Covariant phase space and boundary ambiguities

Let $M$ be a smooth four-manifold and let $\pi :E\to M$ be a smooth fibre bundle. A field configuration is a smooth section $\varphi :M\to E$, so $\pi \circ \varphi ={\mathrm{id}}_M$. For a finite-dimensional smooth manifold $P$, a smooth field family is a smooth map $\mathrm{\Phi }:P\times M\to E$ with $\pi (\mathrm{\Phi }(p,x))=x$ for all $(p,x)$; it gives configurations ${\varphi }_p(x)=\mathrm{\Phi }(p,x)$. For $u\in T_{p}P$, the variation ${\delta }_u{\varphi }_p$ at $x$ is the tangent vector $d(r\mapsto \mathrm{\Phi }(c(r),x)){\mathrm{\mid }}_{r=0}$, where $c(0)=p$ and $c^{\mathrm{\prime }}(0)=u$. In a local fibre chart this is the ordinary derivative of the field components with respect to the parameter. For commuting parameter coordinates $s_1,s_2$, ${\delta }_i\varphi ={\mathrm{\partial }}_{s_i}\varphi$ and ${\delta }_1{\delta }_2\varphi ={\delta }_2{\delta }_1\varphi$.

Smooth fibre bundles and local trivializations

Smooth maps between manifolds

Smooth fibre bundles and local trivializations

$C^r$ and smooth maps between smooth manifolds

The parameter manifold is finite-dimensional. This definition does not assume that the full space of field configurations or its solution subset has a global infinite-dimensional manifold structure.

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be open and star-shaped with respect to $a\in U$, and let $\alpha \in {\mathrm{\Omega }}^k(U)$ be a smooth closed differential form with $1\le k\le n$. Define

$(K\alpha {)}_x(v_1,\dots ,v_{k-1}):={\int }_0^1{t}^{k-1}{\alpha }_{a+t(x-a)}(x-a,v_1,\dots ,v_{k-1})dt\mathrm{.}$

Then $K\alpha$ is a smooth $(k-1)$-form on $U$ and $d(K\alpha )=\alpha$.

Advanced Lectures on General Relativity, §1.3.3

Some Properties of Noether Charge, §4

The conclusion is local: each star-shaped coordinate domain is contractible. It supplies local primitives only and gives no global cohomological conclusion.

The cited general-relativity sections use local exactness for their Noether currents; they do not state this general $k$-form homotopy formula. The displayed argument proves the broader statement here.

Let $P$ be a finite-dimensional smooth manifold, let $M$ be an oriented smooth four-manifold, and let $\mathrm{\Sigma }$ be a fixed finite oriented smooth singular three-chain in $M$. Let ${\alpha }_p$ be a smooth parameter-dependent spacetime three-form, and assume that for every simplex in $\mathrm{\Sigma }$ its pullback coefficient is jointly smooth in the parameter coordinates and the standard simplex coordinates. Define $I(p):={\int }_{\mathrm{\Sigma }}{\alpha }_p\mathrm{.}$ Then $I$ is smooth and, for every $v\in T_{p}P$, $d_P{I}_p(v)={\int }_{\mathrm{\Sigma }}{d}_P{\alpha }_p(v)\mathrm{.}$

The chain is fixed and finite, and the parameter manifold is finite dimensional. No infinite-chain convergence or infinite-dimensional field space structure is part of the lemma.

Let $M$ be an oriented smooth four-manifold and let $\varphi$ be a field configuration in a specified smooth field bundle. A local Lagrangian four-form $\mathcal{L}(\varphi )$ is a smooth four-form whose coordinate coefficients depend smoothly on the field components and finitely many spacetime derivatives at each point. A chosen first-variation decomposition consists of an Euler functional $E_{\varphi }[\eta ]$ with values in four-forms and a three-form $\theta (\varphi ;\eta )$. In local bundle charts, both are smooth finite-order local differential expressions in the field configuration $\varphi$ and its variation $\eta$: their coefficients depend smoothly on the spacetime coordinates, finitely many spacetime derivatives of $\varphi$, and any fixed smooth background fields, and their dependence on $\eta$ is linear in finitely many of its spacetime derivatives. Every smooth one-parameter field family then satisfies

$\delta \mathcal{L}=E_{\varphi }[\delta \varphi ]+d_M\theta (\varphi ;\delta \varphi )\mathrm{.}$

The choice is part of the data; it is not asserted to be unique. The stated smoothness makes its parameter derivatives defined along every smooth finite-dimensional field family. In a parameter family, ${\delta }_i{E}_{\varphi }$ means the derivative of the coefficients of $E_{\varphi }$ with its variation input held fixed. Thus the total derivative of $E_{\varphi (p)}[\eta (p)]$ is $({\delta }_i{E}_{\varphi })[\eta ]+E_{\varphi }[{\delta }_i\eta ]$. For a smooth finite-dimensional field family with parameter exterior derivative $d_P$, the one-form $\mathrm{\Theta }$ on parameter space with values in spacetime three-forms is ${\mathrm{\Theta }}_p(v)=\theta ({\varphi }_p;{\delta }_v{\varphi }_p)$. Define the covariant presymplectic current by $\omega :=-d_P\mathrm{\Theta }$. For commuting parameter coordinates $s_1,s_2$ this is

$\omega ({\delta }_1\varphi ,{\delta }_2\varphi )={\delta }_2\theta (\varphi ;{\delta }_1\varphi )-{\delta }_1\theta (\varphi ;{\delta }_2\varphi )\mathrm{.}$

For a finite oriented smooth singular three-chain $\mathrm{\Sigma }$, define the integrated two-form by ${\mathrm{\Omega }}_{\mathrm{\Sigma }}({\delta }_1\varphi ,{\delta }_2\varphi )={\int }_{\mathrm{\Sigma }}\omega$.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §§2.2–2.3

Covariant field families and variations

A smooth differential $k$-form

The exterior derivative by the invariant vector-field formula

Smooth singular chain and cochain complexes

Integral of a form over a smooth singular simplex

The definition uses finite-dimensional smooth families to specify all variations and their derivatives. It does not assume a global smooth infinite-dimensional configuration or solution manifold. No asymptotic falloff, boundary condition, or uniqueness of $\theta$ is included.

The sign conventions of the cited sources require care. Iyer--Wald's equation (44) uses the ordering in this definition. Compère--Fiorucci's equation (1.25) is written as a field-space form; after the contraction convention of (1.26), their classical first variation has the same $+d\theta$ potential convention as this definition. Their component formula in equation (1.28) is ${\delta }_1\theta ({\delta }_2\varphi )-{\delta }_2\theta ({\delta }_1\varphi )$, the negative of the current defined here. Harlow--Wu also use the classical $+d\mathrm{\Theta }$ convention and define $\omega =\delta (\mathrm{\Theta }-dC)$ in equation (2.23) (with $C=0$ in their scalar example, equations (3.8) and (3.13)). When the potential is matched as $\theta =\mathrm{\Theta }-dC$, standard ordered evaluation of the field-space exterior derivative gives ${\omega }_{\mathrm{H}\mathrm{W}}({\delta }_1,{\delta }_2)={\delta }_1\theta ({\delta }_2\varphi )-{\delta }_2\theta ({\delta }_1\varphi )$, again the negative of the convention here. Translate these signs when using their formulas.

Use the chosen first-variation decomposition and its Iyer–Wald current ordering. For a smooth two-parameter family with commuting variations ${\delta }_1,{\delta }_2$,

$d_M\omega ({\delta }_1\varphi ,{\delta }_2\varphi )=({\delta }_1{E}_{\varphi })[{\delta }_2\varphi ]-({\delta }_2{E}_{\varphi })[{\delta }_1\varphi ]\mathrm{.}$

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §2.2

Covariant presymplectic current

Covariant field families and variations

For a smooth finite-dimensional family of field configurations, let $\mathrm{\Theta }$ be the parameter-space one-form with values in spacetime three-forms from the chosen variational decomposition. For every finite oriented smooth singular three-chain $\mathrm{\Sigma }$, the integrated two-form ${\mathrm{\Omega }}_{\mathrm{\Sigma }}={\int }_{\mathrm{\Sigma }}(-d_P\mathrm{\Theta })$ is closed on the parameter space: $d_P{\mathrm{\Omega }}_{\mathrm{\Sigma }}=0$. This is a field-space identity and requires no equations of motion. If a smooth solution family is specified, its restricted form is also closed.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §2.2

Covariant presymplectic current

Let $\mathcal{L}$ have a chosen first-variation decomposition with Euler functional $E_{\varphi }$. Take a smooth two-parameter field family and its commuting coordinate variations ${\delta }_1,{\delta }_2$. If $E_{\varphi }=0$ and each variation satisfies the linearized equation, meaning $({\delta }_i{E}_{\varphi })[\eta ]=0$ for every smooth test variation $\eta$ with the input held fixed as in the definition, then $d_M\omega =0$. For a finite oriented smooth singular four-chain $C$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, where the ${\mathrm{\Sigma }}_i$ are oriented three-chains and $B$ is the remaining oriented boundary chain,

${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_2}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_1}=-{\int }_B\omega \mathrm{.}$

Thus on-shell closure alone gives independence between cuts only when the side flux vanishes.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §2.2

The chain is finite and compact in the sense of being a finite sum of compact simplex domains. No limit at spatial infinity, falloff assumption, or convergence of an exhaustion is included.

For a fixed smooth two-parameter family of commuting variations, replace the Lagrangian and its selected potential by

${\mathcal{L}}^{\mathrm{\prime }}=\mathcal{L}+d_M\mu ,{\theta }^{\mathrm{\prime }}(\varphi ;\delta \varphi )=\theta (\varphi ;\delta \varphi )+\delta \mu (\varphi ;\delta \varphi )+d_{M}Y(\varphi ;\delta \varphi ),$

where $\mu$ is a smooth local spacetime three-form and $Y(\varphi ;\delta \varphi )$ is a smooth local spacetime two-form, linear in the variation and its spacetime derivatives. Their local coefficients depend smoothly on the spacetime coordinates, finitely many derivatives of $\varphi$ and any fixed smooth background fields, and $Y$ depends linearly on finitely many derivatives of $\delta \varphi$. Then

${\omega }^{\mathrm{\prime }}-\omega =d_M({\delta }_{2}Y({\delta }_1\varphi )-{\delta }_{1}Y({\delta }_2\varphi ))\mathrm{.}$

Consequently, for a finite oriented smooth singular three-chain $\mathrm{\Sigma }$, the integrated form changes by the integral of this two-form over $\mathrm{\partial }\mathrm{\Sigma }$.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.3

Covariant phase space with boundaries, §2.2

Covariant presymplectic current

Covariant field families and variations

Stokes theorem for smooth singular chains

The exterior derivative squares to zero

The exact corner term need not vanish on a hypersurface with boundary. Covariance alone supplies no boundary condition that removes it.

The sign in the displayed corner term follows from the specified current ordering and the explicit variation in step 2.1. Iyer--Wald's equations (43)--(44) give the potential ambiguity and this current ordering, but their printed integrated ambiguity formula (47) has the opposite sign under those conventions; equation (47) is therefore not used for the sign here.

Let $M$ be an oriented smooth four-manifold, let $\pi :E\to M$ be the chosen smooth field bundle, and let $\mathcal{L}(\varphi )$ be a local Lagrangian four-form for its smooth sections, with a chosen first-variation decomposition $\delta \mathcal{L}=E_{\varphi }[\delta \varphi ]+d_M\theta (\varphi ;\delta \varphi )$. Let $\xi$ be a smooth spacetime vector field with local flow ${\mathrm{\Phi }}_t$. Suppose there is a smooth local lift ${\tilde{\mathrm{\Phi }}}_t$ to fibre-preserving local diffeomorphisms of $E$ over ${\mathrm{\Phi }}_t$, with ${\tilde{\mathrm{\Phi }}}_0=\mathrm{i}\mathrm{d}$ and the local flow composition law. On the common domain define its induced pullback action on sections by $(A_t\varphi )(x):={\tilde{\mathrm{\Phi }}}_{-t}(\varphi ({\mathrm{\Phi }}_t(x)))\mathrm{.}$ Write ${\delta }_{\xi }\varphi :={\frac{d}{dt}\mid }_{t=0}{A}_t\varphi ,$ the vertical field variation determined by this smooth family; at each $x$ it lies in $\ker (d\pi {)}_{\varphi (x)}$, the tangent space to the fibre $E_x$. In the tensor-field case where $A_t\varphi ={\mathrm{\Phi }}_t^{\ast }\varphi$, it is the Lie derivative ${\mathcal{L}}_{\xi }\varphi$ defined by the local-flow derivative. Assume the action preserves the chosen field domain, and assume covariance in the precise sense $\mathcal{L}(A_t\varphi )={\mathrm{\Phi }}_t^{\ast }\mathcal{L}(\varphi )$ wherever both sides are defined; thus its derivative satisfies ${\delta }_{\xi }\mathcal{L}={\mathcal{L}}_{\xi }\mathcal{L}$. Define the Noether current three-form

$J_{\xi }:=\theta (\varphi ;{\delta }_{\xi }\varphi )-{\iota }_{\xi }\mathcal{L}\mathrm{.}$

Then

$d_M{J}_{\xi }=-E_{\varphi }[{\delta }_{\xi }\varphi ]\mathrm{.}$

Here a solution means a field configuration $\varphi$ at which the Euler functional vanishes, that is, $E_{\varphi }[\eta ]=0$ for every smooth variation $\eta$ in its domain. Thus $J_{\xi }$ is closed on solutions. For every chart $\chi :U\to V\subseteq {\mathbb{R}}^4$ whose open image $V$ is star-shaped, a solution has a local two-form $Q_{\xi }\in {\mathrm{\Omega }}^2(U)$ with $J_{\xi }{\mathrm{\mid }}_U=d_M{Q}_{\xi }$.

Some Properties of Noether Charge, §4

Advanced Lectures on General Relativity, §1.3.3

Covariant phase space with boundaries, §§2.3–2.4

Covariant field families and variations

Covariant presymplectic current

Closed differential forms are exact on star-shaped coordinate domains

Cartan's magic formula

The Lie derivative of a differential form

The Lie derivative of a tensor field

The pullback of a differential form

The exterior derivative commutes with pullback

The primitive is local. This proposition gives no global Noether charge without a separate cohomological argument. It asserts a classical identity only; it constructs no measure, Hilbert space, or anomaly-free quantization.

The current sign in the cited sources is not uniform. Iyer--Wald's equation (49) and Harlow--Wu's equation (2.44) use the current defined here. Compère--Fiorucci's equation (1.59) uses $J_{\xi }^{\mathrm{C}\mathrm{F}}:={\iota }_{\xi }\mathcal{L}-\theta (\varphi ;{\mathcal{L}}_{\xi }\varphi )$, which is $-J_{\xi }$ in the tensor-field case; its on-shell current identity and local primitive therefore require the same sign conversion. The proof above uses only the explicit local hypotheses stated here.

Let $\mathcal{S}$ be a specified finite-dimensional smooth solution manifold with specified admissible variations and boundary conditions, and let $\mathrm{\Omega }$ be a smooth closed two-form on $\mathcal{S}$, obtained by restricting an integrated form ${\mathrm{\Omega }}_{\mathrm{\Sigma }}$ from the chosen covariant variation data. The pair $(\mathcal{S},\mathrm{\Omega })$ is presymplectic when $\mathrm{\Omega }$ is allowed to be degenerate. At $p\in \mathcal{S}$, its pointwise null space is $\ker {\mathrm{\Omega }}_p:=\{v\in T_p\mathcal{S}:{\mathrm{\Omega }}_p(v,w)=0\text{ for every }w\in T_p\mathcal{S}\}\mathrm{.}$ A presymplectic reduction is defined here only when there is a specified smooth Hausdorff manifold $\mathcal{R}$ and a surjective submersion $q:\mathcal{S}\to \mathcal{R}$ with connected fibers such that $\ker (d{q}_p)=\ker {\mathrm{\Omega }}_p(p\in \mathcal{S}),$ where $\ker (d{q}_p)$ consists of tangent vectors mapped to zero by the differential. This condition makes the quotient fibers exactly the connected directions annihilated by $\mathrm{\Omega }$. The reduction further requires a smooth closed nondegenerate two-form $\bar{\mathrm{\Omega }}$ on $\mathcal{R}$ satisfying $q^{\ast }\bar{\mathrm{\Omega }}=\mathrm{\Omega }$. For a smooth vector field $X$ and smooth function $H$ on a specified phase space, the Hamiltonian convention is ${\iota }_X\mathrm{\Omega }=dH$.

Symplectic form and symplectic manifold

Symplectic reduction of a coisotropic vector subspace

Hamiltonian vector field and Hamiltonian function

Covariant presymplectic current

Closedness of the integrated presymplectic form on smooth field families

Symplectic form and symplectic manifold

Symplectic reduction of a coisotropic vector subspace

Hamiltonian vector field and Hamiltonian function

Poisson bracket on a symplectic manifold

Immersions, submersions, and constant-rank maps

The finite-dimensional proposition for a coisotropic vector subspace applies to tangent linear models only. This definition does not establish a smooth quotient of an infinite-dimensional gravitational solution space. No falloff, boundary condition, constant-rank property, or existence of such a quotient is asserted here. These classical identities construct no measure, Hilbert space, or anomaly-free quantization. Thermal first-law and entropy interpretations are assigned to thermodynamics and general relativity; topological quantum field theory quantization is assigned to quantum field theory.

Let $(\mathcal{S},\mathrm{\Omega })$ be exactly a finite-dimensional smooth solution manifold with its smooth closed two-form as specified in Admissible presymplectic reduction and Hamiltonian generators. Let $M$ be the spacetime and $\pi :E\to M$ the chosen field bundle. For a fixed smooth spacetime vector field $\xi$ with local flow ${\mathrm{\Phi }}_t$, suppose a smooth local lift ${\tilde{\mathrm{\Phi }}}_t$ to fibre-preserving local diffeomorphisms of $E$ over ${\mathrm{\Phi }}_t$ is specified, with the local flow composition law. Define its action on a field section by $(A_t\varphi )(x):={\tilde{\mathrm{\Phi }}}_{-t}(\varphi ({\mathrm{\Phi }}_t(x)))\mathrm{.}$ Assume the joint local action $(t,p)\mapsto A_t(p)$ is smooth, preserves the selected solution manifold and boundary conditions, and induces maps $A_t:\mathcal{S}\to \mathcal{S}$. For $p\in \mathcal{S}$, define the induced phase-space vector field by $X_{\xi }(p):={\frac{d}{dt}\mid }_{t=0}{A}_t(p)\in T_p\mathcal{S}\mathrm{.}$ In particular, the association $\xi \mapsto X_{\xi }$ is specified by the lifted action and its preservation hypotheses; it is not inferred merely from the name of a diffeomorphism generator. Define the one-form ${\alpha }_{\xi }$ on $\mathcal{S}$ by $({\alpha }_{\xi }{)}_p(v):={\mathrm{\Omega }}_p(X_{\xi }(p),v),v\in T_p\mathcal{S}\mathrm{.}$ It is integrable on $\mathcal{S}$ when there is a globally defined smooth function $H_{\xi }:\mathcal{S}\to \mathbb{R}$ with $d_{\mathcal{S}}{H}_{\xi }={\alpha }_{\xi }$, using the convention ${\iota }_{X_{\xi }}\mathrm{\Omega }=d_{\mathcal{S}}{H}_{\xi }$. For a smooth map $f:P\to \mathcal{S}$ from a finite-dimensional smooth parameter manifold and a chart $\chi :U\to V\subseteq {\mathbb{R}}^m$ with $V$ open and star-shaped, if the pulled-back one-form $f^{\ast }{\alpha }_{\xi }$ is closed on $U$, then its coordinate representative $({\chi }^{-1}{)}^{\ast }(f^{\ast }{\alpha }_{\xi })$ has a local primitive on $V$. Here $m\ge 1$; if $m=0$, every one-form on the chart is zero and is locally exact. Global integrability on $\mathcal{S}$ is the stronger exactness condition.

Advanced Lectures on General Relativity, §1.4.2

Hamiltonian vector field and Hamiltonian function

Admissible presymplectic reduction and Hamiltonian generators

Closed differential forms are exact on star-shaped coordinate domains

Hamiltonian vector field and Hamiltonian function

The pullback of a differential form

The exterior derivative commutes with pullback

A diffeomorphism direction is null exactly when ${\iota }_{X_{\xi }}\mathrm{\Omega }=0$. If an admissible boundary charge has nonzero variation, that direction is not null and survives a reduction that quotients only by null directions. Neither covariance nor on-shell closure alone guarantees integrability. The potential ambiguity can change boundary terms in $\mathrm{\Omega }$, so integrability must be checked after the boundary conditions and the choice of $\mu$ and $Y$ in the potential ambiguity are specified.