On A Covariant Hamiltonian Description of Palatini's Gravity on Manifolds with Boundary
OON A COVARIANT HAMILTONIAN DESCRIPTION OFPALATINI’S GRAVITY ON MANIFOLDS WITH BOUNDARY
A. IBORT AND A. SPIVAK
Abstract.
A covariant Hamiltonian description of Palatini’s gravity on mani-folds with boundary is presented. Palatini’s gravity appears as a gauge theorysatisfying a constraint in a certain topological limit. This approach allows theconsideration of non-trivial topological situations.The multisymplectic framework for first-order covariant Hamiltonian field the-ories on manifolds with boundary, developed in [Ib15], enables analysis of thesystem at the boundary. The reduced phase space of the system is determinedto be a symplectic manifold with a distinguished isotropic submanifold corre-sponding to the boundary data of the solutions of the Euler-Lagrange equations.
Contents
1. Introduction 22. The geometry of the covariant phase space for Yang-Mills theories 32.1. A brief account of the multisymplectic formalism for first ordercovariant Hamiltonian Yang-Mills theories on manifolds withboundary 32.2. The fundamental formula 113. The presymplectic formalism at the boundary 113.1. The evolution picture near the boundary 113.2. The presymplectic picture at the boundary and constraints analysis 133.3. The limit λ → a r X i v : . [ m a t h - ph ] M a y A. IBORT AND A. SPIVAK Introduction
Our understanding of the Hamiltonian structure of Gravity has taken half acentury. The initial difficulties faced by Dirac and Bergmann [Be58],[Be81], wereslowly resolved through the work of Arnowit, Deser and Misner [Ar62], all theway to Ashtekar’s formulation [As87]. At least part of the motivation has been toplace the theory of gravity on grounds that will make it suitable for a canonicalquantization scheme.In [Ro06], C. Rovelli illustrated a simple Hamiltonian formulation of GeneralRelativity which is manifestly 4D generally covariant and that drops the referenceto the underlying space-time in Palatini’s formulation of gravity. Rovelli’s proposalis highly geometrical and constructs its space as the 4 + 16 + 24 dimensionalspace (cid:101) C with local coordinates ( x µ , e Iµ , A IJµ ). In a further effort at extracting thegeometrical essence of such space, the variables x µ are dropped (accounting bythe invariance of the theory under global diffeomorphisms) and we are led to a 40dimensional space C [Ro01]. The disappearance of the spacetime manifold M andits coordinates x µ , which survive only as arbitrary parameters on the ‘gauge orbits’of the canonical geometrical structure defined on it, generalizes the disappearanceof the time coordinate in the ADM formalism and is analogous to the disappearanceof the Lagrangian evolution parameter in the Hamiltonian theory of a free particle[Ro01]. It simply means that the general relativistic space- time coordinates arenot directly related to observations.Our program in this paper is similar but our inspiration is the geometricalfoundations of covariant first order Hamiltonian field theories on manifolds withboundary discussed recently in [Ib15]. There the role of a covariant phase spacefor a first order Hamiltonian theory modelled on the affine dual space of the firstjet bundle of the bundle defining the fields of the theory is assessed and the crucialrole played by boundaries as determining symplectic spaces of fields defining theclassical counterpart of the quantum states of the theory is stressed in accordancewith the point of view expressed in [Sc51].Actually a generally covariant notion of instantaneous state, or evolution ofstates and observables in time, make little physical sense. They are always referredto an initial data space-like surface that in the picture presented here, correspondsto the boundary of the space-times of events. Such notion does not really conflictswith diffeomorphism invariance because a diffeomorphism of a smooth manifoldwith smooth boundary restricts to a diffeomorphism of the boundary. Thus, pro-viding that the notion of boundary of a spacetime is incorporated in the basicdescription of the theory, we may still consider diffeomorphism invariance as afundamental notion without contradicting it.The covariant phase space of the theory carries a natural multisymplectic struc-ture which is the exterior differential of a canonical m -form Θ defined on it. This AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 3 geometrical structure has been considered in various guises in the various varia-tional formulations of field theories, however its first use in the present setting isto help to identify the nature of the different fields of the theory. Thus it will bediscussed how the vierbein fields e Iµ correspond to an algebraic constraint imposedin the momenta fields of the theory. The corresponding action will be seen tobe invariant under the group of all automorphisms of the geometrical structureand it will induce the corresponding reduction on the space of gauge fields at theboundary. This reduction process is interpreted as the appropriate setting for the‘elimination’ of the space-time M , i.e., the space of physical classical solutions ofthe theory in the bulk is the moduli space of the space of solutions of the Euler-Lagrange equations with respect to the group of automorphisms whereas, the phasespace of physical degrees of freedom of the theory, associated to its boundary, isthe reduced symplectic manifold of fields at the boundary.We can give C a direct physical interpretation in terms of reference systemstransformations. In the quantum domain, it leads directly to the spin-network tospin-network amplitudes computed in loop quantum gravity.2. The geometry of the covariant phase space for Yang-Millstheories
As discussed in the introduction our approach to Palatini’s gravity will be toconsider it as a constrained first order covariant Hamiltonian field theory on amanifold with boundary obtained as a topological phase of a gauge theory. Wewill review first the geometrical setting for covariant first order Hamiltonian Yang-Mills theories and the topological phase that will interest us.2.1.
A brief account of the multisymplectic formalism for first order co-variant Hamiltonian Yang-Mills theories on manifolds with boundary.
We will review first the basic notions and notations for first order covariant Hamil-tonian field theories (see more details in [Ib15]).2.1.1.
The covariant phase space of Yang-Mills theories.
The fundamental geo-metrical structure of a given first order Hamiltonian theory will be provided by afiber bundle π : E → M with M an m = (1 + d )-dimensional orientable smoothmanifold with smooth boundary ∂M (cid:54) = ∅ and local coordinates adapted to thefibration ( x µ , u a ), a = 1 , . . . , r , where r is the dimension of the standard fiber.Because M is orientable we will assume that a given volume form vol M is se-lected. Notice that it is always possible to chose local coordinates x µ such thatvol M = dx ∧ dx ∧ · · · ∧ dx d .Yang-Mills fields are principal connections A on some principal fiber bundle P → M with structural group G . For clarity in the exposition we are going tomake the assumption that P is trivial (which is always true locally), i.e., P ∼ = M × G → M where (again, for simplicity) G is a Lie group with Lie algebra g . A. IBORT AND A. SPIVAK
Under these assumptions, principal connections on P can be identified with g -valued 1-forms on M , i.e., with sections of the bundle E = T ∗ M ⊗ g −→ M . Localbundle coordinates in the bundle E → M will be written as ( x µ , A aµ ), µ = 1 , . . . , m , a = 1 , . . . , dim g , where A = A aµ ξ a ∈ g with ξ a a basis of the Lie algebra g . Thus,a section of the bundle can be written as(2.1) A ( x ) = A aµ ( x ) dx µ ⊗ ξ a . We will denote by π : J E → E the affine 1-jet bundle of the bundle E π → M .The elements of J E are equivalence classes of germs of sections φ of π , i.e., twosections φ , φ (cid:48) at x ∈ M are equivalent, i.e., represent the same germ, if φ ( x ) = φ (cid:48) ( x )and dφ ( x ) = dφ (cid:48) ( x ). The bundle J E is an affine bundle over E modeled on thevector bundle V E ⊗ E π ∗ ( T ∗ M ). If ( x µ ; u a ), µ = 0 , . . . , d is a bundle chart forthe bundle π : E → M , then we will denote by ( x µ , u a ; u aµ ) a local chart for thejet bundle J E . So in the case of Yang-Mills, local coordinates on J E will bedenoted by ( x, A a , A aµ ).The affine dual of J E is the vector bundle over E whose fiber at ξ = ( x, u )is the linear space of affine maps Aff( J E ξ , R ). The vector bundle Aff( J E, R ),possesses a natural subbundle defined by constant functions along the fibers of J E → E , that we will denote again, with an abuse of notation, as R , then thequotient bundle Aff( J E, R ) / R will be called the covariant phase space bundleof the theory, or the phase space for short. Notice that such bundle, denoted inwhat follows by P ( E ) is the vector bundle with fibre at ξ = ( x, u ) ∈ E given by( V u E ⊗ T ∗ x M ) ∗ ∼ = T x M ⊗ ( V u E ) ∗ ∼ = Lin( V u E, T x M ) and projection τ : P ( E ) → E .Local coordinates on P ( E ) can be introduced as follows: Affine maps on thefibers of J E have the form u aµ (cid:55)→ ρ + ρ µa u aµ where u aµ are natural coordinates onthe fiber over the point ξ in E with coordinates ( x µ , u a ). Thus an affine map oneach fiber over E has coordinates ρ , ρ µa , with ρ µa denoting linear coordinates on T M ⊗ V E ∗ associated to bundle coordinates ( x µ , u a ). Functions constant alongthe fibers are described by the numbers p , hence elements in the fiber of P ( E )have coordinates ρ µa . Thus a bundle chart for the bundle τ : P ( E ) → E is givenby ( x µ , u a ; ρ µa ).The choice of a distinguished volume form vol M in M allows us to identifythe fibers of P ( E ) with a subspace of m -forms on E as follows ([Ca91]): Themap u aµ → ρ µa u aµ corresponds to the m -form ρ µa du a ∧ vol µ where vol µ stands for i ∂/∂x µ vol M . Let (cid:86) m ( E ) denote the bundle of m -forms on E . Let (cid:86) mk ( E ) be thesubbundle of (cid:86) m ( E ) consisting of those m -forms which vanish when k of theirarguments are vertical. So in our local coordinates, elements of (cid:86) m ( E ), i.e., m -form on E that vanish when one of their arguments is vertical, commonly calledsemi-basic 1-forms, have the form ρ µa du a ∧ vol µ + ρ vol M , and elements of (cid:86) m ( E ),i.e., basic m -forms, have the form p vol M . These bundles form a short exactsequence: 0 → (cid:86) m E (cid:44) → (cid:86) m E → P ( E ) → . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 5
Hence (cid:86) m E is a real line bundle over P ( E ) and, for each point ζ = ( x, u, p ) ∈ P ( E ), the fiber is the quotient (cid:86) m ( E ) ζ / (cid:86) m ( E ) ζ .In the case of Yang-Mills, elements of P ( E ) have the form P = P µνa dA aµ ∧ d m − x ν .The bundle (cid:86) m ( E ) carries a canonical m –form which may be defined by ageneralization of the definition of the canonical 1-form on the cotangent bundle ofa manifold. Let σ : (cid:86) m ( E ) → E be the canonical projection, then the canonical m -form Θ is defined byΘ (cid:36) ( U , U , . . . , U m ) = (cid:36) ( σ ∗ U , . . . , σ ∗ U m )where (cid:36) ∈ (cid:86) m ( E ) and U i ∈ T (cid:36) ( (cid:86) m ( E )). As described above, given bundlecoordinates ( x µ , u a ) for E we have coordinates ( x µ , u a , ρ, ρ µa ) on (cid:86) m ( E ) adaptedto them and the point (cid:36) ∈ (cid:86) m ( E ) with coordinates ( x µ , u a ; ρ, ρ µa ) is the m -covector (cid:36) = ρ µa du a ∧ vol µ + ρ vol M . With respect to these same coordinates we have thelocal expression Θ = ρ µa du a ∧ vol µ + ρ vol M , for Θ, where ρ and ρ µa are now to be interpreted as coordinate functions.The ( m + 1)-form Ω = d Θ defines a multisymplectic structure on the manifold (cid:86) m ( E ), i.e.( (cid:86) m ( E ) , Ω) is a multisymplectic manifold. There is some variationin the literature on the definition of multisymplectic manifold. For us, following[Ca91] and [Go98], a multisymplectic manifold is a pair ( X, Ω) where X is a man-ifold of some dimension m and Ω is a form on X of some dimension d ≥
2, andΩ is both closed and nondegenerate. By nondegenerate we mean that if i v Ω = 0then v = 0.We will refer to (cid:86) m E by M ( E ) to emphasize that its status as a multisymplecticmanifold. We will denote the projection M ( E ) → E by ν , while the projection M ( E ) → P ( E ) will be denoted by µ . Thus ν = τ ◦ µ , with τ : P ( E ) → E thecanonical projection.(See figure 1.)A Hamiltonian H on P ( E ) is a section of µ . Thus in local coordinates H ( ρ µa du a ∧ vol µ ) = ρ µa du a ∧ vol µ − H ( x µ , u a , ρ µa )vol M , where H is here a real-valued function.We can use the Hamiltonian section H to define an m -form on P ( E ) by pullingback the canonical m -form Θ from M ( E ). We call the form so obtained theHamiltonian m -form associated with H and denote it by Θ H . Thus if we write thesection defined in local coordinates ( x µ , u a ; ρ, ρ νa ) as(2.2) ρ = − H ( x µ , u a , ρ µa ) , then(2.3) Θ H = ρ µa du a ∧ vol µ − H ( x µ , u a , ρ µa ) vol M . In (2.1) the minus sign in front of the Hamiltonian is chosen to be in keepingwith the traditional conventions in mechanics for the integrand of the action overthe manifold: pdq − Hdt . When the form Θ H is pulled back to the manifold M , A. IBORT AND A. SPIVAK as described in section 2.2.1, the integrand of the action over M will have a formreminiscent of that of mechanics, with a minus sign in front of the Hamiltonian.See equation (2.5).2.1.2. The action and the variational principle.
From here on, in addition to beingan oriented smooth manifold with either a Riemannian or a Lorentzian metric, M has a boundary ∂M . The orientation chosen on ∂M is consistent with the orien-tation on M . Everything in the last section applies. The presence of boundarieswill enable us to enlarge the use to which the multisymplectic formalism can beapplied, starting with eqn. (2 . χ of the theory in the Hamiltonian formalism constitute a class ofsections of the bundle τ : P ( E ) → M . P ( E ) is a bundle over E with projection τ and it is a bundle over M with projection τ = π ◦ τ . The sections that will beused to describe the classical fields in the Hamiltonian formalism are those sections χ : M → P ( E ) ,i.e. τ ◦ χ = id M , such that χ = P ◦ Φ where φ : M → E is a sectionof π : E → M , i.e. π ◦ Φ = id M , and P : E → P ( E ), is a section of τ : P ( E ) → E i.e. τ ◦ P = id P . (See Figure). The sections Φ will be called the configurationsand the sections P the momenta of the theory. In other words u a = Φ a ( x ) and ρ µa = P µa (Φ( x )) will provide local expression for the section χ = P ◦ Φ. We willdenote such a section χ by (Φ , P ) to indicate the iterated bundle structure of P ( E )and we will refer to χ as a double section . M ( E ) J E ⇤ EM@ME @M = i ⇤ Ei ⇤ ( J E ⇤ ) ⇡⇡ ⇡ µ⇡ @M P ⇥⇥ H = h ⇤ ✓h' ( p, ) ⌧ ⌧ H H ⇤ ⇥ i P ( E ) P ( E ) - Figure 1.
Bundles, sections and fields: configurations and momentaWe will denote by F M the space of sections Φ of the bundle π : E → M , thatis Φ ∈ F M , and we will denote by F P ( E ) the space of double sections χ = (Φ , P ).Thus F P ( E ) represents the space of fields of the theory in the first order covariantHamiltonian formalism. It can also be said that χ is a section of P ( E ) along Φ. AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 7
Thus the fields of the theory in the multisymplectic picture for Yang-Mills the-ories are provided by sections (
A, P ) of the double bundle P ( E ) → E → M .The equations of motion of the theory will be defined by means of a varia-tional principle, i.e., they will be characterized as the critical points of an actionfunctional S on F P ( E ) . Such action will be given simply by(2.4) S ( χ ) = (cid:90) M χ ∗ Θ H , In the case of Yang-Mills theories, the action in a first-order covariant Hamiltonianformulation of the theory is given by,(2.5) S YM ( A, P ) = (cid:90) M P µνa dA aµ ∧ dx m − ν − H λ ( A, P )vol M . with Hamiltonian function,(2.6) H λ ( A, P ) = 12 (cid:15) abc P µνa A bµ A cν + λ P µνa P aµν for some λ ≥
0, where the indexes µν ( a ) in P µνa have been lowered (raised) withthe aid of the Lorentzian metric η (the Killing-Cartan form on g , respect.).Of course, as is usual in the derivations of equations of motion via variationalprinciples, we assume that the integral in Eq. (2 .
4) is well defined. It is alsoassumed that the ‘differential’ symbol in equation (2 .
7) below, defined in termsof directional derivatives, is well defined and that the same is true for any othersimilar integrals that will appear in this work.A simple computation leads to,(2.7) d S ( χ )( U ) = (cid:90) M χ ∗ (cid:0) i (cid:101) U d Θ H (cid:1) + (cid:90) ∂M ( χ ◦ i ) ∗ (cid:0) i (cid:101) U Θ H (cid:1) , where U is a vector field on P ( E ) along the section χ , (cid:101) U is any extension of U to a tubular neighborhood of the image of χ , and i : ∂M → M is the canonicalembedding.2.1.3. The cotangent bundle of fields at the boundary.
The boundary term (cid:82) ∂M ( χ ◦ i ) ∗ ( i ˜ U Θ H ) in eq. (2.7) suggests that there is a family of fields at the boundarythat play a special role. Actually, we notice that the field ˜ U being vertical withrespect to the projection τ : P ( E ) → M has the form ˜ U = A a ∂/∂u a + B aµ ∂/∂ρ aµ .Hence we obtain for the boundary term,(2.8) (cid:90) ∂M ( χ ◦ i ) ∗ (cid:0) i (cid:101) U Θ H (cid:1) = (cid:90) ∂M ( χ ◦ i ) ∗ ρ µa A a vol µ = (cid:90) ∂M i ∗ ( P µa A a vol µ )for χ = (Φ , P ).We will assume that there exists a collar around the boundary U (cid:15) ∼ = ( − (cid:15), × ∂M ,and we choose local coordinates x = t ∈ ( − (cid:15), x k , k = 1 , . . . , d , describing A. IBORT AND A. SPIVAK local coordinates for ∂M , such that vol U (cid:15) = dt ∧ vol ∂M . The r.h.s. of eq. (2.8)then becomes,(2.9) (cid:90) ∂M i ∗ ( P µa A a vol µ ) = (cid:90) ∂M p a A a vol ∂M , where p a = P a ◦ i is the restriction to ∂M of the zeroth component of the momentafield P µa .Consider the space of fields at the boundary obtained by restricting the zerothcomponent of sections χ to ∂M , that is the fields of the form (see Figure 1) ϕ a = Φ a ◦ i , p a = P a ◦ i . Notice that the fields ϕ a are nothing but sections of the bundle i ∗ E , the pull-backalong i of the bundle E , while the space of fields p a can be thought of as 1-semibasic d -forms on i ∗ E → ∂M . This statement is made precise in the following: Lemma 2.1.
Given a collar around ∂M , U (cid:15) ∼ = ( − (cid:15), × ∂M and a volume form vol ∂M on ∂M such that vol U (cid:15) = dt ∧ vol ∂M with t the normal coordinate in U (cid:15) ,then the pull-back bundle i ∗ ( P ( E )) is a bundle over the pull-back bundle i ∗ E anddecomposes naturally as i ∗ P ( E ) ∼ = (cid:86) m ( i ∗ E ) ⊕ (cid:86) m − ( i ∗ E ) .Proof. See[Ib15] (cid:3)
If we denote by F ∂M the space of configurations of the theory, ϕ a , i.e., F ∂M =Γ( i ∗ E ), then the space of momenta of the theory p a can be identified with the spaceof sections of the bundle (cid:86) m ( i ∗ E ) → i ∗ E , according to Lemma 2.1. Therefore thespace of fields ( ϕ a , p a ) can be identified with the cotangent bundle T ∗ F ∂M over F ∂M in a natural way, i.e., each field p a can be considered as the covector at ϕ a that maps the tangent vector δϕ a at ϕ a into the number (cid:104) p, δϕ (cid:105) given by,(2.10) (cid:104) p, δϕ (cid:105) = (cid:90) ∂M p a ( x ) δϕ a ( x )vol ∂M . Notice that the tangent vector δϕ at ϕ is a vertical vector field on E along ϕ , andthe section p is a 1-semibasic m -form on E (Lemma 2.1). Hence the contractionof p with δϕ is an ( m − ϕ , and its pull-back ϕ ∗ (cid:104) p, δϕ (cid:105) along ϕ is an( m − ∂M whose integral defines the pairing above, Eq. (2.10).Viewing the cotangent bundle T ∗ F ∂M as double sections ( ϕ, p ) of the bundle (cid:86) m ( i ∗ E ) described by Lemma 2.1, the canonical 1-form α on T ∗ F ∂M can be ex-pressed as,(2.11) α ( ϕ,p ) ( U ) = (cid:90) ∂M p a ( x ) δϕ a ( x ) vol ∂M where U a tangent vector to T ∗ F ∂M at ( ϕ, p ), that is, a vector field on the spaceof 1-semibasic forms on i ∗ E along the section ( ϕ a , p a ), and therefore of the form U = δϕ a ∂/∂u a + δp a ∂/∂ρ a . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 9
Finally, notice that the pull-back to the boundary map i ∗ , defines a natural mapfrom the space of fields in the bulk, F P ( E ) , into the phase space of fields at theboundary T ∗ F ∂M . Such map will be denoted by Π in what follows, that is,Π : F P ( E ) → T ∗ F ∂M , Π(Φ , P ) = ( ϕ, p ) , ϕ = Φ ◦ i, p a = P a ◦ i . With the notations above, by comparing the expression for the boundary termgiven by eq. 2.9, and the expression for the canonical 1-form α , eq. (2.11), weobtain, (cid:90) ∂M ( χ ◦ i ) ∗ ( i ˜ U Θ H ) = (Π ∗ α ) χ ( U ) . In words, the boundary term in eq.(2.7) is just the pull-back of the canonical1-form α at the boundary along the projection map Π.In what follows it will be customary to use the variational derivative notationwhen dealing with spaces of fields. For instance, if F ( ϕ, p ) is a differentiablefunction defined on F ∂M we will denote by δF/δϕ a and δF/δp a functions (if theyexist) such that(2.12) dF ( ϕ,p ) ( δϕ a , δp a ) = (cid:90) ∂M (cid:18) δFδϕ a δϕ a + δFδp a δp a (cid:19) vol ∂M , with U = ( δϕ a , δp a ) a tangent vector at ( ϕ, p ). We also use an extended Einsteinsummation convention such that integral signs will be omitted when dealing withvariational differentials. For instance,(2.13) δF = δFδϕ a δϕ a + δFδp a δp a , may replace dF as in Eq. (2.12). Also in this vein we will write, α = p a δϕ a , and the canonical symplectic structure ω ∂M = − dα on T ∗ F ∂M will be written as, ω ∂M = δϕ a ∧ δp a , by which we mean ω ∂M (( δ ϕ a , δ p a ) , ( δ ϕ a , δ p a )) = (cid:90) ∂M ( δ ϕ a ( x ) δ p a ( x ) − δ ϕ a ( x ) δ p a ( x )) vol ∂M , where ( δ ϕ a , δ p a ) , ( δ ϕ a , δ p a ) are two tangent vectors at ( ϕ a , p a ).2.1.4. Euler-Lagrange’s equations and Hamilton’s equations.
We now examine thecontribution from the first term in dS , eq. (2.7). Notice that such a term can bethought of as a 1-form on the space of fields on the bulk, F P ( E ) . We will call it theEuler-Lagrange 1-form and denote it by EL, thus with the notation of eqn (2 . χ ( U ) = (cid:90) M χ ∗ ( i ˜ U dΘ H ) . A double section χ = (Φ , P ) of P ( E ) → E → M will be said to satisfy the Euler-Lagrange equations determined by the first-order Hamiltonian field theory definedby H , if EL χ = 0, that is, if χ is a zero of the Euler-Lagrange 1-form EL on F P ( E ) .Notice that this is equivalent to(2.14) χ ∗ ( i ˜ U d Θ H ) = 0 , for all vector fields ˜ U on a tubular neighborhood of the image of χ in P ( E ). Theset of all such solutions of Euler-Lagrange equations will be denoted by E L M orjust E L for short.If the metric η on M is just the Minkowski metric so that (cid:112) | η | = 1 or if wechange to normal coordinates on M which we can always find, then the volumeelement takes the form vol M = dx ∧ · · · ∧ dx d . For local coordinates ( x µ , u a , ρ µa )on P ( E ), using eqs. (2.2), (2.3), we then have, i ∂/∂ρ µa dΘ H = − ∂H∂ρ µa d m x + d u a ∧ d m − x µ i ∂/∂u a dΘ H = − ∂H∂u a d m x − dρ µa ∧ d m − x µ . Applying (2.13) to these last two equations we obtain the Hamilton equations forthe field in the bulk :(2.15) ∂u a ∂x µ = ∂H∂ρ µa ; ∂ρ µa ∂x µ = − ∂H∂u a , where a summation on µ is understood in the last equation. Note that had wenot changed to normal coordinates on M , the volume form would not have theabove simple form and therefore there would be related extra terms in the previousexpressions and in equations (2.15).These Hamilton equations are often described as being covariant. This termmust be treated with caution in this context. Clearly, by writing the equations inthe invariant form χ ∗ ( i ˜ U dΘ H ) = 0 we have shown that they are in a sense covariant.However, it is important to remember that the function H is, in general, onlylocally defined; in other words, there is in general no true ‘Hamiltonian function’,and the local representative H transforms in a non-trivial way under coordinatetransformations. When M ( E ) is a trivial bundle over P ( E ), so that there is apredetermined global section, then the Hamiltonian section may be representedby a global function and no problem arises. This occurs for instance when E istrivial over M . In general, however, there is no preferred section of M ( E ) over P ( E ) to relate the Hamiltonian section to, and in order to write the Hamiltonequations in manifestly covariant form one must introduce a connection. (See[Ca91] for a more detailed discussion.) The equations obtained by taking ˜ U to be ∂/∂x µ are consequences of these, and simplyexpress the partial derivatives of H ◦ χ as ‘total’ derivatives of H . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 11
The fundamental formula.
Thus we have obtained the formula that relatesthe differential of the action with a 1-form on a space of fields on the bulk manifoldand a 1-form on a space of fields at the boundary.(2.16) d S χ = EL χ + Π ∗ α χ , χ ∈ F P ( E ) . In the previous equation EL χ denotes the Euler-Lagrange 1-form on the space offields χ = (Φ , P ) with local expression (using variational derivatives):(2.17) EL χ = (cid:18) ∂ Φ a ∂x µ − ∂H∂P µa (cid:19) δP µa − (cid:18) ∂P µa ∂x µ + ∂H∂ Φ a (cid:19) δ Φ a , or, more explicitly:EL χ ( δ Φ , δP ) = (cid:90) M (cid:20)(cid:18) ∂ Φ a ∂x µ − ∂H∂P µa (cid:19) δP µa − (cid:18) ∂P µa ∂x µ + ∂H∂ Φ a (cid:19) δ Φ a (cid:21) vol M . In what follows we will denote by ( P ( E ) , Θ H ) the covariant Hamiltonian fieldtheory with bundle structure π : E → M defined over the m -dimensional manifoldwith boundary M , Hamiltonian function H and canonical m -form Θ H .We will say that the action S is regular if the set of solutions of Euler-Lagrangeequations E L M is a submanifold of F P ( E ) . Thus we will also assume when neededthat the action S is regular (even though this must be proved case by case) andthat the projection Π( E L ) to the space of fields at the boundary T ∗ F ∂M is a smoothmanifold too.3. The presymplectic formalism at the boundary
The evolution picture near the boundary.
We discuss in what followsthe evolution picture of the system near the boundary. As discussed in Section2.1.3, we assume that there exists a collar U (cid:15) ∼ = ( − (cid:15), × ∂M of the boundary ∂M with adapted coordinates ( t ; x , . . . , x d ), where t = x and where x i , i = 1 , . . . , x d define a local chart in ∂M . The normal coordinate t can be used as an evolutionparameter in the collar. We assume again that the volume form in the collar is ofthe form vol U (cid:15) = dt ∧ vol ∂M .If M happens to be a globally hyperbolic space-time M ∼ = [ t , t ] × Σ whereΣ is a Cauchy surface, [ t , t ] ⊂ R denotes a finite interval in the real line, andthe metric has the form − dt + g ∂M where g ∂M is a fixed Riemannian metric on ∂M , then t represents a time evolution parameter throughout the manifold andthe volume element has the form vol M = dt ∧ vol ∂M . Here, however, all we needto assume is that our manifold has a collar at the boundary as described above.Restricting the action S of the theory to fields defined on U (cid:15) , i.e., sections of thepull-back of the bundles E and P ( E ) to U (cid:15) , we obtain,(3.1) S (cid:15) ( χ ) = (cid:90) U (cid:15) χ ∗ Θ H = (cid:90) − (cid:15) d t (cid:90) ∂M vol ∂M (cid:2) P a ∂ Φ a + P ka ∂ k Φ a − H (Φ a , P a , P ka ) (cid:3) . Defining the fields at the boundary as discussed in Lemma 2.1, ϕ a = Φ a | ∂M , p a = P a | ∂M , β ka = P ka | ∂M , we can rewrite (3.1) as S (cid:15) ( χ ) = (cid:90) − (cid:15) d t (cid:90) ∂M vol ∂M [ p a ˙ ϕ a + β ka ∂ k ϕ a − H ( ϕ a , p a , β ka )] . Letting (cid:104) p, ˙ ϕ (cid:105) = (cid:82) ∂M p a ˙ ϕ a vol ∂M denote, as in (2.10), the natural pairing and,similarly, (cid:104) β, d ∂M ϕ (cid:105) = (cid:90) ∂M β ka ∂ k ϕ a vol ∂M , we can define a density function L as,(3.2) L ( ϕ, ˙ ϕ, p, ˙ p, β, ˙ β ) = (cid:104) p, ˙ ϕ (cid:105) + (cid:104) β, d ∂M ϕ (cid:105) − (cid:90) ∂M H ( ϕ a , p a , β ka ) vol ∂M , and then S (cid:15) ( χ ) = (cid:90) − (cid:15) d t L ( ϕ, ˙ ϕ, p, ˙ p, β, ˙ β ) . Notice again that because of the existence of the collar U (cid:15) near the boundaryand the assumed form of vol U (cid:15) , the elements in the bundle i ∗ P ( E ) have the form ρ a d u a ∧ vol ∂M + ρ ka d u a ∧ d t ∧ i ∂/∂x k vol ∂M and, as discussed in Lemma 2.1, thebundle i ∗ P ( E ) over i ∗ E is isomorphic to the product (cid:86) m ( i ∗ E ) × B , where B = (cid:86) m − ( i ∗ E ). The space of double sections ( ϕ, p ) of the bundle (cid:86) m ( i ∗ E ) → i ∗ E → ∂M correspond to the cotangent bundle T ∗ F ∂M and the double sections ( ϕ, β ) ofthe bundle B → i ∗ E → ∂M correspond to a new space of fields at the boundarydenoted by B .We will introduce now the total space of fields at the boundary M which is thespace of double sections of the iterated bundle i ∗ P ( E ) → i ∗ E → ∂M . Followingthe previous remarks it is obvious that M has the form, M = T ∗ F ∂M × F ∂M B = { ( ϕ, p, β ) } . Thus the density function L , Eq. (3.2), is defined on the tangent space T M to the total space of fields at the boundary and could be called accordingly theboundary Lagrangian of the theory.Consider the action A = (cid:82) − (cid:15) L d t defined on the space of curves σ : ( − (cid:15), → M .If we compute d A we obtain a bulk term, that is, an integral on ( − (cid:15), ∂ [ − (cid:15),
0] = {− (cid:15), } . Setting the bulk term equal to zero, we obtain theEuler-Lagrange equations of this system considered as a Lagrangian system on thespace M with Lagrangian function L ,(3.3) dd t δ L δ ˙ ϕ a = δ L δϕ a , AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 13 which becomes,(3.4) ˙ p a = − ∂ k β ka − ∂H∂ϕ a . Similarly, we get for the fields p and β :dd t δ L δ ˙ p a = δ L δp a , dd t δ L δ ˙ β ka = δ L δβ ka that become respectively,(3.5) ˙ ϕ a = ∂H∂p a , and, the constraint equation:(3.6) d ∂M ϕ − ∂H∂β ka = 0 . Thus, Euler-Lagrange equations in a collar U (cid:15) near the boundary, can be under-stood as a system of evolution equations on T ∗ F ∂M depending on the variables β ka ,together with a constraint condition on the extended space M . The analysis ofthese equations, Eqs. (3.4), (3.5) and (3.6), is best understood in a presymplecticframework.3.2. The presymplectic picture at the boundary and constraints analy-sis.
We will introduce now a presymplectic framework on M that will be helpful inthe study of Eqs.(3.4)-(3.6).Let (cid:37) : M −→ T ∗ F ∂M denote the canonical projection (cid:37) ( ϕ, p, β ) = ( ϕ, p ). (SeeFigure 2.) Let Ω denote the pull-back of the canonical symplectic form ω ∂M on T ∗ F ∂M to M , i.e., let Ω = (cid:37) ∗ ω ∂M . Note that the form Ω is closed but degenerate,that is, it defines a presymplectic structure on M . An easy computation showsthat the characteristic distribution K of Ω, is given by K = ker Ω = span (cid:26) δδβ ka (cid:27) . Let us consider the function defined on M , H ( ϕ, p, β ) = −(cid:104) β, d ∂M ϕ (cid:105) + (cid:90) ∂M H ( ϕ a , p a , β ka ) vol ∂M . We will refer to H as the boundary Hamiltonian of the theory. Thus L can berewritten as L ( ϕ, ˙ ϕ, p, ˙ p, β, ˙ β ) = (cid:104) p, ˙ ϕ (cid:105) − H ( ϕ, p, β )and(3.7) S (cid:15) ( ϕ, p, β ) = (cid:90) − (cid:15) [ (cid:104) p, ˙ ϕ (cid:105) − H ( ϕ, p, β )] dt , ' p F @M T ⇤ F @M ( M , ⌦) % ( C , ⌦ ) B ker % ⇤ T ( ',p ) C Figure 2.
The space of fields at the boundary M and its relevant structures.and therefore the Euler-Lagrange equations (3.8) and (3.9) can be written as(3.8) ˙ ϕ a = δ H δp a , ˙ p a = − δ H δϕ a , and(3.9) 0 = δ H δβ ka . Now it is easy to prove the following:
Theorem 3.1.
The solutions to the equations of motion defined by the Lagrangian L over a collar U (cid:15) at the boundary, (cid:15) small enough, are in one-to-one correspon-dence with the integral curves of the presymplectic system ( M , Ω , H ) , i.e., with theintegral curves of the vector field Γ on M satisfying (3.10) i Γ Ω = d H . Proof.
Let Γ = A a δδϕ a + B a δδp a + C a δδβ ka be a vector field on M (notice that we areusing an extension of the functional derivative notation introduced in Section 2.1.3on the space of fields M ). Then because Ω = δϕ a ∧ δp a , we get from i Γ Ω = d H that, A a = δ H δp a , B a = − δ H δϕ a , δ H δβ ka . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 15
Thus, Γ satisfies Eq. (3.10) iff˙ ϕ a = δ H δp a , ˙ p a = − δ H δϕ a , and 0 = δ H δβ ka . (cid:3) Let us denote by C the submanifold of the space of fields M = T ∗ F ∂M ×B definedby eq. (3 . M to the boundary ∂M , are contained in C ; i.e., Π( E L ) ⊂ C . Given initial data ϕ, p and fixing β , existence and uniqueness theorems for initialvalue problems when applied to the initial value problem above, would show theexistence of solutions for small intervals of time, i.e., in a collar near the boundary.However, the constraint condition given by eq. (3.9), satisfied automatically bycritical points of S (cid:15) on U (cid:15) , must be satisfied along the integral curves of the system,that is, for all t in the neighborhood U (cid:15) of ∂M . This implies that consistencyconditions on the evolution must be imposed. Such consistency conditions are justthat the constraint condition eq. (3.9), is preserved under the evolution definedby eqs. (3.8). This is the typical situation that we will find in the analysis ofdynamical problems with constraints and that we are going to summarily analyzein what follows.3.2.1. The Presymplectic Constraints Algorithm (PCA).
Let i denote the canon-ical immersion C = { ( ϕ, p, β ) | δ H δβ = 0 } → M and consider the pull-back of Ω to C , i.e., Ω = i ∗ Ω. Clearly then, ker Ω = ker (cid:37) ∗ ∩ T C . But C is defined as thezeros of the function δ H /δβ . Therefore if δ H /δ β is nondegenerate (notice thatthe operator δ H /δβ ia δβ jb becomes the matrix ∂ H/∂β ia ∂β jb ), by an appropriateextension of the Implicit Function Theorem, we could solve β as a function of ϕ and p . In such case, locally, C would be the graph of a function F : T ∗ F ∂M → B ,say β = F ( ϕ, p ). Collecting the above yields: Proposition 3.2.
The submanifold ( C , Ω ) of ( M , Ω , H ) is symplectic iff H is reg-ular, i.e., ∂ H/∂β ia ∂β jb is non-degenerate. In such case the projection (cid:37) restrictedto C , which we denote by (cid:37) C , is a local symplectic diffeomorphism and therefore (cid:37) ∗ C ω ∂M = Ω . When the situation is not as described above, and β is not a function of ϕ and p , then ( C , Ω ) is indeed a presymplectic submanifold of M and i Γ Ω = d H willnot hold necessarily at every point in C . In this case we would apply Gotay’sPresymplectic Constraints Algorithm [Go78], to obtain the maximal submanifoldof C for which i Γ Ω = d H is consistent and that can be summarized as follows.Consider a presymplectic system ( M , Ω , H ) where M = T ∗ F ∂M × B and, Ω and H are as defined above. Let M = M , Ω = Ω, K = ker Ω , and H = H . Wedefine the primary constraint submanifold M as the submanifold defined by the consistency condition for the equation i Γ Ω = d H , i.e., M = { χ ∈ M | (cid:104) Z ( χ ) , d H ( χ ) (cid:105) = 0 , ∀ Z ∈ K } . Thus M = C . Denote by i : M → M the canonical immersion. Let Ω = i ∗ Ω , K = ker Ω , and H = i ∗ H . We now define recursively the ( k + 1)-th constraintsubmanifold as the consistency condition for the equation i Γ Ω k = d H k , that is, M k +1 = { χ ∈ M k | (cid:104) Z k ( χ ) , d H k ( χ ) (cid:105) = 0 , ∀ Z k ∈ K k } k ≥ , and i k +1 : M k +1 → M k is the canonical embbeding (assuming that M l +1 is aregular submanifold of M k ), and Ω k +1 = i ∗ k +1 Ω k , K k +1 = ker Ω k +1 and H k +1 = i ∗ k +1 H k .The algorithm stabilizes if there is an integer r > M r = M r +1 . Werefer to this M r as the final constraints submanifold and we denote it by M ∞ .Letting i ∞ : M ∞ → M denote the canonical immersion, we define,Ω ∞ = i ∗∞ Ω , K ∞ = ker Ω ∞ , H ∞ = i ∗∞ H . Notice that the presymplectic system ( M ∞ , Ω ∞ , H ∞ ) is always consistent, that is,the dynamical equations defined by i Γ Ω ∞ = d H ∞ will always have solutions on M ∞ . The solutions will not be unique if K ∞ (cid:54) = 0, hence the integrable distribution K ∞ will be called the “gauge” distribution of the system, and its sections (thatwill necessarily close a Lie algebra), the “gauge” algebra of the system.The quotient space R = M ∞ / K ∞ , provided it is a smooth manifold, inherits acanonical symplectic structure ω ∞ such that π ∗∞ ω ∞ = Ω ∞ , where π ∞ : M ∞ → R is the canonical projection. We will refer to it as the reduced phase space of thetheory. Notice that the Hamiltonian H ∞ also passes to the quotient and we willdenote its projection by h ∞ i.e., π ∗∞ h ∞ = H ∞ . Theorem 3.3.
The reduction (cid:101) Π( E L ) of the submanifold of Euler-Lagrange fieldsof the theory is an isotropic submanifold of the reduced phase space R of the theory.Proof. Recall that Π(
E L ) ⊂ C . It is clear that Π( E L ) ⊂ Π( E L (cid:15) ) ⊂ M ∞ where E L (cid:15) = E L U (cid:15) are the critical points of the action S (cid:15) , i.e., solutions of the Euler-Lagrange equations of the theory on U (cid:15) .The reduction (cid:101) Π( E L ) = Π(
E L ) / ( K ∞ ∩ T Π( E L )) of the isotropic submanifoldΠ(
E L ) to the reduced phase space R = M ∞ / K ∞ is isotropic because π ∗∞ ω ∞ = Ω ∞ ,hence π ∗∞ ( ω ∞ | (cid:101) Π( EL ) ) = ( π ∗∞ ω ∞ ) | Π( EL ) = (cid:37) ∗ d α | Π( EL ) = 0. (cid:3) The limit λ → of Yang-Mills theories. Recall equations (2 .
5) and (2 . S YM ,λ ( A, P ) = (cid:90) M P µνa dA aµ ∧ d x m − ν − H λ ( A, P )vol M . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 17 with Hamiltonian function,(3.12) H λ ( A, P ) = 12 (cid:15) abc P µνa A bµ A cν + λ P µνa P aµν for some λ ≥
0, where the indexes µν ( a ) in P µνa have been lowered (raised) withthe aid of the Lorentzian metric η (the Killing-Cartan form on g , respect.).Plugging (2 .
19) into (2 .
18) and expanding the right hand side of (2 . S YM ,λ ( A, P ) = − (cid:90) M (cid:20) P µνa ( ∂ µ A aν − ∂ ν A aµ + (cid:15) abc A bµ A cν ) + λ P µνa P aµν (cid:21) vol M . Using that the curvature, F A = d A A = dA + 12 [ A ∧ A ] = F µν d x µ ∧ d x ν (3.14) = 12 (cid:0) ∂ µ A aν − ∂ ν A aµ + (cid:15) abc A bµ A cν (cid:1) d x µ ∧ d x ν ⊗ ξ a we can rewrite eqn (2 .
20) as S YM ,λ ( A, P ) = − (cid:90) M (cid:20) P µνa F aµν + λ P µνa P aµν (cid:21) vol M . This last expression is the action of the Yang-Mills theory for any given λ ≥ λ →
0, we obtain,(3.15) S Y M, ( A, P ) = (cid:90) M P µνa F aµν vol M , whose equations of motion are given by, F A = 0 , d ∗ A P = 0 . Thus the moduli space of solutions of the Euler-Lagrange equations is given by, M = { F A = 0 , d ∗ A P = 0 } / G M , where G M denotes the group of gauge transformations of the theory.4. Palatini’s Gravity
Palatini’s Yang-Mills.
The primary fields of a theory of gravity a la Palatiniwill be given by principal connections A on a G -principal bundle over a smoothmanifold M with G the Lorentz group O (1 , d ), the group of isometries preservingthe non-degenerate quadratic form Q of signature − + · · · +, with m = 1 + d =dim M .The connections A can be considered as vertical equivariant 1-forms on a prin-cipal fiber bundle with structural group O (1 , d ). The choice of the principal fiberbundle P → M determines a sector of a full theory of gravity where, in additionto the bundle P , we should consider all equivalence classes of principal O (1 , d ) bundles over M . If we fix a topology on M , the corresponding family of classesof principal fiber bundles are in one-to-one correspondence with homotopy classesof maps f : M → B O (1 .d ) , where B O (1 .d ) is the universal classifying space of theLorentz group and the principal fiber bundle corresponding to the map f is givenby P f = f ∗ E O (1 ,d ) , where E O (1 .d ) → B O (1 .d ) is the universal principal O (1 , d ) bun-dle. Thus the fields corresponding to each equivalence class will define connectedcomponents in the space of all fields and we will focus on one of them.4.2. Palatini’s constraint.
Palatini’s constraint determines a subbundle of thecovariant phase space whose sections define a submanifold of the space of fields J F ∗ such that the restriction of the topological sector of SO (1 , F = GL ( τ m , T M ) ⊂ Hom( τ m , T M ) ∼ = τ ∗ m ⊗ T M over M whose fiber at x ∈ M consists on invertible linear maps e ( x ) from τ m ( x ) to T x M and where τ m = M × M m is the trivial bundle over M with fiber the m -dimensionalMinkowski space M m with metric η = diag( − , + · · · , +). Notice that local crosssections of the bundle F can be thought as local frames on M , i.e., if U is an openset on M such that T M | U ∼ = U × R m , then a cross section e : U → τ ∗ m ⊗ T M , definesa map e x := e ( x ) : R m → T x M for each x ∈ U , i.e., a family of linearly independentvectors e I ( x ), I = 0 , , . . . , d , which are the images under e x of the standardorthogonal basis u i on M m , that is η ( u , u ) = − η ( u k , u k ) = 1, k = 1 , . . . , d .With an slight abuse of notation we will denote e x ( u I ) = e I ( x ). Global crosssections e are usually called vierbeins for an arbitrary dimension m , or tetradfields if m = 4. In what follows we will not assume that there are globally definedsections of F (that it may not exist). Notice that given a local cross section e itdefines a Lorentz metric on U by means of g x ( u, v ) = η x ( e − ( u ) , e − ( v )) for any u, v ∈ T x U . The metric g is Lorentz because clearly the vectors e I ( x ) determine anorhonormal basis for g at T x M such that g x ( e I ( x ) , e J ( x )) is diagonal with diagonal( − , + . . . , +).Choosing local coordinates x µ on U we will have that e I = e µI ( x ) ∂/∂x µ willdefined a local vector field on U for each I . With this notation we may also writethe local cross section e as e = e I ⊗ u I = e µI ( x ) ∂/∂x µ ⊗ u I where u I denotes thecanonical dual basis of the standard orthogonal basis u I .Let us recall that we have a distinguished volume form vol M on M , i.e., aglobal section of the determinant bundle det( M ) = Λ m ( T M ). Morevoer there is acanonical section of the bundle det( τ m ) = Λ m ( τ m ) given by vol η = u ∧ u ∧ · · · ∧ u d .Then a linear map e x : τ m ( x ) → T x M defines a pull-back e ∗ (vol M ) = (cid:15) vol η , in otherwords, (cid:15) ( x ) is the determinant of the map e x . In local coordinates: (cid:15) ( x ) = det( e µI ( x )) . Consider the map P : F → P ( E ) defined as: P ( e ) = (cid:15)e ∧ e AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 19 where e ∧ e is defined as the linear map from τ m ∧ τ m to T x M ∧ T x M given by e ∧ e ( u ∧ v ) = e ( u ) ∧ e ( v ). Using the previous notation we may write: P ( e ) = (cid:15)e µI e νJ ∂∂x µ ∧ ∂∂x ν ⊗ u I ∧ u J . Notice that if we write the tensor P ( e ) in the local basis ∂∂x µ ∧ ∂∂x ν ⊗ u I ∧ u J as: P ( e ) = P µνIJ ∂∂x µ ∧ ∂∂x ν ⊗ u I ∧ u J , then P µνIJ = det( e µI ) e [ µ [ I e ν ] J ] , with P µνIJ = − P νµIJ = − P µνJI = P νµJI . We will sometimes use the notation P µνIJ =det( e µI ) e µI ∧ e νJ to indicate the skew symmetry in the pairs of indices IJ and µν .Finally notice that P ( e ) actually lies in P ( E ) as the fiber of P ( E ) at x is givenby T x M ∧ T x M ⊗ so (1 , d ) and τ m ∧ τ m ⊂ so (1 , d ).The image of F under the map P will be called the Palatini subbundle of P ( E )and will be denoted simply by P ( F ) ⊂ P ( E ). Double sections of this bundle arethe fields of the theory we are interested in. Such space of sections will be denotedas P ⊂ J F ∗ M . Notice that a double section ( A, P ) of P is a section of P ( E ) suchthat locally there exists e such that P = (cid:15) e ∧ e .Hence the space of fields of the theory we are constructing can be considered asa submanifold of the space of fields J F ∗ M defined by the range of the map P .4.3. The action.
The action of the topological phase of Yang-Mills given by Eq.(2 .
22) is given by: S Y M, = (cid:90) M P µνIJ F IJµν vol M , with ( A, P ) ∈ J F ∗ M , then if we restrict ( A, P ) to P , the action becomes: S Y M, | P = (cid:90) M (cid:15) e µI e νJ F IJµν vol M , which is exactly Palatini’s action for gravity.The Euler-Lagrange equations of the theory can be obtained by standard meth-ods by computing the differential of S Y M, restricted to P or, alternatively, usingan appropriate version of Lagrange’s multipliers theorem to obtain the criticalpoints of S Y M, restricted to P . We will develop this point of view in the followingsection.4.4. Critical points and Euler-Lagrange equations.
Lagrange’s multipliers theorem.
We will discuss first the version of Lagrange’smultipliers theorem suited to the problem at hand.Theorem: Let M be an affine manifold and let F : M → R be a differentiablefunction. Let D be a smooth manifold and let Φ : D → M be a smooth injectivefunction. Let N = { x ∈ M | ∃ e ∈ D , x = Φ( e ) } . x ∈ N is a critical point of F | N : N → R iff there exists e ∈ D and λ ∈ M ∗ such that ( x, λ, e ) is a critical point of the extended function F : M × M ∗ × D → R given by: F ( x, λ, e ) = F ( x ) + (cid:104) λ, x − Φ( e ) (cid:105) . Proof:Suppose x ∈ N is a critical point of F | N , i.e. d( F | N ) x ( δx ) = 0 for all δx ∈ T x N , or d F x ∈ T x N . Since Φ :
D → N is bijective, there exists e ∈ D such that Φ( e ) = x and for given δe ∈ T e D there exists δx ∈ T Φ( e ) N such thatΦ ∗ ( e )( δe ) = δx , where Φ ∗ ( e ) : T e D → T Φ( e ) N denotes the tangent map to Φ at e ∈ D . It therefore follows that since d( F | N ) x ( δx ) = 0 for all δx ∈ T x N ,(d F )(Φ ∗ ( e )( δe )) = 0 for any δe ∈ T e D .Computing the differential of F , we obtain,(4.1) d F ( x,λ,e ) ( δx, δλ, δe ) = dF x ( δx ) + (cid:104) δλ, x − Φ( e ) (cid:105) + (cid:104) λ, δx − Φ ∗ ( e )( δe ) (cid:105) , where δe ∈ T e D , δx ∈ T x M ∼ = M , δλ ∈ T λ M ∗ ∼ = T ∗ λ M ∼ = M ∗ . The notation (cid:104) λ, x (cid:105) denotes the natural pairing between M and its dual space M ∗ .For ( x, λ, e ) such that x ∈ N is a critical point of F | N , Φ( e ) = x and λ = − d F x ∈ T x N ⊂ T ∗ x M ∼ = M ∗ , it follows from (3 .
1) and from the prior statementsthat d F ( x,λ,e ) ( δx, δλ, λe ) = 0 for all δx, δλ and δe . Thus ( x, λ, e ) is a critical pointof F .Now we prove the other direction of the theorem. Let ( x, λ, e ) be a critical pointof F , i.e. d F ( x,λ,e ) ( δx, δλ, δe ) = 0 for all δx, δλ, δe. In particular, fixing δx = δe = 0,for any δλ , since ( x, λ, e ) is a critical point of F , d F ( x,λ,e ) (0 , δλ,
0) = 0. This impliesby (3 .
1) that x = Φ( e ), thus x ∈ N . For any δx ∈ T x N , since x = Φ( e ) and sinceΦ : D → N is bijective, there exists δe ∈ D such that Φ ∗ ( e )( δe ) = δx . So forour critical point ( x, λ, e ) of F and for any δx ∈ T x N , applying (3 . F ( x,λ,e ) ( δx, δλ, δe ) = d F x ( δx ). Thus for any δx ∈ T x N , d F x ( δx ) = 0, i.e. x isa critical point of F .4.4.2. Critical points.
We apply Lagrange’s multipliers theorem discussed in theprevious section to the following setting. The affine manifold M is the space offields J F ∗ M in the covariant phase space . The manifold D is the manifold of vier-bein fields, i.e, sections e of the bundle F discussed before. The map submanifold N is the submanifold P defined by Palatini’s constraints, i.e., we have the map P : D → J F ∗ given by P ( e ) = (cid:15) e ∧ e . Then, finally, the function F : M → R isthe topological Yang-Mills action functional S Y M, : J F ∗ M → R . AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 21
Then we conclude that critical points of Palatini’s action S P are in correspon-dence with families of critical points of the extended action: S ( A, P, Λ , e ) = S Y M, ( A, P ) + (cid:104) Λ , P − (cid:15) e ∧ e (cid:105) , or, more explicitly:(4.2) S ( A, P, Λ , e ) = (cid:90) M P µνIJ F IJµν + Λ
IJµν (cid:0) P IJµν − (cid:15) e Iµ e Jν (cid:1) vol M . According to Lagrange’s multipliers theorem, the critical points of S have theform ( A, P, Λ , e ) where ( A, P ) is a critical point of S Y M, | P = S P , for all Λ, i.e., P = (cid:15) e ∧ e for some vierbein field e and ( A, e ) is a critical point of: S P = (cid:90) M e µI e Jν F IJµν (cid:15) vol M . Then standard arguments (se for instance [ ? ], [ ? ]) show that the Palatini connec-tion is torsionless and metric with respect to the metric g e defined by the vierbeinfield, that is A is the Levi-Civita connection of the metric g e . Moreover, it satisfiesRicci’s equation: Ric( A ) = 0 . From Eq. (4.1) we also get that if ( x, λ, e ) is a critical point of F , then at x ∈ N we get: d F x ( δx ) = −(cid:104) λ, δx − Φ ∗ ( e ) δe (cid:105) , and δx an arbitrary vector in T x M , that is not necessarily in T x N . This showsthat if ( A, P = P ( e ) , Λ , e ) is a critical point of S , thend S P ( A, P = P ( e ))( δA, δP ) = −(cid:104) Λ , δP − P ∗ ( e ) δe (cid:105) . The canonical formalism near the boundary.
In order to obtain anevolution description for Palatini Gravity and to prepare the ground for canonicalquantization, we need to introduce a local time parameter. We will only assumethat a collar U (cid:15) = ( − (cid:15), × ∂M around the boundary can be chosen and so thata choice of a time parameter t = x can be made near the boundary that wouldbe used to describe the evolution of the system. The fields of the theory wouldthen be considered as fields defined on a given spatial frame that evolve in timefor t ∈ ( − (cid:15), The dynamics of such fields would be determined by the restriction of the Pala-tini action (4.2) to the space of fields on U (cid:15) . Expanding we obtain, S ( A, P, Λ , e ) = (cid:90) U (cid:15) [ P µνIJ F IJµν + Λ
IJµν ( P IJµν − (cid:15)e Iµ e Jν )] vol M = (cid:90) U (cid:15) [ P µνIJ ( −
12 )( ∂ µ A IJν − ∂ ν A IJµ + (cid:15) IJKL,MN A KLµ A MNν ) + Λ
IJµν ( P IJµν − (cid:15)e Iµ e Jν )] vol M = (cid:90) − (cid:15) dt (cid:90) ∂M vol ∂M [ P k IJ ( ∂ A IJk − ∂ k A IJ + (cid:15) IJKL,MN A KL A MNk ) − P kjIJ ( ∂ k A IJj − ∂ j A IJk + (cid:15) IJKL,MN A KLk A MNj ) + 2Λ
IJk ( P IJk − (cid:15)e Ik e J ) + Λ IJkj ( P IJkj − (cid:15)e Ik e Jj )] . In the previous expressions (cid:15) abc denote the structure constants of the Lie algebra g with respect to the basis ξ a , that is [ ξ b , ξ c ] = (cid:15) abc ξ a . Notice that (cid:15) abc A b A c = 0because for fixed a, (cid:15) abc is skew-symmetric. Moreover the indexes µ and a havebeen pushed down and up by using the metric η and the Killing-Cartan form (cid:104)· , ·(cid:105) respectively.In the last equation we used that P is a bivector, i.e., P µνa is skew symmetric in µ and ν and therefore P a = 0, and also P k a P ak = P ia P a i , because P k = − P k , etc.The momenta fields are defined as sections of the bundle P ( E ) and as such areunrestricted. However, because Yang-Mills theories are Lagrangian theories theLegendre transform selects a subspace of the space of momenta that correspondsto fields P , skew symmetric in the indices µ , ν .(For more details see [Ib15].)The previous expression acquires a clearer structure by introducing the appro-priate notations for the fields restricted at the boundary and assuming that theyevolve in time t . Thus the pull-backs of the components of the fields A and P tothe boundary will be denoted respectively as, a ak := A ak | ∂M ; a = ( a ak ) , a a := A a | ∂M ; a = ( a k ) ,p ka := P k a | ∂M ; p = ( p ka ) , p a := P a | ∂M = 0; p = ( p a ) = 0 ,β kia := P kia | ∂M ; β = ( β kia ) . Given two fields at the boundary, for instance p and a , we will denote as usual by (cid:104) p, a (cid:105) the expression, (cid:104) p, a (cid:105) = (cid:90) ∂M p µa a aµ vol ∂M , and the contraction of the inner (Lie algebra) indices by using the Killing-Cartanform and the integration over the boundary is understood. AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 23
Introducing the notations and observations above in the expression for S U (cid:15) weobtain, S U (cid:15) ( A, P, Λ , e ) = (cid:90) − (cid:15) dt (cid:90) ∂M vol ∂M [ p kIJ ( ∂ a IJk − ∂ k a IJ + (cid:15) IJKL,MN a KL a MNk ) − β kjIJ ( ∂ k a IJj − ∂ j a IJk + (cid:15) IJKL,MN a KLk a MNj ) + 2Λ
IJk ( p IJk − (cid:15)e Ik e J ) + Λ IJkj ( β IJkj − (cid:15)e Ik e Jj )] . = (cid:90) − (cid:15) d t L ( a, ˙ a, a , ˙ a , p, ˙ p, β, ˙ β, Λ , ˙Λ , Λ , ˙Λ , e, ˙ e )where L ( a, ˙ a, a , ˙ a , p, ˙ p, β, ˙ β, Λ , ˙Λ , Λ , ˙Λ , e, ˙ e ) = < p, ˙ a − d a a + 2Λ > − < β, F a − Λ > + < Λ , − (cid:15)e ∧ e > + < Λ , − (cid:15)e ∧ e > . Euler-Lagrange equations will have the form: d d t δ L δ ˙ χ = δ L δχ , where χ ∈ P ( E ) and δ/δχ denotes the variational derivative of the functional L .Thus for χ = p we obtain, δ L δ ˙ p = 0 , hence 0 = δ L δp = ˙ a − d a a + 2Λ , and thus,(4.3) ˙ a = d a a − . For χ = a we obtain, δ L δ ˙ a = p, hence ˙ p = δ L δa = d ∗ β + [ p, a ] , that is,(4.4) ˙ p = d ∗ β + [ p, a ] . For χ = a we obtain, δ L δ ˙ a = 0 , hence 0 = δ L δa = ∂∂a < p, d a a > = ∂∂a − < d ∗ a p, a > = − d ∗ a p , that is,(4.5) d ∗ a p = 0 . For χ = β we obtain, δ L δ ˙ β = 0 , hence 0 = δ L δβ = − F a + Λ , that is,(4.6) F a = Λ . For χ = Λ we obtain, δ L δ ˙Λ = 0 , hence 0 = δ L δ Λ = β − (cid:15)e ∧ e, that is,(4.7) β = (cid:15)e ∧ e. For χ = Λ we obtain, δ L δ ˙Λ = 0 , hence 0 = δ L δ Λ = 2 p − (cid:15)e ∧ e , that is,(4.8) p = (cid:15)e ∧ e . For χ = e we obtain, δ L δ ˙ e = 0 , hence 0 = δ L δe = − (cid:15)e Λ − (cid:15)e Λ , that is,(4.9) − e Λ = e Λ . For χ = e we obtain, δ L δ ˙ e = 0 , hence 0 = δ L δe = 2 (cid:15)e Λ i.e.,(4.10) e Λ = 0 . Thus solving for the Euler-Lagrange equations, we have obtained two evolutionequations, (4.3) and (4.4) and six constraint equations (4.5) - (4.10).
AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 25
The presymplectic formalism: Palatini at the boundary and reduc-tion.
As discussed in general in section 3 .
2, we define the extended Hamiltonian, H , so that L = (cid:104) p, ˙ a (cid:105) − H :(4.11) H ( a, a , p, β, Λ , Λ , e ) = < p, − d a a +2Λ > − < β, F a − Λ > + < Λ , − (cid:15)e ∧ e > + (cid:104) Λ , − (cid:15)e ∧ e (cid:105) . Thus the Euler-Lagrange equations can be rewritten as(4.12) ˙ a = δ H δp ; ˙ p = − δ H δa , (4.13) δ H δa = 0; δ H δβ = 0; δ H δ Λ = 0; δ H δ Λ = 0; δ H δe = 0; δ H δe = 0 . We denote again by (cid:37) : M → T ∗ F ∂M the canonical projection (cid:37) ( a, a , p, β ) =( a, a , p ). Let ω ∂M denote the form on the cotangent bundle T ∗ F ∂M , ω ∂M = δa ∧ δp. We will denote again by Ω the pull-back of this form to M along (cid:37) , i.e., Ω = (cid:37) ∗ ω ∂M .Clearly, ker Ω = span { δ/δβ, δ/δa } , and we have the particular form that Thm.3.1 takes here. Theorem 4.1.
The solution to the equation of motion defined by the PalatiniLagrangian (4.2) , are in one-to-one correspondence with the integral curves of thepresymplectic system ( M , Ω , H ) , i.e. with the integral curves of the vector field Γ on M such that i Γ Ω = d H . The primary constraint submanifold M is defined by the six constraint equa-tions, M = { ( a, a , p, β, Λ , e ) |F a = Λ , d ∗ a p = 0 , β = (cid:15)e ∧ e, p = (cid:15)e ∧ e , e Λ = e Λ , e Λ = 0 } . Since Λ = F a , and β is a just a function of e , we have that M ∼ = { ( a, a , p, e ) | d ∗ a p = 0 , p = (cid:15)e ∧ e , e F a = eF a , eF a = 0 } and ker Ω | M ⊃ span { ∂∂a } .Thus M (cid:48) = M / (ker Ω |M ) ∼ = { ( a, p, e ) | d ∗ a p = 0 , p = (cid:15)e ∧ e , e F a = eF a , eF a =0 } . Gauge transformations: symmetry and reduction.
The group of gaugetransformations G , i.e, the group of automorphisms of the principal bundle P overthe identity, is a fundamental symmetry of the theory. Notice that the Palatiniaction is invariant under the action of G . The quotient of the group of gauge transformations by the normal subgroupof identity gauge transformations at the boundary defines the group of gauge transformations at the boundary G ∂M , and it constitutes a symmetry group ofthe theory at the boundary, i.e. it is a symmetry group both of the boundaryLagrangian L and of the presymplectic system ( M , Ω , H ). We may take advantageof this symmetry to provide an alternative description of the constraints found inthe previous section. Proposition 4.2.
The map J : T ∗ F ∂M → g ∗ ∂M given by J ( a, p ) = d ∗ a p is themoment map of the action of the group G ∂M on T ∗ F ∂M where the action of G ∂M on T ∗ F ∂M is by cotangent liftings.Proof. The moment map J : T ∗ F ∂M → g ∗ ∂M is given by, (cid:104)J ( a, p ) , ξ (cid:105) = (cid:104) p, ξ F ∂M (cid:105) = (cid:104) p, d a ξ (cid:105) = (cid:104)− d ∗ a p, ξ (cid:105) , because the gauge transformation g s = exp sξ acts in a as a (cid:55)→ g s · a = g − s ag s + g − s d g s and the induced tangent vector is given by, ξ A ∂M ( a ) = dd s g s · a | s =0 = d a ξ . (cid:3) By the standard Marsden-Weinstein reduction, J − ( ) = { ( a, p ) ∈ T ∗ F ∂M | d ∗ a p =0 } is a coisotropic submanifold of the symplectic manifold T ∗ F ∂M and J − ( ) / G ∂M is symplectic. { ( a, p ) ∈ T ∗ F ∂M | p = e ∧ e } is easily seen to be a symplectic sub-manifold of ( T ∗ F ∂M ) , Ω) , Ω = δa ∧ δp . e F a = eF a and eF a = 0 are coisotropicsubmanifolds of T ∗ F ∂M . This follows from the elementary observation that in asymplectic manifold a subspace defined by a function, φ = 0 is a coisoptropic sub-manifold of the symplectic manifold. Now we need to check that the intersection ofthe coisotropic submanifolds comprising M (cid:48) is a coisotropic submanifold. But thisfollows easily from the fact that the kernel J − ( ) = { ( a, p ) ∈ T ∗ F ∂M | d ∗ a p = 0 } isspanned by the action of the gauge group G ∂M and from the observation that theaction of G ∂M leaves invariant the submanifolds e F a = eF a and eF a = 0. ker J − ( ) is tangent to { ( a, e ) | eF a = 0 } and to { ( a, e ) | e F a = eF a } and is thereforecontained in the tangent spaces of the two surfaces, and vice versa. Thus M (cid:48) isa coisotropic and as described in section 3.2, the reduced space R = M (cid:48) / G ∂M issymplectic and (cid:101) Π( E L ) is an isotropic submanifold of R .5. Conclusions and discussion
Using multisymplectic geometry we have described a Hamiltonian formulationof Palatini’s General Relativity that is simple. Unlike ADM it does not involvelapse and shift operators and it does not require for it’s application the assumptionthat spacetime is topologically R × S where S is space. All we need to assume isthat our spacetime manifold has a boundary and that the boundary has a collar.After the presymplectic constraint analysis, the analysis in the collar providesconsistent solutions of the initial value problem for General Relativity. Unlike AMILTONIAN PALATINI’S GRAVITY WITH BOUNDARIES 27
ADM, we use a formalism that is canonical, i.e. at every step the fundamentalstructures are preserved, both when discussing the constraints introduced from thebulk Palatini constraint P = e ∧ e and when reducing the system by using gaugeinvariance.In a following work we will apply our techniques to study Ashtekar gravity and tothe corresponding quantum aspects. Acknowledgements
A.S. thanks Nicolai Reshetikhin for suggesting to her a problem that motivatedthis work. A.I. was partially supported by the Community of Madrid projectQUITEMAD+, S2013/ICE-2801, and by MINECO grant MTM2014-54692-P.
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E-mail address : [email protected] Dept. of Mathematics, Univ. of California at Berkeley, 903 Evans Hall, 94720Berkeley CA, USA
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