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Covariant phase space for gravity with boundaries: metric vs tetrad formulations

We use covariant phase space methods to study the metric and tetrad formulations of General Relativity in a manifold with boundary and compare the results obtained in both approaches. Proving their equivalence has been a long-lasting problem that we solve here by using the cohomological approach provided by the relative bicomplex framework. This setting provides a clean and ambiguity-free way to describe the solution spaces and associated symplectic structures. We also compute several relevant charges in both schemes and show that they are equivalent, as expected.

Classical GR defined in a space-time without boundary may be alternatively described by the first order Hilbert-Palatini (or Holst) action in terms of tetrads or the Einstein-Hilbert Lagrangian in metric variables. In particular, the corresponding Hamiltonian descriptions are equivalent. Moreover, by employing standard covariant phase space (CPS) methods, it is possible to see that their respective symplectic potentials differ by an exact form. Thus, when this form is integrated over a Cauchy slice Σ, the result vanishes as a consequence of Stokes' theorem. This shows that the presymplectic forms corresponding to the metric and the tetrad formalisms are equivalent in the space of solutions.
The situation changes substantially when boundaries (possible at conformal infinity) are present. Some of the concerns regarding the equivalence of the tetrad and metric formulations may be attributed to the choice of bulk and boundary Lagrangians. In fact, many discrepancies found and discussed in the literature [20][21][22][23] are to be expected since the boundary dynamics is not always explicit and the solution spaces are not fully specified.
To avoid these problems, it is crucial to choose variational principles such that the equivalence of the dynamics they describe, both in the bulk and at the boundaries, is guaranteed. The choice of the bulk and boundary Langrangians will determine the presymplectic structure in the solution space as well as the Noether charges.
The purpose of this paper is to prove that the metric and tetrad formulations are equivalent in the CPS with Dirichlet boundary conditions (BC) and (homogeneous) Neumann BC (the same methods can be applied to other BC). We do this by relying on the relative bicomplex framework, a formalism developed in [15] which is cohomological in nature so no ad hoc choices are required. This formalism will allow us to establish the equality of the metric and tetrad symplectic potentials in the relative cohomology (equality up to a relative exact form as explained in the appendix). Consequently, the respective presymplectic forms in the solution space are equivalent and, as expected, there is a precise correspondence between the Noether charges in both formulations. As a direct result of our discussion, we will consider the asymptotically flat case and rederive the well-known formula for the ADM-energy.
The paper is structured as follows. In the next section, we present some of the geometric concepts relevant for the implementation of the CPS-algorithm developed in [15]. In III, we apply this algorithm to the metric formulation of GR. Section IV presents an analogous study for the tetrad formalism. In Section V both formulations are compared and shown to be equivalent. We present our conclusions in the last section of the paper. We have also included in appendix A a short summary of the relative bicomplex framework together with some computational details.

II.1 M as a space-time
Let M be a connected and oriented n-manifold admitting a foliation by Cauchy hypersurfaces. Without loss of generality, M = I × Σ for some interval I = [t i , t f ], (t i < t f ) and some (n − 1)-manifold Σ with boundary ∂Σ (possibly empty).
where Σ i,f are the "lids" and ∂ L M := I ×∂Σ the "lateral boundary". M = I ×Σ is a manifold with corners ∂Σ i ∪ ∂Σ f , which, as a set are ∂(∂ L M ). The following diagram summarizes relevant notational information about embeddings and the induced geometric objects.
The entries in each 4-tuple in (II.1) are: the manifold, the (non-degenerate) metric or pulledback metric, its associated Levi-Civita connection, and the abstract indices used to describe tensors in the manifold. The arrow labels specify the notation used for embeddings and normal unitary vector fields. Horizontal arrows are associated with future pointing normal unit vector fields and vertical arrows are used for outward pointing normal unit vector fields at the boundary. The notation just introduced is not consistent for the bottom lid Σ i , as it may be thought of as a spacelike hypersurface embedded by ı i or considered as part of ∂M. Σ i often appears as the boundary of M, so we choose the outwards (past pointing) convention in this case. An overline is often used to denote objects that live exclusively at the boundary, such as g, which in index notation reads g αβ . Notice that the embeddings  and  are fixed since the boundaries are also fixed. However, ı and ı, which embed (Σ, ∂Σ) into (M, ∂M ), can be chosen among the Cauchy embeddings satisfying ı(∂Σ) ⊂ ∂ L M.
As (M, g) is oriented, we have the metric volume form vol g that assigns the value 1 to every positive orthonormal basis. We orient Σ and ∂ L M with vol γ and vol g , respectively given by for every U ∈ X(M ). These orientations are the ones for which Stokes' theorem holds in its usual form. Finally, ∂Σ can be oriented as the boundary of Σ. Thus vol γ is given by Notice that if we use the Stokes' theorem from ∂ L M to ∂(∂ L M ) = ∂Σ i ∪ ∂Σ f , a minus sign appears in the integral over ∂Σ i .

II.2 From M -vector fields to F -vector fields
Let F be a space of tensor fields on M (sections of a certain bundle E π → M ) and think of it as an infinite dimensional differential manifold where we assume that the usual differential objects-like tensor fields, the exterior derivative d , the interior derivative , the wedge ∧ ∧, or the Lie derivative -are well defined (see the appendix of [15] for more details about this and other technicalities). As a reminder for the reader, most objects defined over the space of fields are denoted with a double font. Given a vector field ξ ∈ X(M ) on M , let us associate to it a canonical vector field X ξ ∈ (F) on F. To that purpose, we assume that ξ is tangent to ∂ L M but not necessarily to Σ i and Σ f (notice that we can always extend the interval [t i , t f ] to avoid problems at the lids) and take advantage of the fact that a field φ r (r ranging over the number of fields of the theory) can be interpreted in two ways: • As a tensor field on M. That is, a section φ r : M → E r of some bundle E r π → M such that φ r p := φ r (p) ∈ E r p := π −1 (p). In particular, we can take its Lie derivative If F is reasonable enough, as it is the case in this paper, we will have L ξ φ ∈ F.
• As a point of F. A vector field X ξ ∈ (F) is a section of the tangent bundle T F i.e.
(X ξ ) φ := X ξ (φ) ∈ T φ F ∼ = F. The last isomorphism comes from the fact that the field space F is linear in this case. The non-linear case is not as straightforward but, in concrete examples, it is possible to carry out similar constructions.
Taking into account these remarks, we define Notice that d φ r (which appears in the definition of the Lie derivative through Cartan's magic formula) has to be interpreted as the exterior derivative of the evaluation function.

II.3 CPS-Algorithm
This paper relies heavily on the results of [15]. For convenience, a summary has been included in the appendix but we list here the main steps of the CPS-algorithm since it will be used throughout the paper. This algorithm provides an ambiguity-free method to construct a presymplectic structure in the space of solutions canonically associated with the action of the theory.
S does not depend on ı.
It is important to notice that it is not always possible to perform the second step. If b r and θ do not exist (their existence does not depend on the chosen Θ), that means that the theory is not well defined and hence we will have to change F and/or the action (hence (L, )) to ensure that the equality holds. Once we have the presymplectic structure, the following two steps provide important additional information about the theory at hand. 5. Study symmetries; i.e. find out if X ξ is a d-symmetry and obtain the ξ-currents and ξ-charges.
Compare S with the presymplectic structure obtained in the standard Hamiltonian formulation.

III General Relativity in terms of metrics
Step GHY ) consists of the Einstein-Hilbert term with cosmological constant in the bulk and the Gibbons-Hawking-York term on the lateral boundary. R g denotes the g-scalar curvature (following the signs conventions of [24]), vol g and vol g the metric volume forms, and K the extrinsic curvature of ∂ L M ⊂ M given by

Step 1 and 2: Variations
According to the computations given in the appendix, we have and d g := g αβ d g αβ stands for the g-trace of d g αβ (not to be confused with the variation of the determinant of g, which we do not use in this paper). Besides, we use the notation (ι ν d g) β := ν α d g αβ even though d g αβ is not a 2-form.

Step 3: Space of solutions
Once we know the basic variations, we can compute the variation of the action where L = denotes equality up to integrals over the lids. Notice that those integrals are unimportant because the values of the dynamical fields on the lids are kept fixed when

Remark III.2
It is important to realize that a boundary term must be included in the two theories that we consider (i.e. when no a priori BC are imposed on the space of fields and with Dirichlet BC). Otherwise, as explained in section II.3, the second step cannot be performed and thus the space of solutions is not well defined. Nonetheless, it is possible to impose homogeneous Neumann BC a priori and include a different boundary term [25] (which vanishes for dimension 4). However, as we have shown, this is not necessary. Finally, it is worth noting that non-trivial BC may lead to constrained variations, which must be properly handled.

Step 4: Symplectic form
Given a Cauchy embedding ı : Σ → M , we compute the presymplectic structure canonically associated with the action as Using the results of the previous sections and the computations of the appendix, we obtain: As indicated in (II.1), n α is the g-normal to ı(Σ) ⊂ M , ν β is the g-normal to ∂ L M ⊂ M , m α is the pushforward of the g-normal to ı(∂Σ) ⊂ ∂ L M , and µ λ is the pushforward of the γ-normal to ∂Σ ⊂ Σ. We have also used the generalized Kronecker delta Step 5: Charges The Noether currents associated with an arbitrary vector field ξ ∈ X(M ) tangent to the lateral boundary are computed in the appendix. The result is where Q (m) ξ = g dξ is the ξ-Noether potential and (Ẽ (m) ,b (m) ) are the 2-tensors that multiply the top-forms in (E (m) , b (m) ). As usual, g is the g-Hodge dual operator and, in order to ease the notation, we represent also by ξ the 1-form metrically equivalent to the vector field ξ. The ξ-charge is then given by The last expression, when pulled back to Sol(S (m) N , vanishes as expected because the theory is Diff-invariant [15]. However, over Sol(S (m) D , the boundary integral is not necessarily zero as b (m) does not vanish in general. Notice that fixing the metric on the boundary breaks Diff-invariance.
Step 6: comparing with the Hamiltonian formulation If we consider ∂Σ = ∅, the equivalence between the CPS presymplectic form obtained in step 4 and the canonical ADM presymplectic form has been established in [26,27]. The case with boundaries is not as straightforward and will be studied elsewhere.

The asymptotically flat case
As a further application of the CPS-algorithm, let us obtain the ADM energy of an asymptotically flat space-time (see [28,29] for similar approaches). For that purpose, consider M = Σ × R endowed with the Minkowski metric g 0 = −dt 2 + σ. Then Σ is a 3-manifold with Euclidean metric σ and boundary ∂Σ = S 2 r , a 2-sphere of σ-radius r 0 . Take a small perturbation h ∈ T g 0 Met(M ), such that h :=  * h = 0 (compatible with the Dirichlet condition, otherwise the charge is zero as the theory is Diff-invariant). The ξ-charge given by (III.4) restricted to the space of solutions is given by Expanding the integrand in terms of h, and labeling with a circle the quantities associated with g 0 (the "background" objects), we obtain where we omit higher order terms in h as they vanish when r 0 → ∞ if we consider ξ := n and the inertial foliation {{t} × Σ} t (in particular, we take n = ∂ t ).
Using the decomposition of the normal ν α = −ν ⊥ n α + ν α and some of the results of section A.2, we have where we have used that • ν ⊥ = 0, ξ = n and the fact that ξ is tangent to the lateral boundary (hence we can replace h with h, which is zero). We have also used the (not very standard) notation Notice that the previous computation is performed at the point g 0 in the space of fields and at the lateral space-time boundary.
This computation together with • m β = ξ β takes care of the first parenthesis of Q ξ . The last one can also be easily computed by using the variation of the volume given in section A.2.
Let us now deal with the term in the second parenthesis. First notice that when we contract it with ξ α ξ β (in coordinates this would be long but straightforward computation gives the following variations As the result is independent of the Cauchy slice, we introduce coordinates {t, x 1 , x 2 , x 3 } and take, as we mentioned before, a Cauchy slice given by Σ = {t = t 0 }, then Notice that everything is constant along the t-direction (which is the ξ direction because n = ∂ t ). Thus we have where here A, B = 1, 2, 3 label the coordinates. Finally, notice that the integral of h • Kvol σ goes to zero in the limit r 0 → ∞ and ι ξ K = 0 because the normal is constant along the foliation. Putting everything together, we obtain A final comment is now in order. As we can see we have a divergent constant term in the limit r 0 → ∞. To remove it, we introduce the following r 0 -dependent boundary Lagrangian By doing this, the constant term does not appear, the limit is well defined, and we recover the well-known expression for the ADM-energy.

Notation
In order to introduce the tetrad formulation for GR, we will use internal abstract indices as discussed, for instance, in [30][31][32]. For the convenience of the unfamiliar reader, we have included this short subsection with the basic ingredients. The more familiar reader can go directly to equation (IV.2), where we introduce some new notation.
Let V → M be a Minkowski vector bundle i.e with typical fiber V, an n-dimensional vector space with internal indices {I, J, . . .}, endowed with a Minkowskian metric η, which in abstract index notation reads η IJ . This metric allows us to define the η-metric volume form ε, the η-Hodge dual η , and the η-trace Tr η (see [33] for a recent review). The latter is given by Tr η (α ∧ β) = η I 1 J 1 · · · η I 1 Jr α I 1 ···Ir ∧ β J 1 ···Jr -8 -Tetrads are bundle isomorphisms T M → V. In abstract index notation they look like e I α , so it is clear that a tetrad, together with the non-degenerate metrics g αβ and η IJ , provides a way to swap space-time and internal indices. It is also clear from the index notation that tetrads can be identified with sections of the product bundle V ⊗ T * M i.e. elements of Γ(V ⊗ T * M ), the set of V-valued 1-forms over M , although in practice it would be more useful to identify them with V -valued 1-forms over M . This is the case, for instance, if M is parallelizable (an example would be a non-compact manifold admitting a spinor structure [34]). In that case it is enough to work with coframes which, loosely speaking, are the local version of a tetrad. Indeed, a coframe e I α (p) at p ∈ M is a linear ismorphism Equivalently, a coframe can be thought of as an orthonormal basis in T * p M . When M is parallelizable, coframes can be identified with global covector fields but, otherwise, they can only be considered as local covector fields, [32].
It is also useful to introduce objects with more indices, the so-called generalized tensor fields t ∈ Γ(V ⊗p ⊗ V * ⊗q ⊗ T M ⊗r ⊗ T * M ⊗s ), where V * → M is the dual bundle of the Minkowski bundle. In index notation, generalized tensor fields look like t To ease the notation, space-time indices will often be omitted and we will simply write ω I 1 ···Ir . Using the generalized Kronecker delta with internal indices, δ J 1 ···Jm where the symbol ∧ · will be called the ledge (Lie wedge). The previous operation is performed by contracting the last index of A with the first one of B and anti-symmetrizing the remaining ones. Although not necessary for our purposes, it is worth mentioning that Ω k 2 (M ) with the ledge product is a Lie algebra. Moreover, notice that {e ∧ e} IJ = 2e I ∧ e J . The following easy to derive formula will be used in the rest of the paper

Constructing the geometric objects
Consider {e I α } a co-frame, such that ε I 1 ···In e I 1 ∧ · · · ∧ e In defines a volume form on M . We define the Lorentzian e-metric g = Tr η (e ⊗ e) ≡ g αβ = η IJ e I α e J β together with its g-Levi-Civita connection ∇. We can now define the dual frame {E α I }, given by the relations as well as the e-connection ω defined by which can be proven to uniquely determine ω. Notice that the internal indices of ω are antisymmetric once the second one is raised ω IJ µ = −ω JI µ . Notice also that we have the covariant derivative of forms over Ω r s (M ) given by where here and in the following, we will sometimes omit the internal indices when not needed. Let us also denote the associated e-Christoffel as Γ, the e-torsion as T , the ecurvature as R, and the ω-curvature as F , which are given by where Tor and Riem are the torsion and the Riemannian curvature of the g-Levi-Civita connection. In particular Tor = 0. It is interesting to notice that, as mentioned in the introduction of this section, we are using E α I and e I α to transform the space-time indices in Tor and Riem into internal indices in order to define T and R. From the previous equations, the following relations are easy to derive It is also straightforward to show that the following properties hold Step 0: Action Let us consider the following spaces of non-degenerate tetrads on M Finding a useful explicit expression for the boundary Lagrangian requires more work. Let us define the "internal" normal N I := ι ν e I ∈ Ω 0 1 (M ), which satisfies N I e I α = 0 and N I N I = 1, and the "internal" projector γ I J := η I J − N I N J . With these elements at hand, together with is the induced covariant derivative over the boundary given by the pullback connection ω =  * ω. Recall also that e :=  * e which, in turn, allows us to define its dual E α I := g αβ η IJ e J β which satisfies With these ingredients, it is possible to obtain a better expression for (t) by considering which is zero as all four internal indices of ε are projected. This identity allows us to obtain, as already done for instance in [35,36], the alternative expression -11 -

Step 1 and 2: Variations
As before, we rely on the results and computations of the appendix to obtain where E (t)

Step 3: Space of solutions
The solutions derived from S

Step 4: Symplectic form
Given a Cauchy embedding ı : Σ → M , we have Using the results of the previous section, we immediately obtain

Step 5: Charges
A direct computation using the definition of the ξ-current J (t) ξ leads to where we have used ι ξ F = (L ξ − Dι ξ )ω, which follows from Cartan's magic formula. The second term of J (t) ξ vanishes as the torsion De is zero, while the third one is also zero as a consequence of (II.4) and the fact that the only background object is η, which is "constant". Thus we obtain ξ is then exact over the space of solutions with ξ-potential Q (t) ξ . Similarly, we have for the boundary In the last equality, a term may seem to be missing according to the definition of b (t) but, in fact, such term vanishes as a consequence of e I N I = 0. We then conclude that the ξ-charges are given by This expression, when pulled-back to Sol(S N , vanishes as expected because the theory is Diff-invariant [15]. However, over Sol(S D the boundary integral will not be zero in general, as b (t) does not necessarily vanish (fixing the tetrad on the boundary breaks the invariance under diffeomorphisms).
Step 6: comparing with the Hamiltonian formulation

From
Step 6 of the CPS-algorithm in the metric formalism and the following section, the equivalence is assured when no boundaries are present. The case with boundaries is again not as straightforward and it will be studied elsewhere.

V Metric vs Tetrad formulation
We have obtained in the previous sections the presymplectic structure over the space of solutions for both the metric and tetrad formulations together with their ξ-charges. In this section, we prove that the spaces can be naturally mapped, that their symplectic structures are equivalent (without considering the internal gauge freedom), and that the ξ-charges are equal.

V.1 Space of solutions
Let us first show the correspondence between the solution spaces of metric and tetrad gravity. To this end, we define the maps both obtained by assigning the following metric to a given tetrad Of course, in the Dirichlet case the compatibility condition g αβ = η IJ e I α e J β must hold for Φ D to be well defined. On one hand, it is well known that Φ is surjective but not injective. On the other hand, we have that It is easy to check that d e Φ is surjective. Hence, the relation between the spaces of solutions is clear: if e ∈ Sol(S (t) ), then Φ(e) ∈ Sol(S (m) ) and if g ∈ Sol(S (t) ), then every e ∈ Φ −1 ({g}) belongs to Sol(S (t) ).

V.2 Presymplectic structures
Let us now compare ı (m) and ı (t) by looking at their symplectic potentials. For that purpose, we perform the "change of variable" g = Φ(e) := e I e I in (Θ (m) , θ (m) ). First, we notice that instead of working with Θ (m) , it is more convenient to work with its g-dual where W µ := ∇ β d g µβ − ∇ µ d g was obtained in step 1 of section III. Using the variation which follows from the definition of ω IJ µ , and the definition of W µ , we obtain on one hand where (δU) µ := −∇ α U αµ is the codiferential. On the other hand, a standard computation using the definition of the g-Hodge star operator leads to Taking the Hodge dual of (V.2), using (V.1) and (V.3), and the fact that for k-forms we have g g = (−1) k(n−k)+1 Id and g δ = (−1) k d g , we finally obtain Once we have taken care of the bulk terms, we focus on the boundary ones. If we use we can obtain on one hand On the other hand, using (A.8), we compute the following expression α was obtained in the step 2 of section III. Notice that we have made a small abuse of notation because  * ι ν U is a 1-form so an index must be raised with the help of g. Finally, we show that this last expression is in fact zero Putting everything together, we finally obtain the main result of the paper Hence, the symplectic potentials are equal up to a relative exact form. Notice that the fact that the LHS comes from the metric formalism while the RHS comes from the tetrad formalism is not a problem because the LHS is implicitly evaluated at the e-dependent metric Φ(e) := η IJ e I e J .
Taking now the d -exterior derivative of (V.5), integrating over a Cauchy slice (Σ, ∂Σ), and using the relative Stokes' theorem (A.2), we obtain the desired equality of the two presymplectic forms (t) and (m) (see appendix A.1 for a brief account of the relative framework). More specifically, we have A final comment is in order now: the previous formula says that both presymplectic structures are equivalent modulo the gauge freedom given by the kernel of Φ * . This gauge freedom is present in the tetrad formalism and originates in the SO(1, 3)-invariance of Φ but it has no metric counterpart. In particular, this means that (t) has more degenerate directions than (m) . This can be neatly understood by noticing that if we consider a curve Ψ τ ∈ SO(1, 3) and its associated vector V (t) := d dτ 0 Ψ τ · e ∈ T e F (t) , we have is a nonzero vector which belongs to the kernel of (t) (as a consequence of being in the kernel of Ψ * ) so it is a gauge vector field. However, the metric counterpart V (m) := Ψ * V = 0 is not gauge because by definition the zero vector is not gauge. Finally, the equivalence of the charges (III.4) and (IV.8) is obvious as both theories are equivalent, implying that b (m) and b (t) are equivalent as well.

VI Conclusions
In this paper, we have studied the metric and tetrad formulations for general relativity on a manifold with boundary. By considering the appropriate bulk and boundary Lagrangians, we have shown that both theories are equivalent and hence, as one would expect, they are symplectically equivalent in the covariant phase space. Here we have focused on Dirichlet and Neumann BC, but any other BC will give the same results as long as the metric and tetrad actions are in a suitable correspondence.
It has been known for some time that, in the absence of boundaries, the metric symplectic current Ω (m) := d Θ (m) is equal to the tetrad symplectic current Ω (t) := d Θ (t) up to an exact form dA (and thus cohomologically equal). Therefore, their presymplectic forms over the space of solutions are equivalent since the integral of dA over a Cauchy slice Σ is zero according to Stokes' theorem. On the other hand, if one considers a space-time with boundaries, the boundary-free covariant phase space procedure fails. This is due to some ambiguities that arise in the construction of the presymplectic form that hinder the direct comparison between metric and tetrad formulations. This has caused some discrepancies in previous works [20,22,23], but as we have shown, these difficulties arise because the traditional covariant phase space methods are only suited for the boundary-free case.
When boundaries are present, it is necessary to use more sophisticated techniques, like the relative bicomplex framework [15]. Following the ideas of that formalism, we obtain the main result of the paper: the metric symplectic currents (Ω (m) , ω (m) ) := (d Θ (m) , d θ (m) ) and the tetrad symplectic currents (Ω (t) , ω (t) ) := (d Θ (t) , d θ (t) ) are equal up to a relative exact form d (A, a). This implies, in particular, that they are equal in the relative cohomology (see appendix A.1 for the relevant definitions) and that their presymplectic forms over the space of solutions are equivalent. Indeed, the relative Stokes' theorem tells us that the integral of d(A, a) over a relative Cauchy slice (Σ, ∂Σ) is zero. Furthermore, we have proved that the Noether charges are equivalent as expected. Finally, we have applied the covariant phase space methods to the asymptotically flat case to recover the well-known formula for the ADM-energy. which in turn implies that (A.1) for N = ∂M is well defined on relative cohomology because ∂(M, ∂M ) = ∅. We introduce now the space of forms (r,s) (M × F) of degree r in M (horizontal part) and s in F (vertical part). Endowed with the wedge product ∧ ∧, this space becomes a bigraded algebra with two exterior derivatives: the horizontal d, which increases r in one unit, and the vertical d , increasing s in one unit. The wedge product ∧ ∧ restricted to (k, 0)-forms coincides with ∧. We will often abuse notation and use the latter. If we replace (M, d) by the relative pair ((M, N ), d), we can define the relative bicomplex where the loc subscript indicates that we only consider (r, s)-forms which are locally constructed i.e. a form α evaluated at p only depends on p, φ(p), and finitely many of the derivatives of φ at p.

Lagrangians and actions
Definition A. 3 We define a Lagrangian pair as an element of In this work, we have only considered contractible bundles, for which the -equivalence is the same as the cohomological equivalence (a proof was given in [15]). Nevertheless, if the bundles are not contractible, it is still possible to keep track of the ambiguities that arise from the fact that there exist non-zero Lagrangians [(L, )] = 0 whose Euler-Lagrange equations and BC are zero.

Variations
We assume that the action is defined in such a way that it is possible to find Euler-Lagrange equations and boundary equations (E, b), and symplectic potentials (Θ, θ), such that where r labels the fields of the theory φ = (φ 1 , . . . , φ R ) ∈ F. If this is not possible, the theory is ill-posed and we have to change the space of fields F and/or the action S. The symplectic potentials (Θ, θ) are defined up to a relative exact form. The space of solutions is Sol(S) := {φ ∈ F / (E, b)(φ) = 0}