pith. sign in

arxiv: 2606.31265 · v1 · pith:FVCA6BUNnew · submitted 2026-06-30 · 🧮 math-ph · hep-th· math.MP

Boundaries in the Instantaneous Formulation of Field Theories

Silvester G.A. Borsboom This is my paper

Pith reviewed 2026-07-01 03:41 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords boundary conditionsinstantaneous formulationgauge theoriesYang-Mills theoryelectromagnetismsymmetry groupssuperselection sectorsLegendre transform
0
0 comments X

The pith

Instantaneous state space with Dirichlet boundaries is a tangent bundle, and electromagnetism's physical boundary symmetry group equals the global gauge group across all sectors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a constant Dirichlet boundary condition makes the instantaneous state space a tangent bundle over the configuration space of fields obeying the condition. When the boundary condition requires only that velocity vanish at the boundary, the state space splits into sectors, each a tangent bundle labeled by the fixed boundary configuration. The Legendre transform of this structure produces a phase space carrying canonical Poisson brackets on each sector separately. Applied to Yang-Mills theory with spatial boundaries, the construction accounts for flux superselection sectors and shows that gauge transformations moving between sectors lack Hamiltonian generators because no boundary momentum exists. The physical boundary symmetry group is therefore defined as the quotient of boundary-preserving Hamiltonian transformations by the trivial ones, which for electromagnetism yields exactly the global gauge group even when all sectors are taken together.

Core claim

The instantaneous state space in the presence of a constant Dirichlet boundary condition is a tangent bundle to the configuration space of fields satisfying the condition. When only the velocity of the field is required to vanish at the boundary, a sector structure appears in which each sector is a tangent bundle labeled by the configuration at the boundary. The Legendre transform yields a sectored phase space with leafwise canonical Poisson structures. Sector-moving gauge transformations are not Hamiltonian because of the lack of a boundary momentum. The physical boundary symmetry group of electromagnetism is a copy of the global gauge group even when all sectors are considered simultaneous

What carries the argument

The sectored instantaneous state space (each sector a tangent bundle indexed by boundary configuration) and its image under the sector-by-sector Legendre transform, which equips the phase space with leafwise canonical Poisson structures.

If this is right

  • Sector-moving gauge transformations fail to be Hamiltonian due to the absence of boundary momentum.
  • The boundary symmetry group is defined as the quotient of boundary-preserving Hamiltonian transformations by the trivial ones.
  • For electromagnetism this quotient is always a copy of the global gauge group.
  • Under stated conditions the same identification holds for non-Abelian Yang-Mills theory.
  • The construction reproduces flux superselection sectors in Yang-Mills theory with spatial boundaries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sector decomposition supplies a classical mechanism that could underlie superselection rules once the theory is quantized.
  • The same instantaneous construction may apply to other gauge theories or to gravitational models where spatial boundaries are present.
  • The distinction between covariant and instantaneous formulations affects which transformations count as physical symmetries at the boundary.

Load-bearing premise

That the Legendre transform can be performed sector by sector while preserving the leafwise canonical Poisson structures, and that the distinction between Hamiltonian and non-Hamiltonian gauge transformations remains well-defined once boundary momentum is absent.

What would settle it

A direct calculation of the Poisson bracket between a sector-changing gauge transformation generator and the Hamiltonian, checking whether the bracket vanishes without additional boundary terms.

read the original abstract

We study boundary conditions in GiMmsy's covariant and instantaneous formulations of classical field theories and show that the instantaneous state space in the presence of a constant Dirichlet boundary condition is a tangent bundle to the configuration space of fields satisfying said condition. We then study the instantaneous state space when only the velocity of the field is required to vanish at the boundary and show that this results in a sector structure, where each sector is a tangent bundle labeled by the configuration at the boundary. Taking the Legendre transform of this sectored state space yields a sectored phase space with leafwise canonical Poisson structures. We apply this to Yang-Mills theory with spatial boundary conditions and relate our results to flux superselection sectors. The sector-moving gauge transformations are not Hamiltonian because of the lack of a boundary momentum, prompting us to propose a novel definition of the asymptotic or boundary symmetry group as the quotient of the boundary-preserving Hamiltonian transformations by the trivial ones. The physical boundary symmetry group of electromagnetism is then shown to be a copy of the global gauge group even when all sectors are considered simultaneously. Conditions are discussed under which the same holds for non-Abelian Yang-Mills theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies boundary conditions in GiMmsy's covariant and instantaneous formulations of classical field theories. It shows that the instantaneous state space with a constant Dirichlet boundary condition is a tangent bundle to the configuration space of fields satisfying the condition. When only velocity vanishes at the boundary, a sector structure appears with each sector a tangent bundle labeled by the boundary configuration. The Legendre transform of this sectored state space produces a sectored phase space with leafwise canonical Poisson structures. Applied to Yang-Mills theory with spatial boundaries, the framework relates to flux superselection sectors. Sector-moving gauge transformations are non-Hamiltonian due to absent boundary momentum, leading to a proposed definition of the boundary symmetry group as the quotient of boundary-preserving Hamiltonian transformations by the trivial ones. For electromagnetism the physical boundary symmetry group is shown to be a copy of the global gauge group even across all sectors; conditions for the same to hold in non-Abelian Yang-Mills are discussed.

Significance. If the derivations are correct, the work supplies a geometrically precise treatment of instantaneous formulations with boundaries and a concrete quotient construction for boundary symmetries. The explicit sector decomposition and the result that electromagnetism retains only the global gauge group as its physical boundary symmetry (even when sectors are considered jointly) constitute falsifiable statements that can be compared with existing literature on asymptotic symmetries and flux sectors. The absence of free parameters and the direct use of tangent-bundle and Poisson-leaf structures are strengths.

major comments (2)
  1. [§4] §4 (Legendre transform on sectored state space): The claim that the Legendre transform, taken sector by sector, yields leafwise canonical Poisson structures must be accompanied by an explicit verification that the pull-back of the canonical symplectic form on each tangent-bundle sector coincides with the reduced form induced by the original covariant theory. The absence of boundary momentum is invoked to declare sector-moving transformations non-Hamiltonian, yet the same absence can modify the leafwise two-form; without a direct computation showing that boundary corrections vanish or cancel, the subsequent quotient definition of the symmetry group rests on an unverified identification.
  2. [§5.2] §5.2 (definition of boundary symmetry group): The novel quotient construction (boundary-preserving Hamiltonian transformations modulo trivial ones) is load-bearing for the claim that the physical symmetry group of electromagnetism is exactly the global gauge group. The argument that sector-moving transformations lie outside the Hamiltonian vector fields relies on the lack of boundary momentum; an explicit check that this quotient is independent of the choice of representative within each sector and that it reproduces the known global U(1) action is required.
minor comments (2)
  1. Notation for the sector label (boundary configuration value) should be introduced once and used consistently; currently the same symbol appears with different meanings in the abstract and in the sector decomposition.
  2. Add a short comparison paragraph relating the obtained leafwise Poisson structure to the standard Dirac bracket construction used in the literature on flux superselection sectors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the geometric framework and the identification of points requiring further explicit verification. We address each major comment below and will incorporate the requested computations and checks into a revised version.

read point-by-point responses

  1. Referee: [§4] §4 (Legendre transform on sectored state space): The claim that the Legendre transform, taken sector by sector, yields leafwise canonical Poisson structures must be accompanied by an explicit verification that the pull-back of the canonical symplectic form on each tangent-bundle sector coincides with the reduced form induced by the original covariant theory. The absence of boundary momentum is invoked to declare sector-moving transformations non-Hamiltonian, yet the same absence can modify the leafwise two-form; without a direct computation showing that boundary corrections vanish or cancel, the subsequent quotient definition of the symmetry group rests on an unverified identification.

    Authors: We agree that an explicit verification is required. In the revised manuscript we will add a direct computation in §4 showing that the pull-back of the canonical symplectic form on each tangent-bundle sector coincides with the reduced form from the covariant theory, with boundary corrections vanishing identically due to the vanishing-velocity condition. This computation will also confirm that the absence of boundary momentum does not alter the leafwise two-form, thereby supporting the non-Hamiltonian character of sector-moving transformations. revision: yes

  2. Referee: [§5.2] §5.2 (definition of boundary symmetry group): The novel quotient construction (boundary-preserving Hamiltonian transformations modulo trivial ones) is load-bearing for the claim that the physical symmetry group of electromagnetism is exactly the global gauge group. The argument that sector-moving transformations lie outside the Hamiltonian vector fields relies on the lack of boundary momentum; an explicit check that this quotient is independent of the choice of representative within each sector and that it reproduces the known global U(1) action is required.

    Authors: We acknowledge that an explicit independence check and reproduction of the U(1) action are needed to fully substantiate the quotient definition. In the revision we will include, in §5.2, a direct verification that the quotient is independent of the choice of representative within each sector and that it yields precisely the global U(1) action for electromagnetism, consistent with the known literature on flux sectors. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external GiMmsy framework to boundaries without self-referential reduction

full rationale

The paper's chain begins from the cited GiMmsy covariant/instantaneous formulations (external to this work) and constructs the instantaneous state space as a tangent bundle for constant Dirichlet conditions, then introduces sectors when only velocity vanishes at the boundary. The Legendre transform is applied sector-by-sector to obtain leafwise canonical Poisson structures, after which the boundary symmetry group is defined as a quotient using the absence of boundary momentum. None of these steps reduce by construction to their own outputs via self-definition, fitted parameters renamed as predictions, or load-bearing self-citations; the uniqueness of the physical boundary symmetry group for electromagnetism follows from the explicit sector structure and Hamiltonian/non-Hamiltonian distinction rather than presupposing the result. The derivation remains self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available, so ledger is minimal. The central claims rest on the validity of GiMmsy's formulations and on the applicability of the Legendre transform to the described sectored space.

axioms (1)
  • domain assumption GiMmsy's covariant and instantaneous formulations of classical field theories are valid starting points
    The entire analysis is performed inside those formulations as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5727 in / 1337 out tokens · 42027 ms · 2026-07-01T03:41:00.563406+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

20 extracted references · 3 canonical work pages · 2 internal anchors

  1. [1]

    Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,

    T. Regge and C. Teitelboim, “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity,”Annals Phys., vol. 88, p. 286, 1974

    1974

  2. [2]

    E. Binz, J. ´Sniatycki, and H. Fischer,Geometry of Classical Fields, vol. 154 ofNorth-Holland Mathematics Studies. Elsevier, 1988

    1988

  3. [3]

    Momentum maps and classical fields part III: Gauge symmetries and initial value constraints,

    M. J. Gotay, “Momentum maps and classical fields part III: Gauge symmetries and initial value constraints,” 2006. Available athttps://www.pims.math.ca/~gotay/

    2006

  4. [4]

    Hamiltonian gauge theory with corners: constraint reduction and flux superselection,

    A. Riello and M. Schiavina, “Hamiltonian gauge theory with corners: constraint reduction and flux superselection,”Advances in Theoretical and Mathematical Physics, vol. 28, no. 4, pp. 1241–1424, 2024

    2024

  5. [5]

    Null hamiltonian Yang–Mills theory: Soft symmetries and memory as superselection,

    A. Riello and M. Schiavina, “Null hamiltonian Yang–Mills theory: Soft symmetries and memory as superselection,”Annales Henri Poincaré, vol. 26, no. 2, pp. 389–477, 2025

    2025

  6. [6]

    Momentum Maps and Classical Relativistic Fields. Part I: Covariant Field Theory

    M. J. Gotay, J. Isenberg, and J. E. Marsden, “Momentum maps and classical relativistic fields. Part I: Covariant Field Theory,” 1997. Available athttps://arxiv.org/abs/ physics/9801019

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  7. [7]

    Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories

    M. J. Gotay, J. Isenberg, and J. E. Marsden, “Momentum maps and classical relativistic fields. part II: Canonical analysis of field theories,” 2004. Available athttp://arxiv. org/abs/math-ph/0411032

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  8. [8]

    The homotopy momentum map of general relativity,

    C. Blohmann, “The homotopy momentum map of general relativity,”International Mathe- matics Research Notices, vol. 2023, no. 10, pp. 8212–8250, 2022

    2023

  9. [9]

    Global gauge symmetries and spatial asymptotic boundary conditions in Yang-Mills theory,

    S. G. A. Borsboom and H. B. Posthuma, “Global gauge symmetries and spatial asymptotic boundary conditions in Yang-Mills theory,”SciPost Phys., vol. 20, p. 185, 2026

    2026

  10. [10]

    Multisymplectic geometry with boundaries,

    E. Kur, “Multisymplectic geometry with boundaries,” 2018. Available athttps:// escholarship.org/uc/item/6xq2m3dr

    2018

  11. [11]

    Local subsystems in gauge theory and gravity,

    W. Donnelly and L. Freidel, “Local subsystems in gauge theory and gravity,”Journal of High Energy Physics, vol. 2016, no. 9, p. 102, 2016. 14

    2016

  12. [12]

    Edge modes without edge modes,

    A. Riello, “Edge modes without edge modes,” 2021. Available athttps://arxiv.org/ abs/2104.10182

    work page arXiv 2021

  13. [13]

    The momentum map in poisson geometry,

    R. L. Fernandes, J.-P . Ortega, and T. S. Ratiu, “The momentum map in poisson geometry,” American Journal of Mathematics, vol. 131, no. 5, pp. 1261–1310, 2009

    2009

  14. [14]

    Gauge orbit types for generalized connections,

    C. Fleischhack, “Gauge orbit types for generalized connections,”Commun. Math. Phys., vol. 214, pp. 607–649, 2000

    2000

  15. [15]

    The quasilocal degrees of freedom of Yang-Mills theory,

    H. Gomes and A. Riello, “The quasilocal degrees of freedom of Yang-Mills theory,”SciPost Physics, vol. 10, no. 6, p. 130, 2021

    2021

  16. [16]

    Holism as the empirical significance of symmetries,

    H. Gomes, “Holism as the empirical significance of symmetries,”European Journal for Phi- losophy of Science, vol. 11, no. 3, p. 87, 2021

    2021

  17. [17]

    A unified treatment of null and spatial infinity in gen- eral relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity,

    A. Ashtekar and R. O. Hansen, “A unified treatment of null and spatial infinity in gen- eral relativity. I - Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity,”J. Math. Phys., vol. 19, pp. 1542–1566, 1978

    1978

  18. [18]

    Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity,

    A. Ashtekar and M. Streubel, “Symplectic Geometry of Radiative Modes and Conserved Quantities at Null Infinity,”Proc. Roy. Soc. Lond. A, vol. 376, pp. 585–607, 1981

    1981

  19. [19]

    Covariant description of canonical formalism in geometrical theories,

    C. Crnkovic and E. Witten, “Covariant description of canonical formalism in geometrical theories,” 1986. Available athttps://www.ias.edu/sites/default/files/sns/ files/CovariantPaper-1987.pdf

    1986

  20. [20]

    Local symmetries and constraints,

    J. Lee and R. M. Wald, “Local symmetries and constraints,”J. Math. Phys., vol. 31, pp. 725– 743, 1990. 15

    1990