Differential form description of the Noether-Lagrange machinery, vielbein/gauge-field analogies and energy-momentum complexes
aa r X i v : . [ g r- q c ] A p r Differential form description of the Noether-Lagrange machinery, vielbein/gauge-fieldanalogies and energy-momentum complexes
Ermis Mitsou ∗ D´epartement de Physique Th´eorique and Center for Astroparticle Physics,Universit´e de Gen`eve, 24 quai Ansermet, CH–1211 Geneva, Switzerland
We derive the variational principle and Noether’s theorem in generally covariant field theoryin an explicitly coordinate-independent way by means of the exterior calculus over the space-timemanifold. We then focus on the symmetry of active diffeomorphisms, that is, the pushforwards alongthe integral lines of any vector field, and its analogies with internal gauge symmetries. For instance,it is well known that a class of Noether currents associated to a gauge symmetry can be obtainedby taking the partial derivative of the Lagrangian with respect to the corresponding gauge field.Here we show that this relation also holds for the Noether currents associated to diffeomorphismsand the vielbein, but only if one decomposes all forms in the vielbein basis. We also relate thediffeomorphism Noether currents to the matter energy-momentum tensor of General Relativity, toHamiltonian boundary terms and to two known energy-momentum complexes of the vielbein.
I. INTRODUCTION
The aim of this paper is three-fold. First, we pro-pose to review the basic Noether-Lagrange machinery ofgenerally covariant field theory, i.e. the variational prin-ciple and Noether’s theorem, both in terms of currentsand charges, without ever referring to coordinates. Wetherefore employ the framework of exterior calculus overthe space-time manifold M , an explicitly coordinate-independent and global way of displaying and manipulat-ing the field information . Apart from elegance, stickingto this formalism guarantees that general covariance isalways preserved and prevents us from being distractedby coordinate-induced artefacts. Another advantage isthat it allows one to “see” conservation equations just bylooking at the Euler-Lagrange equations.Using some simple identities listed in appendix C, wewill see that many computations in General Relativity(GR) that are usually lengthy when carried out usingcoordinates become quite straightforward in exterior cal-culus. We illustrate this fact by computing the equationsof motion and Noether currents of Palatini vielbein grav-ity coupled to a Yang-Mills (YM) and a Dirac field.This brings us to the second aim of our paper whichis to draw some interesting analogies between vielbeingravity and classical gauge theory. For instance, it iswell known that some Noether currents of YM theorycan be computed by simply taking the partial derivativeof the Lagrangian with respect to the gauge field. We willshow that, if one decomposes all forms in the vielbein ba-sis, then, in full analogy, one obtains the diffeomorphismNoether currents by taking the partial derivative of theLagrangian with respect to the vielbein. ∗ [email protected] From the fibre bundle point of view of field theory, we will beusing local coordinates for the vertical space, our interest beingmainly to get rid of the space-time coordinates.
Finally, we focus on the concept of energy and momen-tum. We relate the Noether diffeomorphism currents ofthe matter Lagrangian to the energy-momentum tensoras defined in GR, that is, the variational derivative of thematter action with respect to the vielbein. We also dis-cuss the relation between the Noether energy charge andthe Hamiltonian definition of energy, i.e. boundary termsin the canonical formalism, and show that a Noether en-ergy does not always have a Hamiltonian analogue. Weconclude by discussing the relation of the Noether diffeo-morphism currents and some known energy-momentumcomplexes for the gravitational field. For instance, inthe case where one considers Møller’s Lagrangian [1] forGR, we show that the Noether currents correspondingto the diffeomorphisms generated by the vielbein andholonomic frames are nothing but the energy-momentumcomplexes of [2, 3] and the one of Møller [1], respectively.The former is a tensor under diffeomorphisms, but trans-forms inhomogeneously under local Lorentz transforma-tions (LLTs), while the latter is non-covariant under bothtransformations.Our study also contains some useful by-products,such as an elegant expression of Møller’s Lagrangian interms of differential forms (14), as well as a coordinate-independent and compact expression for the variation ofthe Hodge scalar product with respect to the vielbein(C13).The organization of the paper goes as follows. In sec-tion II we introduce the notational conventions, symme-tries and fields we are going to use. In section III wederive the variational principle and Noether’s theoremusing only forms, anti-derivations and integration. Insection IV we focus on the Noether currents associatedto diffeomorphisms and their analogies with the case ofinternal gauge symmetries. In section V we discuss therelation the matter energy-momentum tensor, the Hamil-tonian boundary terms and some gravitational energy-momentum complexes. In section VI we summarize.
II. CONVENTIONS AND NOTATIONA. Geometry and groups
Let M be a real parallelizable smooth manifold of di-mension D ≡ d + 1 ≥ k denote the set of k -dimensional orientable closed embedded submanifoldsof M . For U ∈ ∆ D , X ( U ) ≡ Γ( T U ) denotes the space ofvector fields and Ω p ( U ) ≡ Γ (Λ p ( U )) denotes the spaceof p -forms, over U . If the U dependence is omitted thenit means the assertion works for all U ∈ ∆ D .The parallelizability of M implies the existence ofglobal frames, i.e. sets of D vector fields ǫ I ∈ X ( M ),where I = 1 , . . . , D , which form a basis everywhere on M . Given such a frame, there exists a unique set of D vielbein 1-forms e I obeying e I ( ǫ J ) = δ IJ and they arenon-singular e I ∧ . . . ∧ e I D = 0 everywhere on M . TheLorentzian metric field is defined by g ≡ η IJ e I ⊗ e J ,where η IJ is the Minkowski metric with a signature ofmostly pluses. We define the compact notations e I ...I p ≡ e I ∧ . . . ∧ e I p , ˜ e I ...I p ≡ ε I ...I D e I p +1 ...I D / ( D − p )!, i I ...I p ≡ i I . . . i I p and L I ≡ L ǫ I , where for the Levi-Civita sym-bol we use the convention ε ...D = − ε ...D = 1, i I denotes the interior product and L I the Lie derivativewith respect to ǫ I . Replace I → ξ for an arbitrary vectorfield ξ . The Hodge dual is defined by ⋆ α ≡ p ! ˜ e I ...I p i I p ...I α , α ∈ Ω p . (1)We use d to denote the exterior derivative and D for thecovariant exterior derivative with respect to all internalgauge transformations. We let ¯d denote the codifferential¯d α ≡ ( − D ( p − ⋆ d ⋆ α , where α ∈ Ω p . We adopt the“anti-hermitian” convention for the Lie algebra of a Liegroup, i.e. the latter is the exponentiation of the formerwith no i factor. For the case of SU( N ) for example,we have that the fundamental representation of su ( N )is the set of traceless complex N × N matrices obeying α + α † = 0. The basis of su ( N ) is chosen such that (cid:2) T a , T b (cid:3) = f abc T c , Tr (cid:0) T a F T b F (cid:1) = − δ ab (2)where the structure coefficients f abc are totally antisym-metric and the “F” subscript denotes the fundamentalrepresentation. The indices of T a F will be given by greekletters of the end of the alphabet, i.e. the matrix elementsare ( T a F ) στ .For the LLTs we focus on the component connected tothe identity SO (1 , d ). Since we are also going to considerspinors, the actual group that acts on the I indices is thedouble cover Spin(1 , d ). The standard choice of basis of spin (1 , d ) is the one obeying the Lorentz algebra (cid:2) T IJ , T KL (cid:3) = η IL T JK − η JL T IK − η IK T JL + η JK T IL , (3)where T IJ ≡ − T JI . The vector and Dirac representa-tions read (cid:0) T IJ v (cid:1) KL = η IK δ JL − η IK δ JL , T IJ D = 12 γ IJ , (4) respectively, where γ I ...I p ≡ γ [ I . . . γ I p ] and the γ I obeythe Clifford algebra { γ I , γ J } = 2 η IJ . The following iden-tity will be useful later-on γ I γ JK = η IK γ J − η IJ γ K + γ IJK . (5)Choosing a representation where (cid:0) γ I (cid:1) † = η II γ I (no sum-mation) and defining the bar conjugation so that it is aninvolution ¯ ψ ≡ ψ † iγ , we get ¯ γ I ≡ iγ (cid:0) γ I (cid:1) † iγ = − γ I and thus¯ γ I ...I p = ( − p γ I p ...I = ( − p ( p +1) / γ I ...I p . (6)In particular, ¯ γ IJ = − γ IJ so that the Dirac representa-tion of spin (1 , d ) is the set of matrices obeying θ + ¯ θ = 0.We will use greek indices from the beginning of the al-phabet for the elements of these matrices, i.e. (cid:0) γ IJ (cid:1) αβ .Let Diff( M ) denote the group of diffeomorphisms of M , i.e. the group of homeomorphisms from M to M that are diffeomorphisms in any local coordinate system,and let Diff ( M ) denote its component connected tothe identity. Every vector field ξ ∈ X ( M ) generates aone-parameter family of diffeomorphisms { Ξ t } t ∈ I ⊂ R ⊂ Diff ( M ) with I ∋ = id , ˙Ξ t (cid:12)(cid:12)(cid:12) t =0 = ξ . (7)Inversely, by definition of Diff ( M ), every element Ξ ∈ Diff ( M ) sufficiently close to the identity can be thoughof as the t = 1 element of the family generated by some ξ . In analogy with the theory of Lie groups of finite di-mension, we can thus say that X ( M ) is the “Lie algebra”of Diff ( M ). This is the fundamental representation ofDiff ( M ), also known as “passive diffeomorphisms”, butthe ones of interest here are the tensor representations,also known as “active diffeomorphisms”. In that case theΞ ∈ Diff ( M ) acts on a section T of a bundle based on M through the pushforward map T ′ = Ξ ∗ T . Since Ξ isa bijection, we have Ξ ∗ ,t = Ξ ∗− t , where Ξ ∗ is the pull-back and thus the infinitesimal variation involves the Liederivative δT = −L ξ T ⇒ Ξ ∗ = e −L ξ . (8)Thus, the generators in that representation are the Liederivatives and, given the properties of L , obey the fol-lowing algebra [ L ξ , L ξ ′ ] = L [ ξ,ξ ′ ] , (9)where [ ξ, ξ ′ ] is the Lie bracket, the algebraic product of X ( M ). In our global and coordinate-independent set-ting, there is only one privileged basis for the Lie algebra X ( M ), the one given by the frame vectors ǫ I [ ǫ I , ǫ J ] = C KIJ ǫ K . (10)The C IJK = i IJ d e K are known as the “structure coeffi-cients”. B. Fields and Lagrangians
We use totally dimensionless units ~ = c = 8 πG =1. The fields of the theory we are going to con-sider are the vielbein, spin connection and YM fields e I , ω IJ , A a ∈ Ω ( M ), respectively, and the Dirac spinor ψ ασ ∈ Ω ( M ) ⊗ G , where G is the set of complexGrassmann numbers. The non-trivial infinitesimal vari-ations under an SU( N ) transformation with parameter α = α a T a ∈ Ω ( M ) ⊗ su ( N ) are δA a = − D α a ≡ − d α a + f abc α b A c , δψ = α a T a F ψ . (11)Under an LLT with parameter θ = θ IJ T IJ ∈ Ω ( M ) ⊗ spin (1 , d ) we have δe I = θ IJ e J , δψ = 14 θ IJ γ IJ ψδω IJ = − D θ IJ ≡ − d θ IJ + θ KI ω KJ − θ KJ ω KI , (12)and under a diffeomorphism with parameter ξ ∈ X allfields transform as (8). The Lagrangian D -form is givenby L ≡ L g + L m , where L g and L m are the gravitationaland matter parts, respectively. For gravity we are goingto consider two Lagrangians. The first-order one is thePalatini Lagrangian L P ≡
12 Ω IJ ∧ ˜ e IJ , (13)where Ω IJ ≡ d ω IJ + ω KI ∧ ω KJ are the curvature two-forms of the spin connection. The second-order one isthe Møller Lagrangian [1] (see appendix A) L M ≡ − (cid:18) F IJ ∧ ⋆ F JI − F ∧ ⋆ F (cid:19) , (14)where F IJ ≡ e I ∧ d e J and F ≡ F II . In the matter sectorwe have the YM Lagrangian L YM ≡ − g F a ∧ ⋆ F a , (15)where F a = d A a + f abc A b ∧ A c / A a , and the Dirac Lagrangian L D ≡ Re (cid:2) ¯ ψγ I D ψ (cid:3) ∧ ˜ e I = 12 (cid:2) ¯ ψγ I D ψ − D ¯ ψγ I ψ (cid:3) ∧ ˜ e I = 12 (cid:2) ¯ ψγ I d ψ − d ¯ ψγ I ψ (cid:3) ∧ ˜ e I + 14 ω IJ ∧ ˜ e K ¯ ψγ IJK ψ + A a ∧ ˜ e I ¯ ψT a γ I ψ . (16)We have used the exterior covariant derivative in the ap-propriate representationsD ψ ≡ (cid:18) d + 14 ω IJ γ IJ + A a T a (cid:19) ψ , D ¯ ψ ≡ D ψ , (17)the definition of complex conjugation on complex Grass-mann numbers (cid:0) ψ ασ ψ βτ (cid:1) ∗ = ψ βτ ∗ ψ ασ ∗ and (5). III. LAGRANGE - NOETHER FORMALISMA. The variational principle
We start by considering a generic field content, i.e. let φ a ∈ Ω p ( M ) ⊗ K , with a = 1 , . . . , N and K = R , C or G , be a set of N p -form fields. They are given with aLagrangian L = L ( φ, d φ ) ∈ Ω D ( M ) which is local, i.e.the values of L at p ∈ M only depend on the valuesof φ a and d φ a at that same point. We also ask that L be polynomial in φ a , d φ a except for the vielbein. Toour Lagrangian there corresponds an action functional S ≡ R M L and the variational principle goes as follows.We consider a field configuration φ a and an infinitesimalvariation φ a → φ a + δφ a over M . The variation of d φ a being determined by the one of φ a , i.e. δ d φ a = d δφ a , thevariation of the Lagrangian is given by δL = δφ a ∧ ∂L∂φ a + d δφ a ∧ ∂L∂ d φ a . (18)Here the operators ∂∂φ a and ∂∂ d φ a are defined as anti-derivations of degree − p and − ( p + 1), respectively, sat-isfying ∂∂φ a φ b = δ ba , ∂∂ d φ a d φ b = δ ba , (19)they have the same Grassmann degree as φ a and we con-ventionally apply them from the left. The above equa-tions and the fact that they are anti-derivations deter-mine them on all of Ω( M ). Now the variation of theaction being δS = Z M δL = Z M (cid:18) δφ a ∧ ∂L∂φ a + d δφ a ∧ ∂L∂ d φ a (cid:19) , (20)we integrate by parts the second term and find δS = Z M δφ a ∧ (cid:18) ∂L∂φ a − ( − p d ∂L∂ d φ a (cid:19) + Z ∂ M δφ a ∧ ∂L∂ d φ a . (21)To get rid of the boundary term we have to restrict tovariations such that δφ a | ∂ M = 0. The classical solutionsof the theory given by L are the field configurations whichmake δS vanish for all such δφ a and therefore obeyEL a ≡ ∂L∂φ a − ( − p d ∂L∂ d φ a = 0 . (22)These are the Euler-Lagrange equations in exterior cal-culus. Note that they imply that the ( D − p )-form J a ≡ ∂L∂φ a (23)is exact when evaluated on a classical solution and thusd J a = 0. If p = 0, then this is a trivial identity since J a ∈ Ω D ( M ), but if p >
0, then this is a genuine conser-vation equation, valid on classical solutions. It is there-fore not surprising that it will have a direct relation withthe Noether currents in the case where p = 1, as we willsee later on. For later reference, we will call these d -forms J a the “Euler-Lagrange”, or simply, “EL” currents.
1. Example
Let us consider the Palatini Lagrangian for gravityhere. The equation of motion of the vielbein isEL I ≡ ∂L∂e I + d ∂L∂ d e I = − G I + T I = 0 , (24)where G I ≡ − ∂L P ∂e I − d ∂L P ∂ d e I ( C ) = −
12 ˜ e IJK ∧ Ω JK , (25)is the Einstein tensor in first-order vielbein GR and T I ≡ ∂L m ∂e I + d ∂L m ∂ d e I ( C )( C )( C ) = 12 g ( i I F a ∧ ⋆ F a − F a ∧ i I ⋆ F a ) − Re (cid:2) ¯ ψγ J D ψ (cid:3) ∧ ˜ e JI , (26)is the standard definition of the matter energy-momentum tensor in GR, i.e. the variational derivative ofthe matter action with respect to the gravitational field.The equation of motion of the spin connection isEL IJ ≡ ∂L∂ω IJ + d ∂L∂ d ω IJ = 12 D˜ e IJ + 14 ¯ ψγ IJK ψ ˜ e K ( C ) = 12 ˜ e IJK ∧ Θ K + 14 ¯ ψγ IJK ψ ˜ e K = 0 , (27)where Θ I ≡ d e I + ω IJ ∧ e J are the torsion two-forms.The equation of motion of A a givesEL a ≡ ∂L∂A a + d ∂L∂ d A a ( C ) = − g (cid:20) ∂F b ∂A a ∧ ⋆ F b + d (cid:18) ∂F b ∂ d A a ∧ ⋆ F b (cid:19)(cid:21) +˜ e I ¯ ψT a γ I ψ = − g (cid:2) d ⋆ F a + f abc A b ∧ ⋆ F c (cid:3) + ˜ e I ¯ ψT a γ I ψ ≡ − g D ⋆ F a + ˜ e I ¯ ψT a γ I ψ = 0 . (28)Finally, the equation of motion of ¯ ψ ασ isEL ασ ≡ ∂L∂ ¯ ψ ασ − d ∂L∂ d ¯ ψ ασ (29) ( C )( C ) = (cid:20) γ I d ψ + 12 ˆ ω IJ γ J ψ + 14 ω JK γ IJK ψ + A a T a ψ (cid:21) ασ ∧ ˜ e I ( ) = (cid:20) γ I ˆD ψ + 14 K JK γ IJK ψ (cid:21) ασ ∧ ˜ e I = 0 , where we have used (5), ˆD is the covariant derivativeusing the Levi-Civita spin connection ˆ ω [ e ], i.e. ˆD e I = 0,and K IJ ≡ ω IJ − ˆ ω IJ are the contorsion 2-forms. Wehave thus retrieved the Palatini, Yang-Mills and Diracequations of motion from a variational principle withoutever having to introduce coordinates, only some simpleidentities of appendix C. Finally, we have three non-zeroform fields and the corresponding EL currents are J I ≡ ∂L∂e I = EL I , (30) J IJ ≡ ∂L∂ω IJ = ω [ I | K ∧ ˜ e K | J ] + 14 ¯ ψγ IJK ψ ˜ e K , (31) J a ≡ ∂L∂A a = − g f abc A b ∧ ⋆ F c + ˜ e I ¯ ψT a γ I ψ . (32)Note that the one of e I is zero on-shell since the La-grangian we chose does not depend on d e I .Finally, note that equation (27) is equivalent toΘ I = − ¯ ψγ IJK ψ e JK which implies ω IJ = ˆ ω IJ [ e ] + ¯ ψγ IJK ψ e K . We thus retrieve the well-known resultthat L P is classically equivalent to the Einstein-Hilberttheory only if there is no matter coupling to ω IJ . B. Noether’s theorem
1. Currents
A continuous symmetry is a transformation of thefields under a continuous group whose infinitesimal ver-sion makes the Lagrangian transform as δL = − d K , (33)for some d -form K . Equivalently, the transformation isa symmetry if the action is invariant up to a bound-ary term. Noether’s theorem states that to every suchsymmetry there corresponds a d -form J , the Noethercurrent, which is conserved when evaluated on classicalsolutions, i.e. we have an identity d J ∼
EL. In fieldtheory one usually works with the dual current 1-form j ∼ ⋆ J ∈ Ω ( M ), for which the above equation is ex-pressed in terms of the codifferential ¯d j ≡ ∇ µ j µ ∼ EL. Itis however the d -form J which is the natural coordinate-independent representation of a “current”, since the lat-ter must be integrated over a d -volume in order to givea charge. Even the conservation equation is simpler interms of J since d is an anti-derivation while ¯d is not.To determine J , and thus prove the theorem, we com-pute the infinitesimal variation of the Lagrangian but asinduced by the variation of the fields δL = δφ a ∧ ∂L∂φ a + d δφ a ∧ ∂L∂ d φ a (22)= δφ a ∧ EL a + ( − p δφ a ∧ d ∂L∂ d φ a + d δφ a ∧ ∂L∂ d φ a = δφ a ∧ EL a + d (cid:18) δφ a ∧ ∂L∂ d φ a (cid:19) . (34)Equating this with (33) and defining the d -form J ≡ δφ a ∧ ∂L∂ d φ a + K , (35)one gets the desired identity d J = − δφ a ∧ EL a . Eq. (35)is therefore the definition of the Noether current. By thePoincar´e lemma, if d J = 0 on-shell, then there existslocally a ( d − U , called the “superpotential”, suchthat J = d U . In the following examples we will see that,as a consequence of assuming the classical solutions to beglobal, this U form also exists globally.
2. Example
We consider the Noether currents associated with theSU( N ) and Spin(1 , d ) transformations. We will use ashorthand notation π a ≡ − g − ⋆ F a to simplify our ex-pressions. The variations are given in (11) and (12)and the Lagrangian is invariant so K = 0. Thus, thecurrents associated to the transformations generated by α = α a T a ∈ Ω ( M ) ⊗ su ( N ) and θ = θ IJ γ IJ ∈ Ω ( M ) ⊗ spin (1 , d ) are J [ α ] ≡ δA a ∧ ∂L∂ d A a + δψ ασ ∂L∂ d ψ ασ + δ ¯ ψ ασ ∂L∂ d ¯ ψ ασ = (cid:0) d α a + f abc A b α c (cid:1) ∧ π a + α a ˜ e I ¯ ψT a γ I ψ , (36)and S [ θ ] = δe I ∧ ∂L∂ d e I + δω IJ ∧ ∂L∂ d ω IJ (37)+ δψ ασ ∂L∂ d ψ ασ + δ ¯ ψ ασ ∂L∂ d ¯ ψ ασ = − (cid:0) d θ IJ − θ KI ω KJ (cid:1) ∧ ˜ e IJ + 14 θ IJ ˜ e K ¯ ψγ IJK ψ , respectively. The fact that these currents are conservedon-shell for any α and θ is the mark of the redundancyin the apparent number of degrees of freedom in a gaugetheory. We then consider the special cases J a ≡ J [ T a ] = f abc A b ∧ π c + ˜ e I ¯ ψT a γ I ψ = J a , (38) S IJ ≡ S (cid:2) T IJ (cid:3) = ω [ I | K ∧ ˜ e K | J ] + 14 ˜ e K ¯ ψγ IJK ψ = J IJ . As anticipated earlier, the EL currents J a ≡ ∂L∂A a and J IJ ≡ ∂L∂ω IJ are thus nothing but a special subset ofthe Noether currents associated with the groups A and ω gauge, respectively. Moreover, these currents are theones one obtains in the case where the gauge field is ab-sent and the symmetry is only global. Here we see thatby gauging the symmetry we obtain a whole lot of newconserved currents that are indexed by geometric objects,the fields α and θ in the adjoint representation of theirrespective groups.Now, all of these currents are not independent but canbe expressed in terms of J a and J IJ using (36) and (38) J [ α ] = α a J a − d α a ∧ π a , S [ θ ] = θ IJ J IJ −
12 d θ IJ ∧ ˜ e IJ (39) Using the equations of motion J a = − d π a and J IJ = − d˜ e IJ , we get that on classical solutions J [ α ] | EL=0 = − α a d π a − d α a ∧ π a = − d ( α a π a ) , (40) S [ θ ] | EL=0 = − (cid:0) θ IJ d˜ e IJ + d θ IJ ∧ ˜ e IJ (cid:1) = −
12 d (cid:0) θ IJ ˜ e IJ (cid:1) . So the superpotentials exist globally indeed. It is im-portant to understand the difference between J [ α ] and J a . The former is the Noether current associated to acovariant field α and is thus gauge-invariant, as can beseen in (36). On the other hand, the EL current J a cor-responds to the fixed choice α = T a and thus transformsinhomogeneously. Note finally that J [ α ] is R -linear inits argument J [ α + β ] = J [ α ] + J [ β ] , J [ c α ] = c J [ α ] , (41)for all c ∈ R . Of course, what has been said for J [ α ] inthis paragraph holds analogously for S [ θ ] as well.
3. Charges
Consider now a Noether current J evaluated on a givenfield configuration φ a which is not necessarily a classicalsolution. We can construct the corresponding Noethercharge Q contained in a region Σ ∈ ∆ d , that is, define amap Q : ∆ d → R Q (Σ) ≡ Z Σ J . (42)Let us now express the conservation law in terms of Q .We start by choosing a local evolution direction, thatis, a vector field ξ ∈ X ( M ). We then consider the one-parameter subgroup { Ξ t } t ∈ I ⊂ R of Diff ( M ) that it gen-erates, i.e. the one satisfying (7). We also considerthe continuous one-parameter family of submanifoldsΣ t ≡ Ξ t (Σ), the corresponding charges Q ( t ) ≡ Q (Σ t )and also define the “tube” W ≡ [ t ∈ I Σ t . (43)Thus, Q ( t ) is the charge contained in the Σ t hypersur-face and the latter evolves along the flow-lines of ξ . Thevariation of Q ( t ) with respect to t gives˙ Q ( t ) ≡ lim ǫ → ǫ [ Q (Σ t + ǫ ) − Q (Σ t )]= lim ǫ → ǫ "Z Ξ t + ǫ (Σ) J − Z Ξ t (Σ) J = Z Σ lim ǫ → ǫ (cid:2) Ξ ∗ t + ǫ J − Ξ ∗ t J (cid:3) ≡ Z Σ Ξ ∗ t L ξ J = Z Σ t L ξ J = Z Σ t ( i ξ d + d i ξ ) J = Z Σ t i ξ d J + Z ∂ Σ t i ξ J , (44)where Ξ ∗ t is the pullback with respect to Ξ t and we haveused the definition of the Lie derivative as the generatorof pullbacks. Focusing from now on on solutions of theequations of motion for φ a , the first term drops by currentconservation and we are left with˙ Q ( t ) (cid:12)(cid:12)(cid:12) EL=0 = Z ∂ Σ t i ξ J , (45)which is the Noether conservation law in terms of thecharge: the variation of the charge within Σ t is entirelydetermined by the current normal to ξ at the boundary.More precisely, the vector field ξ determines the shape ofthe boundary of W and i ξ J ∼ ⋆ (cid:0) ξ ♭ ∧ j (cid:1) . Therefore, thisexpression captures the part of j which is normal to ∂W .The standard use of this law is in the case where the Σ t are the space-like leaves of a foliation and ξ is the time-like coordinate-induced vector field ∂ t . It then reducesto the fact that the variation of the charge in time insidethe space-region Σ t is equal to the integrated currentflux through the boundary of W at the level t . Thistakes both into account the flow of the current out of thevolume and the fact that the volume itself may vary withtime. Here we see the full reach of Noether’s theoremsince it applies to any vector-field ξ and hypersurface Σ,whether the corresponding family Σ t is a foliation or not,and independently of its space-time interpretation.As shown in the example given above, in the case ofgauge symmetries the superpotential U is a local functionof the fields and their derivatives, and the charge of on-shell configurations can be written Q (Σ) | EL=0 = Z ∂ Σ U . (46)Note that it is crucial that U depends on the derivativeof the gauge field, since this makes Q (Σ) sensitive to thefields in an infinitesimally thick but still d -dimensional re-gion around ∂ Σ. Therefore, the total charge can only becomputed by starting with a compact Σ and then send-ing the boundary to infinity. If Q depended only on thefield values on ∂ Σ, then vanishing boundary conditionsat infinity would simply imply zero total charges.
IV. THE NOETHER CURRENTS OFDIFFEOMORPHISMS
Let us now consider the Noether current associated todiffeomorphisms. All fields transform infinitesimally as δφ a = −L ξ φ a = − i ξ d φ a − d i ξ φ a . (47)General covariance of the theory implies that the La-grangian is a tensor, and more specifically a D -form, sod L = 0 and thus δL = − d i ξ L , which is a total deriva-tive and therefore we have a symmetry. Following thederivation of Noether’s theorem, we have K = i ξ L and δφ a = −L ξ φ a , so the Noether current associated to thediffeomorphism in the ξ direction is P [ ξ ] = −L ξ φ a ∧ ∂L∂ d φ a + i ξ L . (48) This is a generalization of (minus) the covariant Hamilto-nian of ref[4] which corresponds to the case ξ = ∂ t . Justlike in the case of internal symmetries, we can computethe current corresponding to the basis elements of the Liealgebra X ( M ). If we expect to relate this object to theEL current of the vielbein, by analogy with the internalsymmetries, we must give it an I index, so we evaluateit on the ǫ I basis P I ≡ P [ ǫ I ] = −L I φ a ∧ ∂L∂ d φ a + i I L . (49)The problem with this expression is that, unlike in thecase of internal symmetries, the EL current of e I is not aNoether current P I = ∂L∂e I . This can be seen in full gen-erality by noting that the only non-trivial step in com-puting the latter is when ∂∂e I acts on ⋆ , whose solutionis given by (C13). This cannot produce a term ∼ d i I φ a which is present in (49) through L I φ a for forms of non-zero degree. Moreover, if we express P [ ξ ] in terms of the P I using (48) P [ ξ ] = ξ I P I − d ξ I ∧ i I φ a ∧ ∂L∂ d φ a , (50)and compare with (39), we see that all non-zero-formfields appear in the second term. Thus, we cannot yetdeduce the superpotential corresponding to P [ ξ ]. Thisissue is addressed in the following section. A. The anholonomic representation
The problem raised above is related to the fact that,because of the existence of a vielbein, an ambiguity arisesregarding precisely non-zero forms. Should one takethe p -forms φ a as the independent fields, or should onerather consider their components in the vielbein basis φ aI ...I p ≡ i I p ...I φ a ∈ Ω ? We will call the first case the“holonomic representation” while the second one will bethe “anholonomic representation”, since the correspond-ing vector bases ∂ µ and ǫ I have trivial and non-trivialLie brackets, respectively. Using φ a to denote all fields but the vielbein from now on, we have the two theories L hol ≡ L hol (cid:2) φ a , e I (cid:3) , L an ≡ L an h φ aI ...I p , e I i . (51)The holonomic representation is well suited to gauge the-ory since the EL currents and holonomies make use ofthe gauge field 1-forms, not the Lorentz-indexed 0-forms.Moreover, the gauge transformations of the gauge fieldsin terms of the 0-forms make use of the inverse vielbein,so they are less natural. As we will show now, the anholo-nomic representation, however, is well suited for gravitybecause then the vielbein formally behaves as the gaugefield associated to diffeomorphisms, i.e. in total analogywith the properties of the gauge fields we have seen untilnow.The first thing to show of course, is that both represen-tations are classically equivalent, i.e. that their equationsof motion imply one another. This is a priori obvioussince the two choices of independent fields are related bya non-linear but bijective field redefinition, φ a = 1 p ! φ aI ...I p e I ...I p , φ aI ...I p = i I p ...I φ a . (52)We show the equivalence explicitly in appendix B be-cause, in the process, we see that one can choose a mixedrepresentation for the equations of motion. We can com-pute them in the anholonomic one for the vielbein and inthe holonomic one for the rest of the fields. We also showthat the Noether currents are the same, even thoughsome fields have changed Spin(1 , d ) and Diff ( M ) rep-resentations. So let us consider the L an theory, and fornotational simplicity let us absorb the I indices in thegeneric internal index a , i.e. let us write φ ˜ a ≡ φ aI ...I p and keep in mind that now φ ˜ a ∈ Ω ( M ). The key prop-erty of the anholonomic representation is that the Liederivative with respect to ǫ I becomes simply L I | an = i I d , (53)on all fundamental fields, since i I φ ˜ a = 0 and i I e J = δ JI .Considering the ǫ I basis for X ( M ) means that we take L I = i I d as our generators in the active representation ofDiff ( M ). These are nothing but the ǫ I vectors seen asderivations. Then, under an infinitesimal diffeomorphismthe 0-forms transform homogeneously δ ξ φ ˜ a = −L ξ φ ˜ a = − ξ I L I φ ˜ a , (54)while the vielbein is the only field transforming “inhomo-geneously” δ ξ e I = −L ξ e I = − d ξ I − ξ J L J e I . (55)Note the analogy with the transformation of A a in (11)and ω IJ in (12). We have the same inhomogenous part,while the homogeneous one is the Lie derivative in differ-ent respective senses. Here it is the Lie derivative withrespect to the base manifold M , while for the internaltransformations it is the Lie derivative with respect to theSU( N ) and Spin(1 , d ) fibres . Thus, the vielbein formallytransforms as the gauge field associated to Diff ( M ).The Noether currents are the same but are now com-puted using L an , so (49) now reads P I = − i I d e J ∧ ∂L an ∂ d e J − i I d φ ˜ a ∧ ∂L an ∂ d φ ˜ a + i I L an , (57) Indeed, in those cases one can write the transformations in termsof the algebra-valued fields A ≡ A a T a ∈ Ω ( M ) ⊗ su ( N ) and ω ≡ ω IJ T IJ ∈ Ω ( M ) ⊗ spin (1 , d ) where it reads δA = − D α = − d α − [ A, α ] , δω = − D θ = − d θ − [ ω, θ ] , (56)and the commutator with α, θ is nothing but the Lie derivativein the α, θ directions, seen as left-invariant vector fields, on theirrespective group manifolds. and the general Noether current associated to the diffeo-morphism in the ξ direction (50) is P [ ξ ] = ξ I P I − d ξ I ∧ ∂L an ∂ d e I . (58)This result is again analogous to the case of the internalsymmetries, see (39).The antiderivations i I and ∂∂e I now have a lot in com-mon. Since they are equal on vielbein products, andnow all fields have been decomposed into linear combi-nations of vielbein products, they are also equal whenacting on any combination of the fields which does notcontain derivatives, like the potential part of L an for in-stance. For forms containing derivatives however they dodiffer since ∂∂e I d φ ˜ a = 0 = i I d φ ˜ a . (59)Thus, acting with i I on L an we get ∂L an ∂e I plus terms ∼ i I d e J and ∼ i I d φ ˜ a . Since L an is polynomial in d e I and d φ ˜ a , the extra term which is not captured by ∂∂e I issimply i I L an − ∂L an ∂e I = i I d e J ∧ ∂L an ∂ d e J + i I d φ ˜ a ∧ ∂L an ∂ d φ ˜ a . (60)Now, isolating ∂L an ∂e I and using (57), we get P I = ∂L an ∂e I , (61)which is again analogous to the case of internal sym-metries: the EL current of the vielbein J an I ≡ ∂L an ∂e I isnothing but a special case of the Noether currents asso-ciated to the group e I gauges. The only peculiarity isthat one has to go to the anholonomic representation forthis to hold. We finish the comparison with gauge theoryby taking (50), using (61) and evaluating everything onclassical solutions J an I = − d ∂L an ∂ d e I to get the analogue of(41) P [ ξ ] | EL=0 = − ξ I d ∂L an ∂ d e I − d ξ I ∧ ∂L an ∂ d e I = − d (cid:18) ξ I ∂L an ∂ d e I (cid:19) . (62)Thus, the P [ ξ ] too are globally exact when on-shelland we can now compute the superpotential straightfor-wardly.
1. Example
We illustrate the equality P I = ∂L an ∂e I with the YMLagrangian. Computed for example in the holonomicrepresentation P YM I ( ) ≡ −L I A a ∧ ∂L holYM ∂ d A a + i I L holYM( C ) = 1 g (cid:20) L I A a ∧ ∂F b ∂ d A a ∧ ⋆ F b − i I ( F a ∧ ⋆ F a ) (cid:21) = 1 g (cid:20) i I F a ∧ ⋆ F a − F a ∧ i I ⋆ F a + (cid:0) d A aI + f abc A b A cI (cid:1) ∧ ⋆ F a (cid:3) , (63)where A aI ≡ i I A a . To show the equality, we first needthe curvature F a in terms of A aI F a [ A I , e I ] = d A aI ∧ e I + A aI d e I + 12 f abc A bI A cJ e IJ (64)and then we get ∂L sclYM ∂e I ( C ) = − g ∂F a [ A I , e I ] ∂e I ∧ ⋆ F a − g F a ∧ (cid:18) ∂∂e I ⋆ (cid:19) F a (65) ( C ) = 1 g (cid:2)(cid:0) d A aI + f abc A b A cI (cid:1) ∧ ⋆ F a + 12 i I F a ∧ ⋆ F a − F a ∧ i I ⋆ F a (cid:21) = P YM I . V. ENERGY-MOMENTUMA. Relation to the GR matter energy-momentumtensor
Let us compute the general relation between the twodefinitions of the matter energy-momentum tensor thatare the Noether current one P m I = ∂L anm ∂e I (66)and the one of GR, for which it must be noted that it iscomputed in the holonomic representation T I ≡ δS holm δe I . (67)We start with the latter and perform a computation anal-ogous to (B4) for the matter sector only T I = δS holm δe I = δS anm δe I − i I φ a ∧ EL hol a = ∂L anm ∂e I + d ∂L anm ∂ d e I − i I φ a ∧ EL hol a = P m I + d ∂L anm ∂ d e I − i I φ a ∧ EL hol a , (68)i.e. here the φ a are all fields but the vielbein and thespin connection, in the standard representation. Thus,the two energy-momentum tensors are related by a term which is a total derivative when the matter fields are on-shell. Note that the relation is fully general, i.e. it appliesto any matter action.The advantage of the GR definition is that it is fullycovariant, as a consequence of being the variation of afully invariant action. This is not the case of the Noethercurrents as we show here using the Palatini Lagrangianfor gravity in the holonomic representation P [ ξ ] = −L ξ e I ∧ ∂L∂ d e I − L ξ ω IJ ∧ ∂L∂ d ω IJ −L ξ A a ∧ ∂L∂ d A a − L ξ ψ ασ ∧ ∂L∂ d ψ ασ −L ξ ¯ ψ ασ ∧ ∂L∂ d ¯ ψ ασ + i ξ L ( )( ) = ( T I − G I ) ξ I + J [ i ξ A ] + S [ i ξ ω ] . (69)The first two terms are the matter energy momentumtensor as defined in GR and the Einstein tensor (25),in the combination which vanishes on shell. The restcan be expressed as the Noether currents of the internalsymmetries evaluated on the ξ -projection of the corre-sponding gauge fields. Therefore, P [ ξ ] is invariant onlyunder internal gauge transformations whose parametersobey L ξ α a = 0 and L ξ θ IJ = 0. Considering more gen-eral transformations changes P [ ξ ], the most extreme ex-amples being the transformations that bring the fields tothe generalized Weyl gauge i ξ ω IJ = 0 and i ξ A a = 0, inwhich case P [ ξ ] = 0 when on-shell since G I = T I . B. Gravitational energy-momentum complexes
We now show how one can obtain some known energy-momentum complexes for gravity out of the P [ ξ ] cur-rents. For this task the most appropriate Lagrangian isthe Møller one (14) because it is quadratic in d e I . Sincethe latter is already in the scalar representation, we get P I = ∂L M ∂e I ( C )( C ) = − d e J ∧ ⋆ F JI (70)+ 12 F JK ∧ i I ⋆ F KJ + 12 i I F JK ∧ ⋆ F KJ + 12 (cid:20) d e I ∧ ⋆ F − F ∧ i I ⋆ F − i I F ∧ ⋆ F (cid:21) . the corresponding superpotential is U I ≡ − ∂L M ∂ d e I ( C ) = e J ∧ ⋆ F JI − e I ∧ ⋆ F , and the equation of motion reads P I = d U I . An interest-ing property of this current is that it is traceless in fourdimensions e I ∧ P I ( C ) = ( D − L M , (71)in complete analogy with YM theory if one uses the GRenergy-momentum tensor e I ∧ T YM I = ( D − L YM . (72)To make contact with known objects, we express P I and U I in terms of the Levi-Civita connection ˆ ω IJ by usingd e I = − ˆ ω IJ ∧ e J and (C3), (C5) twice P I = 12 (cid:2) ˆ ω JK ∧ ˆ ω LI ∧ ˜ e KJ L − ˆ ω KJ ∧ ˆ ω JL ∧ ˜ e LK I (cid:3) , U I = −
12 ˆ ω JK ∧ ˜ e JKI . (73)These are known as “Sparling’s d -form” and the “Nester-Witten ( d − E ≡ R Σ P and P i ≡ R Σ P i ,for a space-like Σ ∈ ∆ d . An important property of thisenergy-momentum complex, first considered in [2], is thefact that for small enough Σ we have the desired property E ≥ | P i | ≥
0, thanks to a direct relation of P I to theBel-Robinson tensor . To get the general current P [ ξ ],we proceed as in (58) P [ ξ ] = ξ I P I + d ξ I ∧ U I . (74)So the general superpotential is U [ ξ ] ≡ ξ I U I , i.e. P [ ξ ] =d U [ ξ ] when on-shell. The complex considered above cor-responds to the choice ξ = ǫ I . Another famous energy-momentum complex, the one of Møller [1], correspondsto taking a holonomic ξ . Let us therefore define somelocal coordinates x µ , so that the vielbein decomposes e I = e Iµ d x µ , ǫ I = e µI ∂ µ . Then, considering the case ξ = ∂ µ translates into ξ I = e Iµ and P µ ≡ P [ ∂ µ ] = e Iµ P I + d e Iµ ∧ U I . (75)The second term shows that, although U [ ∂ µ ] is a tensorunder diffeomorphisms, this is not the case for P µ . Tomake contact with the usual notation in the literature weuse the dual contravariant density representation of thecurrent/superpotential couple j µ ≡ − eg µν ( ⋆ J ) ν , U µν ≡ − eg µρ g νσ ( ⋆ U ) ρσ , (76)where e ≡ det( e Iµ ). In terms of these the relation becomessimply j µ | EL=0 [ ξ ] = − eg µν ( ⋆ d U [ ξ ]) ν = ∇ ρ U µρ [ ξ ] = ∂ ρ U µρ [ ξ ] , (77)and the conservation equation also makes use of partialderivatives ∂ µ j µ = 0. Note that here ∇ is the Levi-Civita connection, i.e. the one made out of the Christoffel This is shown in [5], but the authors erroneously call this com-plex the “Møller complex”, since this is not the one proposed byMøller in ’61, which is a pseudo-tensor under diffeomorphisms. symbols. This representation is particularly useful in thecontext of energy-momentum pseudo-tensors where onehas a non-trivial dependence on coordinates anyway. Inour case, we have U µν [ ξ ] ≡ − eg µρ g νσ ( ⋆ U [ ξ ]) ρσ , (78)and for ξ = ∂ µ we obtain Møller’s superpotential U νρµ ≡ U νρ [ ∂ µ ] ( C )( C )( C ) = e (cid:0) δ νµ ˆ ω ρ − δ ρµ ˆ ω ν − ˆ ω νρµ (cid:1) , (79)where ˆ ω µνρ ≡ e Jν e Kρ ˆ ω µJK = − e Jν e Kρ ( ∇ µ e τJ ) e τK = − e Jν ∇ µ e ρJ = e Jρ ∇ µ e νJ , (80) ω µ ≡ g νρ ω νρµ and we have used the alternative definitionof the Levi-Civita spin connection ∇ ǫ I ǫ J = − ˆ ω KIJ ǫ K .The so-called “Møller complex” is then given by M νµ ≡ ∂ ρ U νρµ but now this is not equal to ∇ ρ U νρµ because ofthe extra µ index. We therefore retrieve in this repre-sentation as well the fact that M νµ is a pseudo-tensordensity. However, since the superpotential is a tensor,the definition of energy in the Møller case E ≡ − R Σ P t = − R ∂ Σ U [ ∂ t ], where g tt ≡ g ( ∂ t , ∂ t ) <
0, is invariant underspatial diffeomorphisms.
C. Relation to the Hamiltonian energy
The notion of energy is not only present in the diffeo-morphism Noether charges, but also in the Hamiltonianformalism, and in many cases the two definitions agree.Here we show their relation and its limitations. So let usfoliate M with a time coordinate t and use (48) to writethe action in canonical form [4] S ≡ Z M L = Z M d t ∧ i ∂ t L (81)= Z M d t ∧ (cid:20) L ∂ t φ a ∧ ∂L∂ d φ a + P [ ∂ t ] (cid:21) ≡ Z M h ˙ φ a ∧ π a − d t ∧ H i , (82)where we have identified the conjugate momenta ( D − p )-forms and the Hamiltonian d -form π a ≡ ( − d ∂L∂ d φ a ∧ d t , H ≡ −P [ ∂ t ] . (83)Since H is an exact form on-shell, we retrieve what isknown from the ADM formalism, namely, that the bulkpart of the Hamiltonian is zero on-shell, and the bound-ary term is thus the superpotential integrated over ∂ Σ t .Improving P [ ∂ t ] by adding a total derivative on-shell thenamounts to changing that boundary term. In their sem-inal paper [6], the authors used this relation with thecanonical formalism to show that a large class of super-potential complexes actually originates in such Hamil-tonian boundary terms, these in turn being determined0by the boundary conditions one wishes to impose. Thisactually alleviated the discomfort in considering pseudo-tensors, because their non-covariant behaviour under dif-feomorphisms was ultimately related to the breaking ofthe symmetry by the choice of boundary conditions.In our context, it is important to note that this appliesto complexes that are defined using holonomic indices,i.e. energy corresponds to a µ = t component, such asin the case of the Møller complex. More precisely, the t parameter which is singled-out in going to the Hamil-tonian formalism is the same as the parameter of thesymmetry corresponding to P t . Of course, for arbitrarytime-directions ξ one can always choose the foliation suchthat ξ = ∂ t . But if ξ is field dependent ξ = ξ [ φ ], then thecanonical term “ ˙ φ ∧ π ” is polluted by φ and we are outof the canonical formalism. For instance, if ξ = ǫ S = Z M e ∧ i L = Z M e ∧ (cid:18) L φ a ∧ ∂L∂ d φ a [ π ] − H ′ (cid:19) , (84)where H ′ ≡ −P [ ǫ ] ≡ −P , then this is not an action incanonical form since the equations of motion of φ and π have non-trivial ∂e I dependencies. Alternatively, H ′ is atensor under the full diffeomorphism group.So, as could be expected, the tensor P I and the as-sociated charges are intrinsically Lagrangian quantitiessince they refuse to make the minimal compromise of thecoordinate-dependence the canonical formalism requires.We therefore conclude that the Noether currents allowfor more general energy definitions than the Hamiltonianformalism since one has also access to field-dependenttime-translation generators. VI. SUMMARY
In this paper we have shown that for generic fieldmanipulations, including differentiation with respect tofields, one gains in simplicity and efficiency by using ex-terior calculus. As a first application, we have used thisformalism to show the full generality of Noether’s theo-rem, both in terms of currents and charges. We have alsoused it to show the utility of the “anholonomic represen-tation” in which vielbein gravity is seen to share manyformal properties with standard gauge theories. In par-ticular, we have shown that the partial derivative of theLagrangian with respect to the vielbein yields a class ofNoether currents, just as is the case of Yang-Mills the-ory. We have shown how some diffeomorphism Noethercurrents give known energy-momentum complexes of thevielbein. Finally, we have discussed the relation betweenthe Noether energy and other definitions of that notionthat are the matter energy-momentum tensor in GR andHamiltonian boundary terms in the canonical formalism.
ACKNOWLEDGMENTS
I would like to thank Michele Maggiore, Stefano Foffa,Yves Dirian and Xiao Xiao for useful discussions. Thiswork is supported by the Swiss National Science Foun-dation. [1] C. Møller, Annals Phys. (1961) 118.[2] M. Dubois-Violette and J. Madore, Commun. Math. Phys. (1987) 213.[3] L. B. Szabados, Class. Quant. Grav. (1992) 2521.[4] J. M. Nester, Mod. Phys. Lett. A (1991) 2655.[5] L. L. So and J. M. Nester, gr-qc/0612061.[6] C. C. Chang, J. M. Nester and C. -M. Chen, Phys. Rev.Lett. (1999) 1897 [gr-qc/9809040]. Appendix A: The Møller Lagrangian with forms
Here we show the relation between the Einstein-HilbertLagrangian L EH = 12 ˆΩ IJ [ e ] ∧ ˜ e IJ , ˆΩ IJ ≡ dˆ ω IJ + ˆ ω KI ∧ ˆ ω KJ , (A1)and the Møller Lagrangian in the form (14). We firstintegrate by parts L EH = 12 (cid:2) dˆ ω IJ ∧ ˜ e IJ + ˆ ω KI ∧ ˆ ω KJ ∧ ˜ e IJ (cid:3) = L M + d (cid:0) ˆ ω IJ ∧ ˜ e IJ (cid:1) . (A2) where L M ≡ (cid:2) ˆ ω IJ ∧ d˜ e IJ + ˆ ω KI ∧ ˆ ω KJ ∧ ˜ e IJ (cid:3) . (A3)The first thing to notice is that the two terms are propor-tional to each other. Indeed, by definition of the Levi-Civita spin connection, we have d e I = − ˆ ω IJ ∧ e J soˆ ω IJ ∧ d˜ e IJ ( C ) = ˆ ω IJ ∧ ˜ e IJK ∧ d e K = − ˆ ω IJ ∧ ˆ ω KL ∧ e L ∧ ˜ e IJK ( C ) = − ω KI ∧ ˆ ω KJ ∧ ˜ e IJ , and therefore L M = − ˆ ω KI ∧ ˆ ω KJ ∧ ˜ e IJ . We then ex-press this in terms of the components in the vielbein basis ω IJK ≡ i I ω JK and ω I ≡ ω JJI L M = −
12 ˆ ω KAI ˆ ω BKJ e AB ∧ ˜ e IJ ( C ) = −
12 ˆ ω KAI ˆ ω BKJ (cid:0) η AI η BJ − η AJ η BI (cid:1) ˜ e = 12 (cid:0) ˆ ω IJK ˆ ω JKI + ˆ ω I ˆ ω I (cid:1) ˜ e , (A4)1which, given (80), is the original form of Møller’s La-grangian [1]. Now, using again d e I = − ˆ ω IJ ∧ e J and(C4) and (C5), we get e I ∧ d e J ∧ ⋆ (cid:0) e J ∧ d e I (cid:1) = ˆ ω AJK ˆ ω BIL e AIK ∧ ˜ e BJL = (cid:0) ˆ ω IJK ˆ ω IJK + ˆ ω IJK ˆ ω JKI − ˆ ω I ˆ ω I (cid:1) ˜ e ,e I ∧ d e I ∧ ⋆ (cid:0) e J ∧ d e J (cid:1) = ˆ ω AIJ ˆ ω BKL e AIJ ∧ ˜ e BKL = 2 (cid:0) ˆ ω IJK ˆ ω IJK +2 ˆ ω IJK ˆ ω JKI (cid:1) ˜ e , (A5)so, defining the 3-forms F IJ ≡ e I ∧ d e J and F ≡ F II ,one can write it as (14). Appendix B: Equivalence of the holonomic andanholonomic representations
Here we use φ a to denote collectively all fields but thevielbein. Using φ a = 1 p ! φ aI ...I p e I ...I p , (B1)d φ a = 1 p ! h d φ aI ...I p ∧ e I ...I p + φ aI ...I p d e I ...I p i , to translate from one representation to the other we get ∂L an ∂φ aI ...I p = ∂φ b ∂φ aI ...I p ∧ ∂L hol ∂φ b + ∂ d φ b ∂φ aI ...I p ∧ ∂L hol ∂ d φ b = 1 p ! (cid:20) e I ...I p ∧ ∂L hol ∂φ a + d e I ...I p ∧ ∂L hol ∂ d φ a (cid:21) ,∂L an ∂ d φ aI ...I p = ∂φ b ∂ d φ aI ...I p ∧ ∂L hol ∂φ b + ∂ d φ b ∂ d φ aI ...I p ∧ ∂L hol ∂ d φ b = 1 p ! e I ...I p ∧ ∂L hol ∂ d φ a , (B2)so the equations of motion for φ aI ...I p giveEL an ,I ...I p a ≡ ∂L an ∂φ aI ...I p − d ∂L an ∂ d φ aI ...I p = 1 p ! (cid:20) e I ...I p ∧ ∂L hol ∂φ a + d e I ...I p ∧ ∂L hol ∂ d φ a − d (cid:18) e I ...I p ∧ ∂L hol ∂ d φ a (cid:19)(cid:21) = 1 p ! e I ...I p ∧ (cid:18) ∂L hol ∂φ a − ( − p d ∂L hol ∂ d φ a (cid:19) ≡ p ! e I ...I p ∧ EL hol a . (B3)Thus EL a = 0 implies EL I ...I p a = 0. Since this holds forall I . . . I p and the vielbein is a basis, we also have that the converse is true. With the same type of manipula-tions for the vielbein we getEL an I − EL hol I = ∂φ a ∂e I ∧ ∂L hol ∂φ a + ∂ d φ a ∂e I ∧ ∂L hol ∂ d φ a +d (cid:18) ∂ d φ a ∂ d e I ∧ ∂L hol ∂ d φ a (cid:19) = i I φ a ∧ (cid:18) ∂L hol ∂φ a − ( − p d ∂L hol ∂ d φ a (cid:19) ≡ i I φ a ∧ EL hol a (B4)so they are the same when the rest of the fields are on-shell. We have thus shown that both representations areclassically equivalent. Most importantly for us however,the equivalence in the matter sector is independent of thechoice of representation for the vielbein. Thus, one cantake the equations of motion in the holonomic represen-tation for φ a and in the anholonomic one for e I while stilldescribing the same classical physics.Finally, we also show that the Noether currents are thesame even though some fields have changed representa-tion with respect to LLTs and diffeomorphisms. We havethat the infinitesimal variations decompose δφ a = 1 p ! δ (cid:16) φ aI ...I p ∧ e I ...I p (cid:17) = 1 p ! (cid:16) δφ aI ...I p ∧ e I ...I p + φ aI ...I p ∧ δe I ...I p (cid:17) = 1 p ! δφ aI ...I p ∧ e I ...I p + 1( p − φ aI ...I p ∧ δe I ∧ e I ...I p (B5)so that the Noether currents of the anholonomic repre-sentation read J an ≡ δe I ∧ ∂L an ∂ d e I + δφ aI ...I p ∧ ∂L an ∂ d φ aI ...I p + K ( B ) = δe I ∧ ∂ d e J ∂ d e I ∧ ∂L hol ∂ d e J + δe I ∧ ∂ d φ a ∂ d e I ∧ ∂L hol ∂ d φ a + 1 p ! δφ aI ...I p e I ...I p ∧ ∂L hol ∂ d φ a + K (B6)= δe I ∧ ∂L hol ∂ d e I + 1 p ! δφ aI ...I p e I ...I p ∧ ∂L hol ∂ d φ a + 1( p − φ aII ...I p δe I ∧ e I ...I p ∧ ∂L hol ∂ d φ a + K ( B ) = δe I ∧ ∂L hol ∂ d e I + δφ a ∧ ∂L hol ∂ d φ a + K ≡ J hol . Appendix C: Useful identities
1. Vielbein contraction i I k ...I e J ...J l = l !( l − k )! δ [ J I . . . δ J k I k e J k +1 ...J l ] ,i I ...I k ˜ e J ...J l = ˜ e J l ...J I k ...I . (C1)22. Vielbein exterior derivatived e I ...I k = k d e [ I ∧ e I ...I k ] , d˜ e I ...I k = ˜ e I ...I k I k +1 ∧ d e I k +1 . (C2)3. k -dimensional partial decomposition of α ∈ Ω p with k ≤ pe I ...I k ∧ i I k ...I α = p !( p − k )! α . (C3)4. Hodge dual and dual vielbein˜ e I ...I k = ⋆ e I ...I k . (C4)5. Vielbein - dual vielbein product e I ...I k ∧ ˜ e J ...J l = ( − k ( l − k ) l !( l − k )! × δ I [ J . . . δ I k J k ˜ e J k +1 ...J l ] . (C5)6. Square of the Hodge dual. Let α ∈ Ω p , then ⋆ α = − ( − p ( D − p ) α . (C6)7. Integration by parts of i ξ . For all α ∈ Ω p and β ∈ Ω D − p +1 , i ξ α ∧ β = ( − p α ∧ i ξ β . (C7)8. The product α ∧ ⋆ β is symmetric if α, β ∈ Ω p α ∧ ⋆ β = β ∧ ⋆ α . (C8)9. The interior product i ξ is the dual operation of thewedge multiplication by ξ ♭ ≡ g ( ξ, · ) ∈ Ω i ξ ⋆ α = ⋆ (cid:16) α ∧ ξ ♭ (cid:17) , ⋆ i ξ α = ( − d ⋆ α ∧ ξ ♭ . (C9)10. The previous identity implies the following gener-alizations i I k ...I ⋆ α = ⋆ ( α ∧ e I ...I k ) ,⋆ i I k ...I α = ( − kd ( ⋆ α ) ∧ e I ...I k , (C10)11. On the other hand ⋆ ( α ∧ ˜ e I ...I k ) = − ( − ( k − p )( D − k ) (cid:18) kp (cid:19) × e [ I p +1 ...I k i I p ...I ] α . (C11) 12. For all α ∈ Ω p (cid:18) δe I ∧ ∂∂e I i I k ...I (cid:19) α = − k i [ I k δe I i I | I k − ...I ] α . (C12)13. For all α, β ∈ Ω p α ∧ (cid:18) ∂∂e I ⋆ (cid:19) β = α ∧ i I ⋆ β − ( − p i I β ∧ ⋆ α . (C13) Proof . δe I ∧ α ∧ (cid:18) ∂∂e I ⋆ (cid:19) β = ( − p α ∧ (cid:18) δe I ∧ ∂∂e I ⋆ (cid:19) β ( ) = ( − p p ! (cid:20) α ∧ (cid:18) δe I ∧ ∂∂e I ˜ e I ...I p (cid:19) i I p ...I β + α ∧ ˜ e I ...I p (cid:18) δe I ∧ ∂∂e I i I p ...I (cid:19) β (cid:21) ( C ) = ( − p p ! (cid:2) α ∧ δe I ∧ i I ˜ e I ...I p i I p ...I β − p α ∧ ˜ e I ...I p ∧ i I p δe I ∧ i II p − ...I β (cid:3) ( C ) = ( − p (cid:2) ( − p δe I ∧ α ∧ i I ( ⋆β )+( − D p − i I p α ∧ ˜ e I ...I p ∧ δe I ∧ i II p − ...I β (cid:21) = δe I ∧ [ α ∧ i I ⋆ β − ( − p p − i I p (cid:0) α ∧ ˜ e I ...I p (cid:1) ∧ i II p − ...I β (cid:21) . We compute the second term separately i I p (cid:0) α ∧ ˜ e I ...I p (cid:1) ∧ i II p − ...I β = i I p (cid:0) α ∧ ˜ e I p I ...I p − (cid:1) ∧ i I p − ...I I β ( C ) = i I p (cid:0) α ∧ ⋆ e I p I ...I p − (cid:1) ∧ i I p − ...I I β ( C ) = i I p (cid:0) e I p I ...I p − ∧ ⋆ α (cid:1) ∧ i I p − ...I I β = (cid:0) i I p e I p I ...I p − ∧ ⋆ α + ( − p e I p I ...I p − ∧ i I p ⋆ α (cid:1) ∧ i I p − ...I I β ( C ) = ( D − p + 1) e I ...I p − ∧ ⋆ α ∧ i I p − ...I I β +( − p e I p I ...I p − ∧ i I p ⋆ α ∧ i I p − ...I I β = ( − ( p − D − p ) (cid:2) ( D − p + 1) ⋆ α ∧ e I ...I p − ∧ i I p − ...I I β − e I p ∧ i I p ⋆ α ∧ e I ...I p − ∧ i I p − ...I I β (cid:3) ( C ) = ( − ( p − D − p ) ( p − D − p + 1) ⋆ α ∧ i I β − ( D − p ) ⋆ α ∧ i I β ]= ( − ( p − D − p ) ( p − ⋆ α ∧ i I β = ( p − i I β ∧ ⋆ α .⋆ α .