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Variational, Symplectic, and Constraint Methods

Use this chapter when a QFT calculation asks how an action produces local equations and boundary data, how fluctuations are organized around a stationary field, how a two-form generates Hamiltonian evolution, how a continuous symmetry acquires a phase-space generator, or how constraints and gauge directions change the physical phase space. The organizing question is not simply “Lagrangian or Hamiltonian?” It is which space is being varied, which two-form lives on it, and which directions are physical, constrained, or redundant.

There are three independent entrances. The local-variation route begins with integration by parts and does not require the manifold language used later. The Hamiltonian route begins with alternating forms and Poisson brackets. The constraint route begins with a symplectic phase space and asks what remains after restriction, bracket modification, or quotienting. No reader needs to complete all six pages before using one of these routes.

The chapter keeps three facts separate throughout: a bulk equation does not settle the boundary variational problem; a zero direction of a Hessian or a presymplectic form is not automatically gauge redundancy; and a symplectic form is an alternating bilinear form, not an inner product. These distinctions are especially important in gauge field theory, where the status of a transformation can change with support, falloff, or boundary conditions.

The scope is reusable classical mathematics. Developed charge algebras, asymptotic symmetries, boundary-condition physics, gravitational charges, and quantization after reduction require the later treatments linked below. Lagrangian submanifolds and generating functions continue instead to the asymptotic and semiclassical chapter. The final page supplies a bounded covariant phase-space ambiguity map; it is not a general theorem about every infinite-dimensional solution space.

Diagnose · Choose a route · Compare structures · Page guide · Maxwell thread · Review · Continue

Enter through the operation you need

Section titled “Enter through the operation you need”

This overview has no prerequisite. Use the checks below to choose an entry page. They are independent: uncertainty about Lie group actions does not block local field variation, and uncertainty about boundary currents does not block the finite-dimensional symplectic route.

Observable readiness checks and exact repair routes
Check Ready Unsure Repair
Can you vary a first-order local action, integrate by parts, and retain both the admissible variation and the boundary term? Enter local field variation directly. Write the chain-rule term containing a derivative of the variation before setting any boundary value to zero. Use Field Variations and Boundary Terms. If the coordinate calculation is unfamiliar, use Variational and classical-field repair.
Can you form a second variation and distinguish a Jacobi solution from a zero mode in a specified operator domain? Enter the Hessian and fluctuation route. For one differential expression, name the allowed variations, boundary form, domain, and normalizability condition. Complete Field Variations and Boundary Terms, then use Second Variation, Hessians, and Jacobi Operators.
Can you distinguish an alternating two-form from an inner product and solve ι(Xf)ω = d f in a declared sign convention? Enter Hamiltonian geometry. Check ω(v,v) = 0 and recover Xf for one function on (q,p) space. Repair Direct Sums, Tensor Products, and Index Structure, then use Symplectic Forms, Hamiltonian Flows, and Poisson Brackets.
Can you compute the infinitesimal Lie-algebra action and state whether its generators are global, equivariant, and conserved? Enter the moment-map route. Differentiate one group action, then test generator existence, bracket closure, and Hamiltonian invariance separately. Repair Symplectic Forms and Lie Groups, Lie Algebras, and Exponential and Adjoint Maps before Hamiltonian Group Actions and Moment Maps.
Can you distinguish restriction to a constraint surface, a Dirac-bracket modification, and quotienting characteristic leaves? Enter constrained reduction. For one constraint matrix, ask whether it is invertible; if it is singular, identify the kernel of the pulled-back two-form. Use Symplectic Forms, then Constraints, Dirac Brackets, and Symplectic Reduction.
Can you identify a presymplectic kernel while keeping gauge degeneracy, boundary flux, and potential-current ambiguity distinct? Enter the covariant bridge. Write the spacetime degree and field-space degree of the Lagrangian, potential current, and presymplectic current. Repair Field Variations and Symplectic Forms, then use Presymplectic Systems and the Covariant Phase-Space Ambiguity Map.

The readiness repair is deliberately narrower than this chapter: it supports local variation, boundary terms, and the interpretation of basic constraints, but it does not establish competence with moment maps or covariant phase space.

Choose a route from the QFT question

Section titled “Choose a route from the QFT question”

In the table, requires identifies preparation used in the target page’s main argument. Recommended preparation improves fluency but does not block entry. Continue points to the treatment where the mathematical structure acquires its developed physical role.

Goal-to-route choices through variational and phase-space methods
Reader goal Route and preparation Observable result
Derive local field equations without erasing boundary information Begin directly: Field Variations and Boundary Terms. Continue: The Action Principle and Field Equations. Write δS as bulk plus boundary, state the admitted variations, and explain exactly why the surface contribution vanishes, cancels, or yields a boundary condition.
Analyze fluctuations, local extrema, or candidate zero modes Requires: Field Variations. Enter Second Variation, Hessians, and Jacobi Operators, then continue to Saddles and the Semiclassical Expansion. Separate the Hessian bilinear form, the Jacobi differential expression, its boundary form, the chosen operator realization, and the limited stability conclusion.
Construct Hamiltonian evolution or an equal-time Poisson bracket Requires: tensor and index structure. Graded algebra and differential forms are recommended. Enter Symplectic Forms, Hamiltonian Flows, and Poisson Brackets, then Hamiltonian Initial Data and Phase Space. Invert a closed nondegenerate alternating form, recover Hamilton's equations in the declared sign package, and state the boundary assumptions behind functional derivatives.
Turn a continuous phase-space symmetry into generators and conserved quantities Requires: Symplectic Forms and Lie Groups. Enter Hamiltonian Group Actions and Moment Maps, then Continuous Symmetries, Generators, and Charges. Construct the moment-map components, test equivariance, and test conservation without treating those three tasks as one implication.
Reduce a regular constrained Hamiltonian system Requires: Symplectic Forms. Enter Constraints, Dirac Brackets, and Symplectic Reduction, then Gauge Orbits, Gauss Constraints, and Stabilizers. Stabilize the constraint set, separate first- from second-class combinations, compute the intrinsic or Dirac bracket, and quotient only when the regular leaf space exists.
Relate a covariant boundary current to gauge kernels and surface pairings Requires: Field Variations and Symplectic Forms. Constraints and differential forms are recommended. Enter Presymplectic Systems and the Covariant Phase-Space Ambiguity Map, then Surface Charges, Integrability, and Ambiguities. Distinguish field-space closure, on-shell spacetime closure, lateral flux, kernel degeneracy, and the codimension-two ambiguity of an integrated form.

Keep the logical operations distinct

Section titled “Keep the logical operations distinct”

Several pages use the word degenerate, but they do not diagnose the same object. The comparison below prevents a Hessian kernel, a first-class constraint direction, and a presymplectic kernel from being identified by notation alone.

Spaces, structures, operations, and warranted conclusions
Structure Operation and result What does not follow automatically
First variation on an admitted space of fields Differentiate the action S once and integrate by parts: the interior coefficient gives the Euler–Lagrange expression, while the surface term tests the boundary data and boundary action. A bulk solution need not be stationary under the allowed boundary variations, and stationarity does not prove a minimum.
Hessian at a stationary field Differentiate twice to obtain a symmetric bilinear form; integration by parts identifies a Jacobi expression and a boundary form. A local Jacobi solution need not satisfy the operator domain or be normalizable; a Lorentzian sign by itself is not a stability theorem.
Symplectic form on phase space A closed nondegenerate alternating two-form identifies differentials with Hamiltonian vector fields and induces a Poisson bracket. It supplies no norm, positivity, angle, or metric; indeed ω(v,v) = 0 for every v.
Moment map for a group action Each component satisfies dμξ = ι(ξM)ω; equivariance separately controls the generator bracket, and Hamiltonian invariance separately gives conservation. A symplectic action need not have global Hamiltonians, a weak moment map need not be equivariant, and a generator need not be conserved.
Second-class constraint surface An invertible constraint matrix makes the pulled-back two-form nondegenerate; the Dirac bracket computes the intrinsic Poisson bracket after restriction. This is not quotienting by gauge leaves, and the inverse need not exist across a rank-changing locus.
First-class characteristic distribution The kernel of the pulled-back form is quotiented only under constant-rank and smooth-Hausdorff-leaf-space hypotheses. A first-class constraint is not automatically the complete gauge generator, and a mathematical null direction is not automatically physical redundancy.
Presymplectic form on a solution space A closed but possibly degenerate two-form makes Hamiltonian existence conditional and uniqueness modulo its kernel; boundary pairings test proposed gauge directions. On-shell closure does not imply zero lateral flux, hypersurface independence, a smooth quotient, or an integrable surface charge.

The finite-dimensional symplectic, Hamiltonian, moment-map, and reduction statements are developed in Cannas da Silva 2006, Lectures 1, 2, 8, and 18, pp. 3–12, 46, and 105–109; §§ 22.1, 22.4, and 24.1, pp. 133–134, 137–138, and 147; Chapters 23–24, pp. 141–150; and §§ 26.1–26.4, pp. 164–167, Open PDF. The constrained-system distinctions are developed in Date 2010, Chapters 3–4, pp. 14–22, arXiv PDF. Presymplectic compatibility, nonuniqueness, characteristic directions, and regular-quotient assumptions are treated in Gotay and Nester 1979, pp. 272 and 275–278, stable publisher record. The covariant bulk-plus-boundary decomposition, on-shell current, and representative ambiguities are treated in Iyer and Wald 1994, § 3, especially equations (20) and (40)–(47), arXiv PDF pp. 6 and 9–10, Open PDF. The second-variation route from necessary nonnegativity through the Jacobi equation and its limitations is developed in Cristoferi 2016, §§ 5.1 and 8.1–8.5, especially pp. 57, 77–79, and 84–86, author-hosted course-note PDF. Teschl 2014, §§ 2.2 and 9.1–9.2, especially pp. 58–63 and 182–188, author-hosted book page supports the distinction between a formal expression and a self-adjoint realization with boundary conditions. Devoto et al. 2022, §§ 2.2, 2.3, and 3.5.1, especially pp. 9–10, 23–24, and 39–41, stable journal record supplies the Euclidean negative-mode, symmetry-zero-mode, and normalizability checks.

How the six pages fit together

Section titled “How the six pages fit together”

The displayed order is pedagogical, not one compulsory sequence.

  1. Vary the action once. Local integration by parts separates bulk equations from boundary stationarity without requiring global geometry.
  2. Vary around a stationary field again. The Hessian and Jacobi operator organize fluctuations, zero modes, and carefully bounded stability tests.
  3. Move to phase space. A closed nondegenerate two-form defines Hamiltonian vector fields and Poisson brackets without using an inner product.
  4. Add a group action. Moment maps package infinitesimal symmetry generators; existence, equivariance, and conservation remain separate.
  5. Restrict and reduce. Constraints select a surface. Second-class directions lead to an intrinsic or Dirac bracket; first-class characteristic directions may lead to a quotient.
  6. Allow degeneracy and boundaries. Presymplectic geometry exposes compatibility conditions, gauge candidates, lateral flux, and codimension-two representative ambiguities on a chosen solution space.

There are seven direct required preparation links. Five are internal: Field Variations before Second Variation; Symplectic Forms before Moment Maps and before Constraints; and both Field Variations and Symplectic Forms before Presymplectic Systems. Two arrive from other chapters: tensor and index structure before Symplectic Forms, and Lie groups before Moment Maps.

There are four recommended links. Graded algebra and differential forms are recommended before Symplectic Forms; Constraints is recommended before Presymplectic Systems; and differential forms is recommended there again. In particular, Field Variations is not required for the ordinary symplectic page, Moment Maps is not required for Constraints, and Constraints is not hard preparation for the presymplectic page.

Shared conventions and failure checks

Section titled “Shared conventions and failure checks”

The site uses the (+−−−)(+---) metric convention. A directed surface element is fixed by Stokes’ formula,

∫Uddx ∂μVμ=∫∂UdΣμ Vμ.\int_U\mathrm d^d x\,\partial_\mu V^\mu =\int_{\partial U}\mathrm d\Sigma_\mu\,V^\mu.

For a first-order local Lagrangian and a fixed-coordinate variation δϕa=ηa\delta\phi^a=\eta^a, the chapter’s basic split is

δSU=∫Uddx Ea(L)ηa+∫∂UdΣμ πaμηa,πaμ=∂L∂(∂μϕa).\begin{aligned} \delta S_U ={}&\int_U\mathrm d^d x\, \mathcal E_a(\mathcal L)\eta^a\\ &+\int_{\partial U}\mathrm d\Sigma_\mu\, \pi_a^\mu\eta^a, \qquad \pi_a^\mu= \frac{\partial\mathcal L}{\partial(\partial_\mu\phi^a)}. \end{aligned}

Compact support removes the displayed surface integral and licenses the local bulk equation. Fixed boundary values, free boundary values, falloff, and cancellation by a boundary action are different mechanisms and must be named rather than compressed into “drop the boundary term.” The coordinate identity and its scalar and Maxwell checks follow Tong 2006–2007, § 1.1, equations (1.7)–(1.23). Saito 2025, Lecture 5, pp. 3–6, Open PDF develops natural boundary conditions, while Harlow and Wu 2020, §§ 2.2, 3.2, and 3.3, especially arXiv PDF pp. 13–14 and 26–27, Open PDF treats bulk and boundary actions in one variational problem.

Three derivative symbols must not be merged. The symbol δ\delta denotes one directional field variation; δ\boldsymbol{\delta} denotes the field-space exterior derivative on the covariant page. The symbol d\mathrm d is the de Rham exterior derivative on the manifold in context: phase space in the Hamiltonian formulas and spacetime in the covariant formulas. A dot denotes time differentiation. The local covariant identity and the site-sign current are

δL=Eaδϕa+dΘ,ω=−δΘ.\begin{aligned} \boldsymbol{\delta}\boldsymbol L &=\boldsymbol E_a\boldsymbol{\delta}\phi^a +\mathrm d\boldsymbol\Theta, \\ \boldsymbol\omega &=-\boldsymbol{\delta}\boldsymbol\Theta. \end{aligned}

Here L\boldsymbol L, Θ\boldsymbol\Theta, and ω\boldsymbol\omega have spacetime degrees dd, d−1d-1, and d−1d-1, and field-space degrees 00, 11, and 22, respectively. Field-space closure δω=0\boldsymbol{\delta}\boldsymbol\omega=0 is algebraic. Spacetime closure dω=0\mathrm d\boldsymbol\omega=0 requires a solution and admitted linearized solutions.

For a spacetime slab with ∂R=Σ2∪(−Σ1)∪B\partial R=\Sigma_2\cup(-\Sigma_1)\cup B, Stokes’ theorem gives

ΩΣ2−ΩΣ1=−∫Bω.\Omega_{\Sigma_2}-\Omega_{\Sigma_1} =-\int_B\boldsymbol\omega.

On-shell closure alone therefore does not imply hypersurface independence; the lateral flux must vanish or be included in a justified boundary completion. The displayed current formulas also assume commuting, field-independent variations. Field-dependent gauge parameters require the field-space commutator and parameter-variation terms omitted here.

The Hamiltonian sign package is

ω=∑idqi∧dpi,ιXfω=df,{f,g}=ω(Xf,Xg),Xf[g]={g,f},{qi,pj}=δij,f˙=∂tf+{f,H}.\begin{gathered} \omega=\sum_i\mathrm dq^i\wedge\mathrm dp_i, \qquad \iota_{X_f}\omega=\mathrm df, \\ \{f,g\}=\omega(X_f,X_g), \qquad X_f[g]=\{g,f\}, \\ \{q^i,p_j\}=\delta^i{}_j, \qquad \dot f=\partial_t f+\{f,H\}. \end{gathered}

Reversing ω\omega, the Hamiltonian-vector-field equation, or the Poisson bracket can define another consistent convention, but the whole package must move together. Recovering Hamilton’s equations is the quickest invariant check.

For a left action, use the direct infinitesimal field

ξM(x)=ddt∣t=0exp⁡(tξ)⋅x,dμξ=ιξMω.\xi_M(x) = \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \exp(t\xi)\mathbin{\cdot}x, \qquad \mathrm d\mu_\xi=\iota_{\xi_M}\omega.

With this +t+t choice, [ξM,ηM]=−[ξ,η]M[\xi_M,\eta_M]=-[\xi,\eta]_M. An equivariant moment map obeys

{μξ,μη}=μ[ξ,η],\{\mu_\xi,\mu_\eta\}=\mu_{[\xi,\eta]},

in the same convention. Existence, equivariance, and conservation remain separate: for a time-independent moment map, μ˙ξ=−ξM[H]\dot\mu_\xi=-\xi_M[H]. Generator normalization, the pairing with g\mathfrak g, and any additive central constants remain visible.

For second-class constraints χA\chi_A, order the matrix and its inverse as

ΔAB={χA,χB},ΔABΔBC=δAC,\Delta_{AB}=\{\chi_A,\chi_B\}, \qquad \Delta^{AB}\Delta_{BC}=\delta^A{}_C,

so the Dirac bracket is

{F,G}D={F,G}−{F,χA}ΔAB{χB,G}.\{F,G\}_D =\{F,G\} -\{F,\chi_A\}\Delta^{AB}\{\chi_B,G\}.

Its immediate convention check is {F,χA}D=0\{F,\chi_A\}_D=0. Weak equality F≈GF\approx G means equality after restriction to the final constraint surface; it does not license imposing constraints before an ordinary Poisson bracket. Equality on the constraint surface, equality modulo constraints, equality on shell, and equality after the gauge quotient are four different statements.

Finally, for a spacetime (d−1)(d-1)-form ℓ\boldsymbol\ell of field-space degree zero and a spacetime (d−2)(d-2)-form β\boldsymbol\beta of field-space degree one, the representative change is

L′=L+dℓ,Θ′=Θ+δℓ+dβ,ω′=ω−d(δβ),ΩΣ′=ΩΣ−δ∫∂Σβ.\begin{aligned} \boldsymbol L'&=\boldsymbol L+\mathrm d\boldsymbol\ell, \\ \boldsymbol\Theta' &=\boldsymbol\Theta +\boldsymbol{\delta}\boldsymbol\ell +\mathrm d\boldsymbol\beta, \\ \boldsymbol\omega' &=\boldsymbol\omega -\mathrm d(\boldsymbol{\delta}\boldsymbol\beta), \\ \Omega_\Sigma' &=\Omega_\Sigma -\boldsymbol{\delta} \int_{\partial\Sigma}\boldsymbol\beta. \end{aligned}

The final line is a corner contribution, not permission to ignore the boundary action, admitted field space, or boundary degrees of freedom. This representative map is the site-sign version of Iyer and Wald 1994, § 3, equations (40)–(47), arXiv PDF pp. 9–10, Open PDF.

Exact page guide

Section titled “Exact page guide”

Field Variations and Boundary Terms

Section titled “Field Variations and Boundary Terms”

Field Variations and Boundary Terms asks how local field variations produce bulk equations and boundary terms without presupposing global geometry. It has no required or recommended preparation.

Use it to derive the first-variation identity on a coordinate domain, state admissible variations, compare fixed and natural boundary data, and check scalar and Maxwell signs in the (+−−−)(+---) convention. Compactly supported variations prove only the interior Euler–Lagrange equations. Continue to The Action Principle and Field Equations for the developed classical-field use.

Second Variation, Hessians, and Jacobi Operators

Section titled “Second Variation, Hessians, and Jacobi Operators”

Second Variation, Hessians, and Jacobi Operators asks how the second variation diagnoses fluctuations, zero modes, and local stability around a classical solution. It requires Field Variations and Boundary Terms.

Use it to separate the symmetric Hessian form from its Jacobi differential expression, boundary form, and operator realization. A Jacobi solution must also satisfy the boundary domain and normalizability requirements before it is a spectral zero mode; Teschl 2014, §§ 2.2 and 9.1–9.2 supplies the operator-domain distinction. A negative direction has a different interpretation in a Euclidean minimum problem, a Lorentzian action, and a constrained gauge system; Cristoferi 2016, §§ 5.1 and 8.1–8.5, PDF and Devoto et al. 2022, §§ 2.2, 2.3, and 3.5.1 give the necessary-condition and Euclidean QFT qualifications. Continue to Saddles and the Semiclassical Expansion for determinants, collective coordinates, and contour choices.

Symplectic Forms, Hamiltonian Flows, and Poisson Brackets

Section titled “Symplectic Forms, Hamiltonian Flows, and Poisson Brackets”

Symplectic Forms, Hamiltonian Flows, and Poisson Brackets asks how a closed nondegenerate two-form turns functions into Hamiltonian flows and Poisson brackets. It requires Direct Sums, Tensor Products, and Index Structure. Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration and Differential Forms, Integration, Orientation, and Stokes Theorem are recommended.

Use it to distinguish closedness from nondegeneracy, translate the sign package, derive Hamilton’s equations, and pass carefully to functional brackets under stated boundary conditions. The form is alternating rather than positive, and a formal infinite-dimensional expression may still need a declared phase space and differentiable Hamiltonian. Continue to Hamiltonian Initial Data and Phase Space.

Hamiltonian Group Actions and Moment Maps

Section titled “Hamiltonian Group Actions and Moment Maps”

Hamiltonian Group Actions and Moment Maps asks how a moment map encodes symmetry generators and conserved quantities on phase space. It requires Symplectic Forms and Lie Groups, Lie Algebras, and Exponential and Adjoint Maps.

Use it to compute infinitesimal action fields, test whether their contractions with ω\omega are globally exact, assemble the components into μ:M→g∗\mu:M\to\mathfrak g^*, and distinguish weak existence from equivariance and conservation. Central constants and cocycles can obstruct the naive generator algebra. Continue to Continuous Symmetries, Generators, and Charges for currents, quantum charges, Ward identities, and physical qualifications.

Constraints, Dirac Brackets, and Symplectic Reduction

Section titled “Constraints, Dirac Brackets, and Symplectic Reduction”

Constraints, Dirac Brackets, and Symplectic Reduction asks how constraint surfaces, bracket modification, and quotienting produce a reduced physical phase space. It requires Symplectic Forms; Moment Maps is not required.

Use it to stabilize a singular Hamiltonian system, distinguish primary/secondary from first/second class, compute a Dirac bracket for a regular second-class set, and descend a first-class characteristic distribution only when the quotient is smooth. For rr independent first-class and ss independent second-class constraints on a regular 2n2n-dimensional phase space,

dim⁡Mred=2n−2r−s,\dim M_{\mathrm{red}}=2n-2r-s,

but this count fails for reducible, rank-changing, or singular systems. Continue to Gauge Orbits, Gauss Constraints, and Stabilizers for developed gauge-theory orbit structure.

Presymplectic Systems and the Covariant Phase-Space Ambiguity Map

Section titled “Presymplectic Systems and the Covariant Phase-Space Ambiguity Map”

Presymplectic Systems and the Covariant Phase-Space Ambiguity Map asks how degenerate two-forms expose gauge directions, boundary sensitivities, and ambiguities on solution space. It requires Field Variations and Symplectic Forms. Constraints and Differential Forms are recommended.

Use it to solve ιXϖ=dH\iota_X\varpi=\mathrm dH in the presence of a kernel, test whether a regular quotient exists, derive the local current from the first variation, and distinguish on-shell closure from vanishing lateral flux. A gauge transformation is null only when its pairing with every admitted variation vanishes; a nonzero but integrable boundary pairing instead defines a nontrivial Hamiltonian action. Continue to Surface Charges, Integrability, and Ambiguities for the physical charge analysis.

Constrained mechanics and Maxwell across the chapter

Section titled “Constrained mechanics and Maxwell across the chapter”

The finite-dimensional method and the Maxwell example share one geometric decision. A singular Legendre map first produces constraints. After the full set is stabilized, pull the symplectic form back to the constraint surface. An invertible constraint matrix gives a symplectic submanifold whose bracket is computed by the Dirac formula. A constant-rank kernel instead supplies characteristic leaves; if the model identifies them as gauge and the leaf space is smooth, quotienting gives

π∗ωred=ι∗ω.\pi^*\omega_{\mathrm{red}}=\iota^*\omega.

Restriction and quotienting are therefore different operations. Each regular constraint removes one dimension by restriction, while each first-class constraint removes one additional characteristic direction. Date 2010, Chapters 3–4, pp. 14–22, PDF gives the stabilization and class split used here; Cannas da Silva 2006, Chapters 23–24, pp. 141–150, PDF gives the regular symplectic-reduction geometry.

Source-free Maxwell theory shows why the support and boundary clauses are part of that method. On a fixed spatial slice Σ\Sigma, take fields and variations with the regularity and falloff required below. The covariant action first gives

S[A]=−14∫Ud4x FμνFμν,δS=∫Ud4x (∂μFμν)δAν−∫∂UdΣμ FμνδAν.\begin{aligned} S[A]&=-\frac14\int_U\mathrm d^4x\, F_{\mu\nu}F^{\mu\nu}, \\ \delta S ={}&\int_U\mathrm d^4x\, (\partial_\mu F^{\mu\nu})\delta A_\nu -\int_{\partial U}\mathrm d\Sigma_\mu\, F^{\mu\nu}\delta A_\nu. \end{aligned}

Arbitrary compactly supported variations yield ∂μFμν=0\partial_\mu F^{\mu\nu}=0; they do not decide the boundary problem. A variation restricted from the outset to δA=dλ\delta A=\mathrm d\lambda probes the gauge identity rather than deriving every Maxwell equation.

The unreduced Hessian has the formal Jacobi expression

Jνρ=ηνρ□−∂ν∂ρ,Jνρ∂ρλ=0.\mathcal J^{\nu\rho} =\eta^{\nu\rho}\Box -\partial^\nu\partial^\rho, \qquad \mathcal J^{\nu\rho}\partial_\rho\lambda=0.

The second equality identifies a local kernel direction, not yet a physical or spectral conclusion. The parameter must preserve the boundary domain and belong to the subgroup declared redundant; a spectral zero mode must also be admissible and normalizable. This uses the operator-domain distinction in Teschl 2014, §§ 2.2 and 9.1–9.2 and the normalizability warning in Devoto et al. 2022, § 3.5.1.

On an equal-time slice, with Ei=−F0iE^i=-F^{0i}, raise spatial indices with the positive Euclidean slice metric, so ∂i=δij∂j\partial^i=\delta^{ij}\partial_j. The canonical form is

Ω=∫Σd3x δAi∧δEi.\Omega =\int_\Sigma\mathrm d^3x\, \boldsymbol{\delta}A_i \wedge\boldsymbol{\delta}E^i.

For a compactly supported gauge parameter ϵ\epsilon, define

G[ϵ]=−∫Σd3x ϵ ∂iEi=∫Σd3x Ei∂iϵ.\begin{aligned} G[\epsilon] &=-\int_\Sigma\mathrm d^3x\, \epsilon\,\partial_iE^i \\ &=\int_\Sigma\mathrm d^3x\, E^i\partial_i\epsilon. \end{aligned}

The discarded surface term vanishes under the stated support or falloff conditions. Then

ιRϵΩ=δG[ϵ],RϵAi=∂iϵ,RϵEi=0.\iota_{R_\epsilon}\Omega=\boldsymbol{\delta}G[\epsilon], \qquad R_\epsilon A_i=\partial_i\epsilon, \qquad R_\epsilon E^i=0.

Thus G[ϵ]G[\epsilon] is the negative smeared Gauss-law moment-map component for the Abelian action in this controlled setting. This is a formal infinite-dimensional specialization of the finite-dimensional moment-map theorem. Equivariance reduces to {G[ϵ],G[η]}=0\{G[\epsilon],G[\eta]\}=0, while conservation still requires the Hamiltonian to be gauge invariant. The zero level is Gauss’s constraint ∂iEi≈0\partial_iE^i\approx0. The functional G[ϵ]G[\epsilon] generates the spatial field transformation; by itself it is not the complete spacetime gauge generator including A0A_0, which can require primary constraints and time derivatives of the gauge parameter.

On tangent variations satisfying ∂iδEi=0\partial_i\boldsymbol{\delta}E^i=0, the compact-support gauge direction is in the kernel of the pulled-back form. Quotienting it removes longitudinal data. Alternatively, pair Gauss’s constraint locally with Coulomb gauge ∂iAi≈0\partial^iA_i\approx0 and choose boundary conditions for which ∇2\nabla^2 has an inverse. The Dirac bracket becomes

{Ai(x),Ej(y)}D=(δij−∂i∇−2∂j)δ(3)(x−y),\{A_i(\mathbf x),E^j(\mathbf y)\}_D = \left( \delta_i{}^j -\partial_i\nabla^{-2}\partial^j \right) \delta^{(3)}(\mathbf x-\mathbf y),

the transverse projector away from zero momentum. This local agreement does not prove a global gauge slice exists or that reduction commutes with quantization. The canonical role of A0A_0, Gauss’s law, Coulomb gauge, and the transverse projector are reviewed in Tong 2006–2007, §§ 6.2–6.2.1, pp. 127–130, PDF.

The covariant construction makes the missing boundary clause visible. From L=−12F∧⋆F\boldsymbol L=-\tfrac12F\wedge\star F and the site sign, one obtains

Θ=−δA∧⋆F,ω=−δA∧⋆δF,\boldsymbol\Theta =-\boldsymbol{\delta}A\wedge\star F, \qquad \boldsymbol\omega =-\boldsymbol{\delta}A \wedge\star\boldsymbol{\delta}F,

whose pullback gives the same Ω\Omega above. On a slice with boundary and outward spatial normal nin_i, contraction with a field-independent gauge variation gives, after the linearized Gauss law,

(ιRλΩΣ)(δ)=∫∂ΣdS λ δEn,En=niEi.(\iota_{R_\lambda}\Omega_\Sigma)(\delta) = \int_{\partial\Sigma}\mathrm dS\, \lambda\,\delta E^n, \qquad E^n=n_iE^i.

The same local transformation is null when this integral vanishes for every allowed variation, Hamiltonian but nontrivial when the resulting boundary one-form is nonzero and integrable, and not Hamiltonian on the proposed phase space when it is nonintegrable. That trichotomy is the endpoint of this chapter, not a calculation of a boundary charge algebra. The current, on-shell closure, and ambiguity map follow Iyer and Wald 1994, § 3, equations (20) and (40)–(47), arXiv PDF pp. 6 and 9–10, Open PDF. The bounded Maxwell pairing and its dependence on a well-posed boundary variational problem are cross-checked by Harlow and Wu 2020, §§ 2.2 and 3.3, arXiv PDF pp. 13–14 and 26–27, Open PDF. Their presymplectic current has the opposite overall sign and is translated here as a complete package.

Identifying this canonical initial-data space with a covariant solution space also requires a well-posed Cauchy problem and compatible boundary and flux conditions. The chapter checks the common two-form where both descriptions are available; it does not prove that equivalence in general.

One Maxwell system viewed through the six pages
Page viewpoint Calculation Qualification that remains
First variation Arbitrary δAν gives the Maxwell equation plus −∫∂U dΣμ Fμν δAν. The admitted boundary traces and any boundary action decide stationarity.
Second variation The unreduced Jacobi expression annihilates gradients ∂ρλ. Boundary preservation, normalizability, and the declared gauge subgroup decide whether this is a spectral zero mode or redundancy.
Symplectic geometry Ω = ∫Σ δAi ∧ δEi gives the equal-time bracket. The phase space and functional differentiability depend on support, falloff, and boundary conditions.
Moment map G[ε] = ⟨μ, ε⟩ generates Ai ↦ Ai + ∂iε for compactly supported ε. Global generator existence, boundary improvement, equivariance, and conservation are separate tests.
Constraints and reduction The zero level is Gauss's law; quotienting compact-support gauge directions or using the local Coulomb-gauge Dirac bracket leaves transverse data. Zero modes, rank changes, residual transformations, and a failure of a global slice can invalidate the regular picture.
Presymplectic bridge The covariant current reproduces the canonical form, while a gauge contraction reduces to ∫∂Σ λ δEn. The boundary pairing, lateral flux, and corner ambiguity decide whether the transformation is null, Hamiltonian, or obstructed.

Chapter synthesis

Section titled “Chapter synthesis”

Four distinctions organize the chapter.

Bulk stationarity and boundary stationarity are separate. Compactly supported variations isolate the Euler–Lagrange expression. The full variational problem also specifies boundary traces, falloff, boundary actions, and which transformations preserve them.

A Hessian and a symplectic form do different jobs. The Hessian is a symmetric second derivative used for fluctuations and local-extremum tests. The symplectic form is alternating and converts differentials into flows and brackets. Neither is automatically a Hilbert-space inner product.

A kernel has to be typed before it is interpreted. A Jacobi kernel refers to a linearized equation and operator domain. A first-class characteristic kernel arises after restriction to a constraint surface. A covariant presymplectic kernel depends on the admitted solution space and boundary pairing. Only additional physical input identifies a kernel direction as gauge redundancy.

Restriction, bracket modification, and quotienting are not synonyms. A second-class surface carries its own nondegenerate form and Dirac bracket. A first-class surface remains degenerate and may admit a characteristic quotient. A boundary-nonzero gauge transformation can instead act nontrivially and must not be quotiented as though it were null.

A compact working rule is:

Before varying, generating, constraining, or quotienting, state the space, admitted tangent directions, boundary and support conditions, two-form and sign convention, equality being used, regularity hypotheses, and the strongest conclusion those data justify.

Review the chapter

Section titled “Review the chapter”

A successful response meets the stated criterion; repair the first missing step with the linked page.

Reconstruct the first variation. Starting from a first-order local Lagrangian, derive δS=\delta S= bulk plus boundary and name three distinct ways the surface term can vanish or cancel. A successful response states the admitted variations and does not infer boundary data from compactly supported variations. Repair with Field Variations and Boundary Terms.

Separate three zero directions. Compare a Jacobi solution, a first-class characteristic direction, and a vector in the kernel of an integrated presymplectic form. A successful response names the relevant space, domain, boundary conditions, and extra physical statement needed before calling any direction gauge redundancy. Repair with Second Variation, Constraints and Reduction, and Presymplectic Systems.

Translate a Hamiltonian sign package. Replace ιXfω=df\iota_{X_f}\omega=\mathrm df by ιXf′ω=−df\iota_{X_f'}\omega=-\mathrm df and translate XfX_f, the Poisson bracket, the generator action, and the moment-map equation consistently. A successful response preserves {qi,pj}=δij\{q^i,p_j\}=\delta^i{}_j or explicitly declares its replacement and recovers the same Hamilton equations. Repair with Symplectic Forms and Moment Maps.

Classify and reduce a regular constraint set. Given a final constraint matrix, decide whether it has an invertible second-class block, compute the Dirac bracket, identify any remaining characteristic distribution, and state the smooth-quotient hypotheses. A successful response checks {F,χA}D=0\{F,\chi_A\}_D=0 and the degree count without imposing weak equality before ordinary brackets. Repair with Constraints, Dirac Brackets, and Symplectic Reduction.

Trace Maxwell through the chapter. Begin with the covariant action, recover the bulk equation and boundary term, identify the Hessian gradient directions, construct Ω\Omega and G[ϵ]G[\epsilon], impose Gauss’s law, and end with ∫∂Σλ δEn\int_{\partial\Sigma}\lambda\,\delta E^n. A successful response states where compact support, linearized Gauss law, Laplacian invertibility, and boundary conditions enter; it does not infer a quantum theory or surface charge algebra. Repair with the Maxwell thread and the six leaf pages.

Track the covariant ambiguity. Starting with L′=L+dℓ\boldsymbol L' = \boldsymbol L+\mathrm d\boldsymbol\ell and Θ′=Θ+δℓ+dβ\boldsymbol\Theta' = \boldsymbol\Theta+ \boldsymbol{\delta}\boldsymbol\ell+\mathrm d\boldsymbol\beta, derive the change in ω\boldsymbol\omega and ΩΣ\Omega_\Sigma. A successful response gives the spacetime and field-space degrees and distinguishes a representative change from changing the boundary action or admitted field space. Repair with Presymplectic Systems and the Covariant Phase-Space Ambiguity Map.

Continue from here

Section titled “Continue from here”

Within Mathematical Methods:

For developed QFT applications:

References

Section titled “References”