P. K. Townsend, “Massive Gravity in Three Dimensions,” Phys. Rev. Lett. (2009), 201301 [arXiv:0901.1766 [hep-th]].[26] E. A. Bergshoeff, O. Hohm and P. K. Townsend, ‘More on Massive 3D Gravity,” Phys. Rev. D (2009),124042 doi:10.1103/PhysRevD.79.124042 [arXiv:0905.1259 [hep-th]].[27] M. ¨Ozkan, Y. Pang and P. K. Townsend, “Exotic Massive 3D Gravity,” JHEP (2018), 035; [arXiv:1806.04179[hep-th]]. Physics Faculty, Tomsk State University, Lenin ave. 36, Tomsk 634050, Russia.
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