Hamiltonian Gotay-Nester-Hinds analysis of the parametrized unimodular extension of the Holst action
J. Fernando Barbero G., Bogar Díaz, Juan Margalef-Bentabol, Eduardo J. S. Villaseñor
HHamiltonian GNH analysis of the parametrizedunimodular extension of the Holst action
J. Fernando Barbero G., a,b
Bogar Díaz, c,d,b
Juan Margalef-Bentabol e,b and Eduardo J.S. Villaseñor c,b a Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain b Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). UnidadAsociada al Instituto de Estructura de la Materia, CSIC c Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30,28911 Leganés, Spain. d Departamento de Física de Altas Energías, Instituto de Ciencias Nucleares, Universidad NacionalAutónoma de México, Apartado Postal 70-543, Ciudad de México, 04510, México e Institute for Gravitation and the Cosmos & Physics Department, Penn State University, Univer-sity Park, PA 16802, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We give a detailed account of the Hamiltonian GNH analysis of the parame-trized unimodular extension of the Holst action. The purpose of the paper is to derive,through the clear geometric picture furnished by the GNH method, a simple Hamiltonianformulation for this model and explain why it is difficult to arrive at it in other approaches.We will also show how to take advantage of the field equations to anticipate the simpleform of the constraints that we find in the paper.
Keywords:
Holst action; GNH method; Hamiltonian formulation; Unimodular gravity;Parametrized field theory a r X i v : . [ g r- q c ] J a n ontents Z C.1 Rewriting the non-degeneracy condition for the tetrads 25C.2 A determinant computation 25
D Some details about the GNH procedure 26
D.1 Computing the pullback of Ω to M Z IJω Although less popular than the Dirac algorithm [1], the Gotay-Nester-Hinds (GNH)approach to the Hamiltonian formulation of mechanical systems and field theories definedby singular Lagrangians is very powerful and conceptually clean [2–6]. Its geometric un-derpinnings provide a rigorous viewpoint that avoids many of the drawbacks of Dirac’s– 1 –ethod –in particular when applied to field theories– while ultimately giving the samebasic information. Several differences between both approaches should be noted:i) Dirac’s method relies heavily on the language of classical mechanics. For instance,singular Lagrangian systems are characterized as those for which it is impossible towrite all the velocities in terms of momenta; this leads to the ensuing appearance ofconstraints and the need to enforce their stability in order to guarantee the consistencyof time evolution. The GNH method, on the other hand, is based on geometry. Forinstance, the notion of dynamical stability is translated into the requirement that thevector fields that encode the dynamics must be tangent to the phase space submanifolddefined by the constraints. Although the rationale behind Dirac’s approach can berephrased in geometric terms (see [3, 7, 8]), this is non-standard and probably feelsunnatural for many readers.ii) The final descriptions provided by both methods are different, although it is possiblein practice to go back and forth from one to the other. Dirac’s method is designedto produce a Hamiltonian description in the full phase space . This is useful for quan-tization because the canonical symplectic structure is retained and, hence, the ideaof turning Poisson brackets into commutators can be implemented as in standardquantum mechanics. The presence of first class constraints is taken into account byusing their quantized version to select physical subspaces of the full Hilbert space ofthe system, while second class constraints are taken care of either by solving them orusing the so-called Dirac brackets. The arena of the GNH approach is the primaryconstraint submanifold endowed with a presymplectic form obtained by pulling backthe canonical symplectic structure of the full phase space. Even though the setting isslightly different, the constraints obtained with the Dirac method can also be foundand, conceivably, quantized in a similar way.iii) From a practical point of view, the emphasis on geometry characteristic of the GNHmethod has some unexpected consequences. In particular, the possibility of altogetheravoiding the use of Poisson brackets when dealing with the tangency conditions men-tioned in i) is instrumental in circumventing the difficulties that crop up, for instance,when field theories are defined in spatial regions with boundary. Another consequenceof the shift in perspective is the possibility of incorporating the, often subtle, func-tional analytic issues relevant for field theories that originate in the fact that theirconfiguration spaces are infinite dimensional manifolds.iv) Finally, it must be pointed out that the differences between both methods sometimeslead to insights within one of them that are difficult to arrive at in the other. In fact,this paper illustrates an instance of this phenomenon.The main purpose of this work is to apply the GNH method to the study of theHolst [9] action and some interesting generalizations of it, in particular, its parametrizedunimodular version. Unimodular gravity is an alternative approach to general relativitywith some interesting features, in particular regarding the role of the cosmological constant.– 2 –ts Hamiltonian analysis in metric variables is well known [10]. However, and despite someclaims to the effect [11], a similar analysis in terms of tetrads starting from the Holstaction has not been performed yet. The Holst action has several features that make thestudy of its parametrized unimodular version quite attractive. In particular, it involves theImmirzi parameter and leads to the real Ashtekar formulation. On its turn, parametrizedunimodular gravity is interesting because parametrization offers some useful insights on theproblem of time [12, 13]. Given the differences between the metric and tetrad formulationsfor general relativity, we deem it interesting to understand the Hamiltonian formulation ofparametrized unimodular general relativity in the context of the Holst action. The hope–ultimately realized– is that the analysis will provide an interesting perspective on theHamiltonian formulation of unimodular gravity. As we will see, the final description givenby the GNH approach –one of the results of the present paper– is concise and clean (see[14] for a short partial summary involving just the Holst action). It naturally leads to thereal Ashtekar formulation of general relativity and illuminates some issues related to therole of the time gauge and the Immirzi parameter, both at the classical and quantum level[14].The paper is structured as follows. After this introduction, we devote section 2 to adiscussion of the action principle used in the paper (the parametrized, unimodular version ofthe Holst action), the field equations and the Lagrangian formulation. Section 3 contains adetailed discussion of the GNH analysis. The very simple final form of the constraints in theHamiltonian formulation that we find suggests a streamlined approach to the Hamiltoniantreatment of field theories linear in first order time derivatives. We discuss it in section4. The literature on the Hamiltonian treatment of the Holst action and how the Ashtekarformulation can be derived from it is quite extensive. In order to put in perspective theresults presented here we provide an appraisal of the main works on this subject in section5. Although it is difficult to be exhaustive, we do try to provide a balanced assessment ofthe most important results and their relation to the present work. In section 6 we give ourconclusions and some comments. The contrast between the simple Hamiltonian formulationthat we find here and the long computations necessary to arrive at it is somehow striking.We discuss this issue in the conclusions. The paper ends with several appendices where wegive a number of auxiliary results and some computational details.
We consider the following generalization of the Holst action S ( e , ω , Λ , Φ ) = ∫ R × Σ (( ∗ ( e I ∧ e J ) + εγ e I ∧ e J ) ∧ F IJ + Λ ( Φ ∗ vol − (cid:15) IJKL e I ∧ e J ∧ e K ∧ e L )) . In this expression Σ is a closed (i.e. compact without boundary), orientable, 3-dimensionalmanifold. This implies that Σ is parallelizable and, hence, globally-defined frames exist.The cotetrads e I are 1-forms, ω IJ is a so ( , ) -valued connection 1-form with curvature F IJ ∶= d ω IJ + ω IK ∧ ω KJ . It satisfies the identity DF IJ = , where D is defined by suitably– 3 –xtending D α I = d α I + ω IJ ∧ α J . The tetrads are required to be non-degenerate, i.e., (cid:15) IJKL e I ∧ e J ∧ e K ∧ e L is a volume form in R × Σ . They also have to satisfy the conditionthat, for all τ ∈ R , the hypersurfaces { τ }× Σ are spacelike as measured by the metric e I ⊗ e I .The Levi-Civita symbol (cid:15) IJKL is totally antisymmetric and chosen to satisfy (cid:15) = + . Inmore abstract terms, it should be interpreted as a volume form in so ( , ) . The internalindices I, J, . . . take the values , , , . When needed, these indices will be raised andlowered with the invariant metric η in so ( , ) that, in an appropriate basis, can be writtenas η = diag ( ε, + , + , + ) , with ε = − . We have included ε to keep track of the spacetimesignature and facilitate the extension of our results to the Euclidean case. The dual ∗ of V IJ is ∗ V IJ ∶= (cid:15) IJ KL V KL . Let M be a 4-dimensional manifold diffeomorphic to R × Σ . The volume form vol on M is defined by a fixed, background (i.e., non-dynamical) metric g on M . The inclusionof this fiducial metric is not necessary but it will allow us to reuse some computations from[34, 35], in which case we have to restrict slightly the diffeomorphisms Φ ∈ Diff ( R × Σ , M) .Indeed, we consider the diffeomorphisms such that for all τ ∈ R , { τ } × Σ is a Φ ∗ g -spacelikehypersurface. The action depends on Φ only through the Φ ∗ vol term.The scalar field Λ ∈ C ∞ ( R × Σ ) is dynamical and not the cosmological constant (atleast at this stage). It plays the role of a Lagrange multiplier enforcing the parametrizedunimodularity condition (cid:15) IJKL e I ∧ e J ∧ e K ∧ e L = Φ ∗ vol . (2.1)Notice that the action is defined on R × Σ . This notwithstanding, as M is diffeomorphicto R × Σ , it is straightforward to change the viewpoint and define it on M , in which casethe dynamical diffeomorphisms would take M to another manifold N .The Immirzi parameter is denoted as γ ( ≠ ). It is convenient to introduce the invariant SO ( , ) tensor P IJKL ∶= ( (cid:15) IJKL + εγ η IK η JL − εγ η JK η IL ) . (2.2)Its main properties, including the form of its inverse [ P − ] IJKL (which exists only if γ ≠ ε )can be found in Appendix C.By using P IJKL we can rewrite the action as S ( e , ω , Λ , Φ ) = ∫ R × Σ ( P IJKL e I ∧ e J ∧ F KL + Λ ( Φ ∗ vol − (cid:15) IJKL e I ∧ e J ∧ e K ∧ e L )) . (2.3)The field equations are obtained by varying this action with respect to the dynamicalvariables e I , ω IJ , Λ and Φ . The variations with respect to the tetrads e I and the connection ω IJ give the equations e [ I ∧ De J ] = , (2.4a) P IJKL e J ∧ F KL −
13 Λ (cid:15)
IJKL e J ∧ e K ∧ e L = . (2.4b)– 4 –he variations with respect to Λ give the unimodularity condition (2.1). As the fieldequations of parametrized theories imply that the dynamical diffeomorphisms are alwaysarbitrary, this condition only restricts the possible values of the tetrads.Finally, the variations with respect to the dynamical diffeomorphisms Φ (which we knowfrom the parametrization procedure that they do not add any additional conditions) canbe written in terms of Lie derivatives £ V Φ [the variation of a diffeomorphism Φ ∶ R × Σ → M can be represented as a vector field V Φ ∈ X (M) ] as D ( Φ ,V Φ ) S = − ∫ M £ V Φ ( Φ − ∗ Λ ) vol . (2.5)The vanishing of the integral in (2.5) for every V Φ implies that d ( Φ − ∗ Λ ) = and, hence, Φ − ∗ dΛ = . Since Φ is a diffeomorphism we conclude that dΛ = . This last field equationtells us that Λ is actually a constant (an integration constant as usually stated in thetraditional literature on this subject, see [10, 12]). As a consequence, and given its role in(2.4b), Λ becomes a cosmological constant, of an arbitrary magnitude, through a dynamicalmechanism. The equation dΛ = is, as we mentioned before, redundant because it can beobtained from (2.4b) by taking = D ( P IJKL e J ∧ F KL −
13 Λ (cid:15)
IJKL e J ∧ e K ∧ e L ) , (2.6)using the identity DF IJ = , the fact that, for non-degenerate tetrads, (2.4a) is equivalentto De I = (see [15]), and the non-degeneracy of (cid:15) IJKL e I ∧ e J ∧ e k ∧ e L .Notice that computing the covariant differential of De I = we get the identity F IJ ∧ e J = , hence, plugging this into (2.4b), the γ -dependent terms drop out and (2.4a) and (2.4b)turn into the field equations given by the usual Hilbert-Palatini action with a cosmologicalconstant. In order to obtain the Lagrangian from the action (2.3), we need to perform a 3+1decomposition. The manifold R × Σ is naturally foliated by the 3-dimensional hypersurfaces { τ }× Σ with τ ∈ R . The tangent vectors to the parametrized curves c p ∶ R → R × Σ ∶ τ ↦ ( τ, p ) ,with p ∈ Σ , define a vector field ∂ τ ∈ X ( R × Σ ) . For each τ ∈ R we introduce the embedding τ ∶ Σ → R × Σ ∶ p ↦ ( τ, p ) .In terms of these geometric elements, we can write for any 4-form L∫ R × Σ L = ∫ R d τ ∫ Σ ∗ τ ı ∂τ L . If the preceding integral is the action for a particular field theory, L is a Lagrangian 4-form and the real function τ ↦ ∫ Σ ∗ τ ı ∂τ L can often be interpreted as being determined by a Lagrangian L ∶ T Q → R (a real functionin the tangent bundle T Q of a configuration space Q ) evaluated on curves in Q .– 5 –iven differential forms of arbitrary degree in R × Σ , it is convenient to build otherdifferential forms adapted to the foliation defined by the Cartesian product R × Σ . If α ∈ Ω p ( R × Σ ) , with p = , . . . , , we define its transverse and tangent parts α t ∶= ı ∂ τ α ∈ Ω p − ( R × Σ ) ,α ∶= α − d τ ∧ α t ∈ Ω p ( R × Σ ) , (2.7)leading to the decomposition α = α + d τ ∧ α t . Notice that one can also perform the for-mer decomposition using the normal to the foliation n , as in [34, 35], in which case theHamiltonian turns out to be zero everywhere in phase space. However, in this case, it iseasier to break the objects with the fixed foliation using ∂ τ . Besides, in this case, the com-parison with the unparametrized version is straightforward [14]. Obviously, ı ∂ τ α t = and ı ∂ τ α = . The basic objects used to find the Lagrangian corresponding to the action (2.3)are obtained with the help of the previous decomposition for e I and ω IJ . We also need tofind out how to perform a 3+1 decomposition of the dynamical diffeomorphisms Φ and thetype of dynamical objects obtained by doing this. This is explained in appendix B.The first result coming from the 3+1 decomposition is the characterization of theconfiguration space of the system. In the present case, it consists of the scalar fields e I t , ω I t J , Λ ∈ C ∞ ( Σ ) , the 1-forms e I , ω IJ ∈ Ω ( Σ ) and the g -spacelike embeddings X ∶ Σ ↪ M . The points in the tangent bundle of the configuration space of our system aredenoted as v q (where q = ( e I t , e I , ω IJ t , ω IJ , Λ , X ) denotes a point in Q ) with components ( v e t , v e , v ω t , v ω , v Λ , v X ) that can be interpreted as velocities. We have v e t , v ω t , v Λ ∈ C ∞ ( Σ ) , v e , v ω ∈ Ω ( Σ ) and v X ∈ Γ ( X ∗ T M) (i.e. the velocity associated with the embedding X is avector field along the map X ). In terms of these objects the Lagrangian can be written as L ( v q ) = ∫ Σ [ P IJKL ( e K t e L ∧ F IJ + e K ∧ e L ∧ ( v ωIJ − Dω t IJ )) (2.8) + ε Λ n X ( v X ) vol γ X −
13 Λ (cid:15)
IJKL e I t ( e J ∧ e K ∧ e L )] . where the curvature is F JI ∶= d ω JI + ω KI ∧ ω JK , and the embedding-dependent objects n X and γ X are defined in appendix B. As we can see, the Lagrangian only depends on the v ω and v X components of the velocity. Notice also that L is linear in v ω and v X , a fact that itis not completely obvious a priori as far as the dynamical diffeomorphisms are concerned. The canonical momenta are obtained from the fiber derivative
F L ∶ T Q → T ∗ Q deter-mined by the Lagrangian L . They are p e t ( w Ie t ) ∶= ⟨ F L ( v q ) , ( w Ie t , , , , , )⟩ = , (3.1a) p e ( w Ie ) ∶= ⟨ F L ( v q ) , ( , w Ie , , , , )⟩ = , (3.1b) p ω t ( w IJω t ) ∶= ⟨ F L ( v q ) , ( , , w IJω t , , , )⟩ = , (3.1c)– 6 – ω ( w IJω ) ∶= ⟨
F L ( v q ) , ( , , , w IJω , , )⟩ = ∫ Σ P IJKL w IJω ∧ e K ∧ e L , (3.1d) p Λ ( w Λ ) ∶= ⟨ F L ( v q ) , ( , , , , w Λ , )⟩ = , (3.1e) p X ( w X ) ∶= ⟨ F L ( v q ) , ( , , , , , w X )⟩ = ∫ Σ εn X ( w X ) Λ vol γ X , (3.1f)where we denote ⟨ p , w ⟩ ∶= p ( w ) .In the present case the fiber derivative F L is obviously not onto. As it is not a diffeo-morphism between
T Q and T ∗ Q , our action defines a singular system.As we can see, the momenta are all independent of the velocities. The conditions(3.1) can be interpreted as primary constraints that characterize the image of F L , usuallyknown as the primary constraint submanifold M . As all the momenta can be written interms of the configuration variables e I t , e I , ω IJ t , ω IJ , Λ and X , the submanifold M canbe parametrized by these objects, in fact, M is the configuration space of the system. Forthis reason we will often refer to the configuration variables when talking about points in M . On M the Hamiltonian is defined by the condition H ○ F L = E , where the energy is E ∶ T Q → R ∶ v q ↦ ⟨ F L ( v q ) , v q ⟩ − L ( v q ) . In the present case we have H = ∫ Σ ( P IJKL ( e I ∧ e J ∧ Dω KL t − e I t e J ∧ F KL ) +
13 Λ (cid:15)
IJKL e I t ( e J ∧ e K ∧ e L )) . (3.2)Notice that H does not depend neither on the momenta (it is defined only on M ) nor theembeddings X . In order to get a Hamiltonian description for the dynamics of a singular Lagrangiansystem, it is necessary to identify a maximal submanifold M of the primary constraintsubmanifold M and vector fields Z ∈ X ( M ) tangent to M such that the following conditionholds ( ı Z ω − dl H )∣ M = . (3.3)In the previous expression ω denotes the pullback of the canonical symplectic form Ω in T ∗ Q to M and dl denotes the differential in phase space. A very good way to solve (3.3)is to follow the procedure introduced by Gotay, Nester, and Hinds in [2–4].Vector fields in the phase space discussed here have the form Z = ( Z e t , Z e , Z ω t , Z ω , Z Λ , Z X ; Z e t , Z e , Z ω t , Z ω , Z Λ , Z X ) , (3.4)where the boldface components are associated with the “momenta directions” in phase spaceand the components in the “field directions” have the following internal index structure ( Z Ie t , Z Ie , Z IJω t , Z IJω , Z Λ , Z X ) . Given Z , Y ∈ X ( T ∗ Q ) and the canonical symplectic form Ω we have Ω ( Z , Y ) = Y e t ( Z e t ) − Z e t ( Y e t ) + Y e ( Z e ) − Z e ( Y e ) + Y ω t ( Z ω t ) − Z ω t ( Y ω t ) – 7 – Y ω ( Z ω ) − Z ω ( Y ω ) + Y Λ ( Z Λ ) − Z Λ ( Y Λ ) + Y X ( Z X ) − Z X ( Y X ) . (3.5)Remember that ı Z Ω ( Y ) = Ω ( Z , Y ) . From here on we will work on M . The pullback of Ω to the primary constraint submanifold M is given by (see appendix D.1) ω ( Z , Y ) = (3.6) ∫ Σ P IJKL ( Z IJω ∧ Y Ke − Y IJω ∧ Z Ke ) ∧ e L + ( Z ⊥ X ( Y Λ − ı Y ⊺ X dΛ ) − Y ⊥ X ( Z Λ − ı Z ⊺ X dΛ )) vol γ X . On the other hand dl H ( Y ) = ∫ Σ [ Y IJω ∧ ( − D ( P IJKL e K t e L ) + P IKLM ω K t J e L ∧ e M ) (3.7) + Y Ie ∧ ( P IJKL ( e J ∧ Dω KL t + e J t F KL ) − Λ (cid:15) IJKL e J t e K ∧ e L )− Y IJω t D ( P IJKL e K ∧ e L ) + Y Λ (cid:15) IJKL e I t ( e J ∧ e K ∧ e L )+ Y Ie t ( − P IJKL e J ∧ F KL +
13 Λ (cid:15)
IJKL ( e J ∧ e K ∧ e L ))] . Requiring now that ω ( Z , Y ) = dl H ( Y ) for all vector fields Y ∈ X ( M ) we get:1) Conditions involving the components of Z . Z ⊥ X vol γ X = (cid:15) IJKL e I t ( e J ∧ e K ∧ e L ) , (3.8a) Z Λ = ı Z ⊺ X dΛ , (3.8b) Z ⊥ X dΛ = , (3.8c) Z [ Ie ∧ e J ] = D ( e [ I t e J ] ) − [ P − ] IJMN P MKLP ω N t K e L ∧ e P , (3.8d) P IJKL e J ∧ Z KLω = P IJKL ( e J ∧ Dω KL t + e J t F KL ) − Λ (cid:15) IJKL e J t e K ∧ e L . (3.8e)2) Secondary constraints e [ I ∧ De J ] = , (3.9a) P IJKL e J ∧ F KL −
13! Λ (cid:15)
IJKL e J ∧ e K ∧ e L = . (3.9b)Before we analyze in detail the conditions on Z and the secondary constraints we make acouple of comments. First, in order to find the components of the Hamiltonian vector field Z we have to solve equations (3.8a)-(3.8e). Their solutions will give us the components of Z in terms of the configuration variables. As the equations are inhomogeneous, they may notbe solvable in the whole of M . If this is the case, new secondary constraints will arise thatwe will duly have to take into account. It may also happen that some of the components of Z are left arbitrary (a feature characteristic of gauge theories). Indeed, it is straightforwardto see that this is the case since there are no conditions involving Z Ie t or Z IJω t . Second, as weare only interested in the values of Z on the final constraint submanifold M , we can takeadvantage of the constraints to simplify the expressions for the components of Z . This isspecially useful when checking the tangency of Z to M .– 8 – .2.1 Conditions on the components of Z The conditions (3.8a)-(3.8c) are easy to analyze. To begin with, it is important to pointout that diffeomorphisms must be interpreted as curves of embeddings in this setting (seeAppendix B), hence, we must demand that Z ⊥ X ≠ at every point in Σ . This can be easilyachieved by restricting the configuration variables to satisfy (cid:15) IJKL e I t ( e J ∧ e K ∧ e L ) ≠ , (3.10)everywhere on Σ because then, (3.8a) implies Z ⊥ X ≠ for all p ∈ Σ . Notice that, roughlyspeaking, (3.10) defines an open subset of our initial configuration space. A further conse-quence of (3.10) is that (3.8c) is then equivalent to dΛ = –a secondary constraint– whichmeans that the scalar field Λ has the same value on all the points of Σ although, in prin-ciple, it could depend on the evolution parameter. Notice, however, that condition (3.8b)implies Z Λ = and, hence, ˙Λ = Z Λ = , i.e. Λ must be also constant under evolution . Wethen conclude that Λ is constant in M in agreement with the result obtained from the fieldequations. In fact, this result is also a consequence of the dynamics of the theory and canbe found without invoking the parametrization (analogous to (2.6)). As we can see, theparametrized unimodularity condition can be easily implemented in the connection-triadformalism discussed here and leads to conclusions similar to those of [12]. In summary,if we restrict our configuration variables to satisfy (3.10), the conditions (3.8a)-(3.8c) areequivalent to Z ⊥ X = ( (cid:15) IJKL e I t ( e J ∧ e K ∧ e L ) vol γ X ) , (3.11a) dΛ = , (3.11b) Z Λ = , (3.11c)where we have used the notation introduced in appendix A.Let us discuss now conditions (3.8d) and (3.8e). As we can see, both of them can beinterpreted as linear inhomogeneous equations for the 1-forms Z Ie and Z IJω , respectively.The fact that they are inhomogeneous means that they may not be solvable in all of M .In other words, additional secondary constraints may appear.By using identity (C.2) of appendix C, condition (3.8d) can be written in the followingform, independent of γ and ε : Z [ Ie ∧ e J ] = D ( e [ I t e J ] ) − ω t K [ I e J ] ∧ e K . (3.12)These are six equations for the four 1-forms Z Ie , ( I = , . . . , ) or, counting components, 18equations for 12 unknowns. It is convenient to write them in the form Ξ [ I ∧ e J ] = e [ I t De J ] , (3.13)with Ξ I ∶= Z Ie − De I t − e K ω I t K . – 9 –n order to solve these equations it helps to split them in two groups: one correspondingto I = i and J = j with i , j = , . . . , and the other to I = and J = j with j = , . . . , . Wethus get (cid:15) ijk e j ∧ Ξ k = (cid:15) ijk A jk , (3.14a) Ξ ∧ e j − Ξ j ∧ e = A j , (3.14b)with A ij ∶= e t [ i De j ] , (3.15a) A j ∶= e t0 De j − e t j De . (3.15b)It is important to notice that in the previous expressions De i means De i = d e i + ω iJ ∧ e J = d e i + ω ij ∧ e j + ω i ∧ e , (analogously, for other objects of this type).Equation (3.14a) can always be solved for Ξ k without having to impose any conditionson the inhomogeneous term (see appendix C). However, equation (3.14b) can only be solvedfor Ξ when the following condition holds [here Ξ i is the solution to (3.14a)] Ξ i ∧ e ∧ e j + A i ∧ e j + Ξ j ∧ e ∧ e i + A j ∧ e i = , (3.16)(see appendix C). This gives the new secondary constraints (cid:15) klm e k t ( D il E jm + D jl E im ) − e ( D ij + D ji ) + e i t D ○ j + e j t D ○ i = , (3.17)in terms of the objects introduced in appendix A.Let us discuss now how to solve for Z IJω in (3.8e) and, in particular, whether newconstraints arise in the process. In terms of components we have now 12 equations and 18unknowns. By defining T IJ ∶= P IJKL ( − Z KLω + Dω KL t ) the equations (3.8e) become e J ∧ T IJ = − P IJKL e J t F KL +
12 Λ (cid:15)
IJKL e J t e K ∧ e L . (3.18)In the following we will denote the r.h.s. of the previous equation as C I . In components,equations (3.18) are e i ∧ T i = C , (3.19a) e j ∧ T ij = C i + e ∧ T i , (3.19b)with C = − (cid:15) ijk e i t F jk − εγ e i t F i +
12 Λ (cid:15) ijk e i t e j ∧ e k , (3.20) C i = (cid:15) ijk e F jk + ε ⋅ (cid:15) ijk e j t F k + εγ e F i + εγ e j t F ji + Λ (cid:15) ijk e j t e ∧ e k −
12 Λ (cid:15) ijk e e j ∧ e k . As shown in Appendix C, equations (3.19a) and (3.19b) can always be solved with noconditions coming from their inhomogeneous terms, so no new secondary constraints appearhere. Notice that to solve the latter we have to dualize and consider T ij = (cid:15) ijk T k .– 10 – .2.2 Simplifying the constraints Before solving the equations for the components of Z and checking the tangency of Z to thesubmanifold of M determined by the secondary constraints, it is useful to look at theseconstraints in detail and simplify them as much as possible.The constraints (3.9a) can be split in the following set of conditions (cid:15) ijk e j ∧ De k = , (3.21a) De ∧ e i − De i ∧ e = . (3.21b)In terms of the objects introduced in appendix A, these are equivalent to D ij − D ji = , (3.22a) D ○ i − D i ○ = . (3.22b)It is very important to point out that, when (3.22a) and (3.22b) hold, the secondaryconstraints (3.17) are equivalent to (see appendix D.2) ( e − (cid:15) klm e k t E lm ) D ( ij ) = . (3.23)As discussed in appendix C.1, the non-degeneracy condition for the tetrads (3.10) impliesthat the term in parenthesis in (3.23) is different from zero at every point of Σ , hence, thenew secondary constraints can be written in the pleasingly concise form D ( ij ) = . (3.24)We prove now an important result: Proposition 1
The constraints D ( ij ) = and e [ I ∧ De J ] = are equivalent to the condition De I = . Proof. ⇐ This is obvious as De i = implies D ij ∶= ( e i ∧ De j w ) = . ⇒ To begin with, notice that, as a consequence of (3.22a) and (3.22b), the conditions D ( ij ) = together with e [ I ∧ De J ] = are equivalent to D ij = and D i ○ − D ○ i = . Now,according to (C.12c), D ij = implies De i = . From this and the definition of D i ○ , we imme-diately obtain D i ○ = so that (3.22b) implies D ○ i = and, hence, De = as a consequenceof (C.12d).As a trivial –but nonetheless reassuring– check, notice that (3.9a) and (3.24) provide 12conditions “per point”, the same as De I = . Notice also that we have shown that D ○ i = D i ○ = on the secondary constraint submanifold.The condition De I = can be used to simplify the constraints (3.9b). To this end,notice that De I = implies F IJ ∧ e J = and, hence, the γ -dependent terms in (3.9b)– 11 –anish. Summarizing, we have shown that the secondary constraints found up to this point–ultimately the whole set of constraints– can be written as dΛ = , (3.25a) De I = , (3.25b) (cid:15) IJKL e J ∧ ( F KL −
13 Λ e K ∧ e L ) = . (3.25c)In terms of the objects introduced in appendix A they can be written as dΛ = , (3.26a) D ij = , (3.26b) D ○ i = D i ○ = , (3.26c) − (cid:15) ijk F ijk = , (3.26d) (cid:15) ijk ( Λ E jk + F j ○ k − F ○ jk ) = . (3.26e) Up to this point, we have restricted ourselves to study the secondary constraints comingeither from (3.3) or as conditions for the solvability of the equations for the componentsof Z . There is, though, an additional consistency requirement which is central to theGNH approach: we must ensure the tangency of Z to the submanifold of M defined bythe secondary constraints. The tangency conditions can be easily obtained by computingthe directional derivatives of the constraints in the direction of Z . As a side remark, it isimportant to notice that this step does not involve the presymplectic form ω (in other words,Poisson brackets play no role here). The tangency conditions take the form of additionallinear and homogeneous equations involving the components of Z and the variables e I t , ω I t J , e I , ω IJ , Λ and X . In the present case it is useful to derive them from (3.25). They are d Z Λ = , (3.27a) DZ Ie + Z IωJ ∧ e J = , (3.27b) (cid:15) IJKL ( Z Je ∧ ( F KL − Λ e K ∧ e L ) + e J ∧ DZ KLω − Z Λ e J ∧ e K ∧ e L ) = . (3.27c)These must be considered together with (3.8e), (3.11c), (3.12) and taking into accountthat the constraints (3.25) must hold. So far, we have found a set of constraints (3.25), defining a submanifold M of M wherethe dynamical variables are forced to live, and the conditions (3.8e), (3.11c), (3.12), and(3.27) that the components of the restriction of the Hamiltonian vector fields to M mustsatisfy. The latter have different origins: some of them come directly from the resolutionof ω ( Z , Y ) = dl H ( Y ) on M whereas the rest appear as tangency conditions. Notice that,despite their different origins, we have to consider all these equations together in order– 12 –o get the final form of the components of the Hamiltonian vector field that defines thedynamics of our model.There are several possibilities now:• There are no solutions for the components of Z so the theory is inconsistent (obviouslynot the case here).• The equations can be solved with no extra conditions on the configuration variables.Their solutions then give the components of the Hamiltonian vector field that encodesthe dynamics on the final constraint submanifold.• New consistency conditions appear. These should be added to the secondary con-straints together with the corresponding tangency conditions to start the processagain.Let us find out what happens in the present case. To begin with, we immediately seethat (3.11c) implies (3.27a) and we can remove Z Λ from the remaining equations. Next,it is convenient to solve (3.12) for Z Ie or, equivalently, (3.13). When the constraints holdthese equations can be written as (cid:15) ijk e j ∧ Ξ k = , (3.28a) Ξ ∧ e j − Ξ j ∧ e = . (3.28b)Now, by using the results of appendix C we immediately see that the (unique) solutionof (3.28a) and (3.28b) is Ξ I = , which tells us that Z Ie = De I t − ω I t J e J . (3.29)We can use now (3.29) to simplify (3.8e), (3.27b) and (3.27c). To this end we D -differentiate(3.29) and use the constraints to get DZ Ie = F IJ e J t − Dω I t J ∧ e J , which, plugged into the tangency condition (3.27b) gives F IJ e J t + ( Z IωJ − Dω I t J ) ∧ e J = . (3.30)When this condition holds (3.8e) is equivalent to (cid:15) IJKL ( e J ∧ ( Z KLω − Dω KL t ) − e J t ( F KL − Λ e K ∧ e L )) = . (3.31)Remember that we also have the tangency condition (cid:15) IJKL ( Z Je ∧ ( F KL − Λ e K ∧ e L ) + e J ∧ DZ KLω ) = . (3.32)At this point the only task left is to solve (3.30), (3.31) and (3.32) for Z IJω . In orderto do this we first show that by D -differentiating (3.31), using (3.29) and the secondaryconstraints, equation (3.32) holds. This is a direct computation that we give in some detail– 13 –n appendix D.3. Now, in order to find the Z IJω we only have to consider (3.30) and (3.31).An important point that we have to address in the first place is the consistency of theseequations: can they always be solved or should we introduce extra secondary constraintsin order to guarantee their solvability?As we have shown above, the constraints De I = are equivalent to e ( i ∧ De j ) = and e [ I ∧ De J ] = , hence, the tangency conditions of the Hamiltonian vector field to thesecondary constraint submanifold can also be written in the form e ( i ∧ DZ j ) e + e ( i ∧ Z j ) ω K ∧ e K = , (3.33a) e [ I ∧ DZ J ] e + e [ I ∧ Z J ] ω K ∧ e K = , (3.33b)which, on account of (3.29), become e ( i ∧ F j ) K e K t + e ( i ∧ S j ) K ∧ e K = , (3.34a) e [ I ∧ F J ] K e K t + e [ I ∧ S J ] K ∧ e K = , (3.34b)where we have introduced the notation S IJ ∶= Z IωJ − Dω I t J . By defining F IJ Λ ∶= F IJ − Λ e I ∧ e J ,condition (3.31) can be written as e [ J ∧ S KL ] − e [ J t F KL ] Λ = , which implies e J ∧ ( e [ J ∧ S KL ] − e [ J t F KL ] Λ ) = . (3.35)Taking into account that e J ∧ F JK Λ ∶= e J ∧ ( F JK − Λ e J ∧ e K ) = –remember that theconstraint De I = implies F IJ ∧ e J = – (3.35) is simply e [ K ∧ S L ] J ∧ e J − e t J e J ∧ F KL + Λ e t J e J ∧ e K ∧ e L = , which, as a consequence of the constraint (3.25c), can be immediately seen to be equivalentto (3.34b). We then conclude that the problem of finding the components of Z IωJ on thesecondary constraint submanifold reduces to that of solving (3.31) together with (3.34a).Let us look now at equation (3.31). By separately considering I = and I = i , it can besplit into e i ∧ S i = (cid:15) ijk e i t F jk Λ , (3.36a) (cid:15) ijk e j ∧ S k = e ∧ S i + (cid:15) ijk e j t F k Λ − (cid:15) ijk e F jk Λ , (3.36b)where S i ∶= (cid:15) ijk S jk . As shown in appendix C, equation (3.36a) can always be solved andthe solution written in the form S i = τ ij e j + σ i , (3.37)where τ ij ∈ C ∞ ( Σ ) with τ ij = τ ji but, otherwise arbitrary, and σ i a concrete function of thedynamical fields which can be computed by using equation (C.9). Plugging S i into (3.36b)and using (C.4) we find S k = τ ij E kj e i + η k , (3.38)– 14 –here the concrete form of η k is not specially illuminating so we do not give it here [ofcourse, it can be obtained by using (C.4)]. As we can see, equation (3.31) can always besolved but its solutions depend on the arbitrary objects τ ij . At this point, all that is leftto do is plugging (3.37) and (3.38) into (3.34a) and study the resulting equations for τ ij . Astraightforward computation tells us that these equations are ( δ ik δ jl − δ ij δ kl + ε E ( ik E j ) l ) τ kl + ξ kl = , (3.39)where ξ kl is another concrete function of the dynamical fields. Now, as we show in appendixC.2, the × matrix M ( ij )( kl ) ∶= δ ik δ jl − δ ij δ kl + ε E ( ik E j ) l , (3.40)is always invertible for the field configurations that we are considering in the paper and,hence, it is always possible to solve for all the components of τ ij . One these are known, wecan plug them into the expressions for S i and S k and, finally, obtain Z IJω . The completeexpressions are long and not specially illuminating, so we will not give them here. Of coursethey are simpler in the time gauge.
Here we show how the constraints and Hamiltonian vector fields obtained above by usingthe GNH method can be directly obtained from the field equations (see [16, 17]). This isinteresting because knowing in advance that extra secondary constraints are expected andhaving an idea of their form can be useful. As discussed in section 2, the field equations ofthe -dimensional action (2.3) are equivalent to De I = , (4.1a) (cid:15) IJKL e J ∧ ( F KL −
13 Λ e K ∧ e L ) = , (4.1b) Φ ∗ vol − (cid:15) IJKL e I ∧ e J ∧ e K ∧ e L = , (4.1c) dΛ = , (4.1d)where the last equation is redundant as explained in section 1. Let us write e I = e I + d τ e I t , ω I J = ω IJ + d τ ω I t J . with e I t ∶= ı ∂ τ e I and ω I t J ∶= ı ∂ τ ω I J . Plugging this decomposition into (4.1a) gives = De I = d e I + ω IJ ∧ e J + ( d e I t + ω IJ e J t − ω I t J e J ) ∧ d τ . Pulling this back to Σ τ , we get (remember that ∗ τ d τ = ) d ∗ τ e I + ∗ τ ω IJ ∧ ∗ τ e J = , (4.2)– 15 –nd pulling back to to Σ τ after taking the interior product with ı ∂ τ we find ∗ τ £ ∂ τ e I = dd τ ( ∗ τ e I ) = d ∗ τ e I t + ∗ τ ω IJ ∗ τ e J t − ∗ τ ω I t J ∗ τ e J . (4.3)By performing the substitutions ∗ τ e I → e I , ∗ τ e I t → e I t , dd τ ( ∗ τ e I ) → Z Ie , (4.4) ∗ τ ω IJ → ω IJ , ∗ τ ω I t J → ω I t J , in (4.2) and (4.3) we get De I = ,Z Ie = De I t − ω I t J e J . An analogous computation for (4.1b) gives (cid:15)
IJKL e J ∧ ( F KL −
13 Λ e K ∧ e L )= (cid:15) IJKL e J ∧ ( d ω KL + ω KM ∧ ω ML −
13 Λ e K ∧ e L )+ (cid:15) IJKL e J t d τ ∧ ( d ω KL + ω KM ∧ ω ML −
13 Λ e K ∧ e L )− (cid:15) IJKL d τ ∧ e J ∧ (− d ω KL t − ω KM ω ML t + ω LM ω K t M −
23 Λ e K t e L ) = . Pulling this back to Σ τ we get (cid:15) IJKL ∗ τ e J ∧ ( d ∗ τ ω KL + ∗ τ ω KM ∧ ∗ τ ω ML − ∗ τ Λ ∗ τ e K ∧ ∗ τ e L ) = , (4.5)and pulling back to to Σ τ after taking the interior product with ı ∂ τ we find − (cid:15) IJKL ∗ τ e J ∧ dd τ ( ∗ τ ω KL ) + (cid:15) IJKL ∗ τ e J t ( d ∗ τ ω KL + ∗ τ ω KM ∧ ∗ τ ω ML − ∗ τ Λ ∗ τ e K ∧ ∗ τ e L )+ (cid:15) IJKL ∗ τ e J ∧ ( d ∗ τ ω KL t + ∗ τ ω KM ∗ τ ω ML t − ∗ τ ω LM ∗ τ ω K t M + ∗ τ Λ ∗ τ e K t ∗ τ e L ) = . (4.6)After some straightforward manipulations and using (4.4) together with ∗ τ £ ∂ τ ω IJ = dd τ ( ∗ τ ω IJ ) → Z IωJ , (4.7)these expressions translate into (cid:15) IJKL e J ∧ ( F KL −
13 Λ e K ∧ e L ) = ,(cid:15) IJKL ( e J ∧ ( Z KLω − Dω KL t − Λ e K t e L ) − e J t F KL ) = . Considering now (4.1c) and proceeding as in the previous cases, we get ∗ τ ı ∂ τ ( Φ ∗ vol ) − ∗ τ ( (cid:15) IJKL e I t e J ∧ e K ∧ e L ) = , – 16 –hich, taking into account (B.1), the previous substitutions, and Φ τ → X , ˙Φ τ → Z X , (4.8)leads to (3.8a).Finally, the same procedure applied to (4.1d) gives d ∗ τ Λ = , dd τ ∗ τ Λ = , which immediately translate –taking into account that we have Z ⊥ X ≠ – into dΛ = and Z Λ = by using ∗ τ Λ → Λ , dd τ ( ∗ τ Λ ) → Z Λ . (4.9)As we can see, we have been able to obtain the full set of constrains and equationsfor the Hamiltonian vector fields found by using the GNH method. Although we do notexpect this to happen always (i.e. not all the constraints may be obtained by pulling backthe field equations), by enforcing appropriate tangency requirements it should be possibleto arrive at the same final description given by the GNH procedure, once the suitablepresymplectic form (obtained by pulling back the canonical symplectic structure to theprimary constraint submanifold) is included. This could be a convenient, alternative methodto find the Hamiltonian description for singular field theories linear in time derivatives. The purpose of this section is to explain how our paper fits in the extensive literature onthe Hamiltonian formulation for general relativity as derived from the Holst action [9]. Inthe more than twenty five years since the publication of Holst’s paper, a number of authorshave looked at this question from different perspectives. Although we do not intend to beexhaustive, we will try to mention the most representative papers and compare their resultswith ours when relevant.As a general comment, the main difference between the approach that we have followedhere and the vast majority of the works on this subject stems from our use of the GNHmethod instead of Dirac’s. As a consequence, our point of view is “much more geometric”.Although geometry plays a role also in Dirac’s approach –for instance in the classificationof constraints as first or second class–, and satisfactory geometrizations of Dirac procedurealready exist [7], the essence of the procedure relies heavily on the interpretation of thePoisson brackets as generators of time evolution. In contrast with this, the central consis-tency requirement in the GNH method is the tangency of the Hamiltonian vector fields tothe constraint submanifold. If boundaries are present this criterion is specially appropriate.At variance with Dirac’s approach, Hamiltonian vector fields play a central role because themain goal of the Dirac method is, usually, to find the constraints and study their Poissonbrackets. – 17 –orks can be classified attending to several features: their use or not of the time gauge,the implementation of the full SO ( , ) symmetry or only the SU ( ) one, the treatment ofsecond class constrains (solving them or not) and the more or less strict adherence to theDirac algorithm as originally formulated. In many cases, the discussion of the role of theImmirzi parameter also plays a central role. A rough classification of papers according tothese criteria is:• Papers where the Hamiltonian analysis relies heavily in the use of the time gauge:[9, 18–20].• Papers where no time gauge is used in the Hamiltonian analysis: [21, 22].• Papers where the full SO ( , ) invariance is retained: [21–23].• Papers where the constraint De I = is identified and plays a central role [15, 19, 24].• Papers that address the treatment of second class constraints, either by solving themor introducing Dirac brackets: [18, 20, 21, 23, 25–27].• The paper that, in our opinion, adheres to the letter of Dirac’s algorithm in a moreclear way is [20].• Works introducing constraints quadratic in momenta: [18, 28, 29].• Some incomplete analyses of the Hamiltonian formulation for unimodular gravity interms of tetrads can be found in [11, 30].The actual implementation of Dirac’s algorithm is often subtle and it is important tofollow it to the letter to avoid conceptual mistakes. A reason for this is the fact that sec-ondary constraints may appear as conditions for the stability of the primary constraints oras conditions for the stability of other secondary constraints (that can show up as consis-tency conditions for the solvability of the equations for the Lagrange multipliers introducedin the definition of the total Hamiltonian). The analysis of the latter, in particular, is oftenunpleasant as they tend to be quite complicated. This is probably the reason why the sim-ple form of the constraints that we give here has eluded most Hamiltonian analysis of theHolst action (in particular the fine one appearing in [20], which follows Dirac’s method tothe letter). Very often, these difficulties are alleviated by introducing the time gauge. Thisis a standard way to arrive at the Ashtekar formulation, that can be ultimately justified byinvoking the Lorentz invariance of the action and the possibility of adapting the tetrads toany given spacetime foliation. Notice, however, that from a general perspective an actualgauge fixing can only be (safely) performed after the whole set of constraints and the finalform of the Hamiltonian vector fields has been determined . This is probably one of thereasons why a number of authors have discussed the possibility of dispensing with the timegauge.The constraints De I = have appeared in the literature, most notably, in [15, 19].In [19], the authors check that this condition is compatible with the dynamics defined by– 18 –he Holst action by looking at the Hamiltonian dynamics in the time gauge. The approachfollowed in that paper relies on the geometric interpretation of De I = as a vanishing torsioncondition whose compatibility with the dynamics is verified. In our approach, that conditionis obtained by using the GNH method to obtain the Hamiltonian dynamics without usingthe time gauge.The paper [15] merits special attention. There, the authors discuss the Hamiltoniandescription of general relativity from the Holst action by relying on a geometric methodintroduced by Kijowski and Tulczyjew [31]. The frame fields are subject to non-degeneracyconditions equivalent to the ones that we have been naturally compelled to introduce here.Despite the apparent differences in approach, there seems to be a clear correspondencebetween the results of [15] and ours (if we leave aside the part of our analysis involvingparametrization and unimodularity). We list some of them:• The splitting of the conditions contained in De I = as structural and residual con-straints is similar to our decomposition as e [ I ∧ De J ] = and D ( ij ) = . Notice inparticular that in order to write (3.24), it is necessary to choose an internal time-likevector in order to “make the splitting into and i , j indices”.• The part of the presymplectic form on the primary constraint submanifold (3.6) cor-responding to the triads and the spin connection is essentially equation (4.7) of [15].• The final form of the constraints.Our results very strongly suggest that the results described here can be found by follow-ing Dirac’s approach although, arguably, the necessary computations may be quite involved.It is not that the Dirac approach leads to a complicated reduced space formulation but,rather, that it is difficult to suspect that the nice formulation furnished by the constraints(3.25) can be actually found by manipulating the expressions that appear in the imple-mentation of the Dirac algorithm. To a certain extent, this is also true within the GNHapproach. Of course, the arguments relying on the derivation of the constraints from thefield equations are very helpful in this regard and provide a very useful guide. We have studied in detail the Hamiltonian formulation for parametrized unimodular grav-ity derived from a suitable modification of the Holst action on a 4-dimensional manifolddiffeomorphic to R × Σ , with Σ closed. We have relied on the GNH method which, owing toits clear geometric foundations, is superior to the more traditional Dirac approach, at leastfor the purposes of this paper. This is, probably, one of the reasons why we have been ableto find a simple way to describe the purely gravitational sector of the theory.The unimodularity condition has been incorporated in the action by introducing abackground volume form (associated with a fixed background metric) and demanding itto be equal to the volume form defined by the tetrads. The presence of this geometricbackground structure has allowed us to parametrize the model in a non-trivial way by– 19 –ntroducing dynamical diffeomorphisms. As we have shown the resulting theory reproducesthe behavior of metric parametrized unimodular gravity.It is somehow surprising that arriving at the concise formulation that we have discussedhere requires a non-negligible effort, even when the GNH approach is used. However, inhindsight, it is obvious that such a formulation must exist, as shown by the argumentpresented in section 4. As we have shown, the secondary constraints (3.17) can be simplifiedto the form (3.24), precisely when the non-degeneracy of the tetrads holds. This is a veryneat and sensible result which is probably harder to arrive at by using Dirac’s approach.As described in detail in [14], the real Ashtekar formulation for general relativity canbe readily derived from the results presented here by using the time gauge fixing. This is, ofcourse, to be expected as the Holst action is known to lead to the Ashtekar formulation. Ina sense, the formulation described here can be thought of as the Lorentz invariant precursorof the real Ashtekar formulation. By itself, it has some interesting features:• The constraints have a very simple form in the Lorentzian case and are independentof the Immirzi parameter γ .• Only the (pre)symplectic form depends on γ . Although the presymplectic form isarguably more complicated than the usual one (in fact, it is not written in canonicalform), it is not inconceivable that it can be used for quantization. This is an interestingproblem that should be looked at.• The internal symmetry group in this case is the full Lorentz group. The fact that itis not compact may present technical difficulties for quantization.• The Hamiltonian formulation for the Palatini action can be easily found from thisone in the γ → ∞ limit.• There is no need to introduce constraints quadratic in momenta. Actually, only theprimary constraints that appear when the fiber derivative is computed involve themomenta and they do that in a very special way. First, they are linear in momentaand, second, they do not involve the velocities (this is a consequence of the factthat the Lagrangian is linear in the velocities because the action depends linearly onderivatives).An interesting side-product of the present work is the idea of relying on the fieldequations to arrive at the Hamiltonian formulation, at least for first order theories andtheories with actions linear in velocities such as the one discussed here. We would liketo add that this is hardly a new idea. Similar ideas can be found in the literature, forinstance, it is well known that the equations of motion provide Lagrangian constraints forsingular Lagrangian systems [16, 32] and, among them, the so called projectable constraints can be taken to the cotangent bundle of the configuration space as the usual constraints inthe Hamiltonian framework (see the preceding papers and also [17, 33]). This method mayprove to be specially fruitful when dealing with boundaries in generalizations of gravitationalactions written in terms of tetrads. – 20 – cknowledgments
The authors wish to thank Marc Basquens for useful comments. This work has beensupported by the Spanish Ministerio de Ciencia Innovación y Universidades-Agencia Estatalde Investigación/FIS2017-84440-C2-2-P grant. Bogar Díaz was partially supported by aDGAPA-UNAM postdoctoral fellowship and acknowledges support from the CONEX-Plusprogramme funded by Universidad Carlos III de Madrid and the European Union’s Horizon2020 programme under the Marie Sklodowska-Curie grant agreement No. 81538. JuanMargalef-Bentabol is supported by the Eberly Research Funds of Penn State, by the NSFgrant PHY-1806356 and by the Urania Stott fund of Pittsburgh foundation UN2017-92945.
A Glossary and Notation
Although we have introduced some of the notation used in the paper whenever it wasrelevant, in order to facilitate its reading we have left a number of definitions for thisAppendix. ● Given a volume form vol in a differentiable manifold M and a top-form α , it is alwayspossible to find a smooth function f ∈ C ∞ (M) such that α = f ⋅ vol . We will often denotesuch function as ( α vol ) . Although this seems cumbersome at first, the unwieldy parentheses are a good reminder ofthe fact that we are dealing with a scalar density. ● We introduce now the simplified notation used to write the solutions to the equations for Z Ie and Z e and in the tangency analysis. Let w be the volume form on Σ w = (cid:15) ijk e i ∧ e j ∧ e k , (remember that we are working with non-degenerate frames in Σ ). We define E ij ∶= ( e i ∧ e j ∧ e w ) , D ○ i ∶= ( ( De ) ∧ e i w ) , D i ○ ∶= ( ( De i ) ∧ e w ) , D ij ∶= ( e i ∧ De j w ) , F i ○ j ∶= ( e i ∧ F j w ) , F ○ ij ∶= ( e ∧ F ij w ) , F ○○ i ∶= ( e ∧ F i w ) , F ijk ∶= ( e i ∧ F jk w ) . Notice that E ij , F ○ ij , F ijk , F i ○ j and F ○○ i are antisymmetric in the last pair of indices.– 21 – Diffeomorphisms and embeddings as dynamical variables
We give here a few details about the use of diffeomorphisms and embeddings as dy-namical variables. Interested readers are referred to [6, 34, 35] for details. Given the4-dimensional manifolds R × Σ and M , we use as dynamical variables diffeomeorphisms Φ ∶ R × Σ → M such that for every τ ∈ R the embeddings Φ τ ∶= Φ ○ τ ∶ Σ ↪ M havespacelike images (i.e. Φ τ ( Σ ) ⊂ M is g -spacelike). Here τ ∶ Σ → R × Σ ∶ p ↦ ( τ, p ) . Wewill denote the space of such embeddings as Emb s ( Σ , M) . The diffeomorphisms that weconsider in the paper can be loosely interpreted as curves of embeddings of this type.We denote the tangent map associated with Φ as T Φ . The diffeomorphisms that weuse are such that the field ˙Φ ∶= T Φ .∂ τ is transverse to Φ τ ( Σ ) for all τ ∈ R . When restrictedto a particular embedding Φ τ , ˙Φ defines its instantaneous velocity ˙Φ τ .Given X ∈ Emb s ( Σ , M) we can build a vector field over X consisting of future directed,unit normals that we denote as n X . Notice that n X ( p ) ∈ T X ( p ) M for each p ∈ Σ . Now, ifwe have Y X ∈ Γ ( X ∗ T M) we can expand it as Y X = Y ⊥ X n X + T X.Y ⊺ X , where Y ⊥ X = g ( n X , Y X ) =∶ n X ( Y X ) is a smooth real function on Σ , Y ⊺ X ∈ X ( Σ ) and T X isthe tangent map of X .A useful result –that we give without proof– is ∗ τ ı ∂ τ ( Φ ∗ vol ) = ˙Φ ⊥ τ vol γ Φ τ , (B.1)where ˙Φ ⊥ τ = εn Φ τ ( ˙Φ τ ) and vol γ Φ τ ∈ Ω ( Σ ) is the volume form associated with the metric γ Φ τ ∶= Φ ∗ τ g on Σ : vol γ Φ τ ( v , v , v ) ∶= vol ( n Φ τ , T Φ .v , T Φ .v , T Φ .v ) , for v , v , v ∈ X ( Σ ) . Equation (B.1) is used in sections 2.2 and 4. C Useful mathematical results ● The invariant SO ( , ) tensor P IJKL can be inverted whenever γ ≠ ε in the sense that P IJKL [ P − ] KLMN = δ [ MI δ N ] J . The inverse is [ P − ] IJKL ∶= γ ( ε − γ ) (− εγ ⋅ (cid:15) IJKL + η IK η JL − η JK η IL ) , (C.1)as can be checked by a direct computation. ● The tensor P IJKL satisfies DP IJKL = . ● For any H IJ ∈ C ∞ ( Σ ) antisymmetric in I and J the following identity holds [ P − ] IJNQ P NKLM H QK = H [ I [ L δ J ] M ] . (C.2)– 22 – roof. [ P − ] IJNQ P NKLM H QK = H [ I [ L δ J ] M ] + γ ( ε − γ ) ( (cid:15) IJK [ L H KM ] − (cid:15) K [ ILM H J ] K ) . The last term of this expression can be shown to vanish by computing its dual (cid:15)
NQIJ ( (cid:15) IJK [ L H KM ] − (cid:15) KILM H JK )= ε ( δ NK δ Q [ L H KM ] − δ QK δ N [ L H KM ] )+ ε ( δ JK δ NL δ QM + δ NK δ QL δ JM + δ QK δ JL δ NM − δ NK δ JL δ QM − δ JK δ QL δ NM − δ QK δ NL δ JM ) H KJ = . ∎● In the following we give the solutions to several types of inhomogeneous linear equationsinvolving differential forms in a 3-dimensional manifold Σ . We use the notation explainedin appendix A. The proofs are quite direct so they are left to the reader.i) Let us consider the system of equations (cid:15) ijk e j ∧ z k = u i , (C.3)where the unknowns are z k ∈ Ω ( Σ ) , with Σ a three-dimensional manifold, the triads e i ∈ Ω ( Σ ) are such that w ∶= ( (cid:15) ijk e i ∧ e j ∧ e k )/ is a volume form in Σ and the u i ∈ Ω ( Σ ) are given 2-forms. Then the solutions are z k = ( e i ∧ u i w ) e k − ( e k ∧ u i w ) e i . (C.4)Even though (C.3) is inhomogeneous it can be solved for any given u i .ii) Let us consider the system of equations α ∧ e i = β i , (C.5)where α ∈ Ω ( Σ ) is the unknown, e i ∈ Ω ( Σ ) are such that w ∶= ( (cid:15) ijk e i ∧ e j ∧ e k )/ is avolume form in Σ , and the β i ∈ Ω ( Σ ) are given 2-forms. The equations (C.5) can be solvedif and only if the inhomogeneous term β i satisfies the condition β i ∧ e j + β j ∧ e i = , (C.6)in which case the solution is α = (cid:15) ijk ( β i ∧ e j w ) e k . (C.7)iii) Let us consider the equation e i ∧ z i = u , (C.8)where the unknowns are z k ∈ Ω ( Σ ) , the triads e i ∈ Ω ( Σ ) are such that w ∶= ( (cid:15) ijk e i ∧ e j ∧ e k )/ is a volume form in Σ and u ∈ Ω ( Σ ) is a given 2-form. Then the solutions are z i = − (cid:15) ijk ( e j ∧ uw ) e k + ζ ij e j , (C.9)– 23 –ith ζ ij ∈ C ∞ ( Σ ) satisfying ζ ij = ζ ji but, otherwise, arbitrary. ● The following identity is useful e = (cid:15) ijk E ij e k . (C.10)To prove it we expand e = λ i e i and plug it into (cid:15) jkl E jk e l = (cid:15) jkl ( e j ∧ e k ∧ e w ) e l , to get (cid:15) jkl E jk e l = λ i e i = e . With the help of (C.10), it is straightforward to prove the following relations D i ○ = (cid:15) klm E kl D mi , (C.11a) F ○ ij = (cid:15) klm E kl F mij , (C.11b) F ○○ i = (cid:15) jkl E jk F l ○ i . (C.11c)Other useful identities are F ij = (cid:15) klm F kij e l ∧ e m , (C.12a) F i = (cid:15) jkl F j ○ i e k ∧ e l , (C.12b) De i = (cid:15) jkl D ji e k ∧ e l , (C.12c) De = (cid:15) ijk D ○ i e j ∧ e k . (C.12d)These identities are all proven in the same way, so we will just show that (C.12c) holds. Proposition 2 De i = (cid:15) jkl D ji e k ∧ e l is equivalent to D ij = ( e i ∧ De j w ) . Proof. ⇒ De i = (cid:15) jkl D ji e k ∧ e l implies e m ∧ De i = (cid:15) jkl D ji e m ∧ e k ∧ e l = (cid:15) jkl D ji (cid:15) mkl w = D mi w ,whence, D ij = ( e i ∧ De j w ) . ⇐ Let us write De i = (cid:15) jkl H ji e k ∧ e l , then e m ∧ De i = (cid:15) jkl H ji e m ∧ e k ∧ e l = (cid:15) jkl H ji (cid:15) mkl w = H mi w which implies H ji = D ji , so that De i = (cid:15) jkl D ji e k ∧ e l . ● F IJ ∧ e J = is equivalent to F j ○ j = , (C.13a) F ○ i ○ + F jij = . (C.13b)– 24 – .1 Rewriting the non-degeneracy condition for the tetrads It is interesting to take a close look at the condition (3.10). To begin with, it isimportant to notice that, in terms of the objects defined in the preceding section, it can bewritten as e − (cid:15) klm e k t E lm ≠ , (C.14)because ε IJKL e I t e J ∧ e K ∧ e L = ( e − (cid:15) klm e k t E lm ) w . The meaning of (C.14) as a non-degeneracy condition for the tetrads is obvious if we writethem in matrix form as e I = [ e e i t e e i ] , (C.15)and remember that e = (cid:15) ijk e i E jk =∶ λ i e i . (C.16)From (C.15), we can write the 4-metric as g = [ εe e + e t i e i t εe e ⊺ + e t i e i ⊺ εe e + e t i e i εe e ⊺ + e i e i ⊺ ] , (C.17)and conclude that a necessary and sufficient condition for the 3-metric q ∶= εe e ⊺ + e i e i ⊺ tobe definite positive is + ελ i λ i > . This can be seen by writing q ab = ( δ ij + ελ i λ j ) e ai e bj andnoting that the quadratic form δ ij + ελ i λ j is positive definite if and only if + ελ i λ i > . C.2 A determinant computation
We compute here the determinant of the × matrix obtained by grouping the indicesof M ( ij )( kl ) ∶= δ ik δ jl − δ ij δ kl + ε E ( ik E j ) l , as the symmetrized pairs ( ij ) and ( kl ) . To simplify the computations and interpret theresult, it is useful to write M ( ij )( kl ) in terms of the λ i defined in (C.16) as δ ( ik δ j ) l ( + ελ m λ m ) − δ ij δ kl ( + ελ m λ m ) + εδ ij λ k λ l + ελ i λ j δ kl − ελ ( i δ j ) k λ l − ελ ( i δ j ) l λ k . The determinant of M ( ij )( kl ) can be obtained with the help of any computer algebra package.It has the simple expression − ( + ελ m λ m ) , which is equivalent to − ( + ε E ij E ij ) . As can be seen from the discussion in C.1, the condition that the 3-metric εe e ⊺ + e i e i ⊺ bepositive definite implies that the determinant of M ( ij )( kl ) is different from zero.– 25 – Some details about the GNH procedure
D.1 Computing the pullback of Ω to M In order to do this, we have to compute (3.5) for vector fields Y , Z which are tangent tothe primary constraint submanifold M . Some of the components of these fields on M canbe obtained from the primary constraints by computing their directional derivatives andrequiring them to be zero, so by plugging these particular components into the canonicalsymplectic form we get the sought for pull-back. The only unfamiliar directional derivativesare those involving embedding-dependent objects (which only show up in the definition of p X ). In order to compute them, one has to use variations as in [34]. In the present case,from the definition of the momenta (3.1) we get p e t —→ Z e t (⋅) = , (D.1a) p e —→ Z e (⋅) = , (D.1b) p ω t —→ Z ω t (⋅) = , (D.1c) p ω —→ Z ω (⋅) =∫ Σ P IJKL (⋅) ∧ Z Ke ∧ e L , (D.1d) p Λ —→ Z Λ (⋅) = , (D.1e) p X —→ Z X (⋅) =∫ Σ (− Λ £ e X (⋅) Z ⊥ X + εZ Λ n X (⋅) + εn X (⋅) Λdiv γ X Z X ⊺ ) vol γ X , (D.1f)where n X (⋅) and e X (⋅) are embedding dependent objects (which are carefully discussed in[34]).If we demand that the components of Z ∈ X ( T ∗ Q ) take the values given by (D.1), theresulting vector field will be tangent to M so its restriction to M , that will be denoted as Z , will be a vector field on M .Taking into account that Y α X = Y ⊥ X n αX + ( T X ) αa Y ⊺ a X (here ( T X ) αa denotes the tangentmap of the embedding X and the indices a and α refer to Σ and M respectively) we seethat εn X ( Y X ) = Y ⊥ X ,e X ( Y X ) = Y ⊺ X , and, hence, Z X ( Y X ) = ∫ Σ (− Λ £ Y ⊺ X Z ⊥ X − £ Z ⊺ X ( Λ Y ⊥ X ) + Z Λ Y ⊥ X ) vol γ X . Plugging the previous results into (3.5), we finally get ω ( Z , Y ) = ∫ Σ ( P IJKL ( Z IJω ∧ Y Ke − Y IJω ∧ Z Ke ) ∧ e L + Z ⊥ X ( Y Λ − ı Y ⊺ X dΛ ) vol γ X − Y ⊥ X ( Z Λ − ı Z ⊺ X dΛ ) vol γ X ) , which, being closed but degenerate, is a presymplectic form on M .– 26 – .2 Rewriting the new secondary constraints (3.17)The constraints (3.22a) and (3.22b) can be used to rewrite (3.17) in the much simplerand suggestive form (3.23). To this end, we first use (3.22a) and (3.22b) to rewrite (3.17)as (cid:15) klm e k t ( D il E jm + D jl E im ) − e D ij + e i t D j ○ + e j t D i ○ = . We then use (C.11a) to transform the last two terms of the preceding expression to get (cid:15) klm e k t D il E jm + (cid:15) klm e j t E kl D mi + (cid:15) klm e k t D jl E im + (cid:15) klm e i t E kl D mj − e D ij = . (D.2)The obvious identity = (cid:15) klm D i [ m e k t E jl ] , is equivalent to ( (cid:15) klm e k t E lm ) D ij − (cid:15) klm e j t D ik E lm + (cid:15) klm e k t D im E jl = , (D.3)which, together with (3.22a), allows us to simplify (D.2) to the form (3.23). Notice that,with the help of (3.22a), we can write (D.3) also in the form ( (cid:15) klm e k t E lm ) D ij − (cid:15) klm e i t D jk E lm + (cid:15) klm e k t D jm E il = . (D.4) D.3 Some results useful to obtain Z IJω
We show that by D -differentiating (3.31) and using (3.29) and the secondary con-straints, equation (3.32) holds. Indeed, by differentiating (3.31) and using the constraint De I = and the Bianchi identity DF IJ = we get (cid:15) IJKL e J ∧ DZ KLω = F KM ω ML t − F LM ω MK t − (cid:15) IJKL De J t ( F KL − Λ e K ∧ e L ) . Plugging this now into (3.32) and using (3.29), we obtain (cid:15)
IJKL ω J t M ( e [ L ∧ F MK ] −
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