A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity
aa r X i v : . [ g r- q c ] J a n A review on geometric formulations for classicalfield theory: the Bonzom-Livine model for gravity
Jasel Berra–Montiel , , Alberto Molgado , and ÁngelRodríguez–López Facultad de Ciencias, Universidad Autonoma de San Luis PotosiCampus Pedregal, Av. Parque Chapultepec 1610, Col. Privadas del Pedregal, SanLuis Potosi, SLP, 78217, Mexico Dual CP Institute of High Energy Physics, Colima, Col, 28045, MexicoE-mail: [email protected] , [email protected] , [email protected] Abstract.
Motivated by the study of physical models associated with GeneralRelativity, we review some finite-dimensional, geometric and covariant formulationsthat allow us to characterize in a simple manner the symmetries for classical field theoryby implementing an appropriate fibre-bundle structure, either at the Lagrangian, themultisymplectic or the polysymplectic levels. In particular, we are able to formulateNoether’s theorems by means of the covariant momentum maps and to systematicallyintroduce a covariant Poisson-Hamiltonian framework. Also, by focusing on the spaceplus time decomposition for a generic classical field theory and its relation to thesegeometric formulations, we are able to successfully recover the gauge content andthe true local degrees of freedom for the theory. In order to illustrate the relevanceof these geometric frameworks, we center our attention to the analysis of a modelfor 3-dimensional theory of General Relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of thesystem which for this model turn out to be equivalent to the vanishing torsioncondition and the Einstein equations. We also concentrate on the analysis of thegauge symmetries of the system in order to obtain the Lagrangian covariant momentummap associated with the theory and, consequently, its corresponding Noether currents.Next, within the multisymplectic approach, we aim our attention to describing howthe gauge symmetries of the model yield covariant canonical transformations on thecovariant multimomenta phase-space, thus giving rise to the existence of a covariantmomentum map. Besides, we analyze the physical system under consideration withinthe De Donder-Weyl canonical theory implemented at the polysymplectic level, thusestablishing a relation from the covariant momentum map to the conserved currents ofthe theory within this covariant Hamiltonian approach. Finally, after performing thespace plus time decomposition of the space-time manifold, and taking as a startingpoint the multisymplectic formulation, we are able to recover both the extendedHamiltonian and the gauge structure content that characterize the gravity model ofour interest within the instantaneous Dirac-Hamiltonian formulation.Keywords: Multisymplectic formalism, polysymplectic framework, covariant momen-tum map, Immirzi parameter, Einstein equations, gauge theory. review on geometric formulations for classical field theory AMS classification scheme numbers: 70S15, 70S10, 83C05, 53D20.
1. Introduction
A useful method for studying classical field theory is the instantaneous Dirac-Hamiltonian approach as it is able to reveal the true local degrees of freedom andgauge symmetries for a given classical field theory [1]. However, as it is well known, inorder to identify a temporal direction, the instantaneous Dirac-Hamiltonian formulationfor classical field theory starts by performing a foliation of the space-time manifold intoCauchy surfaces, thus concealing the true covariant nature of the theory under analysis.Hence, when implementing a suitable canonical quantization scheme, the instantaneousDirac-Hamiltonian formalism gives rise to non-covariant approaches for quantum fieldtheory [2]. By contrast, based on the De Donder-Weyl canonical theory [3, 4],the multisymplectic approach provides a finite-dimensional, geometric and covariantHamiltonian-like formulation for classical field theory [5, 6, 7, 8]. Roughly speaking,after identifying the fields of the theory as sections of an appropriate fibre-bundle, themultisymplectic formalism starts by introducing the covariant multimomenta phase-space, namely, a finite-dimensional manifold locally constructed by associating to eachfield variable of the system a set of multimomenta (or polymomenta) variables, whichare noting but a covariant extension of the standard instantaneous momenta variablesimplemented within the instantaneous Dirac-Hamiltonian formulation [2]. Particularly,the covariant multimomenta phase-space is endowed with a multisymplectic ( n + 1)-form which is considered as fundamental as it allows not only to obtain the correct fieldequations but also to describe the symmetries of a given classical field theory. In thisregard, we have that within the multisymplectic framework the symmetries of a classicalfield theory give rise to covariant canonical transformations, that is, transformations onthe covariant multimomenta phase-space that preserve the multisymplectic ( n +1)-form.Thus, for the case of infinitesimal covariant canonical transformations generated by theaction of the symmetry group of the theory on the covariant multimomenta phase-space, we can construct the so-called covariant momentum map which allows not onlyto extend Noether’s theorems to the De Donder-Weyl Hamiltonian formulation in anatural way but also to induce the momentum map that characterizes the system fromthe instantaneous Dirac-Hamiltonian perspective [5, 8, 9]. In fact, for classical fieldtheories with localizable symmetries, the latter issue is particularly important since, inlight of the second Noether theorem, one may see that the admissible space of Cauchydata for the evolution equations of the system is determined by the zero level set ofthe momentum map. Hence, according to [5, 9, 10], for this kind of classical fieldtheories, the vanishing of the momentum map gives rise to the complete set of first-classconstraints of the theory, in Dirac’s terminology [1]. In that sense, taking as a startingpoint the multisymplectic approach, we have a natural way to recover the constrainedstructure for a given singular Lagrangian system.It is important to mention that, on the covariant multimomenta phase-space, one review on geometric formulations for classical field theory n + 1)-form on a subspace of thecovariant multimomenta phase-space referred to as the polymomenta phase-space.Such canonical form, known as the polysymplectic form, encodes the relevant physicaldata of a given classical field theory in order to construct a well-defined Poisson-Gerstenhaber bracket for a set of appropriately prescribed differential Hamiltonianforms, thus allowing us to analyze an arbitrary classical field theory in a covariantPoisson-Hamiltonian framework [20, 21, 22]. Some physically motivated examples forwhich the multisymplectic and polysymplectic formalisms have been applied may beencountered in references [5, 8, 9, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Despitetheir mathematical elegance, from our point of view, the analysis of the gauge contentfor a given classical field theory from the perspective of such geometric formulationshas been rarely exploited, especially when considering certain highly non-trivial gaugemodels associated with General Relativity. Therefore, our main aim is to apply thoseformalisms to the description of the gauge structure associated with the Bonzom-Livinemodel for gravity, in particular, by capitalizing the involved symmetries through theintroduction of the covariant momentum map.Since the novel work by Achúcarro and Townsend [35], it is widely known thatthe 3-dimensional Einstein theory of General Relativity can be reformulated as aChern-Simons gauge theory. Such reformulation has been extensively studied at theclassical and quantum levels in reference [36], for example, and may be thought ofas a 3-dimensional analogue of the gauge theory interpretation of the Hamiltonianconstraint equations of 4-dimensional gravity developed by Ashtekar [37, 38], which inturn is a fundamental theoretical framework for the Loop Quantum Gravity approach[39]. Indeed, based on Witten’s reformulation of 3-dimensional gravity as a Chern-Simons gauge theory, Bonzom and Livine have introduced an action that correspondsto a formulation of 3-dimensional Einstein theory of General Relativity that involvesan arbitrary Immirzi-like parameter [40], which may be regarded as a 3-dimensionalanalogue of the Holst action [41]. In brief, the so-called Immirzi parameter correspondsto a quantization ambiguity that arises within the Loop Quantum Gravity approach, aspointed out in [42, 43]. In particular, it has been shown that such ambiguity is relatedto a one-parameter family of canonical transformations on the phase-space of GeneralRelativity that can not be implemented unitarily at the quantum level, as discussed indetail in [44]. Bearing this in mind, the Holst action includes the Immirzi parameter review on geometric formulations for classical field theory n + 1)-formassociated with the system. Then, we focus our attention on describing how thesymmetries of the theory give rise to infinitesimal covariant canonical transformationson the covariant multimomenta phase-space, which, analogously to the Lagrangian case,allows us to construct the covariant momentum map associated with the extendedgauge symmetry group of the model. Besides, at the polysymplectic level, we startby defining the polymomenta phase-space. Thus, in order to carry out a consistentpolysymplectic formulation of the Bonzom-Livine model for gravity, we implementthe proposal developed in [48] to analyze singular Lagrangian systems within thepolysymplectic approach. Following Dirac’s terminology, we thus find that on thepolymomenta phase-space the theory of our interest is characterized by a set of second-class constraint ( n − n − review on geometric formulations for classical field theory
2. Geometric-covariant formalisms for classical field theory
In this section, we will start by introducing the appropriate notation that will behelpful in order to describe the Lagrangian and multisymplectic formalisms as finite-dimensional, geometric and covariant formulations for the characterization of classicalfield theory. Particularly, we will focus our attention on the study of symmetries withinthese frameworks, issue that will be fundamental to formulate Noether’s theorems andto analyze the gauge content of a given classical field theory. Also, we will introduce thepolysymplectic approach, where we will concentrate on the description of the Poisson-Gerstenhaber bracket and its algebraic properties, which will help us to inspect aclassical field theory from a covariant Poisson-Hamiltonian perspective. Finally, we willprovide a brief description of the process to perform the space plus time decompositionfor a generic classical field theory at both the Lagrangian and multisymplectic level,which will eventually allow us to recover the instantaneous Dirac-Hamiltonian analysisof the system under consideration. For this general purpose, we will review the mainideas of the Lagrangian, multisymplectic and polysymplectic formalisms as developed inreferences [2, 5, 6, 7, 8, 9, 10, 20, 21, 49]. Hence, we would like to encourage the readerto examine those references for further technical details. It is important to mention review on geometric formulations for classical field theory C ∞ . Before proceeding with our discussion, we will set the notation that we will usethroughout the rest of the article, however, we will refer the reader to [10, 50, 51] for adetailed account on geometric aspects of fibre-bundles. To start, let M be a manifold.We introduce the triplet ( E, π ME , M ) to denote a fibre-bundle, where M stands for thebase space, E corresponds to the total space (a fibre manifold), and π ME : E → M represents the projector map. At p ∈ M , the subset π − ME ( p ) of E , denoted by E p , iscalled the fibre of E over p . Unless otherwise specified, we introduce E M to denote theset of sections of π ME , that is, the set of maps ˜ κ : M → E satisfying the condition π ME ◦ ˜ κ = Id M , where Id M : M → M corresponds to the identity map on M . Avoidingcumbersome notation, henceforward, a fibre-bundle ( E, π ME , M ) will be just referred toas π ME .In particular, given the tangent bundle ( T M, π
M T M , M ), we introduce(Λ k T M, π M Λ k T M , M ) to be the k-th exterior power of the tangent bundle of M , and wedefine X k ( M ) as the set of sections of π M Λ k T M , namely the collection of k -multivectorfields on M . Thus, for ξ ∈ X k ( M ), we denote by ξ ( p ) ∈ Λ k T p M the k -tangent vectorat p ∈ M assigned by ξ . Besides, given the cotangent bundle ( T ∗ M, π
M T ∗ M , M ), weintroduce (Λ k T ∗ M, π M Λ k T ∗ M , M ) to be the k -th exterior power of the cotangent bundleof M , and we identify Ω k ( M ) as the set of sections of π M Λ k T ∗ M , that is, the family of k -forms on M . Thence, for α ∈ Ω k ( M ), we denote by α ( p ) ∈ Λ k T ∗ p M the k -cotangentvector at p ∈ M assigned by α . In addition, we define Ω ( M ) as the set of functions on M . Bearing this in mind, for π ME a fibre-bundle, we define ( V E, π
E VE , E ) as thevertical tangent bundle of π ME , where the fibre over e ∈ E is defined as V e E := n ζ ∈ T e E (cid:12)(cid:12)(cid:12) T e π ME ( ζ ) = 0 ∈ T π ME ( e ) M o , (1)being T e π ME : T e E → T π ME ( e ) M the tangent map of π ME at e ∈ E . Here, we identify X V ( E ) as the set of sections of π E VE , that is, the collection of vector fields on E tangentto the fibres of π ME [50]. Further, we define (Λ kr T ∗ E, π E Λ kr T ∗ E , E ) as the vector bundleof horizontal ( k ; r )-forms over E , where the fibre over e ∈ E is given byΛ kr T ∗ e E := (cid:26) ω ∈ Λ k T ∗ e E (cid:12)(cid:12)(cid:12)(cid:12) ζ y · · · ζ r y ω = 0 , ∀ ζ , · · · , ζ r ∈ V e E (cid:27) . (2)Thus, we introduce Ω kr ( E ) to denote the set of sections of π E Λ kr T ∗ E , namely the familyof horizontal ( k ; r )-forms on E .Finally, being ( E, π ME , M ) and ( F, π NF , N ) a pair of fibre-bundles, we define abundle morphism from π ME to π NF as a pair of maps (Φ , ¯Φ), such that Φ : E → F and ¯Φ : M → N satisfy the relation π NF ◦ Φ = ¯Φ ◦ π ME [50]. Henceforth, we review on geometric formulations for classical field theory p : E p → F ¯Φ( p ) to denote the restriction of Φ to the subset E p . Thus,a map f : M → N induces a bundle morphism from π M T M to π N T N denoted by(
T f, f ), where
T f : T M → T N is called the tangent map of f . At p ∈ M , given v ∈ T p M , T p f ( v ) ∈ T f ( p ) N is known as the pushforward of v by f and is defined as T p f ( v ) [ g ] := v [ g ◦ f ], being g ∈ Ω ( N ) a function on N [10].It is important to mention that we will implement the above described basicstructures for any manifold or any fibre-bundle to be considered in the following sections. In the present subsection, we will introduce the basic geometric objects to developthe Lagrangian formulation of classical field theory within the fibre-bundle formalismfollowing, as close as possible, references [2, 5, 6, 8, 9, 10, 23, 50, 51]. In particular, wewill focus on the emergence of the symmetries for a generic classical field theory withinthis geometric-covariant Lagrangian formulation. As previously mentioned, our mainmotivation to do this is to enunciate the celebrated Noether’s theorems, which will bea cornerstone for our study.To start, let X be an n -dimensional space-time manifold without boundary andlocally represented by ( x µ ), µ = 0 , . . . , n −
1. Then, given a classical field theory,we define the covariant configuration space of the system as a finite-dimensional fibre-bundle (
Y, π XY , X ), where the classical fields can be identified with Y X , namely the setof sections of π XY [5]. Note that, for x ∈ X , the fibre Y x can be regarded as an m -dimensional manifold locally represented by ( y a ), a = 1 , . . . , m , which allows to identify( x µ , y a ) as an adapted coordinate system on Y . Hence, a section φ ∈ Y X can be locallyrepresented as ( x µ , φ a ( x µ )).As discussed in the literature, the natural arena for describing a first order classicalfield theory within the Lagrangian approach corresponds to the first order jet bundle( J Y, π XJ Y , X ). In order to define it here, we start by introducing the affine jet bundle( J Y, π
Y J Y , Y ) first, where the fibre over y ∈ Y is defined as J y Y := n κ ∈ L ( T x X, T y Y ) (cid:12)(cid:12)(cid:12) T y π XY ◦ κ = Id T x X o , (3)being L ( T x X, T y Y ) the set of all linear mappings κ : T x X → T y Y and x = π XY ( y ) ∈ X the base point associated with y ‡ . In local coordinates an element κ ∈ J y Y can bewritten as κ := dx µ ⊗ ∂∂x µ + y aµ ∂∂y a ! , (4)which allows to identify ( x µ , y a , y aµ ) as an adapted coordinate system on J Y .Consequently, the composition map π XJ Y := π XY ◦ π Y J Y gives rise to the first orderjet bundle ( J Y, π XJ Y , X ). In addition, it is not difficult to see that the tangent map ofa section φ ∈ Y X at x ∈ X , T x φ : T x X → T φ ( x ) Y , is an element of J φ ( x ) Y , and therefore ‡ In fact, π Y J Y is an affine bundle since J y Y is an affine space modeled on the vector space L ( T x X, V y Y ), being V y Y the vertical tangent space over y ∈ Y [2, 6, 50]. review on geometric formulations for classical field theory φ ∈ Y X a section j φ ∈ J Y X of π XJ Y known as the firstjet prolongation of φ , which can be locally represented as ( x µ , φ a ( x µ ) , φ aν ( x µ )), where wehave adopted the notation φ aν ( x µ ) := ∂ ν φ a ( x µ ), being ∂ µ := ∂/∂x µ the partial derivativewith respect to the base space coordinate x µ .Bearing this in mind, the action associated with the system can be written as S [ φ ] := Z X ( j φ ) ∗ L , (5)where the Lagrangian density of the theory, L : J Y → Λ n T ∗ X , is locally given by L := L ( x µ , y a , y aµ ) d n x , with L : J Y → R denoting the Lagrangian function of thesystem § . We would like to emphasize that, from our point of view, the main geometricobjects of interest within the geometric-covariant Lagrangian formulation are the so-called Poincaré-Cartan forms, Θ ( L ) ∈ Ω n ( J Y ) and Ω ( L ) ∈ Ω n +1 ( J Y ), which locallyread Θ ( L ) := ∂L∂y aµ dy a ∧ d n − x µ + L − ∂L∂y aµ y aµ ! d n x , (6a)Ω ( L ) := dy a ∧ d ∂L∂y aµ ! ∧ d n − x µ − d L − ∂L∂y aµ y aµ ! ∧ d n x , (6b)where d n − x µ := ∂ µ y d n x denotes the basis for the π XJ Y -horizontal ( n −
1; 1)-forms on J Y . One may straightforwardly check that the Poincaré-Cartan ( n + 1)-form is simplydefined through the relation Ω ( L ) = − d Θ ( L ) .The set of differential forms (6) is relevant as it contains all the information ofthe theory under consideration. In fact, it is not difficult to see that, for any section φ ∈ Y X , the Poincaré-Cartan n -form satisfies the relation ( j φ ) ∗ Θ ( L ) = ( j φ ) ∗ L , whichallows us to rewrite the action (5) in terms of the Poincaré-Cartan form (6a) and asection j φ ∈ J Y X (see [2] for details). Moreover, the Poincaré-Cartan ( n + 1)-formis necessary in order to write the Euler-Lagrange field equations of the system in aninvariant fashion [5, 6, 8]. In other words, we have that, given a critical point of theaction principle (5), φ ∈ Y X , and an arbitrary vector field on J Y , namely V ∈ X ( J Y ),the condition ( j φ ) ∗ (cid:16) V y Ω ( L ) (cid:17) = 0 , (7)is completely equivalent to the Euler-Lagrange field equations of the theory, namely, ∂L∂y a (cid:16) x µ , φ a , φ aµ (cid:17) − ∂∂x µ ∂L∂y aµ (cid:16) x µ , φ a , φ aµ (cid:17)! = 0 . (8)Now, we are interested in studying the symmetries of the classical field theory(5) within the geometric-covariant Lagrangian formalism, and in particular the actionof a Lie group on the covariant configuration space associated with the system. To § Note that the pullback of the Lagrangian density with a section j φ ∈ J Y X gives rise to an n -formon X . review on geometric formulations for classical field theory G a Lie group and g its corresponding Lie algebra. Then, for all η ∈ G , we say that the action of η on π XY is given by a π XY -bundle automorphism( η Y , η X ), where the maps η Y : Y → Y and η X : X → X satisfy the relation π XY ◦ η Y = η X ◦ π XY . Locally, these transformations read η X ( x µ ) := η µ X ( x ν ) and η Y ( x µ , y a ) := ( η µ X ( x ν ) , η a Y ( x ν , y b )). Subsequently, given the infinitesimal generator of η , ξ η ∈ g , we introduce ξ Xη ∈ X ( X ) and ξ Yη ∈ X ( Y ) to denote the infinitesimal generatorsassociated with the transformations η X and η Y , respectively. Thus, since ξ Xη and ξ Yη mustsatisfy the condition T y π XY ( ξ Yη ( y )) = ξ Xη ( x ) for y ∈ Y and x = π XY ( y ) ∈ X , then weget the local expressions ξ Xη := ξ µ ( x ν ) ∂ µ and ξ Yη := ξ µ ( x ν ) ∂ µ + ξ a ( x ν , y b ) ∂ a , where ∂ a := ∂/∂y a denotes the partial derivative with respect to the fibre coordinate y a . Asa result, by considering y ∈ Y and x = π XY ( y ) ∈ X , we have that the action of η on J y Y may be thought of as the map η J yY : J y Y → J η Y ( y ) Y , which is explicitly given by η J yY ( κ ) := T y η Y ◦ κ ◦ ( T x η X ) − , ∀ κ ∈ J y Y . Here, T y η Y : T y Y → T η Y ( y ) Y correspondsto the tangent map of η Y at y , while ( T x η X ) − : T η X ( x ) X → T x X stands for the tangentmaps of η − X at η X ( x ). Therefore, the induced action of η on J Y , η J Y : J Y → J Y ,locally reads η J Y ( x µ , y a , y aµ ) := (cid:16) η µ X ( x ν ) , η a Y ( x ν , y b ) , (cid:16) ∂ σ η a Y ( x ν , y b ) + ∂ c η a Y ( x ν , y b ) y cσ (cid:17) ∂ µ (cid:16) ( η − X ) σ ( x ν ) (cid:17)(cid:17) , where we have introduced η − X ( x µ ) := ( η − X ) µ ( x ν ) to denote the local representation of η − X . In consequence, a straightforward calculation shows that the infinitesimal generator ξ J Yη ∈ X ( J Y ) associated with the transformation η J Y is explicitly given by ξ J Yη := ξ µ ( x ν ) ∂ µ + ξ a ( x ν , y b ) ∂ a + (cid:16) ∂ µ ξ a ( x ν , y b ) + ∂ c ξ a ( x ν , y b ) y cµ − ∂ µ ξ σ ( x ν ) y aσ (cid:17) ∂ µa , (9)with ∂ µa := ∂/∂y aµ denoting the partial derivative with respect to the fibre coordinate y aµ . Indeed, ξ J Yη := j ξ Yη corresponds to the so-called first jet prolongation of thevector field ξ Yη (see for instance [5, 50, 51]). Henceforward, for all η ∈ G , the pair (cid:16) η J Y , ξ J Yη (cid:17) consisting of a fibre-preserving transformation η J Y : J Y → J Y and itscorresponding infinitesimal generator ξ J Yη ∈ X ( J Y ) will be just referred to as aninfinitesimal transformation on J Y .Thus, in the spirit of [5, 8], we say that G is the symmetry group of theory if forall η ∈ G , the associated infinitesimal transformation (cid:16) η J Y , ξ J Yη (cid:17) on J Y satisfies thecondition L ξ J Yη Θ ( L ) = dα ( L ) η , (10)where L ξ J Yη symbolizes the Lie derivative along the vector field ξ J Yη ∈ X ( J Y ), while α ( L ) η ∈ Ω n − ( J Y ) denotes a π XJ Y -horizontal ( n −
1; 1)-form on J Y locally representedby α ( L ) η = α νη ( x µ , y a ) d n − x ν . (11)From now on, for all η ∈ G , the pair (cid:16) ξ J Yη , α ( L ) η (cid:17) consisting of a vector field ξ J Yη ∈ X ( J Y ) and a π XJ Y -horizontal ( n −
1; 1)-form α ( L ) η ∈ Ω n − ( J Y ) related throughcondition (10) will be referred to as a Noether symmetry [8]. review on geometric formulations for classical field theory h· , ·i : g ∗ × g → R , it is possible to seethat, the action of G on J Y has an associated Lagrangian covariant momentum map[52], that is, a map J ( L ) : J Y → g ∗ ⊗ Λ n − T ∗ J Y such that for all ξ η ∈ g , d J ( L ) ( ξ η ) = ξ J Yη y Ω ( L ) , (12)where J ( L ) ( ξ η ) := h J ( L ) , ξ η i ∈ Ω n − ( J Y ) corresponds to the ( n − J Y explicitly given by J ( L ) ( ξ η ) = ξ J Yη y Θ ( L ) − α ( L ) η . (13)In particular, the Lagrangian covariant momentum map is relevant as it allows toconstruct conserved currents for the solutions of the Euler-Lagrange field equationsof the system. In other words, given ξ η ∈ g and φ ∈ Y X a critical point of the actionprinciple (5), the quantity defined as J ( L ) ( ξ η ) := ( j φ ) ∗ J ( L ) ( ξ η ) , (14)corresponds to a conserved current of the theory [8]. To see this, it is enough toconsider that a critical point of the action (5), φ ∈ Y X , satisfies the condition (7)and therefore we get that ( j φ ) ∗ (cid:16) ξ J Yη y Ω ( L ) (cid:17) = d (cid:16) ( j φ ) ∗ J ( L ) ( ξ η ) (cid:17) = 0. Concisely, theabove result corresponds to the formulation of the first Noether theorem within thegeometric-covariant Lagrangian formalism [5, 6, 8, 10, 51].Further, in order to adapt our geometric formalism to gauge field theories, we willintroduce the notion of localizable symmetries. To this end, we will closely follow thedefinition of this kind of symmetries as presented in [10, 52, 53, 54]. Thus, a set C LS ofpairs (cid:16) ξ J Y , α ( L ) (cid:17) consisting of a vector field ξ J Y ∈ X ( J Y ) and a π XJ Y -horizontal( n −
1; 1)-form α ( L ) ∈ Ω n − ( J Y ) (locally represented as in (11)) is said to be a collectionof localizable symmetries if the following conditions are satisfied: ( i ) C LS forms a vectorspace. ( ii ) Each pair (cid:16) ξ J Y , α ( L ) (cid:17) ∈ C LS is a Noether symmetry. ( iii ) For every pair (cid:16) ξ J Y , α ( L ) (cid:17) ∈ C LS and any two open sets U , U ⊂ X with disjoint closures there existsa pair (cid:16) ˜ ξ J Y , ˜ α ( L ) (cid:17) ∈ C LS satisfying ξ J Y ( β ) = ˜ ξ J Y ( β ) α ( L ) ( β ) = ˜ α ( L ) ( β ) ∀ β ∈ π − XJ Y ( U ) , ˜ ξ J Y ( β ) = 0˜ α ( L ) ( β ) = 0 ∀ β ∈ π − XJ Y ( U ) . (15)Basically, the above definition states that one may deform a Noether symmetry to zeroin the fibres of J Y over a certain region of the base space X .According to [5, 9, 10, 52, 53, 54], the presence of localizable symmetries in aclassical field theory gives rise to trivial Lagrangian Noether charges. In other words,we know that, given Σ t a Cauchy surface of X (a compact oriented ( n − X without boundary), (cid:16) ξ J Y , α ( L ) (cid:17) ∈ C LS a localizable symmetry and review on geometric formulations for classical field theory φ ∈ Y X a critical point of the action principle (5), the associated Lagrangian Noethercharge vanishes, namely Q ( L )Σ t ( ξ ) := Z Σ t ( j φ ◦ i t ) ∗ J ( L ) ( ξ ) = 0 , (16)where i t : Σ t → X denotes the inclusion map. In fact, this last corresponds to the secondNoether theorem, which states that the vanishing of the Lagrangian Noether chargesassociated with each of the localizable symmetries of the theory is a necessary conditionto extend j φ ◦ i t : Σ t → J Y (a section j φ ∈ J Y X restricted to a Cauchy surfaceΣ t ) to a solution of the Euler-Lagrange field equations of the system. In consequence,the latter is one of the main sources of constraints on the space of Cauchy data for theevolution equations of the classical field theories with localizable symmetries [10].In the following subsection, we will introduce the multisymplectic framework, whichwill allow us to study classical field theory in a covariant Hamiltonian-like formulation.In particular, we will focus our attention on describing how the symmetries of a genericclassical field theory arise within the multisymplectic approach. Next, we will give a brief description of the multisymplectic formalism, which providesa finite-dimensional, geometric and covariant Hamiltonian-like formulation for classicalfield theory. As in the previous subsection, we will pay special attention to the study ofthe symmetries of a given classical field theory within this covariant framework. Thesesymmetries will allow us to introduce Noether’s theorems within the context of the DeDonder-Weyl canonical theory.In order to develop a covariant Hamiltonian-like formulation for an arbitraryclassical field theory, we first need to construct a suitable arena where such formulationwill take place. To this end, given a covariant configuration space for the classicalfield theory (5) characterized by the fibre-bundle (
Y, π XY , X ), we define ( Z ⋆ :=Λ n T ∗ Y, π
Y Z ⋆ , Y ) as the covariant multimomenta phase-space associated with thesystem, namely the vector bundle of horizontal ( n ; 2)-forms over Y . In local coordinates,an element Ξ ∈ Z ⋆y can be written asΞ := p d n x + p µa dy a ∧ d n − x µ , (17)which allows us to identify ( x µ , y a , p, p µa ) as an adapted coordinate system on Z ⋆ .Additionally, the composition map π XZ ⋆ := π XY ◦ π Y Z ⋆ yields the bundle structure( Z ⋆ , π XZ ⋆ , X ). Here, in agreement with [20, 21], the fibre coordinates ( p µa ) will bereferred to as the polymomenta, while the fibre coordinate p will be related with thecovariant Hamiltonian of the theory. In fact, it is important to mention that thecovariant multimomenta phase-space may be identified with the affine dual jet bundle( J Y ⋆ , π Y J Y ⋆ , Y ), that is, the affine dual bundle corresponding to the affine jet bundle π Y J Y [2] (see also [50, 51] for technical details about dual bundles). This identificationmay be realized by noticing that the spaces J Y ⋆ and Z ⋆ are canonically isomorphic as review on geometric formulations for classical field theory Y [5]. As we will see below, the covariant multimomenta phase-spacewill allow us to introduce a well-defined arena for a covariant Hamiltonian analysis ofthe classical field theory of our interest.In particular, we would like to emphasize that, the vector space Z ⋆ is endowed witha canonical n -form Θ ( M ) ∈ Ω n ( Z ⋆ ), which is locally given by k Θ ( M ) := p d n x + p µa dy a ∧ d n − x µ . (18)This canonical n -form is also known as the multisymplectic potential since it inducesthe so-called multisymplectic ( n + 1)-form through the relation Ω ( M ) := − d Θ ( M ) ∈ Ω n +1 ( Z ⋆ ), or specificallyΩ ( M ) = dy a ∧ dp µa ∧ d n − x µ − dp ∧ d n x , (19)thus defining the pair (cid:16) Z ⋆ , Ω ( M ) (cid:17) as a multisymplectic manifold [2, 5, 6, 8].Besides, given L : J Y → R the Lagrangian function that characterizes the classicalfield theory (5), we have that the affine jet bundle and the covariant multimomentaphase-space may be related through the covariant Legendre transformation F L : J Y → Z ⋆ , namely the bundle map over Y locally defined by F L ( x µ , y a , y aµ ) := x µ , y a , p = L − ∂L∂y aµ y aµ , p µa = ∂L∂y aµ ! . (20)Thus, by using the covariant Legendre transformation (20), we may induce informationof the classical field theory under study from the affine jet bundle to the covariantmultimomenta phase-space and vice versa [5, 6, 8]. For instance, it is not difficult to seethat, we can obtain the Poincaré-Cartan forms (6) by means of the following relationsΘ ( L ) = F L ∗ Θ ( M ) , Ω ( L ) = F L ∗ Θ ( M ) . Now, it is important to mention that the covariant multimomenta phase-spacehas as a subbundle the vector bundle (Λ n T ∗ Y, π Y Λ n T ∗ Y , Y ), that is, the vector bundleof horizontal ( n ; 1)-forms over Y , which in turn allows us to introduce a new vectorbundle over Y given by the quotient bundle P := Z ⋆ / Λ n T ∗ Y . (21)In local coordinates an element ϑ ∈ P y can be written as ϑ := p µa dy a ∧ d n − x µ , (22) k Here we must clarify that, even though the local representations of Ξ and Θ ( M ) may seemindistinguishable, however, they may not be confused as the former corresponds to an arbitrary elementof the fibre of Z ⋆ over a point y ∈ Y , while the latter stands for an ( n + 1)-form on Z ⋆ , regarding Z ⋆ as a manifold. review on geometric formulations for classical field theory x µ , y a , p µa ) as an adapted coordinate system on P .Consequently, the composition map π XP := π XY ◦ π Y P gives rise to the so-calledpolymomenta phase-space (
P, π XP , X ) [20, 21, 22]. Hence, a section ̺ ∈ P X of π XP covering φ = π Y P ( ̺ ) ∈ Y X can be locally represented by ( x µ , φ a ( x µ ) , π νa ( x µ )). Inaddition, the definition of quotient bundle (21) yields the bundle structure ( Z ⋆ , π P Z ⋆ , P ),where a section h ∈ Z ⋆P is said to be a Hamiltonian section if it locally reads h ( x µ , y a , p µa ) = ( x µ , y a , p = − H ( x µ , y a , p µa ) , p µa ) , (23)being H the denominated Hamiltonian function associated with h [8].Transitively, the affine jet bundle and the quotient bundle (21) are related throughthe covariant Legendre map F DW L : J Y → P , namely the bundle map over Y locallygiven by F DW L ( x µ , y a , y aµ ) := x µ , y a , p µa = ∂L∂y aµ ! , (24)which allows us to define the denominated De Donder-Weyl Hamiltonian section h DW ∈ Z ⋆P , that is, a section of π P Z ⋆ satisfying the condition h DW ◦ F DW L = F L and whoseHamiltonian function is identified with the so-called De Donder-Weyl Hamiltonian [6],specifically H DW ( x µ , y a , p µa ) := p µa y aµ − L (cid:16) x µ , y a , y aµ (cid:17) . (25)Bearing this in mind, we have that, it is possible to induce differential forms on P bypulling-back differential forms on Z ⋆ with a Hamiltonian section h ∈ Z ⋆P . For instance,the De Donder-Weyl forms, Θ ( P ) := h ∗ DW Θ ( M ) ∈ Ω n ( P ) and Ω ( P ) := h ∗ DW Ω ( M ) ∈ Ω n +1 ( P ), which explicitly readΘ ( P ) = p µa dy a ∧ d n − x µ − H DW d n x , (26a)Ω ( P ) = dy a ∧ dp µa ∧ d n − x µ + dH DW ∧ d n x . (26b)In a similar fashion to the Poincaré-Cartan forms (6) within the Lagrangian formalism,the De Donder-Weyl forms (26) will be the main geometric objects of interest withinthe covariant Hamiltonian-like formulation for the classical field theory. In fact, it is notdifficult to see that we may recover the Poincaré-Cartan forms by means of the followingexpressions Θ ( L ) = F DW L ∗ Θ ( P ) , Ω ( L ) = F DW L ∗ Ω ( P ) . From our point of view, the De Donder-Weyl forms are completely relevant as, onthe one side, we may rewrite the action principle (5) in terms of the n -form (26a) anda section ̺ ∈ P X (see for instance [2]) while, on the other side, the De Donder-Weyl( n + 1)-form (26b) is necessary in order to write the De Donder-Weyl-Hamilton fieldequations of the system in an invariant fashion [6]. In other words, let us consider ̺ ∈ P X a section locally represented by ( x µ , φ a ( x µ ) , π νa ( x µ )). Then, we have that for review on geometric formulations for classical field theory P , namely W ∈ X ( P ), a section ̺ ∈ P X satisfying thecondition ̺ ∗ (cid:16) W y Ω ( P ) (cid:17) = 0 , (27)corresponds to a solution of the De Donder-Weyl-Hamilton field equations of the theory,namely ∂ µ φ a = ∂H∂p µa ( x µ , φ a , π µa ) ,∂ µ π µa = − ∂H∂y a ( x µ , φ a , π µa ) . (28)Of course, for a non-singular Lagrangian system, that is, a classical field theorycharacterized by a Lagrangian function obeying the regularity conditiondet ∂ L∂y aµ ∂y bν ! = 0 , (29)the Euler-Lagrange field equations (8) are completely equivalent to the De Donder-Weyl-Hamilton field equations (28). To see this, we start by emphasizing that for anon-singular Lagrangian system the covariant Legendre map (24) is a diffeomorphism[8]. Therefore, given V ∈ X ( J Y ) and W := T F DW L ( V ) ∈ X ( P ) a pair of arbitraryvector fields, a straightforward calculation shows that, if a section j φ ∈ J Y X is asolution of the Euler-Lagrange field equations (8), then the section ̺ := F DW L◦ j φ ∈ P X is a solution of the De Donder-Weyl Hamilton field equations (28), and vice versa.Specifically ̺ ∗ (cid:16) W y Ω ( P ) (cid:17) = ( j φ ) ∗ (cid:16) T F DW L − ( W ) y F DW L ∗ Ω ( P ) (cid:17) , = ( j φ ) ∗ (cid:16) V y Ω ( L ) (cid:17) , = 0 . Here, we would like to mention that the covariant Legendre map (24), the De Donder-Weyl Hamiltonian (25) and the De Donder-Weyl-Hamilton field equations (28) are thepillars of the De Donder-Weyl canonical theory which appeared as early as 1935 in [3, 4].Now, we will analyze the symmetries of the classical field theory (5) within themultisymplectic approach. In particular, we will focus our attention on the actionof the symmetry group of the system on the covariant multimomenta phase-space.For this purpose, we will start by introducing the notion of covariant canonicaltransformations, which will allow us to construct the so-called covariant momentummap and, consequently, the conserved currents for the solutions of the De Donder-Weyl-Hamilton field equations of the theory.To start, let ( Z ⋆ , π Y Z ⋆ , Y ) be the covariant multimomenta phase-space associatedwith the classical field theory defined by the action (5). Then, in accordance with[5, 6], a π XZ ⋆ -bundle automorphism (Φ Z⋆ , Φ X ), namely a pair of maps Φ Z⋆ : Z ⋆ → Z ⋆ and Φ X : X → X satisfying the condition π XZ ⋆ ◦ Φ Z⋆ = Φ X ◦ π XZ ⋆ , is said to be acovariant canonical transformation if it preserves the multisymplectic ( n + 1)-form (19), review on geometric formulations for classical field theory ∗ Z⋆ Ω ( M ) = Ω ( M ) . Therefore, if Φ Z⋆ is the vector flow associated with avector field ξ Z ⋆ Φ ∈ X ( Z ⋆ ), it is clear that L ξ Z⋆ Φ Ω ( M ) = 0 . (30)Bearing this in mind, we are now in the position to describe how fibre-preserving transformations on the covariant configuration space, for instance, thosegenerated by the symmetry group of the theory, transitively induce covariant canonicaltransformations on the covariant multimomenta phase-space associated with the system.To see this, let us consider G the symmetry group of the theory. Then, given ( η Y , η X ) the π XY -bundle automorphism associated with an element η ∈ G , we introduce η Z⋆ : Z ⋆ → Z ⋆ to be the canonical lift of η Y to Z ⋆ . Thus, since π Y Z ⋆ is a vector bundle of differentialforms over Y , at y ∈ Y , being Ξ : Λ n T y Y → R an element of Z ⋆y , we have that, η Z⋆y : Z ⋆y → Z ⋆η Y ( y ) is defined by η Z⋆y (Ξ)( V , · · · , V n ) = Ξ (( T y η Y ) − ( V ) , · · · , ( T y η Y ) − ( V n )), ∀ V , · · · , V n ∈ T η Y ( y ) Y , namely η Z⋆y (Ξ) := η − Y ∗ Ξ. Therefore, given the infinitesimalgenerators of the transformations η X and η Y , ξ Xη ∈ X ( X ) and ξ Yη ∈ X ( Y ), respectively,we get that the vector field ξ Z ⋆ η ∈ X ( Z ⋆ ) on Z ⋆ explicitly defined by ξ Z ⋆ η := ξ µ ( x ν ) ∂ µ + ξ a ( x ν , y b ) ∂ a − (cid:16) p ∂ µ ξ µ ( x ν ) + p µa ∂ µ ξ a ( x ν , y b ) (cid:17) ∂ p − (cid:16) p µc ∂ a ξ c ( x ν , y b ) − p σa ∂ σ ξ µ ( x ν ) + p µa ∂ σ ξ σ ( x ν ) (cid:17) ∂ aµ , (31)corresponds to the infinitesimal generator of the transformation η Z ⋆ , where we haveintroduced the notation ∂ p := ∂/∂p and ∂ aµ := ∂/∂p µa to denote the partial derivativewith respect to the fibre coordinates p and p µa , respectively. In fact, it is notdifficult to show that the canonical lift preserves the multisymplectic potential (18),namely η ∗ Z ⋆ Θ ( M ) = Θ ( M ) , and hence L ξ Z⋆η
Θ = 0. As a result, we have that,the canonical lifts associated with π XY -bundle automorphisms give rise to covariantcanonical transformations [5].Besides, we know that, if G corresponds to the symmetry group of the theory,then for all η ∈ G there is a corresponding Noether symmetry ( ξ J Yη , α ( L ) η ), namelya pair consisting of a vector filed ξ J Yη ∈ X ( J Y ) and a horizontal ( n −
1; 1)-form α ( L ) η ∈ Ω n − ( J Y ) related through condition (10), being ξ J Yη := j ξ Yη the first jetprolongation of ξ Yη ∈ X ( Y ). Then, for any η ∈ G , we define the so-called α ( M ) η -lift of ξ Yη ∈ X ( Y ) to Z ⋆ as the unique vector field ξ αη ∈ X ( Z ⋆ ) that projects by means of T π
Y Z ⋆ : T Z ⋆ → T Y onto ξ Yη and also satisfies the condition L ξ αη Θ ( M ) = dα ( M ) η , (32)where α ( M ) η ∈ Ω n − ( Z ⋆ ) denotes a π XZ ⋆ -horizontal ( n −
1; 1)-form on Z ⋆ locallyrepresented as α ( M ) η = α νη ( x µ , y a ) d n − x ν . (33)Thus, the vector field ξ αη not only contains the information associated with the canonicallift (31) but also with the π XZ ⋆ -horizontal ( n −
1; 1)-form α ( M ) η . Here, it is important review on geometric formulations for classical field theory α ( L ) η ∈ Ω n − ( J Y ) and α ( M ) η ∈ Ω n − ( Z ⋆ ) are the same, and therefore α ( L ) η = F L ∗ α ( M ) η . As pointed outin [8], the relevance of the α ( M ) -lifts lies in the fact that, for η ∈ G , the condition T β F L ( ξ J Yη ( β )) = ξ αη ( F L ( β )) holds, being T β F L : T β J Y → T F L ( β ) Z ⋆ the tangent mapof F L at β ∈ J Y .With all this in mind, given η ∈ G , we introduce (cid:16) η αZ ⋆ , ξ αη (cid:17) to represent the actionof η on Z ⋆ , where η αZ ⋆ : Z ⋆ → Z ⋆ stands for the vector flow associated with the vectorfield ξ αη ∈ X ( Z ⋆ ). Therefore, since the α ( M ) -lifts give rise to covariant canonicaltransformations, the action of G on Z ⋆ has an associated covariant momentum map[5], that is, a map J ( M ) : Z ⋆ → g ∗ ⊗ Λ n − T ∗ Z ⋆ such that for all ξ η ∈ g , d J ( M ) ( ξ η ) = ξ αη y Ω ( M ) , (34)where J ( M ) ( ξ η ) := h J ( M ) , ξ η i ∈ Ω n − ( Z ⋆ ) corresponds to the ( n − Z ⋆ explicitly given by J ( M ) ( ξ η ) = ξ αη y Θ ( M ) − α ( M ) η . (35)In fact, for all ξ η ∈ g , by pulling-back the covariant momentum map with the De Donder-Weyl Hamiltonian section h DW ∈ Z ⋆P , namely J ( P ) ( ξ η ) := h ∗ DW J ( M ) ( ξ η ) ∈ Ω n − ( P ), wecan construct a conserved current for the solutions of the De Donder-Weyl-Hamiltonfield equations [6, 8]. In other words, we have that, for all ξ η ∈ g and for any solutionof the De Donder-Weyl-Hamilton field equations (28), ̺ ∈ P X , the quantity defined by J ( P ) ( ξ η ) := ̺ ∗ J ( P ) ( ξ η ) , (36)corresponds to a conserved current of the system. To see this, we start by emphasizingthat, it is possible to write J ( L ) ( ξ η ) = F L ∗ J ( M ) ( ξ η ). Furthermore, we know that, for anon-singular Lagrangian system the covariant Legendre map (24) is a diffeomorphism,and hence if a section ̺ ∈ P X is a solution of the De Donder-Weyl-Hamilton fieldequations (28), then the section j φ := F DW L − ◦ ̺ ∈ J Y X is a solution of the Euler-Lagrange field equations (8). Thus, by considering the relation h DW = F L ◦ F DW L − , wefind that d (cid:16) ( j φ ) ∗ J ( L ) ( ξ η ) (cid:17) = d (cid:16) ̺ ∗ ( F L ◦ F DW L − ) ∗ J ( M ) ( ξ η ) (cid:17) , = d (cid:16) ̺ ∗ h ∗ DW J ( M ) ( ξ η ) (cid:17) , = d (cid:16) ̺ ∗ J ( P ) ( ξ η ) (cid:17) , = 0 . The latter in light of the first Noether theorem, which states that, for j φ ∈ J Y X asolution of the Euler-Lagrange field equations (8), quantity (14) defines a conservedcurrent of the system.Finally, we have that, for localizable symmetries, given Σ t a Cauchy surface of X and ̺ ∈ P X a solution of the De Donder-Weyl-Hamilton field equations (28), eachHamiltonian Noether charge must vanish [5, 9, 10], namely Q ( P )Σ t ( ξ ) := Z Σ t ( ̺ ◦ i t ) ∗ J ( P ) ( ξ ) = 0 , (37) review on geometric formulations for classical field theory i t : Σ t → X denotes the inclusion map. As we will see in subsequent subsections,after performing the space plus time decomposition of the space-time manifold onwhich a classical field theory is defined, the covariant momentum map may inducethe momentum map that characterizes the system under consideration within theinstantaneous Dirac-Hamiltonian formulation. Thus, for classical field theories withlocalizable symmetries, the second Noether theorem establishes that, the vanishing ofthe momentum map on the space of Cauchy data for the evolution equations of thesystem will be related to the set of first-class constraints of the theory, as discussed in[5, 9, 10].In the following subsection, we will give a brief description of the polysymplecticformalism, which endows the polymomenta phase-space with a Poisson-Gerstenhaberbracket that allows to describe the dynamics of a given classical field theory within acovariant Poisson-Hamiltonian framework. Next, we will introduce the polysymplectic approach for classical field theory, whichis a covariant Poisson-Hamiltonian framework. To this end, we will closely follow thedescription of the geometric and algebraic structures inherent to the aforementionedformalism as presented in [15, 17, 18, 20, 21, 22, 30]. Additionally, we will includea summary of the proposal developed in [48] to analyze singular Lagrangian systemswithin the polysymplectic approach.To begin with, let (
P, π XP , X ) be the polymomenta phase-space associated with theclassical field theory described by the action (5). Then, given ( x µ , y a , p µa ) an adaptedcoordinate system on P , we define the so-called polysymplectic structure, namelyΩ V := h dp µa ∧ dy a ∧ ̟ µ mod Ω n +12 ( P ) i , (38)where ̟ µ := ∂ µ y ̟ denotes the basis for the horizontal ( n − P ,while ̟ := dx ∧ · · · ∧ dx n − stands for the horizontal ( n ; 1) volume form on P .From now on, we will consider only horizontal forms with respect to the projectormap π XP . Note that, we have implemented an equivalence class of forms in thepolysymplectic structure (38) as an alternative for the introduction of a connection onthe multimomenta phase-space in order to identify Ω V as a vertical part of the negativeof the multisymplectic ( n + 1)-form (19). In fact, these two approaches are equivalent asthe fundamental structures inherent to the polysymplectic formalism, for instance thePoisson-Gerstenhaber bracket, have been shown to be independent of both the choiceof representatives in the equivalence class and the choice of a connection [14, 21, 22].From the physical point of view, the relevance of the polysymplectic form (38) liesin the fact that we can appropriately define by means of it both a set of Hamiltonianvector fields and forms and a Poisson-Gerstenhaber bracket on the polymomenta phase-space. To see this, let us consider p X ∈ X p ( P ) a vertical p -multivector field, that is, review on geometric formulations for classical field theory p -multivector field such that p X y θ = 0, ∀ θ ∈ Ω p ( P ). Then, a horizontal ( n − p ; 1)-form n − p F ∈ Ω n − p ( P ) is said to be a Hamiltonian ( n − p )-form if there exists a vertical p -multivector field p X F ∈ X p ( P ) satisfying the condition p X F y Ω V = d V n − p F , (39)where we have introduced the vertical exterior derivative d V : Ω pq ( P ) → Ω p +1 q +1 ( P ),namely d V θ := h dθ mod Ω p +1 q ( P ) i , (40)being d : Ω pq ( P ) → Ω p +1 q +1 ( P ) the exterior derivative on P , while θ ∈ Ω pq ( P ) stands foran arbitrary horizontal ( p ; q )-form. In consequence, we have that, when n − p > n − p ; 1)-form is Hamiltonian since equation (39) imposes astrong integrability condition [11, 21]. Henceforward, we denote by Ω p H ( P ) the set ofHamiltonian p -forms on the polymomenta phase-space.Bearing this in mind, we may introduce the so-called Poisson-Gerstenhaber bracket { [ · , · ] } : Ω n − p H ( P ) × Ω n − q H ( P ) → Ω n +1 − ( p + q )H ( P ), and thus given a pair of Hamiltonianforms, n − p F ∈ Ω n − p H ( P ) and n − q G ∈ Ω n − q H ( P ), it is explicitly defined by { [ n − p F , n − q G ] } := ( − p p X F y q X G y Ω V , (41)where p X F ∈ X p ( P ) and q X G ∈ X q ( P ) denote the Hamiltonian vector fields associatedwith the Hamiltonian forms n − p F and n − q G , respectively. It is worth noting that, for n + 1 > p + q , the operation (41) among a pair of Hamiltonian forms induces a newHamiltonian form, and hence the set of Hamiltonian forms Ω p H ( P ) defines an algebraunder the Poisson-Gerstenhaber bracket (41) [14, 20, 21, 22].In addition, it is possible to show that, given n − p F ∈ Ω n − p H ( P ), n − q G ∈ Ω n − q H ( P ) and n − r H ∈ Ω n − r H ( P ) a set of arbitrary Hamiltonian forms, the Poisson-Gerstenhaber bracket(41) satisfies the following algebraic properties:(i) Graded-commutation { [ n − p F , n − q G ] } = − ( − | F || G | { [ n − q G , n − p F ] } , (42)where | F | := p − | G | := q − n − p F and n − q G under the bracket structure (41), respectively.(ii) Graded Jacobi identity { [ n − p F , { [ n − q G , n − r H ] } ] } = { [ { [ n − p F , n − q G ] } , n − r H ] } + ( − | F || G | { [ n − q G , { [ n − p F , n − r H ] } ] } . (43)(iii) Graded Leibniz rule { [ n − p F , n − q G • n − r H ] } = { [ n − p F , n − q G ] } • n − r H + ( − q | F | n − q G • { [ n − p F , n − r H ] } , (44) review on geometric formulations for classical field theory • : Ω n − q H ( P ) × Ω n − r H ( P ) → Ω n − ( q + r )H ( P ), is given by n − q G • n − r H := ∗ − (cid:18) ∗ n − q G ∧ ∗ n − r H (cid:19) , (45)being ∗ : Ω n − q ( X ) → Ω q ( X ) the Hodge dual operator defined on the base space X .In other words, since for n > q + r the operation (45) among a pair of Hamiltonianforms gives rise to a new Hamiltonian form, we have that, the set of Hamiltonian formsΩ p H ( P ) not only defines an algebra but also a Gerstenhaber algebra with respect to thePoisson-Gerstenhaber bracket (41) and the co-exterior product (45) [22].Here, we would like to emphasize that the subset of Hamiltonian ( n − n − ( P ) will play a primordial role within the polysymplectic formalism: on the onehand, the subset of Hamiltonian ( n − n − n − n − ( P ).Besides, note that, it is possible to define a set of canonically conjugate variablesfor the Poisson-Gerstenhaber bracket (41), specifically { [ p µa ̟ µ , y b ̟ ν ] } = δ ba ̟ ν , (46)which allows us to express the De Donder-Weyl-Hamilton field equations (28) in acovariant Poisson-Hamiltonian fashion. In other words, we have that, given ̺ ∈ P X asection locally represented by ( x µ , φ a ( x µ ) , π νa ( x µ )), the De Donder-Weyl-Hamilton fieldequations (28) can be written as ∂ µ φ a = ̺ ∗ { [ H DW , y a ̟ µ ] } ,∂ µ π µa = ̺ ∗ { [ H DW , p µa ̟ µ ] } . (47)In fact, it is possible to write the equation of motion of an arbitrary Hamiltonian( n − d • : Ω p H ( P ) → Ω p − ( n − ( X ) which, given ̺ ∈ P X a solution of the De Donder-Weyl-Hamilton field equations (28) and p F ∈ Ω p ( P )a Hamiltonian p -form, is defined as d • p F := s m ( n − p )! (cid:18) ̺ ∗ d (cid:18) p F • dx µ ∧ · · · ∧ dx µ n − p (cid:19)(cid:19) • ̟ µ ··· µ n − p , (48)being ̟ µ ··· µ n − p := ∂ µ y · · · ∂ µ n − p y ̟ the basis for the horizontal ( p ; 1)-forms on P , while s m := ± X [17]. Consequently, review on geometric formulations for classical field theory n − n − F ∈ Ω n − ( P ) explicitly reads d • n − F = − s m ( − n ̺ ∗ { [ H DW , n − F ] } + d H • n − F , (49)where the horizontal co-exterior derivative, d H • : Ω p H ( P ) → Ω p − ( n − ( X ), is simplygiven by d H • p F := s m ( n − p )! (cid:18) ̺ ∗ ∂ σ (cid:18) p F • dx µ ∧ · · · ∧ dx µ n − p (cid:19) dx σ (cid:19) • ̟ µ ··· µ n − p . (50)In light of this, it is possible to see that by means of the subset of Hamiltonian ( n − n − ( P ), the De Donder-Weyl Hamiltonian (25), the Poisson-Gerstenhaberbracket (41), and the equation of motion (49), we can study classical field theory in acovariant Poisson-Hamiltonian framework. In particular, from the physical viewpointthis is very convenient as, commonly, a Poisson-Hamiltonian framework may be seen asthe first step towards a quantization of the De Donder-Weyl canonical theory [15, 17, 18].Before ending the present subsection, we will briefly introduce the proposaldeveloped in [48] to analyze singular Lagrangian systems within the polysymplecticapproach. For this purpose, we begin by mentioning that a singular Lagrangian systemmay be simply defined as one that does not satisfy the regularity condition (29). Inthis case, the covariant Legendre map (24) is not a diffeomorphism since it is notinvertible, and therefore we have certain conditions emerging from the definition ofthe polymomenta p µa = ∂L/∂y aµ , namely C ( k ) µP ( y a , p νa ) ≈ , (51)which, using Dirac’s terminology [1], will be referred to as primary constraints. Here,the weak equality symbol ≈ is implemented to specify that a certain relation is validonly on the constraint surface, that is, the surface on the polymomenta phase-spacedelimited by the constraints of the system. Henceforward, for convenience, instead ofusing the primary constraints (51) explicitly, we will consider the associated primaryconstraint ( n − C ( k ) P := C ( k ) µP ̟ µ ¶ (where the index k runs over the completeset of primary constraint ( n − f H DW : P → R , which is given by f H DW := H DW + u ( k ) • C ( k ) P , (52)where u ( k ) stands for a set of Lagrange multiplier (1 ; 1)-forms enforcing the primaryconstraint ( n − ¶ For simplicity, all the constraints of the theory are assumed to be linearly independent andHamiltonian. review on geometric formulations for classical field theory n − d • C ( k ) P ≈ , (53)which extends to the polysymplectic framework the analogous concept within Dirac’sapproach. In particular, it is possible to see that, by using the total De Donder-WeylHamiltonian (52) and the equation of motion (49), the consistency conditions (53) caneither be trivially satisfied, impose further restrictions on the Lagrange multipliers u ( k ) ,or they may give rise to new relations independent of both the primary constraints (51)and the Lagrange multipliers u ( k ) . In the latter case, following Dirac’s terminology, thesenew relations will be referred to as secondary constraints. Thus, if there are secondaryconstraints, writing them as ( n − C ( k ′ ) S := C ( k ′ ) µS ̟ µ (where the index k ′ runs overthe set of secondary constraint ( n − u ( k ) or obtainingnew tertiary constraints. In the case we generate further constraints, we must continueapplying the consistency conditions until we either fix all of the Lagrange multipliers u ( k ) or whenever these conditions are trivially satisfied. Hence, after the process of generatingfurther constraints is finished, we will have a complete set of constraint ( n − {C ( l ) } (where the index l runs over the complete set of primary, secondary, tertiary, etc.,constraint ( n − p -form p F ∈ Ω p H ( P ) is a first-classHamiltonian p -form if its Poisson-Gerstenhaber bracket with each constraint ( n − { [ p F , C ( l ) ] } ≈ . (54)If a Hamiltonian p -form is not first-class, then it will be termed a second-classHamiltonian p -form. Thus, in light of the above definition, we can separate the completeset of constraint ( n − {C ( l ) } into subsets of first- and second-class constraint( n − {A ( i ) } and {B ( i ) } , respectively (where the index i runs over the subset of either first- or second-class constraint ( n − n − B ( i , j ) := { [ B ( i ) , B ( j ) ] } (55)does not vanish on the constraint surface + . The latter because the subset {B ( i ) } consists + Here, the ( n − B ( i,j ) is assumed to be of constant rank on the constraintsurface. review on geometric formulations for classical field theory n − B ( i , j ) is a non-singular( n −
1; 1)-form valued matrix, we can construct a (1 ; 1)-form valued matrix B − i , j ) satisfying B − i , k ) ∧ B ( k , j ) = δ ji ̟ , (56)where δ ji denotes the Kronecker delta. In this way, we have that the (1 ; 1)-form valuedmatrix B − i , j ) may be thought of as the inverse of the ( n − B ( i , j ) .Bearing this in mind and maintaining the analogy with the standard Diracformalism for constrained systems, we are able to define a Dirac-Poisson bracket forHamiltonian 0- and ( n − F a Hamiltonian 0- or ( n − G a Hamiltonian ( n − { [ F, G ] } D := { [ F, G ] } − s m { [ F, B ( i ) ] } • (cid:16) B − i , j ) ∧ { [ B ( j ) , G ] } (cid:17) , (57)which, by construction, eliminates second-class constraint ( n − n − In the present subsection, closely following references [5, 7, 8, 9, 10, 23, 49], we willbriefly present the description of the space plus time decomposition of the geometric-covariant Lagrangian and multisymplectic formulations for classical field theory. Forthis purpose, we will start by introducing a slicing of the space-time manifold X , andsubsequently describing the space plus time decomposition of any fibre-bundle over it,which will allow us to define the elemental ingredients of the instantaneous Lagrangianand Dirac-Hamiltonian formulations for classical field theory. Finally, we will relatethe multisymplectic framework with the instantaneous Dirac-Hamiltonian approach byinvoking a reduction process.To begin with, let us consider Σ a compact ( n − X is defined as a diffeomorphismbetween R × Σ and X , namely ¯Ψ : R × Σ → X . Note that, for t ∈ R , ¯Ψ t := ¯Ψ ( t, · ) : Σ → X corresponds to an embedding. Here, we denote by X := { ¯Ψ t | t ∈ R } and Σ t ⊂ X theset of all embeddings of Σ into X and the image of Σ by ¯Ψ t ∈ X , respectively. In addition, review on geometric formulations for classical field theory ∂ t := ∂/∂t ∈ X ( R × Σ) the generator of time translations ( t, u ) ( t + s, u ) on R × Σ, we define ζ X := T ¯Ψ ( ∂ t ) ∈ X ( X ) as the infinitesimal generator of ¯Ψ, whichby definition is everywhere transverse to Σ t [9]. In particular, we say that, ( x µ ) is acoordinate system adapted to ¯Ψ t if the Cauchy surface Σ t is locally given by a level setof the coordinate x . Thereby, a coordinate system on Σ t can be simply denoted by( x i ), i = 1 , . . . , n − X . To illustrate this, let us consider( K, π XK , X ) a fibre-bundle. Then, given ¯Ψ a slicing of X , a compatible slicing of K isdefined as a fibre-bundle ( K Σ , π Σ K Σ , Σ) and a bundle diffeomorphism Ψ : R × K Σ → K such that the following diagram commutes R × K Σ K R × Σ X Ψ¯Ψ (58)where the vertical arrows stand for fibre-bundle projections [9]. Observe that, for t ∈ R ,Ψ t := Ψ ( t, · ) : K Σ → K defines an embedding. Here, we denote by K t ⊂ K the imageof K Σ by Ψ t . As in the previous case, given ∂ t := ∂/∂t ∈ X ( R × K Σ ), we define ζ K := T Ψ( ∂ t ) ∈ X ( K ) as the infinitesimal generator of Ψ, which by construction iseverywhere transverse to K t and also projects into ζ X by means of the tangent map T π XK : T K → T X . In light of this, the pairs (Σ t , ζ X ) and ( K t , ζ X ) will be referred toas an infinitesimal and compatible slicing of π XK , which in turn defines a one-parametricgroup of π XK -bundle automorphisms [9]. Additionally, we introduce ( K t , π Σ t K t , Σ t ) tobe the restriction of π XK to the Cauchy surface Σ t , where π Σ t K t := π XK (cid:12)(cid:12)(cid:12) Σ t : K t → Σ t denotes the corresponding projector map. Thus, given ( k a ) ( a = 1 , . . . , m ′ ) a set offibre-coordinates on K , we can identify ( x i , k a ) as an adapted coordinate system on K t .Now, let us consider K t the set of sections of π Σ t K t , namely the set of sections of π XK restricted to the Cauchy surface Σ t . Then, the collection K Σ defined by K Σ : = [ ¯Ψ t ∈ X K t , (59)gives rise to an infinite-dimensional fibre-bundle ( K Σ , π X K Σ , X ) as K t (the fibre over¯Ψ t ∈ X ) defines an infinite-dimensional manifold. At ¯Ψ t ∈ X , a local section of π X K Σ can be understood as the identification of an element of the fibre over ¯Ψ t , and thereforea section of π X K Σ induces a section of π XK , and conversely [49].Here, in agreement with [5, 7, 9], K t is assumed to be an infinite dimensional smoothmanifold. At κ ∈ K t , we introduce the set of functions( κ a ) explicitly given by κ a := k a ◦ κ to denote a coordinate system on K t , and thus these functions depend on the coordinateson the Cauchy surface Σ t and belong to the chosen functional space [49]. The tangentand cotangent bundles over K t can be defined as follows. To begin with, let us consider( V K t , π K t V K t , K t ) the restriction of π K V K (the vertical tangent bundle of π XK ) to K t , review on geometric formulations for classical field theory π K t V K t := π K V K (cid:12)(cid:12)(cid:12) K t : V K t → K t and V K t := π − K V K ( K t )to represent the projector map and the total space, respectively. Then, the tangentspace to K t at κ ∈ K t is defined as T κ K t := n ˙ κ : Σ t → V K t (cid:12)(cid:12)(cid:12) π K t V K t ◦ ˙ κ = κ o . Notethat, in adapted coordinates, an element ˙ κ ∈ T κ K t can be written as˙ κ := ˙ κ a ∂∂k a , (60)where the functions ˙ κ a depend on the coordinates on the Cauchy surface Σ t . As usual,the tangent bundle is given by T K t := [ κ ∈ K t T κ K t . (61)Consequently, we introduce ( κ a , ˙ κ a ) to denote a coordinate system on T K t , where thefunctions κ a and ˙ κ a belong to the chosen functional space and take values on the Cauchysurface Σ t .Besides, let us consider ( V ∗ K t , π K t V ∗ K t , K t ) the restriction of π K V ∗ K (the dualbundle of π K V K ) to K t , where we have introduced π K t V ∗ K t : π K V ∗ K (cid:12)(cid:12)(cid:12) K t : V ∗ K t → K t and V ∗ K t := π − K V ∗ K ( K t ) to denote the projector map and the total space, respectively.Then, given the tensor product bundle π ⊗ t : V ∗ K t ⊗ π ∗ Σ t K t (Λ n − T ∗ Σ t ) → K t , thecotangent space to K t at κ ∈ K t is defined as T ∗ κ K t := n τ : Σ t → V ∗ K t ⊗ π ∗ Σ t K t (Λ n − T ∗ Σ t ) (cid:12)(cid:12)(cid:12) π ⊗ t ◦ τ = κ o , where the pullback bundle π ∗ Σ t K t ( π Σ t Λ n − T ∗ Σ t ) : π ∗ Σ t K t (Λ n − T ∗ Σ t ) → K t may be identified with the subbundle of horizontal ( n −
1; 1)-forms over K t , namely π K t Λ n − T ∗ K t : Λ n − T ∗ K t → K t (see reference [50] for detailsabout tensor products of vector bundles and pullback bundles). Observe that, in adaptedcoordinates, an element τ ∈ T ∗ κ K t can be expressed as τ := τ a dk a ⊗ d n − x , (62)where the functions τ a depend on the coordinates on the Cauchy surface Σ t . Naturally,the cotangent bundle is defined by T ∗ K t := [ κ ∈ K t T ∗ κ K t . (63)Analogously to the previous case, we identify ( κ a , τ a ) as a coordinate system on T ∗ K t ,where the functions κ a and τ a belong to the chosen functional space and take valueson the Cauchy surface Σ t . Hence, the natural pairing between elements ˙ κ ∈ T κ K t and τ ∈ T ∗ κ K t is locally given by integration, specifically h ˙ κ , τ i := Z Σ t ˙ κ y τ . (64)Finally, we would like to emphasize that, it is possible to induce differential formson K t from those defined on K [7, 8, 10, 49]. In other words, we have that for all( q + n − K , α ∈ Ω q + n − ( K ), there is an associated q -form α t ∈ Ω q ( K t ) on review on geometric formulations for classical field theory K t whose dual pairing with a set of q tangent vectors V k ∈ T σ K t at σ ∈ K t explicitlyreads α t ( σ ) ( V , · · · , V q ) := Z Σ t σ ∗ ( V q y · · · V y α ) , (65)where the contraction is taken along the image of each of the tangent vectors V k , k = 1 , . . . , q .As discussed below, the above space plus time decomposition of fibre-bundles willallow us to relate the finite- and infinite-dimensional formulations for classical fieldtheory. To illustrate this, let (Σ t , ζ X ) and ( Y t , ζ Y ) be a G -slicing of the covariantconfiguration space π XY , namely an infinitesimal and compatible slicing of π XY suchthat for some ξ η ∈ g , the vector field ζ Y ∈ X ( Y ) contains the information of ξ Yη ∈ X ( Y ), and also its corresponding first jet prolongation, ζ J Y := j ζ Y ∈ X ( J Y ),defines an infinitesimal symmetry of the theory. As pointed out in [7], the vector field ζ Y ∈ X ( Y ) may be thought of as an evolution direction on Y . Consequently, weintroduce ( Y t , π Σ t Y t , Σ t ) to be the restriction of π XY to the Cauchy surface Σ t , and wedenote by ( x i , y a ) an adapted coordinate system on Y t . Furthermore, we define Y t asthe set of sections of π Σ t Y t , that is, the set of sections of π XY restricted to the Cauchysurface Σ t . Observe that, for all ϕ ∈ Y t a section of π Σ t Y t there is a section φ ∈ Y X of π XY such that ϕ := φ ◦ i t , where i t : Σ t → X denotes the inclusion map. Besides, given Y t the t -instantaneous configuration space of the classical field theory (5), we identify T Y t as the t -instantaneous space of velocities of the system where the instantaneousLagrangian formulation of the theory will take place. Here, we introduce ( ϕ a , ˙ ϕ a ) todenote a coordinate system on T Y t , where the functions ϕ a and ˙ ϕ a depend on thecoordinates on the Cauchy surface Σ t and belong to the chosen functional space. Inparticular, by considering φ ∈ Y X a section such that ϕ := φ ◦ i t ∈ Y t , the temporalderivative of the field variables is given by˙ ϕ a := ( L ζ Y φ ) a (cid:12)(cid:12)(cid:12) Σ t = (cid:16) T φ ◦ ζ X − ζ Y ◦ φ (cid:17) a (cid:12)(cid:12)(cid:12) Σ t , (66)where L ζ Y φ denotes the Lie derivative of the section φ along the vector field ζ Y , while T φ : T X → T Y represents tangent map of φ [9]. Note that, L ζ Y φ (cid:12)(cid:12)(cid:12) Σ t ∈ T ϕ Y t correspondsto a vertical vector field on the image of ϕ , while ( L ζ Y φ ) a (cid:12)(cid:12)(cid:12) Σ t stands for the componentsof such a vector field.Now, let us consider (( J Y ) t , π Y t ( J Y ) t , Y t ) the restriction of π Y J Y to Y t , where wehave implemented π Y t ( J Y ) t := π Y J Y (cid:12)(cid:12)(cid:12) Y t : ( J Y ) t → Y t and ( J Y ) t := π − Y J Y ( Y t ) todenote the projector map and the total space, respectively. Then, given ( x i , y a , y aµ ) anadapted coordinate system on ( J Y ) t , we introduce an affine bundle map β ζ Y : ( J Y ) t → J ( Y t ) × V Y t over Y t , which is locally defined by β ζ Y (cid:16) x i , y a , y aµ (cid:17) := ( x i , y a , y ai , ˙ y a ), where( J ( Y t ) , π Y t J ( Y t ) , Y t ) denotes the affine jet bundle of π Σ t Y t ∗ . In fact, the latter is ∗ In adapted coordinates, ¯ κ ∈ J y ( Y t ) an element of the fibre over ¯ y ∈ Y t can be locally written as¯ κ := dx i ⊗ (cid:18) ∂∂x i + y ai ∂∂y a (cid:19) , which allows us to identify ( x i , y a , y ai ) as an adapted coordinate systemon J ( Y t ). review on geometric formulations for classical field theory β ζ Y induces an isomorphism between( J Y ) t (the set of sections of π XJ Y restricted to the Cauchy surface Σ t ) and T Y t , asdiscussed in [7, 9].Observe that, being j φ ∈ J Y X the first jet prolongation of φ ∈ Y X , we can write β ζ Y ( j φ ◦ i t ) = ( j ϕ, ˙ ϕ ), where j ϕ , the first jet prolongation of ϕ := φ ◦ i t ∈ Y t ,corresponds to a section of π Σ t J ( Y t ) : J ( Y t ) → Σ t . As a result, we can obtain theinstantaneous Lagrangian density of the theory, L t, ζ Y : J ( Y t ) × V Y t → Λ n − T ∗ Σ t ,by means of L t, ζ Y ( j ϕ, ˙ ϕ ) := ( j φ ◦ i t ) ∗ (cid:16) ζ X y L (cid:17) and, consequently, the correspondinginstantaneous Lagrangian functional L t, ζ Y : T Y t → R , which in adapted coordinatesexplicitly reads L t, ζ Y ( ϕ, ˙ ϕ ) := Z Σ t L (cid:16) j ϕ, ˙ ϕ (cid:17) ζ d n − x , (67)where L represents the covariant Lagrangian function of the system, ζ corresponds tothe component of the generator ζ X along ∂ and d n − x denotes the ( n − t . In light of this, the instantaneous Legendre transformationis defined as a bundle map F L t, ζ Y : T Y t → T ∗ Y t over Y t locally given by F L t, ζ Y ( ϕ a , ˙ ϕ a ) := (cid:18) ϕ a , π a := ∂L∂ ˙ ϕ a (cid:19) , (68)which allows us to identify the t -primary constraint set of the theory P t, ζ Y ⊂ T ∗ Y t , thatis, the submanifold on the t -instantaneous phase-space of the system, T ∗ Y t , characterizedby the image of the instantaneous Legendre transformation (68). As usual, in adaptedcoordinates, the non-degenerate symplectic structure ω t ∈ Ω ( T ∗ Y t ) on T ∗ Y t is simplydefined as ω t ( ϕ, π ) = Z Σ t dϕ a ∧ dπ a ⊗ d n − x . (69)Now, we will perform the space plus time decomposition for the multimomentaphase-space, and consequently introduce a reduction process to relate the multisym-plectic and the instantaneous Dirac-Hamiltonian formalisms. For this purpose, letus consider ( Z ⋆t , π Y t Z ⋆t , Y t ) the restriction of π Y Z ⋆ to Y t , where we have introduced π Y t Z ⋆t : π Y Z ⋆ (cid:12)(cid:12)(cid:12) Y t : Z ⋆t → Y t and Z t := π − Y Z ⋆ ( Y t ) to denote the projector map andthe total space, respectively. Furthermore, we define Z ⋆t as the set of sections of π Σ t Z ⋆t := π Σ t Y t ◦ π Y t Z ⋆t , namely the set of sections of π XZ ⋆ restricted to the Cauchysurface Σ t , which in fact is a presymplectic manifold. The latter since the multisym-plectic structure Ω ( M ) ∈ Ω n +1 ( Z ⋆ ) on Z ⋆ induces by means of (65) a presymplecticstructure Ω t ∈ Ω ( Z ⋆t ) on Z ⋆t , that is, a closed 2-form with a non-trivial kernel [7].Thus, in order to identify T ∗ Y t with the quotient of Z ⋆t by the kernel of the presym-plectic 2-form Ω t , we introduce a vector bundle map R t : Z ⋆t → T ∗ Y t over Y t definedby h R t ( σ ) , V i := Z Σ t ϕ ∗ ( V y σ ) , (70)where σ ∈ Z ⋆t stands for a section such that ϕ := π Y Z ◦ σ ∈ Y t , while V ∈ T ϕ Y t denotes a tangent vector to Y t at ϕ♯ . Note that, in adapted coordinates, we can write ♯ Of course, the contraction in (70) is taken along the images of σ and V . review on geometric formulations for classical field theory R t ( σ ) = ( p a ◦ σ ) dy a ⊗ d n − x , which implies that ker R t := n σ ∈ Z ⋆t (cid:12)(cid:12)(cid:12) p a ◦ σ = 0 o .According to [7], given the non-degenerate symplectic structure on T ∗ Y t , ω t ∈ Ω ( T ∗ Y t ),the relation R ∗ t ω t = Ω t holds, and hence ker T R t = ker Ω t , where T R t : T Z ⋆t → T T ∗ Y t represents the tangent map of R t . In fact, the latter is relevant as the induced quotientmap Z ⋆t / ker T R t → T ∗ Y t defines a symplectic diffeomorphism, as discussed in [9].Additionally, we identify N t := n σ ∈ Z ⋆t (cid:12)(cid:12)(cid:12) σ = F L ◦ j φ ◦ i t o as the subset of Z ⋆t thatprojects by means of R t onto the t -primary constraint set of the theory P t, ζ Y . Indeed,this arises from the fact that for σ ∈ N t it is possible to write R t ( σ ) = ∂L∂y a (cid:16) j ϕ, ˙ ϕ (cid:17) dy a ⊗ d n − x (71)which corresponds to an element of P t, ζ Y . Henceforward, a section σ ∈ N t that projectsby means of R t onto ( ϕ, π ) ∈ P t, ζ Y will be referred to as an holonomic lift of ( ϕ, π ) [7].Bearing this in mind, we are in the position to study the action of G on Z ⋆t and T ∗ Y t . In order to do so, we start by remembering that, G acts on Z ⋆ by covariantcanonical transformations, and therefore there is an associated covariant momentummap J ( M ) : Z ⋆ → g ∗ ⊗ Λ n − T ∗ Z ⋆ . In particular, the latter is important as it induceson Z ⋆t the so-called energy-momentum map E t : Z ⋆t → g ∗ , namely h E t ( σ ) , ξ η i := Z Σ t σ ∗ J ( M ) ( ξ η ) , (72)where σ ∈ Z ⋆t and ξ η ∈ g . Now, we introduce G t := n η ∈ G (cid:12)(cid:12)(cid:12) η X (Σ t ) = Σ t o to denote thesubset of G that acts on Σ t by diffeomorphisms. In addition, we define g t ⊂ g as the Liealgebra of G t . Observe that, for all η ∈ G t , the map η t := η X (cid:12)(cid:12)(cid:12) Σ t may be identified as anelement of the group of diffeomorphisms on Σ t , Diff (Σ t ). Hereinafter, given ξ η ∈ g t theinfinitesimal generator of η ∈ G t , we introduce ξ Xη t ∈ X ( X ) to denote the infinitesimalgenerator of η t ∈ Diff (Σ t ), which is a vector field tangent to the Cauchy surface Σ t , andtherefore, in adapted coordinates, it must satisfy the condition ξ = 0 on Σ t , being ξ the component of such a vector field along ∂ . In fact, as discussed in [9], the action of G t on Z ⋆t preserves the presymplectic structure Ω t , and in consequence the restrictionof the energy-momentum map E t to the subspace g t gives rise to the correspondingmomentum map, specifically J t := E t (cid:12)(cid:12)(cid:12) g t : Z ⋆t → g ∗ t . Furthermore, since ( T ∗ Y t , ω t ) canbe thought of as the quotient of the presymplectic space ( Z ⋆t , Ω t ) by the kernel of T R t ,we have that, the momentum map J t : T ∗ Y t → g ∗ t associated with the action of G t on T ∗ Y t is given by h J t ( ϕ, π ) , ξ η i := hJ t ( σ ) , ξ η i , (73)where ξ η ∈ g t , ( ϕ, π ) ∈ T ∗ Y t and σ ∈ R − t { ( ϕ, π ) } ⊂ Z ⋆t .Besides, since ζ J Y := j ζ Y (the first jet prolongation of ζ Y ) and its correspondingvector flow define an infinitesimal symmetry of the system, we know that the α ( M ) -liftof ζ Y to Z ⋆ , ζ Z ⋆ ∈ X ( Z ⋆ ), satisfies the relation L ζ Z⋆ Ω ( M ) = 0, and hence there is an( n − H ( M ) ζ Z⋆ ∈ Ω n − ( Z ⋆ ) on Z ⋆ such that ζ Z ⋆ y Ω ( M ) = d H ( M ) ζ Z⋆ , which in turn review on geometric formulations for classical field theory H t , ζ Y : P t, ζ Y → R on the t -primary constraint set by means of H t, ζ Y ( ϕ, π ) := − Z Σ t σ ∗ H ( M ) ζ Z⋆ , (74)being ( ϕ, π ) ∈ P t and σ ∈ R − t { ( ϕ, π ) } ∩ N t . In fact, it is possible to show that the abovefunctional is nothing but the instantaneous Hamiltonian of the theory, as discussed indetail in references [7, 9, 49].Finally, we would like to emphasize that, for a classical field theory with localizablesymmetries, the second Noether theorem extended to the multisymplectic approachestablishes that, the vanishing of the momentum map (73) gives rise to the set offirst-class constraints that characterizes the system within the instantaneous Dirac-Hamiltonian formulation [5, 9, 10]. In other words, we have that, for a classicalfield theory with localizable symmetries, the admissible space of Cauchy data for theevolution equations of the system is determined by the zero level set of the momentummap (73), namely J − t (0) := n ( ϕ, π ) ∈ T ∗ Y t (cid:12)(cid:12)(cid:12) h J t ( ϕ, π ) , ξ η i = 0 , ∀ ξ η ∈ g t o . (75)Bearing this in mind, both the momentum map (73) and the functional (74) will befundamental for our discussion in the subsequent section.
3. Geometric-covariant analysis of the Bonzom-Livine model for gravity
In the present section, we will study the Bonzom-Livine model for gravity within thegeometric-covariant Lagrangian, multisymplectic and polysymplectic formulations forclassical field theory. For this purpose, we will start by introducing a mathematicaldescription of the physical system of our interest, the Bonzom-Livine model for gravity,which consists of a formulation for 3-dimensional gravity that includes an arbitraryImmirzi-like parameter. Here, we will discuss how the different formalisms introducedin section 2 allow us to obtain a more general understanding of the features of the theoryat the classical level. In this regard, we will not only derive the field equations of themodel but we will also analyze its gauge symmetries. Additionally, after performingthe space plus time decomposition of the space-time manifold on which the system isdefined, we will describe the manner in which it is possible to recover the instantaneousDirac-Hamiltonian analysis of the theory by considering the multisymplectic formulationas a starting point.
As mentioned before, the Bonzom-Livine model for gravity corresponds to a formulationof 3-dimensional Einstein theory of General Relativity that includes an arbitraryImmirzi-like parameter [40]. From the point of view of physics, the aforementionedgravity model is particularly interesting since, in close analogy to the role played by review on geometric formulations for classical field theory X be a 3-dimensional Riemannian manifold without boundary †† .Then, given t a Lie algebra (or a vector subspace of a Lie algebra), we identifyΩ p ( X, t ) := Ω p ( X ) ⊗ t as the set of t -valued p -form on X . In addition, let us consider { T a } a basis for t and ˜ β : t ⊗ t → R a symmetric bilinear form. Thus, for anypair of t -valued forms on X , µ = µ a ⊗ T a ∈ Ω p ( X, t ) and λ = λ a ⊗ T a ∈ Ω q ( X, t ),we define the following operations [ µ, λ ] := ( µ a ∧ λ b ) ⊗ [ T a , T b ], dµ := dµ a ⊗ T a and˜ β ( µ ∧ λ ) := µ a ∧ λ b ˜ β ( T a , T b ), where [ · , · ] denotes the Lie bracket. Now, let us considerthe Lie group H := SO(3) and h := so (3) its corresponding Lie algebra. Thus, byfollowing the Palatini formalism, we introduce the h -connection ω and the co-frame field e to be the fundamental fields of the 3-dimensional Einstein theory of General Relativity[55]. As we will see below, the Bonzom-Livine model for gravity arises from the factthat, by combining the h -connection and the co-frame field e into a single connection,it is possible to reformulate 3-dimensional gravity with cosmological constant Λ as aChern-Simons theory with gauge group G , which may be identified with SO(4), ISO(3)or SO(3 ,
1) depending on whether Λ is correspondingly positive, zero or negative, asdiscussed in [36].To illustrate this, we start by mentioning that, on the one hand, given { J a , P a } abasis of g (the Lie algebra of G ), we can write[ J a , J b ] = ǫ abc J c , [ J a , P b ] = ǫ abc P c , [ P a , P b ] = Λ ǫ abc J c , (76)where ǫ abc stands for the 3-dimensional Levi-Civita alternating symbol ( a, b, c = 1 , , β i : g ⊗ g → R on the Lie algebra (76) ( i = 1 , β ( J a , P a ) = δ ab , β ( J a , J a ) = 0 , β ( P a , P a ) = 0 ,β ( J a , P a ) = 0 , β ( J a , J a ) = δ ab , β ( P a , P a ) = Λ δ ab , (77)being δ ab the Kronecker delta. Note that, the bilinear form β is non-degenerated forall Λ, while β is non-degenerated only for Λ = 0. On the other hand, we have that,by regarding H as a closed subgroup of G , we can perform the symmetric splitting g = h ⊕ p , where the supplement space p ⊂ g (a vector subspace of g ) satisfies thecommutation relations [ h , p ] ⊂ p and [ p , p ] ⊂ h (see references [51, 56] for details aboutsymmetric splittings of Lie algebras). Here, the Lie algebra h (as subalgebra of g ) andthe supplement space p are spanned by J a and P a , respectively.In light of this, given the bilinear form β BL := β + (cid:16) γ q | Λ | (cid:17) − β and thinking ofthe h -connection ω ∈ Ω ( X, h ) and the co-frame field e ∈ Ω ( X, p ) as parts of a Cartan †† From now on, we will set the dimension of the space-time manifold to n = 3 and also we will fix thesignature of the metric to s m = 1. review on geometric formulations for classical field theory A := ω + e ∈ Ω ( X, g ) [56], the Bonzom-Livine model for gravity (as aChern-Simons theory with gauge group G ) is defined by S BL [ A ] := Z X β BL (cid:16) A ∧ dA + 13 A ∧ [ A, A ] (cid:17) , (78)where γ represents an arbitrary parameter [40, 45]. Observe that, in terms of the h -connection ω and the co-frame field e , it is possible to write S BL [ e, ω ] := Z X " e a ∧ F a [ ω ] + Λ3 ǫ abc e a ∧ e b ∧ e c + 1 γ q | Λ | (cid:18) ω a ∧ d ω a + 13 ǫ abc ω a ∧ ω b ∧ ω c + s | Λ | e a ∧ d ω e a (cid:19) , (79)where s = − , , F [ ω ] := dω + 12 [ ω, ω ] and d ω := d + [ ω, · ] denote the curvature h -valued 2-form and the covariantexterior derivative associated with the h -connection ω , respectively. As we will seein the subsequent subsections, the action principle (79) describes the 3-dimensionalEinstein theory of General Relativity with γ playing an analogous role to that of theImmirzi parameter in 4-dimensional gravity. Hence, we will refer to γ as the Immirzi-likeparameter.Finally, we would like to emphasize that, when considering the close relationshipof the Bonzom-Livine model for gravity with the Chern-Simons gauge theory, it isnot surprising that, the action principle (79) defines a topological field theory whichis invariant under both gauge transformations and diffeomorphisms. However, thesymmetries of the system are not all independent since the space-time diffeomorphismscan be generated through a combination of the gauge transformations associated withthe so-called H -gauge and translational symmetries of the theory, which are alsostrongly related to the first-class constraints that arise within the instantaneous Dirac-Hamiltonian analysis of the model [40, 46]. Therefore, in what follows, we will focusour attention to the study of the gauge symmetries of the system. In this regard, wehave that, on the one hand, the gauge transformations associated with the H -gaugesymmetry of the theory read e → e θ := e + [ e, θ ] ,ω → ω θ := ω + d ω θ , (80)where θ ∈ Ω ( X, h ) denotes an arbitrary h -valued function on X . On the other hand,the gauge transformations associated with the translational symmetry of the model aregiven by e → e χ := e + d ω χ ,ω → ω χ := ω + [ e, χ ] , (81)being χ ∈ Ω ( X, p ) an arbitrary p -valued function on X . As discussed in [40], thetranslational symmetry, also known as topological symmetry [55], implies that e is a review on geometric formulations for classical field theory G , such that the elements of the Lie subalgebra h give rise to the H -gauge symmetry,whereas the elements of the vector subspace p induce the translational symmetry.Next, we will describe within the geometric-covariant Lagrangian formalism thefeatures of the gauge and diffeomorphism invariant topological field theory (79) puttingspecial attention to analyzing the H -gauge and translational symmetries of the system. In this subsection, our main aim is to analyze the Bonzom-Livine model for gravity (79)from the point of view of the geometric-covariant Lagrangian formulation for classicalfield theory. Here, we will not only obtain the field equations of the system but we willalso study its gauge symmetries, which will be fundamental in order to construct theassociated Noether currents.To start, let us consider Y := (cid:16) T ∗ X ⊗ p (cid:17) ⊕ (cid:16) T ∗ X ⊗ h (cid:17) . Then, we define (cid:16) Y, π XY , X (cid:17) as the covariant configuration space of the Bonzom-Livine model for gravity. This sincethe dynamical fields of such physical system can be understood as local sections of π XY . In particular, given ( x µ ) a coordinate system on X , we identify ( x µ , a aµ , b aµ ) as anadapted coordinate system on Y . Thus, a section φ ∈ Y X can be locally representedby ( x µ , e aµ , ω aµ ). Additionally, being ( J Y, π
Y J Y , Y ) the affine jet bundle over Y , wedenote by ( x µ , a aµ , b aµ , a aµν , b aµν ) an adapted coordinate system on J Y . Then, a section j φ ∈ J Y X of π XJ Y , the first jet prolongation of a section φ ∈ Y X , can be locallyexpressed as ( x µ , e aµ , ω aµ , ∂ µ e aν , ∂ µ ω aν ). In light of this, the action principle (79) can berewritten as S BL [ e, ω ] = Z X d n x ( j φ ) ∗ L BL , with the Lagrangian function, L BL : J Y → R , explicitly given by L BL := ǫ µνσ a aµ F a νσ + Λ3 ǫ abc a aµ a bν a cσ + 1 γ q | Λ | (cid:18) b a µ b aνσ + 13 ǫ abc b aµ b bν b cσ + Λ a a µ D ν a aσ (cid:19) , (82)where F aµν := b aµν − b aνµ + ǫ abc b bµ b cν stands for the components of the curvature h -valued2-form F , D µ a aν := a aµν + ǫ abc b bµ a cν denotes the local representation of the covariantderivative d ω , the symbol ǫ µνσ stands for the 3-dimensional Levi-Civita tensor, while ǫ abc comes from the structure constants of the Lie algebra (76). Nevertheless, as mentionedin section 2, within the geometric-covariant Lagrangian approach, the Lagrangianfunction is not the main object of interest but the so-called Poincaré-Cartan forms,Θ ( L )BL ∈ Ω n ( J Y ) and Ω ( L )BL := − d Θ ( L )BL ∈ Ω n +1 ( J Y ), which by construction contain all review on geometric formulations for classical field theory n -form explicitly readsΘ ( L )BL := ǫ µνσ δ ab s q | Λ | γ a bσ da aν + 2 a bσ + 12 γ q | Λ | b bσ db aν ∧ d n − x µ + ǫ abc a aµ b bν b cσ + Λ3 a aµ a bν a cσ + 1 γ q | Λ | (cid:18) b aµ b bν b cσ + Λ b aµ a bν a cσ (cid:19) d n x . (83)As we will see below, the Poincaré-Cartan forms will allow us to obtain not only thecorrect field equations of the system but also the conserved currents associated with the H -gauge and translational symmetries of the model.For simplicity, in order to obtain the dynamical equations of the Bonzom-Livinemodel for gravity, let us consider W ∈ X ( J Y ) an arbitrary vector field on J Y locallyrepresented by W := W aµ ∂∂a aµ + W aµ ∂∂b aµ . (84)By direct calculation, it is not difficult to see that, W y d Θ ( L )BL = ǫ µνσ s q | Λ | γ W aµ + W aµ (cid:18) − δ ab da bν ∧ d n − x σ + ǫ abc b bν a cσ d n x (cid:19) + W aµ + 1 γ q | Λ | W aµ (cid:18) − δ ab db bν ∧ d n − x σ + ǫ abc (cid:16) b bν b cσ + Λ a bν a cσ (cid:17) d n x (cid:19) . Besides, we know that, given φ ∈ Y X a critical point of the action principle (79), thecondition j φ ∗ (cid:16) W y Ω ( L )BL (cid:17) = 0 must hold, and hence we can write ǫ µνσ δ ab s q | Λ | γ W aµ + W aµ D ν e bσ + W aµ + 1 γ q | Λ | W aµ (cid:18) F bνσ + Λ ǫ bcd e cν e dσ (cid:19) = 0 . In particular, by taking into account that W is an arbitrary vector field on J Y , theabove relation gives rise to the following set of equations s q | Λ | γ D [ µ e aν ] + 12 (cid:18) F aµν + Λ ǫ abc e bµ e cν (cid:19) = 0 ,D [ µ e aν ] + 12 γ q | Λ | (cid:18) F aµν + Λ ǫ abc e bµ e cν (cid:19) = 0 , (85)which correspond to the field equations of the Bonzom-Livine model for gravity obtainedin [40]. Of course, it is not difficult to show that, for γ = s , relations (85) are completelyequivalent to the vanishing torsion condition and the Einstein equations [36, 55, 56],namely D [ µ e aν ] = 0 , (86a) F aµν + Λ ǫ abc e bµ e cν = 0 . (86b) review on geometric formulations for classical field theory γ ambiguity inherent to the Bonzom-Livine model forgravity, the action principle (79) is able to reproduce the same field equations of the3-dimensional Einstein theory of General Relativity, as long as the condition γ = s holds.Now, we will analyze the gauge symmetries associated with the extended gaugesymmetry group of the theory at the Lagrangian level. For this purpose, given ξ θ ∈ h and ξ χ ∈ p , we start by defining ξ Yθ ∈ X ( Y ) and ξ Yχ ∈ X ( Y ) as the infinitesimalgenerators of the gauge transformation (80) and (81), respectively. In local coordinatesthese vector fields are explicitly given by ξ Yθ := ǫ abc a bµ θ c ∂∂a aµ + D µ θ a ∂∂b aµ ,ξ Yχ := D µ χ a ∂∂a aµ + Λ ǫ abc a bµ χ c ∂∂b aµ . (87)Furthermore, we know that fibre-preserving transformations on the covariant configu-ration space of a given classical field theory induce fibre-preserving transformations onthe corresponding affine jet bundle. In particular, by using formula (9), it is possible tosee that, the first jet prolongations, ξ J Yθ ∈ X ( J Y ) and ξ J Yχ ∈ X ( J Y ), associatedwith the infinitesimal generators (87) can be locally written as ξ J Yθ := ǫ abc a bµ θ c ∂∂a aµ + D µ θ a ∂∂b aµ + ǫ abc (cid:16) a bν ∂ µ θ c + a bµν θ c (cid:17) ∂∂a aµν + (cid:16) D ν ( ∂ µ θ a ) + ǫ abc b bµν θ c (cid:17) ∂∂b aµν , (88a) ξ J Yχ := D µ χ a ∂∂a aµ + Λ ǫ abc a bµ χ c ∂∂b aµ + (cid:16) D ν ( ∂ µ χ a ) + ǫ abc b bµν χ c (cid:17) ∂∂a aµν + Λ ǫ abc (cid:16) a bν ∂ µ χ c + a bµν χ c (cid:17) ∂∂b aµν . (88b)Bearing this in mind, we are in the position to study the action of the gaugesymmetries of the Bonzom-Livine model for gravity on its associated Poincaré-Cartan n -form (83). To begin with, let us consider the case of the H -gauge symmetry. Bydirect calculation, we have that, for all ξ θ ∈ h , this gauge symmetry preserves thePoincaré-Cartan n -form up to an exact form, namely L ξ J Yθ Θ ( L )BL = dα ( L ) θ , (89)where α ( L ) θ ∈ Ω n − ( J Y ) represents a π XJ Y -horizontal ( n − J Y explicitlygiven by α ( L ) θ := 1 γ q | Λ | ǫ µνσ δ ab ∂ σ θ a b bν d n − x µ . (90)In a similar manner, a straightforward calculation shows that, for all ξ χ ∈ p , thetranslational symmetry of the theory also preserves the Poincaré-Cartan n -form up review on geometric formulations for classical field theory L ξ J Yχ Θ ( L )BL = dα ( L ) χ , (91)being α ( L ) χ ∈ Ω n − ( J Y ) a π XJ Y -horizontal ( n − J Y locally representedby α ( L ) χ := ǫ µνσ ǫ abc χ a (cid:16) b bν b cσ − Λ a bν a cσ (cid:17) + δ ab ∂ σ χ a s q | Λ | γ a bν + 2 b bν d n − x µ . (92)Here, we would like to emphasize that, on the one hand, the π XJ Y -horizontal( n − H -gauge and translational symmetries of the Bonzom-Livine model forgravity are localizable symmetries. To illustrate this, let us consider the case of the H -gauge symmetry since the same arguments are valid for the translational symmetry. Tostart, note that, for all θ ∈ Ω ( X, h ), both the vector field (88a) and the π XJ Y -horizontal( n − θ and their partial derivatives, andtherefore the collection C H LS of pairs (cid:16) ξ J Yθ , α ( L ) θ (cid:17) is a vector space. In addition, weknown that each pair (cid:16) ξ J Yθ , α ( L ) θ (cid:17) ∈ C H LS corresponds to a Noether symmetry since forany θ ∈ Ω ( X, h ) the relation (89) holds. Now, for an arbitrary element θ ∈ Ω ( X, h ),let us consider U , U ⊂ X two open sets with disjoint closures and ˜ θ ∈ Ω ( X, h ) anelement such that ˜ θ = θ on U but ˜ θ = 0 on U . Then, it is clear that, the pairs (cid:16) ξ J Yθ , α ( L ) θ (cid:17) , (cid:16) ξ J Y ˜ θ , α ( L )˜ θ (cid:17) ∈ C H LS satisfy condition (15), and proceeding in the same wayfor any pair of open sets with disjoint closures U , U ⊂ X , it is possible to see that,the H -gauge symmetry of the gravity model (79) satisfies the properties of a localizablesymmetry.Hence, the action of G on J Y has an associated Lagrangian covariant momentummap, namely J ( L ) : J Y → g ∗ ⊗ Λ n − T ∗ J Y . In particular, by taking into accountthe splitting g = h ⊕ p , we can obtain the local representation of such a map, that is, J ( L ) ( ξ ) ∈ Ω n − ( J Y ) for an arbitrary ξ ∈ g , as follows. To start, let us consider ξ θ ∈ h and ξ χ ∈ p . Then, by considering (13), and in light of relations (89) and (91), it ispossible to write J ( L ) ( ξ θ ) : = ǫ µνσ (cid:18) δ ab ∂ ν θ a a bσ − ǫ abc θ a b bν a cσ (cid:19) + 1 γ q | Λ | δ ab ∂ ν θ b b bσ − ǫ abc θ a (cid:18) b bν b cσ + Λ a bν a cσ (cid:19)! d n − x µ ,J ( L ) ( ξ χ ) : = ǫ µνσ s q | Λ | γ δ ab ∂ ν χ a a bσ − ǫ abc χ a b bν a cσ ! + 2 δ ab ∂ ν χ b b bσ − ǫ abc χ a (cid:18) b bν b cσ + Λ a bν a cσ (cid:19) d n − x µ . (93) review on geometric formulations for classical field theory ξ θ ∈ h , ξ χ ∈ p and φ ∈ Y X a solution of the field equations(85), the ( n − X explicitly defined by J ( L ) ( ξ θ ) : = ǫ µνσ (cid:18) δ ab ∂ ν θ a e bσ − ǫ abc θ a ω bν e cσ (cid:19) + 1 γ q | Λ | δ ab ∂ ν θ b ω bσ − ǫ abc θ a (cid:18) ω bν ω cσ + Λ e bν e cσ (cid:19)! d n − x µ , J ( L ) ( ξ χ ) : = ǫ µνσ s q | Λ | γ δ ab ∂ ν χ a e bσ − ǫ abc χ a ω bν e cσ ! + 2 δ ab ∂ ν χ b ω bσ − ǫ abc χ a (cid:18) ω bν ω cσ + Λ e bν e cσ (cid:19) d n − x µ , (94)are nothing but the Noether currents associated with the H -gauge and translationalsymmetries of the Bonzom-Livine model for gravity, respectively. Observe that, byimposing the field equations (85), the integral of each of the Noether currents (94)over a Cauchy surface Σ t vanishes, that is, relation (16) holds. Naturally, the latter isconsistent with the fact that the H -gauge and translational symmetries of the theory arelocalizable symmetries, and therefore the second Noether theorem establishes that, forall φ ∈ Y X a solution of the Euler-Lagrange field equation of the theory, the LagrangianNoether charges of the system must vanish.In the following subsection, we will analyze the gravity model (79) within themultisymplectic approach paying special attention to the study of the gauge symmetriesof the theory, which will be fundamental for our discussion. In the present subsection, we will perform the multisymplectic analysis of the Bonzom-Livine model for gravity. In particular, we will focus our attention on studying the actionof the extended gauge symmetry group of the theory on its corresponding covariantmultimomenta phase-space. Here, we will describe how the H -gauge and translationalsymmetries of the system give rise to covariant canonical transformations, which in turnwill allow us to obtain the covariant momentum map associated with the extended gaugesymmetry group of the model.To start, let us consider ( Z ⋆ , π Y Z ⋆ , Y ) the covariant multimomenta phase-spaceassociated with the theory. Then, as described in subsection 2.3, given ( x µ , a aµ , b aµ )an adapted coordinate system on Y , we introduce ( x µ , a aµ , b aµ , p, p µνa , ¯ p µνa ) to denote anadapted coordinate system on Z ⋆ . Thus, the canonical and multisymplectic forms ofthe Bonzom-Livine model for gravity, Θ ( M )BL ∈ Ω n ( Z ⋆ ) and Ω ( M )BL ∈ Ω n +1 ( Z ⋆ ), can be review on geometric formulations for classical field theory ( M )BL := p µνa da aν ∧ d n − x µ + ¯ p µνa db aν ∧ d n − x µ + p d n x , (95a)Ω ( M )BL := da aν ∧ dp µνa ∧ d n − x µ + db aν ∧ d ¯ p µνa ∧ d n − x µ − dp ∧ d n x . (95b)Now, in order to study the H -gauge and translational symmetries of the theory atthe multisymplectic level, given ξ θ ∈ h and ξ χ ∈ p , we start by identifying ξ αθ ∈ X ( Z ⋆ )and ξ αχ ∈ X ( Z ⋆ ), that is, the α ( M ) -lifts of the vector fields (87) to Z ⋆ , as the infinitesimalgenerators of the gauge transformations (80) and (81) on the covariant multimomentaphase-space, respectively. In local coordinates these vector field explicitly read ξ αθ := ǫ abc a bµ θ c ∂∂a aµ + D µ θ a ∂∂b aµ − ǫ abc θ b ¯ p µνc − γ q | Λ | ǫ µνσ δ ab ∂ σ θ b ∂∂ ¯ p µνa − ǫ abc θ b p µνc ∂∂p µνa − (cid:18) ǫ abc a bν ∂ µ θ c p µνa + D ν ( ∂ µ θ a )¯ p µνa (cid:19) ∂∂p , (96a) ξ αχ := D µ χ a ∂∂a aµ + Λ ǫ abc a bµ χ c ∂∂b aµ − (cid:18) ǫ abc χ b p µνc − ǫ µνσ δ ab D σ χ b (cid:19) ∂∂ ¯ p µνa − Λ ǫ abc χ b ¯ p µνc − ǫ µνσ γ q | Λ | δ ab ∂ σ χ b + 2 ǫ abc χ b a cσ ∂∂p µνa − D ν ( ∂ µ χ a ) p µνa + Λ ǫ abc a bν ∂ µ χ c ¯ p µνa − ǫ µνσ ǫ abc ∂ µ χ a (cid:18) b bν b cσ − Λ a bν a cσ (cid:19) ∂∂p . (96b)With this in mind, it is not difficult to see that, for all ξ θ ∈ h , the H -gauge symmetryof the model acts on Z ⋆ through covariant canonical transformations, specifically L ξ αθ Θ ( M )BL = dα ( M ) θ , (97)being α ( M ) θ ∈ Ω n − ( Z ⋆ ) a π XZ ⋆ -horizontal ( n − Z ⋆ locally given by α ( M ) θ := 1 γ q | Λ | ǫ µνσ δ ab ∂ σ θ a b bν d n − x µ . (98)In a similar manner, we have that, for all ξ χ ∈ p , the translational symmetry of thesystem also acts on Z ⋆ through covariant canonical transformations since the followingrelation holds L ξ αχ Θ ( M )BL = dα ( M ) χ , (99)where α ( M ) χ ∈ Ω n − ( Z ⋆ ) denotes a π XZ ⋆ -horizontal ( n − Z ⋆ locallyrepresented as α ( M ) χ := ǫ µνσ ǫ abc χ a (cid:16) b bν b cσ − Λ a bν a cσ (cid:17) + δ ab ∂ σ χ a s q | Λ | γ a bν + 2 b bν d n − x µ . (100) review on geometric formulations for classical field theory G on Z ⋆ has an associated covariant momentum map,namely, J ( M ) : Z ⋆ → g ∗ ⊗ Λ n − T ∗ Z ⋆ . In fact, as in the Lagrangian case, by consideringthe splitting g = h ⊕ p , we can express the local representation of such a map, that is, J ( M ) ( ξ ) ∈ Ω n − ( Z ⋆ ) for an arbitrary ξ ∈ g , as follows. To begin with, let us consider ξ θ ∈ h and ξ χ ∈ p . Then, by taking into account (35), and in light of relations (97) and(99), it is possible to write J ( M ) ( ξ θ ) := D ν θ a ¯ p µνa + ǫ abc a bν θ c p µνa − γ q | Λ | ǫ µνσ δ ab ∂ σ θ a b bν d n − x µ ,J ( M ) ( ξ χ ) := D ν χ a p µνa + Λ ǫ abc a bν χ c ¯ p µνa − ǫ µνσ ǫ abc χ a (cid:18) b bν b cσ − Λ a bν a cσ (cid:19) + δ ab ∂ σ χ a s q | Λ | γ a bν + 2 b bν d n − x µ . (101)Note that the Lagrangian covariant momentum map (93) can be recovered by pulling-back the covariant momentum map (101) with the covariant Legendre transformation(20), which establishes a link between both geometric structures. Finally, we would liketo mention that, as we will see in the subsequent subsections, the covariant momentummap associated with the extended gauge symmetry group of the model will allow us notonly to construct conserved currents for the solutions of the De Donder-Weyl-Hamiltonfield equations of the system but it will also be fundamental to obtain, on the space ofCauchy data, the first-class constrained structure that characterizes the theory withinthe instantaneous Dirac-Hamiltonian formulation.Next, we will carry out the polysymplectic analysis of the gravity model (79), wherewe will emphasize the relevance of the Immirzi-like parameter inherent to the systemwithin the De Donder-Weyl Hamiltonian formulation. In this subsection, we will study the Bonzom-Livine model for gravity from the pointof view of the polysymplectic formalism. To this end, since the gravity model of ourinterest corresponds to a singular Lagrangian system, we will make use of the algorithmto study this kind of systems within the polysymplectic approach. Here, we will notonly obtain the correct De Donder-Weyl-Hamilton field equations of the theory but wewill also describe how the covariant momentum map associated with the extended gaugesymmetry group of the system gives rise to the conserved currents of the model withinthe De Donder-Weyl canonical theory.To begin with, let us consider the quotient bundle (
P, π
Y P , Y ), as described insubsection 2.4. Then, given ( x µ , a aµ , b aµ ) an adapted coordinate system on Y , we denoteby (cid:16) x µ , a aµ , b aµ , p µνa , ¯ p µνa (cid:17) an adapted coordinate system on P . Thus, a section ρ ∈ P X of π XP (the polymomenta phase-space of the theory) can be locally represented as review on geometric formulations for classical field theory x µ , e aµ , ω aµ , π µνa , ¯ π µνa ). Now, in order to describe the gravity model (79) in a covariantHamiltonian-like formulation, we proceed to apply the covariant Legendre map (24),which allows us to write p µνa := ∂L BL ∂a aµν = s q | Λ | γ ǫ µνσ δ ab a bσ , ¯ p µνa := ∂L BL ∂b aµν = 2 ǫ µνσ δ ab a bσ + 12 γ q | Λ | b bσ . (102)Note that, on the one hand, the set of relations (102) gives rise to the primaryconstraints of the Bonzom-Livine model for gravity which properly identified as( n − C (1) νa := p µνa − s q | Λ | γ ǫ µνσ δ ab a bσ ̟ µ ≈ , (103a) C (2) νa := ¯ p µνa − ǫ µνσ δ ab a bσ + 12 γ q | Λ | b bσ ̟ µ ≈ . (103b)On the other hand, we have that, on the primary constraint surface P PCSBL ⊂ P , namelythe surface on P characterized by the vanishing of the primary constraint ( n − P PCSBL we are able to choose a more suitable set of canonicallyconjugate variables for which the commutation relations under the Poisson-Gerstenhaberbracket are given by { [ p µνa ̟ µ , a b [ σ ̟ ρ ] ] } = δ ba δ ν [ σ ̟ ρ ] , { [ ¯ p µνa ̟ µ , b b [ σ ̟ ρ ] ] } = δ ba δ ν [ σ ̟ ρ ] . (104)In particular, this choice of canonically conjugate variables for the bracket structureassociated with the physical model under analysis is related to the fact that thedynamical information of a gauge theory whose fields are Lie algebra-valued 1-formsis contained in the anti-symmetric part of the polymomenta variables, as discussed in[27, 51]. Thus, the set of canonically conjugate variables introduced in (104) will allow usto consistently describe the system of our interest within the polysymplectic approach,as we will see throughout this subsection.Bearing this in mind, it is not difficult to see that, on the primary constraint surface P PCSBL , the De Donder-Weyl Hamiltonian associated with the Bonzom-Livine model forgravity can be written as H DW := − ǫ µνσ ǫ abc a aµ b bν b cσ + Λ3 a aµ a bν a cσ + 1 γ q | Λ | (cid:18) b aµ b bν b cσ + Λ a aµ b bν a cσ (cid:19) . (105)However, since the model under consideration corresponds to a singular Lagrangiansystem, we know that the De Donder-Weyl-Hamilton field equations generated by the review on geometric formulations for classical field theory H DW := H DW + λ ( i ) aν • C ( i ) νa , (106)where λ ( i ) aν := λ ( i ) aµν dx µ ∈ Ω ( P ) denotes a set of Lagrange multiplier (1 ; 1)-formsenforcing the primary constraint ( n − H DW as thetotal De Donder-Weyl Hamiltonian. Consequently, to determine if the set of constraint( n − n − n − ǫ νσρ ǫ abc s q | Λ | γ a bσ b cρ + 12 (cid:18) b bσ b cρ + Λ a bσ a cρ (cid:19) + δ ab s q | Λ | γ λ (1) bσρ + λ (2) bσρ ≈ , (107)while a straightforward calculation shows that the consistency condition associated withthe primary constraint ( n − ǫ νσρ ǫ abc a bσ b cρ + 12 γ q | Λ | (cid:18) b bσ b cρ + Λ a bσ a cρ (cid:19) + δ ab λ (1) bσρ + 1 γ q | Λ | λ (2) bσρ ≈ . (108)Observe that, relations (107) and (108) impose restrictions on the Lagrange multiplier(1 ; 1)-forms, which implies that there are no secondary constraints, and therefore the setof constraint ( n − γ = s , the consistencyconditions allow us to explicitly fix the components of the Lagrange multiplier (1 ; 1)-forms, namely λ (1) a [ µν ] = − ǫ abc b b [ µ a cν ] ,λ (2) a [ µν ] = − ǫ abc (cid:16) b bµ b cν + Λ a bµ a cν (cid:17) . (109)Now, we are in the position to classify the set of constraint ( n − n − n − C ( i , j ) µνab := { [ C ( i ) µa , C ( j ) νb ] } , which explicitly reads C ( i , j ) µνab = 2 s q | Λ | γ
11 1 γ q | Λ | δ ab ǫ µνσ ̟ σ . (110) review on geometric formulations for classical field theory n − P PCSBL , it is clear that, according to definition (54), the set of constraint ( n − g = h ⊕ p . Then, given ξ θ ∈ h and ξ χ ∈ p , it is possible to see that, by taking the pullback of the local representation(101) with the De Donder-Weyl Hamiltonian section h DW ∈ Z ⋆P , we can obtain thelocal representation of the covariant momentum map on the polymomenta phase-space,namely J ( P ) ( ξ ) := h ∗ DW J ( M ) ( ξ ) ∈ Ω n − ( P ) for an arbitrary ξ ∈ g , which in terms of thehorizontal ( n −
1; 1)-forms on P can be written as J ( P ) ( ξ θ ) := D ν θ a ¯ p µνa + ǫ abc a bν θ c p µνa − γ q | Λ | ǫ µνσ δ ab ∂ σ θ a b bν ̟ µ ,J ( P ) ( ξ χ ) := D ν χ a p µνa + Λ ǫ abc a bν χ c ¯ p µνa − ǫ µνσ ǫ abc χ a (cid:18) b bν b cσ − Λ a bν a cσ (cid:19) + δ ab ∂ σ χ a s q | Λ | γ a bν + 2 b bν ̟ µ . (111)Observe that, for all ξ θ ∈ h and ξ χ ∈ p , the set of horizontal ( n −
1; 1)-forms (111)corresponds to a set of Hamiltonian ( n − P . In other words, we have that,for each horizontal ( n −
1; 1)-form of the set (111) there exists a vertical 1-multivectorfield on P satisfying condition (39). From our point of view, the latter is particularlyrelevant since this will allow us to study the induced covariant momentum map (111) bymeans of the Poisson-Gerstenhaber bracket inherent to the polysymplectic formalism.In particular, a straightforward calculation shows that, for all ξ θ ∈ h and ξ χ ∈ p , theinduced covariant momentum map (111) satisfies the following commutation relations { [ C (1) νa , J ( P ) ( ξ θ )] } = ǫ abc θ b C (1) νc , { [ C (2) νa , J ( P ) ( ξ θ )] } = ǫ abc θ b C (2) νc , { [ C (1) νa , J ( P ) ( ξ χ )] } = Λ ǫ abc χ b C (2) νc , { [ C (2) νa , J ( P ) ( ξ χ )] } = ǫ abc χ b C (1) νc , (112)which, in light of definition (54), implies that the local representation of the inducedcovariant momentum map (111) is characterized by a set of first-class Hamiltonian( n − P . In addition, we have that, for all ξ θ , ξ ˜ θ ∈ h and ξ χ , ξ ˜ χ ∈ p , thefollowing algebraic structure holds { [ J ( P ) ( ξ θ ) , J ( P ) ( ξ ˜ θ )] } = J ( P ) (cid:16) [ ξ θ , ξ ˜ θ ] (cid:17) − γ q | Λ | d (cid:16) ǫ µνσ δ ab θ a ∂ µ ˜ θ b ̟ νσ (cid:17) , { [ J ( P ) ( ξ θ ) , J ( P ) ( ξ χ )] } = J ( P ) (cid:16) [ ξ θ , ξ χ ] (cid:17) − d (cid:16) ǫ µνσ δ ab θ a ∂ µ χ b ̟ νσ (cid:17) , { [ J ( P ) ( ξ χ ) , J ( P ) ( ξ ˜ χ )] } = J ( P ) ([ ξ χ , ξ ˜ χ ]) − s q | Λ | γ d (cid:16) ǫ µνσ δ ab χ a ∂ µ ˜ χ b ̟ νσ (cid:17) . (113) review on geometric formulations for classical field theory n − F a Hamiltonian 0- or ( n − G a Hamiltonian ( n − { [ F, G ] } D := { [ F, G ] } − { [ F, C ( i ) µa ] } • (cid:16) C − i , j ) abµν ∧ { [ C ( j ) νb , G ] } (cid:17) , (114)where C − i , j ) abµν stands for the (1 ; 1)-form valued matrix satisfying the condition C − i , k ) acµλ ∧ C ( k , j ) λνcb = δ ji δ ab δ νµ ̟ . (115)Of courser, the Latin indices i , j and k run over the complete set of constraint ( n − n − C ( i,j ) µνab is given by (110). Thus, bydirect calculation, it is possible to show that, the (1 ; 1)-form valued matrix C − i , j ) abµν explicitly reads C − i , j ) abµν := − γ s − γ ) γ q | Λ | − − s q | Λ | γ δ ab ǫ µνσ dx σ . (116)Now, once we have constructed the Dirac-Poisson bracket associated with thesystem, we are in the position to study the Bonzom-Livine model for gravity on theconstraint surface P PCSBL . For this purpose, we start by calculating the commutationrelations of the canonically conjugate variables of the theory (104) under bracketstructure (114). In this regard, it is possible to write { [ p µνa ̟ µ , a b [ σ ̟ ρ ] ] } D = 12 s ( s − γ ) δ ba δ ν [ σ ̟ ρ ] , { [ ¯ p µνa ̟ µ , b b [ σ ̟ ρ ] ] } D = 12 s − γ ( s − γ ) δ ba δ ν [ σ ̟ ρ ] . (117)Furthermore, a straightforward computation shows that, the commutation relationsamong the field variable ( n − { [ a a [ ν ̟ µ ] , a b [ σ ̟ ρ ] ] } D = − q | Λ | γ ( s − γ ) δ ab ǫ [ ν | σρ ̟ µ ] , { [ b a [ ν ̟ µ ] , b b [ σ ̟ ρ ] ] } D = − s q | Λ | γ ( s − γ ) δ ab ǫ [ ν | σρ ̟ µ ] , { [ b a [ ν ̟ µ ] , a b [ σ ̟ ρ ] ] } D = 14 γ ( s − γ ) δ ab ǫ [ ν | σρ ̟ µ ] . (118) review on geometric formulations for classical field theory n − { [ p µνa ̟ µ , p ρσb ̟ ρ ] } D = − s γ q | Λ | ( s − γ ) δ ab ǫ νσρ ̟ ρ , { [ p µνa ̟ µ , ¯ p ρσb ̟ ρ ] } D = − s ( s − γ ) δ ab ǫ νσρ ̟ ρ , { [ ¯ p µνa ̟ µ , ¯ p ρσb ̟ ρ ] } D = − γ q | Λ | s ( s − γ ) δ ab ǫ νσρ ̟ ρ . (119)Note that, all the above commutation relations explicitly depend on the parameter γ , which implies that, after introducing the Dirac-Poisson bracket (114) within thepolysymplectic formulation, the Immirzi-like parameter inherent to the Bonzom-Livinemodel for gravity (79) not only modifies the fundamental commutation relations of thetheory but it also alters its canonical polysymplectic structure to a non-commutativeone. In fact, this last issue stands as the polysymplectic counterpart to the analogousresult obtained through the canonical analysis of the system developed in [40].Next, we will study the De Donder-Weyl-Hamilton field equations associated withthe system. To do so, we start by emphasizing that, on the constraint surface P PCSBL , thecorrect field equations of the gravity model (79) can be obtained in a covariant Poisson-Hamiltonian framework by means of the set of canonically conjugate variables (104),the De Donder-Weyl Hamiltonian (105) and the Dirac-Poisson bracket (114). In otherwords, we have that, given ̺ ∈ P X a section of π XP , the De Donder-Weyl-Hamiltonfield equations of the Bonzom-Livine model for gravity are given by ∂ [ µ e aν ] = ̺ ∗ { [ H DW , a a [ ν ̟ µ ] ] } D = − ǫ abc ω b [ µ e cν ] , (120a) ∂ µ π µνa = ̺ ∗ { [ H DW , p µνa ̟ µ ] } D = s q | Λ | γ ǫ abc ǫ νσρ ω bσ e cρ , (120b) ∂ [ µ ω aν ] = ̺ ∗ { [ H DW , b a [ ν ̟ µ ] ] } D = − ǫ abc (cid:16) ω bµ ω cν + Λ e bµ e cν (cid:17) , (120c) ∂ µ ¯ π µνa = ̺ ∗ { [ H DW , ¯ p µνa ̟ µ ] } D = ǫ abc ǫ νσρ e bσ ω cρ + 12 γ q | Λ | ω bσ ω cρ + s q | Λ | e bσ e cρ . (120d)In particular, it is not difficult to see that, equation (120a) can be explicitly written as D [ µ e aν ] = 0 , (121)giving rise to the vanishing torsion condition. In a similar manner, equation (120c)allows us to obtain F aµν + Λ ǫ abc e bµ e cν = 0 , (122)which implies that the curvature is constant and, particularly, given by the cosmologicalconstant. Of course, relations (121) and (122) are nothing but the Einstein equations, review on geometric formulations for classical field theory ξ χ ∈ h and ξ χ ∈ p . Then, on the constraint surface P PCSBL and given ̺ ∈ P X a solution of theDe Donder-Weyl-Hamilton field equations (120), it is possible to write d • J ( P ) ( ξ θ ) = ̺ ∗ { [ H DW , J ( P ) ( ξ θ )] } D + d H • J ( P ) ( ξ θ ) , = ̺ ∗ ǫ µνσ ǫ a [ bc ǫ ad ] e s q | Λ | γ a bµ a cν b dσ + b bµ b cν a dσ + Λ a bµ a cν a dσ + 1 γ q | Λ | b bµ b cν b dσ θ e , = 0 , (123)where we have implemented the Jacobi identity for the structure constants of the Liealgebra (76). In a similar manner, the following relation holds d • J ( P ) ( ξ χ ) = ̺ ∗ { [ H DW , J ( P ) ( ξ χ )] } D + d H • J ( P ) ( ξ χ ) , = ̺ ∗ ǫ µνσ ǫ a [ bc ǫ ad ] e s q | Λ | γ (cid:16) b bµ b cν a dσ + Λ a bµ a cν a dσ (cid:17) + Λ a bµ a cν b dσ + b bµ b cν b dσ χ e , = 0 , (124)which implies that, on the constraint surface P PCSBL , the Hamiltonian conserved currents(36) of the Bonzom-Livine model for gravity can be obtained simply by pulling-backthe induced covariant momentum map (111) with ̺ ∈ P X being a solution of the DeDonder-Weyl-Hamilton field equations (120).Next, we will derive the instantaneous Dirac-Hamiltonian analysis of the gravitymodel (79) by performing the space plus time decomposition of the correspondingmultisymplectic formulation. In the present subsection, we will carry out the space plus time decomposition for theBonzom-Livine model for gravity (79). Here, our main aim is to describe how both thefirst-class constraints and the extended Hamiltonian of the theory can be obtained bymeans of certain geometric objects defined on the covariant multimomenta phase-spaceassociated with the system. To this end, we will implement the elementary objects review on geometric formulations for classical field theory t a Cauchy surface characterized by a level set of thelocal coordinate x , specifically x − t = 0, for some t ∈ R . Subsequently, we identify ζ X := ∂ ∈ X ( X ) as the infinitesimal generator of the slicing of the space-time manifold X . Thus, for some ξ θ ∈ h and ξ χ ∈ p , we introduce (Σ t , ζ X ) and ( Y t , ζ Y ) to be a G -slicingof the covariant configuration space π XY , where the vector field ζ Y ∈ X ( Y ) is locallydefined as ζ Y := ∂ + ξ Yθ + ξ Yχ , (125)being ξ Yθ ∈ X ( Y ) and ξ Yχ ∈ X ( Y ) the vector fields locally represented by (87). Fromnow on, ζ Y will be understood as a temporal direction on Y for the Bonzom-Livinemodel for gravity. Consequently, we identify ( Y t , π Σ t Y t , Σ t ) as the restriction of π XY tothe Cauchy surface Σ t , and we denote by ( x i , a aµ , b aν ) an adapted coordinate system on Y t . Thus, given Y t the set of sections of π Σ t Y t , we define T Y t as the t -instantaneousspace of velocities of the theory. Here, by considering ϕ := φ ◦ i t ∈ Y t for some φ ∈ Y X ,we introduce ( e aµ , ω aµ , ˙ e aµ , ˙ ω aµ ) to denote an adapted coordinate system on T Y t , where thetemporal derivative of the field variables of the model, according to relation (66), isexplicitly given by ˙ e aµ := ∂ e aµ − ǫ abc e bµ θ c − D µ χ a , ˙ ω aµ := ∂ ω aµ − D µ θ a − Λ ǫ abc e bµ χ c . (126)Bearing this in mind, we are in the position to perform the space plus timedecomposition for the gravity model (79) at the Lagrangian level. For this purpose,we begin by introducing (( J Y ) t , π Y t ( J Y ) t , Y t ) to be the restriction of π Y J Y to Y t .Then, the jet decomposition map β ζ Y : ( J Y ) t → J ( Y t ) × V Y t over Y t locally reads β ζ Y (cid:16) x i , a aν , b aν , a aµν , b aµν (cid:17) := (cid:16) x i , a aν , b aν , a aiν , b aiν , ˙ a aν , ˙ b aν (cid:17) . (127)Hereinafter, the Latin indices i and j will denote spatial indices ( i, j = 1 , L BL t, ζ Y : T Y t → R , can be written as L BL t, ζ Y ( e, ω, ˙ e, ˙ ω ) := Z Σ t d n − x ǫ ij δ ab s q | Λ | γ e bj (cid:18) ˙ e ai + ǫ ac d e ci θ d + D i χ a (cid:19) + e bj + 1 γ q | Λ | ω bj (cid:18) ˙ ω ai + Λ ǫ ac d e ci χ d + D i θ a (cid:19) + 2 e a s q | Λ | γ D i e bj + 12 (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) + 2 ω a D i e bj + 12 γ q | Λ | (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) , (128)where we have performed some integration by parts and avoided terms on the boundaryof the Cauchy surface Σ t . review on geometric formulations for classical field theory π µa := ∂L BL ∂ ˙ e aµ (cid:16) e, ω, ˙ e, ˙ ω (cid:17) = s q | Λ | γ δ µi ǫ ij δ ab e bj , ¯ π µa := ∂L BL ∂ ˙ ω aµ (cid:16) e, ω, ˙ e, ˙ ω (cid:17) = δ µi ǫ ij δ ab e bj + 1 γ q | Λ | ω bj , (129)being L BL ( e, ω, ˙ e, ˙ ω ) the Lagrangian function of the model after the space plus timedecomposition, that is, the integrand in (128). Note that, the set of relations (129)gives rise to the primary constraint surface of the Bonzom-Livine model for gravity P t, ζ Y ⊂ T ∗ Y t , specifically P t, ζ Y := n ( e, ω, π, ¯ π ) ∈ T ∗ Y t (cid:12)(cid:12)(cid:12) γ a = 0 , Υ ia = 0 , ¯ γ a = 0 , ¯Υ ia = 0 o , (130)where the primary constraints of the theory are explicitly defined by γ a := π a ≈ , (131a)Υ ia := π ia − s q | Λ | γ ǫ ij δ ab e bj ≈ , (131b)¯ γ a := ¯ π a ≈ , (131c)¯Υ ia := ¯ π ia − ǫ ij δ ab e bj + 1 γ q | Λ | ω bj ≈ . (131d)Of course, as pointed out in [1], the presence of primary constraints in a classical fieldtheory is related with the fact that the instantaneous Legendre transformation of thesystem is not invertible. Therefore, the gravity model (79) corresponds to a singularLagrangian system.Next, we will implement the space plus time decomposition of the theory of ourinterest at the multisymplectic level. To do so, we start by introducing ( Z ⋆t , π Y t Z ⋆t , Y t )to denote the restriction of π Y Z ⋆ to Y t . In addition, we define Z ⋆t as the set of sectionsof π Σ t Z ⋆t := π Σ t Y t ◦ π Y t Z ⋆t , which is related with T ∗ Y t , the t -instantaneous phase-spaceof the model, through the vector bundle map R t : Z ⋆t → T ∗ Y t over Y t , as discussedin subsection 2.5. Besides, we identify ζ Z ⋆ ∈ X ( Z ⋆ ) as the α ( M ) -lift of (125) to Z ⋆ ,namely ζ Z ⋆ := ∂ + ξ αθ + ξ αχ , (132)where the vector fields ξ αθ ∈ X ( Z ⋆ ) and ξ αχ ∈ X ( Z ⋆ ) are locally given by (96). As wewill see below, the vector field (132) will play an important role within our study.As previously mentioned, we are interested in describing how the covariantmomentum map associated with the extended gauge symmetry group of the Bonzom-Livine model for gravity allows to obtain the complete set of first-class constraints inDirac’s terminology of the system. To illustrate this, it is important to emphasize that, review on geometric formulations for classical field theory G (the extended gauge symmetry group of the theory) acts on π XY by means of verticalbundle automorphisms, which implies that, the action of G on T ∗ Y t is well-defined, andhence there is an associated momentum map, specifically J t : T ∗ Y t → g ∗ . Then, in orderto obtain the local representation of such a momentum map, we can proceed as follows.To start, we would like to mention that, in light of the splitting g = h ⊕ p , any element ξ η ∈ g can be written as ξ η := ξ θ + ξ χ , for some ξ θ ∈ h and ξ χ ∈ p . Now, let us consider σ ∈ Z ∗ t a section such that R t ( σ ) = ( e, ω, π, ¯ π ) ∈ T ∗ Y t . Then, by taking into accountrelation (73), we know that, the momentum map associated with the action of G on T ∗ Y t is locally given by D J t ( e, ω, π, ¯ π ) , ξ η E := Z Σ t σ ∗ J ( M ) ( ξ η ) , (133)where J ( M ) ( ξ η ) := J ( M ) ( ξ θ ) + J ( M ) ( ξ χ ) ∈ Ω n − ( Z ⋆ ). In particular, by using the localrepresentation (101) and after identifying π νa = p νa ◦ σ and ¯ π νa = ¯ p νa ◦ σ , we find that D J t ( e, ω, π, ¯ π ) , ξ η E = Z Σ t d n − x (cid:16) D χ a + ǫ abc e b θ c (cid:17) π a + (cid:16) D θ a + Λ ǫ abc e b χ c (cid:17) ¯ π a − θ a D i ¯ π ia + ǫ abc e bi π ic + 1 γ q | Λ | ǫ ij δ ab ∂ i ω bj − χ a D i π ia + Λ ǫ abc e bi ¯ π ic + ǫ ij δ ab s q | Λ | γ ∂ i e bj + F bij − Λ ǫ bc d e ci e dj , where we have performed some integral by parts and avoided terms on the boundary ofthe Cauchy surface Σ t . Observe that, in terms of the primary constraints of the model(131), the momentum map (133) explicitly reads D J t ( e, ω, π, ¯ π ) , ξ η E = Z Σ t d n − x (cid:16) D χ a + ǫ abc e b θ c (cid:17) γ a + (cid:16) D θ a + Λ ǫ abc e b χ c (cid:17) ¯ γ a − θ a D i ¯Υ ia + ǫ abc e bi Υ ic + 2 ǫ ij δ ab D i e bj + 12 γ q | Λ | (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) − χ a D i Υ ia + Λ ǫ abc e bi ¯Υ ic + 2 ǫ ij δ ab s q | Λ | γ D i e bj + 12 (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) . Hence, it is not difficult to see that, after introducing the set of parameters λ a := D χ a + ǫ abc e b θ c ,λ a := − χ a , ¯ λ a := D θ a + Λ ǫ abc e b χ c , ¯ λ a := − θ a , (134)it is possible to write D J t ( e, ω, π, ¯ π ) , ξ η E = Z Σ t d n − x λ a γ a + ¯ λ a ¯ γ a + λ a Γ a + ¯ λ a ¯Γ a , review on geometric formulations for classical field theory a and ¯Γ a are defined byΓ a := D i Υ ia + Λ ǫ abc e bi ¯Υ ic + 2 ǫ ij δ ab s q | Λ | γ D i e bj + 12 (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) , ¯Γ a := D i ¯Υ ia + ǫ abc e bi Υ ic + 2 ǫ ij δ ab D i e bj + 12 γ q | Λ | (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) . (135)Here, it is worth noting that, the local representation (133) exactly coincides with thegenerator of infinitesimal gauge transformations of the Bonzom-Livine model for gravityobtained by means of Dirac’s algorithm in [46].Moreover, it is important to remember that the gauge symmetries of the gravitymodel (79) correspond to localizable symmetries, which implies that, according to[5, 9, 10], the admissible space of Cauchy data for the evolution equations of the theoryis determined by the zero level set of the momentum map (133), specifically J BL − t (0) := n ( e, ω, π, ¯ π ) ∈ T ∗ Y t (cid:12)(cid:12)(cid:12) h J t ( e, ω, π , ¯ π ) , ξ η i = 0 , ∀ ξ η ∈ g o . (136)In this regard, we have that, by using the local representation (133) and since parameters(134) depend on arbitrary functions on the Cauchy surface Σ t , it is possible to write J BL − t (0) = (cid:26) ( e, ω, π, ¯ π ) ∈ T ∗ Y t (cid:12)(cid:12)(cid:12)(cid:12) γ a = 0 , ¯ γ a = 0 , Γ a = 0 , ¯Γ a = 0 (cid:27) , which is nothing but the surface on the t -instantaneous phase-space defined by thecomplete set of first-class constraints that characterizes the Bonzom-Livine model forgravity within the instantaneous Dirac-Hamiltonian formulation as discussed in [46].In other words, the vanishing of the momentum map (133) yields the complete set offirst-class constraints of the gravity model (79), namely γ a ≈ , (137a)¯ γ a ≈ , (137b)Γ a ≈ , (137c)¯Γ a ≈ . (137d)Therefore, by following Dirac’s terminology, the primary constraints (131b) and (131d),which do not belong to set (137), must be second-class constraints. Furthermore, it isnot difficult to see that by considering the second-class primary constraints (131b) and(131d) as strong identities, the first-class constraints (137c) and (137d) reduce to thefollowing weak identitiesΦ a := 2 ǫ ij δ ab s q | Λ | γ D i e bj + 12 (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) ≈ , ¯Φ a := 2 ǫ ij δ ab D i e bj + 12 γ q | Λ | (cid:18) F bij + Λ ǫ bc d e ci e dj (cid:19) ≈ , (138) review on geometric formulations for classical field theory ξ θ ∈ h and ξ χ ∈ p ,the vector field (132) satisfies the condition L ζ Z⋆ Ω ( M )BL = 0, and hence there is an ( n − H ( M ) ζ Z⋆ := ζ Z ⋆ y Θ ( M )BL − α ( M ) θ − α ( M ) χ ∈ Ω n − ( Z ⋆ ) on Z ⋆ such that ζ Z ⋆ y Ω ( M )BL = dH ( M ) ζ Z⋆ ,being α ( M ) θ ∈ Ω n − ( Z ⋆ ) and α ( M ) χ ∈ Ω n − ( Z ⋆ ) the π XZ ⋆ -horizontal ( n −
1; 1)-forms on Z ⋆ locally represented by (98) and (100), respectively. Now, for some section φ ∈ Y X , letus consider σ := F L ◦ j φ ◦ i t ∈ N t the canonical lift of an element ( e, ω, π, ¯ π ) ∈ P t, ζ Y .Then, according to [7, 9, 49], the instantaneous Hamiltonian of the system can beobtained by means of (74), together with relation ζ Z ⋆ y Ω ( M ) = d H ( M ) ζ Z⋆ . It is importantto mention that, contrary to the geometric formulation implemented here, the standardprocedure for studying relation (74) consists of imposing the primary constraints of thesystem as strong identities. However, in our case, by following Dirac’s algorithm [1], wewill treat such constraints as weak identities, which will eventually allow us to obtainthe extended Hamiltonian for the gravity model (79). To illustrate this, we begin byintroducing the functional H BL t, ζ Y : P t, ζ Y → R , which is given by H BL t, ζ Y ( e, ω, π, ¯ π ) := − Z Σ t σ ∗ H ( M ) ζ Z⋆ . (139)Subsequently, a straightforward calculation shows that after identifying π νa = p νa ◦ σ and ¯ π νa = ¯ p νa ◦ σ , it is possible to write H BL t, ζ Y ( e, ω, π, ¯ π ) = Z Σ t d n − x ˙ e a π a + ˙ ω a ¯ π a + ∂ e ai π ia − s q | Λ | γ ǫ ij δ ab e bj + ∂ ω ai ¯ π ia − ǫ ij δ ab e bj + 1 γ q | Λ | ω bj − δ ab e a s q | Λ | γ D i e bj + 12 (cid:16) F bij + Λ ǫ bc d e ci e dj (cid:17) − δ ab ω a D i e bj + 12 γ q | Λ | (cid:16) F bij + Λ ǫ bc d e ci e dj (cid:17) + θ a D i ¯ π ia + ǫ abc e bi π ic + 1 γ q | Λ | ǫ ij δ ab ∂ i ω bj + χ a D i π ia + Λ ǫ abc e bi ¯ π ic + δ ab s q | Λ | γ ∂ i e bj + F bij − Λ ǫ bc d e ci e dj , where, once again, we have performed some integrations by parts and avoided terms onthe boundary of the Cauchy surface Σ t . Note that, on the one hand, in terms of the review on geometric formulations for classical field theory H BL t, ζ Y ( e, ω, π, ¯ π ) = Z Σ t d n − x ˙ e a γ a + ˙ ω a ¯ γ a + ∂ e ai Υ ia + ∂ ω ai ¯Υ ia + χ a Γ a + θ a ¯Γ a − δ ab e a s q | Λ | γ D i e bj + 12 (cid:16) F bij + Λ ǫ bc d e ci e dj (cid:17) − δ ab ω a D i e bj + 12 γ q | Λ | (cid:16) F bij + Λ ǫ bc d e ci e dj (cid:17) . On the other hand, we know that, for γ = s , the field equations of the system (85) giverise to the following set of relations ∂ e ai = D i e a − ǫ abc ω b e ci ,∂ ω ai = ∂ i ω a − ǫ abc (cid:16) ω b ω ci + Λ e b e ci (cid:17) , (140)which allows us to identify the fixed Lagrange multipliers that enforce the second-classprimary constraints (131b) and (131d) into the functional (139), respectively. Hence,by considering φ a section that satisfies relations (140), we can write H BL t, ζ Y ( e, ω, π, ¯ π ) = Z Σ t d n − x (cid:26) ˙ e a γ a + ˙ ω a ¯ γ a + χ a Γ a + θ a ¯Γ a − e a Γ a − ω a ¯Γ a (cid:27) ≈ , where we have again performed some integration by parts and avoided terms on theboundary of the Cauchy surface Σ t . Thereby, it is not difficult to see that, the functional(139) is nothing but the extended Hamiltonian that characterizes the Bonzom-Livinemodel for gravity within the instantaneous Dirac-Hamiltonian formulation. Observethat the extended Hamiltonian of the system is a linear combination of the first-classconstraints (137), which reflects the general covariance of the classical field theory underconsideration. In addition, it is important to highlight that under the gauge fixing e a ≈ ω a ≈ t -instantaneous reduced phase-space, as discussed in [46].Finally, we can compute the number of local degrees of freedom of the system.In this regard, we start by emphasizing that, on the one hand, the number of linearindependent canonical variables of the theory is N CV := 2 ·
18. On the other hand,we know that, the number of linear independent first-class constraints of the systemcorresponds to N FCC := 4 ·
3, while the number of linear independent second-classconstraints of the model is N SCC := 2 ·
6. Therefore, according to [1], the numberof local degrees of freedom of the Bonzom-Livine model for gravity corresponds to N BL DF := N CV − N FCC − N SCC = 0, thus implying that the gravity model (79) is atopological field theory. review on geometric formulations for classical field theory
4. Conclusions
In this paper, we have reviewed the Lagrangian, multisymplectic and polysymplecticgeometric and covariant formalisms for the analysis of gauge field theories at theclassical level. By judiciously identifying the fields with sections of appropriate fibre-bundles, one is able to implement a classical field theory in a finite-dimensional setup.As discussed above, such geometric formalisms are particularly transparent in themanner they characterize the symmetries for classical field theory by analyzing theaction of the symmetries on the Poincaré-Cartan forms at the Lagrangian level or thecanonical forms at the multisymplectic level. In particular, this allowed us to formulateNoether’s theorems by means of the covariant versions of the momentum map andalso to identified the Noether currents and charges in a succinct manner. Importantto mention is the fact that at the multisymplectic level the symmetries of the theorywere associated with covariant canonical transformations in a straightforwardly way.Besides, by considering the polymomenta phase-space, we were able to introduce acovariant Poisson-Hamiltonian framework based on a well-defined Poisson-Gerstenhabergraded bracket. We were also able to undertake the analysis of theories characterizedby singular Lagrangian systems through the implementation of a Dirac-Poisson bracketin the polymomenta phase-space, thus recovering the correct field equations. Further,by suitably enforcing a space plus time decomposition of the background manifold,we studied the manner in which one may recover the complete gauge content and thetrue degrees of freedom for a classical field theory with localizable symmetries solelyby considering the admissible space of Cauchy data for the evolution equations ofthe system which turned out to be determined by the zero level set of the inducedmomentum map. In consequence, we consider that the geometric formulations reviewedhere provide a legitimate way to analyze the dynamics and constrained structure forsingular Lagrangian systems associated with classical field theories.Motivated by the application of such geometric formalisms to physical modelsthat mimic certain aspects of General Relativity, here we have studied the Bonzom-Livine model for gravity within the geometric-covariant Lagrangian, multisymplecticand polysymplectic formulations for classical field theory. At the Lagrangian level, wehave obtained the field equations of the system, which have been shown to be equivalentto the vanishing torsion condition and the Einstein equations, as discussed in [40].Besides, we have analyzed the so-called H -gauge and translational symmetries of themodel, which not only preserve the Poincaré-Cartan n -form of the theory up to an exact review on geometric formulations for classical field theory H -gauge and translational symmetries of the theory act on thecorresponding covariant multimomenta phase-space by infinitesimal covariant canonicaltransformations, thus demonstrating the existence of a covariant momentum map, whichin turn is related with the Lagrangian covariant momentum map of the system by meansof the covariant Legendre transformation. Further, at the polysymplectic level, we havefound that the Bonzom-Livine model for gravity corresponds to a singular Lagrangiansystem. As a consequence, in order to carry out a consistent polysymplectic formulationof the gravity model of our interest, we have implemented the algorithm proposed in[48] to study singular Lagrangian systems within the polysymplectic framework. Inparticular, we have shown that, on the polymomenta phase-space, the Bonzom-Livinemodel for gravity is characterized by a set of second-class constraint ( n − n − n − G -slicing of the covariant configuration space of thetheory generated by a vector field that can be thought of as an evolution direction of review on geometric formulations for classical field theory t -instantaneous phase-space of the theory. In particular, we have realized that,since the symmetries of the model are localizable symmetries, the zero level set ofthe induced momentum map corresponds to the surface on the t -instantaneous phase-space determined by the complete set of first-class constraints of the system, whicharise within the instantaneous Dirac-Hamiltonian analysis of the model as describedin [46]. In addition, we have discussed how the α ( M ) -lift of the generator of the G -slicing of the covariant configuration space of the model gives rise to an infinitesimalcovariant canonical transformation, thus resulting in the existence of an ( n − t -instantaneous phase-space.By taking into account all of these results, we have shown that, the geometric-covariant Lagrangian, multisymplectic and polysymplectic formalisms for classicalfield theory enable us to describe in a geometric, consistent and elegant way thefeatures of the Bonzom-Livine model for gravity. In particular, we have discussed howthe instantaneous Dirac-Hamiltonian analysis of the system can be straightforwardlyrecovered by performing the space plus time decomposition of its correspondingmultisymplectic formulation. Also, we have described how the algorithm to studysingular Lagrangian systems within the polysymplectic approach is necessary in order toobtain a consistent polysymplectic formulation of the gravity model of our interest. Inthis regard, it is worth noting that, such an algorithm has been scarcely explored in theliterature, see for instance [29, 30, 57]. However, based on the results of the present work,we can conclude that the proposal developed in [48] to study singular Lagrangian systemswithin the polysymplecic formalism is completely adequate for the analysis of physicallymotivated classical field theories. In addition, as far as we know, our study provides thefirst non-trivial example where the Dirac-Poisson bracket (57) is explicitly implementedin the context of a model related with General Relativity. Thus, our work may shed somelight on a deeper understanding of such bracket structure, which resulted fundamentalin order to reproduce the correct De Donder-Weyl-Hamilton field equations of the theoryunder consideration. Finally, we would like to mention that our results may be relevantat the quantum level by considering the so-called pre-canonical quantization approachfor the De Donder-Weyl canonical theory [15, 17, 18, 19]. Indeed, such quantizationprogram is strongly based on the polysymplectic formalism for classical field theory,and thus our study may constitute a first step towards the quantum analysis of 3-dimensional gravity with Immirzi-like parameter from the aforementioned quantization review on geometric formulations for classical field theory Acknowledgments
The authors thank Alexis Tepale-Luna for discussions and collaboration. The authorswould also like to acknowledge financial support from CONACYT-Mexico under theproject CB-2017-283838.
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