Presymplectic Systems and the Covariant Phase-Space Ambiguity Map

A closed two-form may have directions that pair to zero with every allowed variation. Those kernel directions make Hamiltonian evolution conditional rather than automatic: a Hamiltonian vector field exists only when the Hamiltonian is constant along the kernel, and any solution is unique only modulo that kernel. In a gauge field theory the same geometry appears on the space of classical solutions, but degeneracy alone does not identify a physical redundancy. Boundary conditions decide whether a gauge transformation is truly null or instead has a nonzero boundary pairing.

The covariant construction starts with the boundary term in the first variation of the Lagrangian. Its field-space exterior derivative is a local presymplectic current. That current is closed in field space by construction and closed in spacetime only on solutions and linearized solutions. After integration over a hypersurface it is independent of the hypersurface only when lateral symplectic flux vanishes or is otherwise accounted for. Finally, the usual freedom to change the Lagrangian and its boundary-potential current can leave a codimension-two term at the edge of the hypersurface. These are the kernel, boundary, and ambiguity mechanisms behind the principal question.

Required background. Field Variations and Boundary Terms supplies the bulk–boundary first-variation identity, while Symplectic Forms, Hamiltonian Flows, and Poisson Brackets supplies the symplectic sign package and the kernel test for two-forms.

Helpful background. Constraints, Dirac Brackets, and Symplectic Reduction supplies regular characteristic quotients, while Differential Forms, Integration, Orientation, and Stokes Theorem supplies the spacetime-form calculus.

Spacetime and field-space exterior derivatives

Two exterior derivatives occur, and they do different jobs:

• $\mathrm d$ is the exterior derivative on spacetime;
• $\boldsymbol{\delta}$ is the exterior derivative on a space of fields.

For the local calculations below,

$$\mathrm d^2=\boldsymbol{\delta}^{\,2}=0, \qquad \boldsymbol{\delta}\,\mathrm d =\mathrm d\,\boldsymbol{\delta}.$$

On biforms, $\wedge$ denotes the exterior product in the relevant factor: $\mathrm d$ obeys the spacetime-degree Leibniz rule and $\boldsymbol{\delta}$ obeys the field-space-degree Leibniz rule. Thus the wedge in $\boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a$ is a field-space wedge, while $F\wedge\star F$ is a spacetime wedge. Mixed expressions carry both degrees.

We first use commuting, field-independent variations. If $V_1$ and $V_2$ are general field-space vector fields, the exterior derivative of a field-space one-form includes the commutator term

$$\begin{aligned} (\boldsymbol{\delta}\boldsymbol\Theta)(V_1,V_2) ={}&V_1[\boldsymbol\Theta(V_2)] -V_2[\boldsymbol\Theta(V_1)]\\ &-\boldsymbol\Theta([V_1,V_2]). \end{aligned}$$

That last term cannot be dropped for field-dependent gauge parameters. Their developed treatment lies beyond the present construction.

The sign is inherited from the earlier canonical page:

$$\boldsymbol\omega =-\boldsymbol{\delta}\boldsymbol\Theta, \qquad \Omega_\Sigma =-\boldsymbol{\delta}\Theta_\Sigma.$$

For commuting variations this means

$$\begin{aligned} \boldsymbol\omega(\delta_1,\delta_2) ={}& \delta_2[\boldsymbol\Theta(\delta_1)] -\delta_1[\boldsymbol\Theta(\delta_2)]. \end{aligned}$$

Some sources define the current with the opposite order of the two variations. Every current, flux, and boundary-pairing sign then reverses together; mixing the two packages would be inconsistent.

Presymplectic geometry as a compatibility problem

Let $P$ be a finite-dimensional smooth manifold. A presymplectic form $\varpi$ is a closed two-form,

$$\mathrm d\varpi=0,$$

which is allowed to be degenerate. Its kernel at $x$ is

$$\begin{aligned} K_x &=\ker\varpi_x\\ &=\left\{ v\in T_xP: \varpi_x(v,w)=0 \text{ for every }w\in T_xP \right\}. \end{aligned}$$

Some authors build constant rank into the word “presymplectic.” Here a form of constant rank will be called regular presymplectic. That hypothesis makes the spaces $K_x$ into a smooth vector subbundle $K\subset TP$. Without it, the dimension of the kernel can jump.

Introduce the bundle map

$$\begin{aligned} \varpi^\flat:TP&\longrightarrow T^*P,\\ v&\longmapsto\iota_v\varpi. \end{aligned}$$

Pointwise finite-dimensional linear algebra gives

$$\operatorname{im}\varpi_x^\flat=K_x^\circ,$$

where $K_x^\circ$ is the annihilator of the kernel. Therefore the Hamiltonian equation

$$\iota_X\varpi=\mathrm dH$$

can be solved at $x$ only if

$$\mathrm dH_x(Y)=0 \qquad \text{for every }Y\in K_x.$$

Under the constant-rank hypothesis this condition also gives a smooth local solution. If $X_0$ is one solution, then every other solution has the form

$$X=X_0+Y, \qquad Y\in\Gamma(K).$$

Thus degeneracy creates two logically separate effects: failure of existence when $\mathrm dH$ does not annihilate the kernel, and nonuniqueness when it does.

A three-dimensional model

Take

$$P=\mathbb R^3_{q,p,z}, \qquad \varpi=\mathrm dq\wedge\mathrm dp.$$

The kernel is

$$K=\operatorname{span}\!\left\{ \frac{\partial}{\partial z} \right\}.$$

For

$$X =a\frac{\partial}{\partial q} +b\frac{\partial}{\partial p} +c\frac{\partial}{\partial z},$$

contraction gives

$$\iota_X\varpi =a\,\mathrm dp-b\,\mathrm dq.$$

Comparing this with

$$\mathrm dH =H_q\,\mathrm dq +H_p\,\mathrm dp +H_z\,\mathrm dz$$

shows that a solution exists exactly when $H_z=0$. Then

$$a=H_p, \qquad b=-H_q,$$

while $c$ remains arbitrary. For example, $H=z$ has no Hamiltonian vector field. For

$$H=\frac12(q^2+p^2),$$

all solutions are

$$X =p\frac{\partial}{\partial q} -q\frac{\partial}{\partial p} +c\frac{\partial}{\partial z}.$$

The $z$ direction is invisible to $\varpi$. It may be quotiented only after the physical model declares translations in $z$ redundant.

When a smooth quotient exists

For sections $Y,Z\in\Gamma(K)$, closedness gives

$$\mathcal L_Y\varpi =\mathrm d(\iota_Y\varpi) +\iota_Y(\mathrm d\varpi) =0.$$

Consequently,

$$\iota_{[Y,Z]}\varpi =\mathcal L_Y(\iota_Z\varpi) -\iota_Z(\mathcal L_Y\varpi) =0.$$

The regular kernel is therefore involutive and has local characteristic leaves. If the leaf space $P/K$ is a smooth Hausdorff manifold and the projection $\pi:P\to P/K$ is a surjective submersion, there is a unique two-form $\varpi_{\mathrm{red}}$ satisfying

$$\pi^*\varpi_{\mathrm{red}}=\varpi.$$

It is closed and nondegenerate. A Hamiltonian descends precisely when it is constant along the characteristic leaves. These regularity conditions are substantive: a leaf space can fail to be Hausdorff, and a variable-rank kernel need not even define a vector bundle. Gotay and Nester 1979, pp. 272–279 give the regular presymplectic reduction statement and its constraint-theory interpretation.

A compact near miss is

$$\varpi=x\,\mathrm dx\wedge\mathrm dy \quad\text{on }\mathbb R^3_{x,y,z}.$$

It is closed, but its rank is two for $x\ne0$ and zero for $x=0$. The kernel jumps from one dimension to three, so the regular quotient theorem does not apply. One should expect a stratified or singular description rather than a single smooth reduced manifold.

Most importantly, the mathematical kernel is not by itself a definition of gauge equivalence. A physical theory must specify which null directions relate descriptions of the same state, which preserve the admitted boundary data, and which may instead change measurable boundary information.

From a first variation to a current on solution space

Let $M$ be an oriented $d$-dimensional Lorentzian spacetime, and fix a spacetime region and its boundary. Let $\mathcal F$ be a chosen space of smooth field histories $\phi^a$ obeying specified boundary conditions. The following calculus is formal unless a suitable infinite-dimensional manifold and convergence properties have been established.

The relevant objects have two degrees:

Object	Spacetime degree	Field-space degree
$\boldsymbol L$	$d$	$0$
$\boldsymbol E_a\boldsymbol{\delta}\phi^a$	$d$	$1$
$\boldsymbol\Theta$	$d-1$	$1$
$\boldsymbol\omega$	$d-1$	$2$

The first variation has the local form

$$\boldsymbol{\delta}\boldsymbol L =\boldsymbol E_a\boldsymbol{\delta}\phi^a +\mathrm d\boldsymbol\Theta.$$

Here $\boldsymbol E_a=0$ are the Euler–Lagrange equations, while $\boldsymbol\Theta$ is a spacetime $(d-1)$-form and a one-form on field space. Applying $\boldsymbol{\delta}$ once more gives

$$\begin{aligned} 0 =\boldsymbol{\delta}^{\,2}\boldsymbol L &= \boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a +\mathrm d( \boldsymbol{\delta}\boldsymbol\Theta)\\ &= \boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a -\mathrm d\boldsymbol\omega. \end{aligned}$$

Therefore

$$\boxed{ \mathrm d\boldsymbol\omega = \boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a. }$$

Let $\mathcal S\subset\mathcal F$ be a patch of the solution space. A tangent vector $\delta\phi\in T_\phi\mathcal S$ must satisfy both the linearized Euler–Lagrange equations and the linearized boundary conditions. On a background solution, for two such tangent vectors,

$$\begin{aligned} (\mathrm d\boldsymbol\omega)(\delta_1,\delta_2) ={}& \delta_1\boldsymbol E_a\,\delta_2\phi^a\\ &-\delta_2\boldsymbol E_a\,\delta_1\phi^a =0. \end{aligned}$$

This is spacetime closure of the current. It is an on-shell, linearized-equation statement.

There is a different closure statement:

$$\boldsymbol{\delta}\boldsymbol\omega =-\boldsymbol{\delta}^{\,2}\boldsymbol\Theta =0.$$

This is field-space closure. It follows from nilpotence and does not assert anything about flux through a spacetime boundary. Confusing these two statements is one of the most common errors in the formalism. Iyer and Wald 1994, § 3, equations (20) and (40)–(47) develop the covariant current and its on-shell closure.

For a fixed oriented hypersurface $\Sigma$, define

$$\Theta_\Sigma =\int_\Sigma\boldsymbol\Theta, \qquad \Omega_\Sigma =\int_\Sigma\boldsymbol\omega.$$

If field-space differentiation can be exchanged with integration and $\Sigma$ is not itself field dependent, then

$$\Omega_\Sigma =-\boldsymbol{\delta}\Theta_\Sigma, \qquad \boldsymbol{\delta}\Omega_\Sigma=0.$$

The result is generally presymplectic rather than symplectic. Its kernel must still be determined using the chosen solution space and boundary conditions.

The sign agrees with the equal-time scalar convention. When

$$\Theta_\Sigma =\int_\Sigma\mathrm d^{d-1}x\, \pi\,\boldsymbol{\delta}\phi,$$

one obtains

$$\begin{aligned} \Omega_\Sigma &=-\boldsymbol{\delta}\Theta_\Sigma\\ &=\int_\Sigma\mathrm d^{d-1}x\, \boldsymbol{\delta}\phi \wedge\boldsymbol{\delta}\pi, \end{aligned}$$

exactly as on the canonical phase-space page.

Hypersurface dependence is a flux question

Let $R$ be an oriented spacetime slab whose boundary is

$$\partial R =\Sigma_2\cup(-\Sigma_1)\cup B.$$

Here $\Sigma_1$ and $\Sigma_2$ are homologous spacelike hypersurfaces, and $B$ is the lateral boundary with its induced orientation. On a solution and for admitted linearized variations, Stokes’ theorem gives

$$\begin{aligned} 0 =\int_R\mathrm d\boldsymbol\omega &= \int_{\Sigma_2}\boldsymbol\omega -\int_{\Sigma_1}\boldsymbol\omega +\int_B\boldsymbol\omega. \end{aligned}$$

Hence

$$\boxed{ \Omega_{\Sigma_2}-\Omega_{\Sigma_1} =-\int_B\boldsymbol\omega. }$$

On-shell closure of the local current is therefore not enough to make $\Omega_\Sigma$ independent of $\Sigma$. One also needs zero lateral symplectic flux, appropriate falloff, or a justified boundary completion whose contribution cancels that flux. The variations on both slices must belong to the same boundary-value problem. Harlow and Wu 2020, §§ 2.2 and 3.3 analyze this boundary-sensitive completion and the associated flux condition.

This distinction separates three questions:

1. Is $\boldsymbol\omega$ closed in spacetime for the chosen variations?
2. Does its flux through $B$ vanish?
3. Is the resulting $\Omega_\Sigma$ nondegenerate after justified reduction?

A “yes” to one does not imply a “yes” to either of the others.

The covariant phase-space ambiguity map

The local first-variation identity does not select a unique representative. Let $\boldsymbol\ell$ be a spacetime $(d-1)$-form and field-space zero-form, and let $\boldsymbol\beta$ be a spacetime $(d-2)$-form and field-space one-form. Consider

$$\boldsymbol L' =\boldsymbol L+\mathrm d\boldsymbol\ell,$$

with the compatible potential-current change

$$\boldsymbol\Theta' = \boldsymbol\Theta +\boldsymbol{\delta}\boldsymbol\ell +\mathrm d\boldsymbol\beta.$$

Using the site sign and commuting differentials,

$$\begin{aligned} \boldsymbol\omega' &=-\boldsymbol{\delta}\boldsymbol\Theta'\\ &=\boldsymbol\omega -\mathrm d( \boldsymbol{\delta}\boldsymbol\beta). \end{aligned}$$

The $\boldsymbol{\delta}\boldsymbol\ell$ term cancels because $\boldsymbol{\delta}^{\,2}=0$. On a fixed hypersurface,

$$\begin{aligned} \Theta_\Sigma' ={}& \Theta_\Sigma +\boldsymbol{\delta} \int_\Sigma\boldsymbol\ell +\int_{\partial\Sigma}\boldsymbol\beta,\\ \Omega_\Sigma' ={}& \Omega_\Sigma -\boldsymbol{\delta} \int_{\partial\Sigma}\boldsymbol\beta. \end{aligned}$$

For commuting variations, one may write

$$\begin{aligned} (\boldsymbol{\delta}\boldsymbol\beta)_{12} ={}& \delta_1[\boldsymbol\beta(\delta_2)] -\delta_2[\boldsymbol\beta(\delta_1)],\\ \boldsymbol\omega'_{12} ={}& \boldsymbol\omega_{12} -\mathrm d( \boldsymbol{\delta}\boldsymbol\beta)_{12}. \end{aligned}$$

The consequences are different for the two changes:

• Adding $\mathrm d\boldsymbol\ell$ leaves the bulk Euler–Lagrange equations and the local presymplectic current unchanged under the compatible shift. It can nevertheless change the action’s boundary term, the admissible boundary data, and therefore the actual solution space.
• Adding $\mathrm d\boldsymbol\beta$ to the potential current is invisible after integration over a closed $\Sigma$. When $\partial\Sigma\ne\varnothing$, it changes $\Omega_\Sigma$ by a codimension-two field-space exact term.
• A globally closed but nonexact representative cannot always be described by a single global $\boldsymbol\beta$. Topology can therefore add a separate global question.

This is an ambiguity of representatives only while the field space, boundary action, boundary conditions, and corner degrees of freedom are held fixed. Changing any of those is a physical change of the problem, not merely a relabeling. A bounded variational principle may also supply a genuine corner potential whose contribution is needed to make the symplectic flux well controlled; that input must not be confused with the freely chosen $\boldsymbol\beta$ above.

Gauge kernel versus boundary pairing

Let $R_\epsilon$ be the field-space vector generated by a field-independent gauge parameter $\epsilon$. Before quotienting, verify that it preserves the equations, the allowed field configurations, and the boundary conditions. In many local gauge theories the current identity has the schematic on-shell form

$$\left. \iota_{R_\epsilon}\boldsymbol\omega \right|_{\mathcal S} =\mathrm d\boldsymbol k_\epsilon,$$

where $\boldsymbol k_\epsilon$ is a spacetime $(d-2)$-form and a field-space one-form. Integration gives

$$\iota_{R_\epsilon}\Omega_\Sigma =\int_{\partial\Sigma}\boldsymbol k_\epsilon.$$

This produces three distinct outcomes:

• If the boundary integral vanishes for every admitted variation, then $R_\epsilon\in\ker\Omega_\Sigma$ and is a candidate gauge redundancy.
• If it is nonzero but is an exact field-space one-form, the transformation is Hamiltonian and acts nontrivially on the phase space. It must not be quotiented as a null direction.
• If the boundary one-form is not integrable, no Hamiltonian for that transformation exists on the proposed phase space without further boundary or flux input.

Even the first case still requires a physical identification: accidental zero modes can lie in a mathematical kernel. Conversely, invariance of $\Omega_\Sigma$ under a gauge transformation is weaker than degeneracy along its orbit. A two-form can be invariant under a transformation while pairing that transformation nontrivially with other variations. Assanioussi et al. 2024, §§ 2.2–2.3 and 3.1–3.3 separate gauge invariance, degeneracy, and nonzero boundary pairings.

Bounded Maxwell theory

Consider source-free Maxwell theory on an oriented four-dimensional Minkowski region, with the site’s $(+---)$ metric convention. Let a spacelike slice $\Sigma$ have smooth boundary $\partial\Sigma$. The fields, their variations, and the boundary conditions are assumed smooth enough for Stokes’ theorem, and tangent variations obey the linearized Maxwell equations and the same linearized boundary conditions. The background metric and its Hodge star are fixed, so $\boldsymbol{\delta}(\star F)=\star\boldsymbol{\delta}F$.

With $F=\mathrm dA$,

$$\boldsymbol L =-\frac12F\wedge\star F.$$

Its variation is

$$\boldsymbol{\delta}\boldsymbol L =-\boldsymbol{\delta}A\wedge\mathrm d\star F -\mathrm d( \boldsymbol{\delta}A\wedge\star F),$$

so

$$\boldsymbol\Theta =-\boldsymbol{\delta}A\wedge\star F.$$

The site-sign presymplectic current is

$$\boldsymbol\omega =-\boldsymbol{\delta}A \wedge\star\boldsymbol{\delta}F.$$

Let

$$\operatorname{vol} =\mathrm dt\wedge\mathrm dx^1 \wedge\mathrm dx^2\wedge\mathrm dx^3, \qquad (\mathrm d^3x)_\mu =\iota_{\partial_\mu}\operatorname{vol}.$$

With this orientation, write

$$\boldsymbol\Theta =\boldsymbol\Theta^\mu(\mathrm d^3x)_\mu, \qquad \boldsymbol\omega =\boldsymbol\omega^\mu(\mathrm d^3x)_\mu.$$

The component currents, using the boundary current from the variation page, are

$$\boldsymbol\Theta^\mu =-F^{\mu\nu}\boldsymbol{\delta}A_\nu, \qquad \boldsymbol\omega^\mu = \boldsymbol{\delta}F^{\mu\nu} \wedge\boldsymbol{\delta}A_\nu.$$

On two commuting variations,

$$\begin{aligned} \boldsymbol\omega^\mu(\delta_1,\delta_2) ={}& \delta_1F^{\mu\nu}\delta_2A_\nu\\ &-\delta_2F^{\mu\nu}\delta_1A_\nu. \end{aligned}$$

The linearized equations $\partial_\mu\delta_iF^{\mu\nu}=0$, together with antisymmetry of $\delta_iF^{\mu\nu}$, give

$$\partial_\mu \boldsymbol\omega^\mu(\delta_1,\delta_2) =0.$$

This is the Maxwell instance of on-shell spacetime closure.

Now choose a future-oriented $t=\text{constant}$ slice, give $\partial\Sigma$ its outward induced orientation, and set

$$E^i=-F^{0i}.$$

The pullback to $\Sigma$ gives

$$\Theta_\Sigma =\int_\Sigma\mathrm d^3x\, E^i\boldsymbol{\delta}A_i,$$

and therefore

$$\Omega_\Sigma =\int_\Sigma\mathrm d^3x\, \boldsymbol{\delta}A_i \wedge\boldsymbol{\delta}E^i.$$

This round trip fixes the signs against the canonical convention rather than borrowing them piecemeal from a source with the opposite current sign.

Let $n_i$ be the outward unit normal to $\partial\Sigma$ within $\Sigma$ and write $E^n=n_iE^i$.

Let a field-independent gauge parameter $\lambda$ generate

$$R_\lambda A_i=\partial_i\lambda, \qquad R_\lambda E^i=0.$$

For an admitted tangent variation $\delta$,

$$\begin{aligned} (\iota_{R_\lambda}\Omega_\Sigma)(\delta) &=\Omega_\Sigma(R_\lambda,\delta)\\ &=\int_\Sigma\mathrm d^3x\, (\partial_i\lambda)\,\delta E^i\\ &=-\int_\Sigma\mathrm d^3x\, \lambda\,\partial_i\delta E^i +\int_{\partial\Sigma}\mathrm dS\, \lambda\,\delta E^n. \end{aligned}$$

The linearized Gauss law removes the bulk term, leaving

$$\boxed{ (\iota_{R_\lambda}\Omega_\Sigma)(\delta) = \int_{\partial\Sigma}\mathrm dS\, \lambda\,\delta E^n. }$$

If $\partial\Sigma$ is empty, if $\lambda$ vanishes there, or if the allowed boundary variations force this integral to vanish, then $R_\lambda$ is a null direction. With fixed pullback of $A$ at a timelike boundary, preservation of the boundary data requires the tangential derivative of $\lambda$ to vanish, so $\lambda$ may be constant on each boundary component. Linearized Gauss law makes one common constant on all components null. If there are two or more components, however, independent constants can pair with varying relative electric fluxes, so the transformation need not be null.

The example isolates the central fact: the same local formula $A\mapsto A+\mathrm d\lambda$ can be a redundancy for one class of boundary data and a nontrivial phase-space transformation for another. Whether the remaining boundary one-form is integrable, how a finite quantity is chosen, and what algebra it obeys are additional physical questions.

Where the construction stops

The formulas above are conditional. Stop and refine the setup in any of the following situations:

• the rank of the presymplectic form changes, so the kernel distribution is singular;
• the characteristic leaf space is non-Hausdorff, stratified, or otherwise not a smooth manifold;
• the proposed tangent variations do not satisfy the same linearized equations and boundary conditions;
• the lateral symplectic flux is nonzero, so different hypersurfaces give different two-forms;
• a corner ambiguity has not been fixed for a bounded surface;
• a gauge transformation fails to preserve the admitted boundary data or has a nonzero boundary pairing;
• a boundary one-form is mistaken for an integrable Hamiltonian without checking field-space closure and global topology;
• the gauge parameter depends on the dynamical fields, so commutator and parameter-variation terms enter;
• formal differentiation, integration, or quotienting has not been justified on the chosen infinite-dimensional function space.

Nothing here constructs a Poisson inverse on the unreduced space, proves that reduction commutes with quantization, supplies an edge-mode prescription, or determines a charge algebra.

Exercises

1. Test existence and uniqueness

For

$$\varpi=\mathrm dq\wedge\mathrm dp \quad\text{on }\mathbb R^3_{q,p,z},$$

find the obstruction to solving $\iota_X\varpi=\mathrm dH$. Compare $H=z$ with $H=(q^2+p^2)/2$.

Solution

Writing $X=a\partial_q+b\partial_p+c\partial_z$ gives

$$\iota_X\varpi=a\,\mathrm dp-b\,\mathrm dq.$$

There is no $\mathrm dz$ component, so solvability requires $H_z=0$. Thus $H=z$ is obstructed. For $H=(q^2+p^2)/2$,

$$a=p, \qquad b=-q,$$

and $c$ is arbitrary. Existence is controlled by annihilation of the kernel, while uniqueness fails precisely along $\partial_z$.

2. Locate the failed regularity hypothesis

Why does

$$\varpi=x\,\mathrm dx\wedge\mathrm dy$$

not define a regular characteristic quotient on all of $\mathbb R^3$?

Solution

For $x\ne0$, the form has rank two and kernel $\operatorname{span}\{\partial_z\}$. At $x=0$, it vanishes and its kernel is the full three-dimensional tangent space. The kernel dimension jumps, so it is not a vector subbundle and the constant-rank foliation argument does not apply across the plane $x=0$.

3. Separate the two closure statements

Starting from

$$\boldsymbol{\delta}\boldsymbol L =\boldsymbol E_a\boldsymbol{\delta}\phi^a +\mathrm d\boldsymbol\Theta, \qquad \boldsymbol\omega =-\boldsymbol{\delta}\boldsymbol\Theta,$$

derive both $\mathrm d\boldsymbol\omega$ and $\boldsymbol{\delta}\boldsymbol\omega$. Which one controls hypersurface dependence?

Solution

Applying $\boldsymbol{\delta}$ to the first identity gives

$$0 =\boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a -\mathrm d\boldsymbol\omega,$$

so

$$\mathrm d\boldsymbol\omega =\boldsymbol{\delta}\boldsymbol E_a \wedge\boldsymbol{\delta}\phi^a.$$

It vanishes on a solution when both variations solve the linearized equations. Separately,

$$\boldsymbol{\delta}\boldsymbol\omega =-\boldsymbol{\delta}^{\,2}\boldsymbol\Theta=0.$$

The spacetime equation controls flux and hypersurface dependence. The field-space equation says that the integrated two-form is closed.

4. Reconstruct the corner ambiguity

Under

$$\boldsymbol\Theta' =\boldsymbol\Theta +\boldsymbol{\delta}\boldsymbol\ell +\mathrm d\boldsymbol\beta,$$

show which term can change $\Omega_\Sigma$.

Solution

Nilpotence removes the $\boldsymbol\ell$ contribution:

$$\begin{aligned} \boldsymbol\omega' &=-\boldsymbol{\delta}\boldsymbol\Theta'\\ &=\boldsymbol\omega -\mathrm d( \boldsymbol{\delta}\boldsymbol\beta). \end{aligned}$$

Stokes’ theorem then gives

$$\Omega_\Sigma' =\Omega_\Sigma -\boldsymbol{\delta} \int_{\partial\Sigma}\boldsymbol\beta.$$

The $\boldsymbol\beta$ term disappears on a closed hypersurface but survives at a corner. The total derivative in the Lagrangian can still change the boundary variational problem even though it cancels from the local current.

5. Decide when Maxwell gauge motion is null

Starting from

$$\Omega_\Sigma =\int_\Sigma\mathrm d^3x\, \boldsymbol{\delta}A_i \wedge\boldsymbol{\delta}E^i,$$

contract with $R_\lambda A_i=\partial_i\lambda$ and $R_\lambda E^i=0$.

Solution

For a tangent variation $\delta$,

$$\begin{aligned} (\iota_{R_\lambda}\Omega_\Sigma)(\delta) ={}& -\int_\Sigma\mathrm d^3x\, \lambda\,\partial_i\delta E^i\\ &+\int_{\partial\Sigma}\mathrm dS\, \lambda\,\delta E^n. \end{aligned}$$

The linearized Gauss law removes the bulk term. The transformation is null only when the remaining boundary integral vanishes for every allowed variation. A boundary-nonzero $\lambda$ is therefore not automatically a redundancy.

Synthesis and physical handoff

A presymplectic kernel turns Hamiltonian motion into an existence-and-nonuniqueness problem. On covariant solution space, the Lagrangian supplies a field-space-closed current whose spacetime closure requires the equations and their linearization. Integrating over a hypersurface exposes two further dependencies: lateral flux can make the form slice dependent, and an exact shift of the potential current can leave a corner term. Gauge directions become quotiented redundancies only after their boundary pairing vanishes and a regular physical quotient exists.

For surface-charge variations, differentiability, integrability, conservation versus flux, reference choices, improvement ambiguities, and field-dependent parameters, continue to Surface Charges, Integrability, and Ambiguities.

References

• Mehdi Assanioussi, Jerzy Kowalski-Glikman, Ilkka Mäkinen, and Ludovic Varrin, “On the Covariant Formulation of Gauge Theories with Boundaries”, arXiv:2312.01918v2, 2024, §§ 2.2–2.3 and 3.1–3.3, PDF pp. 8–10 and 13–15. This review cross-checks the bidegree bookkeeping and the distinction between gauge invariance, degeneracy, and nonzero boundary pairings.
• Mark J. Gotay and James M. Nester, “Presymplectic Hamilton and Lagrange Systems, Gauge Transformations and the Dirac Theory of Constraints”, in Group Theoretical Methods in Physics, Lecture Notes in Physics 94, Springer, 1979, pp. 272–279. The material on pp. 272 and 275–278 supports the existence/nonuniqueness split, characteristic directions, regular quotient assumptions, and the warning that the physical identification of gauge directions is additional input.
• Daniel Harlow and Jie-qiang Wu, “Covariant Phase Space with Boundaries”, JHEP 10 (2020) 146, doi:10.1007/JHEP10(2020)146, §§ 2.2 and 3.3, arXiv PDF pp. 13–14 and 26–27. This boundary-focused source supports the dependence on a well-posed boundary variational principle and the bounded Maxwell check. It uses the opposite overall presymplectic-current sign, translated here as a whole.
• Vivek Iyer and Robert M. Wald, “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy”, Physical Review D 50 (1994), 846–864, doi:10.1103/PhysRevD.50.846, § 3, especially equations (20) and (40)–(47), arXiv PDF pp. 6 and 9–10. This is the structural source for the covariant first variation, current, on-shell closure, and ambiguity map.