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Boundaries in the Instantaneous Formulation of Field Theories

This paper investigates boundary conditions in GiMmsy's instantaneous formulation of field theories, demonstrating that they induce a sector structure in the state space and lead to a novel definition of boundary symmetry groups, which for electromagnetism reduces to the global gauge group even when accounting for flux superselection sectors in Yang-Mills theory.

Original authors: Silvester G. A. Borsboom

Published 2026-07-01

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CC BY 4.0

Original authors: Silvester G. A. Borsboom

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe the weather in a giant, invisible room. In physics, this "room" is the universe, and the "weather" is the fields (like electromagnetic fields) that fill it. Usually, physicists describe this weather by looking at how it changes over time and space all at once. But sometimes, it's easier to take a snapshot of the room at a single moment, look at the state of the weather right then, and ask: "If I know where the wind is now, and how fast it's moving, can I predict the future?"

This paper is about what happens when you put walls around that room.

The Problem: The Walls Change the Rules

In a normal room without walls, if you want to describe the state of the wind, you just need to know two things for every point in the room:

Where is the wind? (The position/configuration)
How fast is it moving? (The velocity)

In physics, the collection of all possible "position + speed" combinations is called the state space. Usually, this looks like a smooth, continuous sheet where you can slide from one state to another freely.

But, what if the room has walls? And what if those walls have special rules?

Rule A (The Sticky Wall): The wind must stop completely at the wall. It can't move.
Rule B (The Invisible Wall): The wind can be anything at the wall, but it must stop moving right at the edge.

The author, Silvester Borsboom, uses a mathematical framework (called the "GiMmsy" framework) to figure out what the "state space" looks like under these rules.

The Discovery: The "Sector" Puzzle

1. The Sticky Wall (Constant Dirichlet Condition)
Imagine you paint the walls of your room a specific color, and the wind must always match that color exactly.

The Result: The state space is still a nice, smooth sheet (mathematically, a "tangent bundle"). It's just a slightly smaller sheet than before because the wind is forced to match the wall. Everything works normally.

2. The Invisible Wall (Velocity Vanishes, Position Free)
Now, imagine the wall is invisible. The wind can be red, blue, or green at the wall, but it must be perfectly still right at the edge.

The Result: This is where things get weird. The state space breaks apart.
The Analogy: Imagine a library. In a normal library, you can walk from the "History" section to the "Science" section freely. But in this new "Invisible Wall" library, there is a magical force field. If you are holding a red book, you are trapped in the "Red Room." You can move around inside the Red Room, but you cannot walk into the "Blue Room" without breaking the rules.
The Physics: The state space becomes a collection of separate "sectors." Each sector corresponds to a specific color (or configuration) of the wind at the wall. You can move freely within a sector, but you cannot jump between sectors using normal physics.

The Consequence: The "Ghost" Symmetry

In physics, we often look for symmetries—things you can do to the system that don't change the laws of physics. For example, rotating a sphere looks the same from all angles.

When the state space is broken into these separate sectors, a new problem arises:

Some "moves" (gauge transformations) try to change the color of the wind at the wall (moving you from the Red Room to the Blue Room).
Because there is no "momentum" (a physical quantity like push or pull) associated with the wall itself, the laws of physics say these moves are impossible to perform using standard energy. They are "non-Hamiltonian."
The Metaphor: Imagine trying to push a car that has no engine and no wheels. You can't make it move. Similarly, the author argues that if a transformation doesn't have a "momentum" to carry it, it shouldn't be considered a "real" physical change.

The New Definition of "Physical"

The paper proposes a new way to define what counts as a "physical" symmetry in a room with walls:

Old Way: Any symmetry that preserves the boundary rules is physical.
New Way (The Author's Proposal): Only symmetries that carry momentum are physical.
- If a transformation moves you between sectors (changes the wall color) but has no momentum, it's not a real physical change. It's just a mathematical trick.
- If a transformation stays within a sector (keeps the wall color the same) but changes the wind inside, and it does have momentum (like an electric flux), then it is a real physical symmetry.

The Big Win: Electromagnetism

The author tests this idea on Electromagnetism (light and electricity).

Even though the math gets complicated with many different "sectors" (different possible wall configurations), the result is surprisingly simple.
When you filter out all the "ghost" moves that have no momentum, the only physical symmetry group left is the Global Gauge Group.
In plain English: For electricity, the only "real" symmetry that matters is the global rotation of the electric field. Even with all the complex walls and sectors, the universe simplifies back to this one fundamental rule.

Summary

This paper is a rigorous mathematical investigation into how boundaries (walls) change the rules of physics.

Boundaries break the state space into separate, isolated "rooms" (sectors).
Moving between rooms is physically impossible without a specific "momentum" to drive it.
Therefore, only symmetries that stay within a room (or have momentum) should be considered real physical laws.
When applied to electricity, this logic confirms that the fundamental symmetry of the universe is preserved, even when we account for these complex boundary effects.

The author essentially says: "If you can't push it (no momentum), it's not a real move."

Problem Statement
The paper addresses the rigorous formulation of boundary conditions within the instantaneous (3+1) framework of classical field theories, specifically extending the covariant and instantaneous formalism developed by GiMmsy (Gotay, Isenberg, Marsden, Montgomery, Sniatycki, and Yasskin). While the interaction between boundaries and localizable symmetry groups is well-studied in canonical Hamiltonian formulations, a complete instantaneous formulation—including the instantaneous Lagrangian, the Legendre transform, and the resulting phase space structure in the presence of boundaries—has lacked a rigorous foundation. The central problem is to characterize the instantaneous state space and phase space when boundary conditions restrict the field configurations and/or their velocities, particularly in cases where the boundary conditions do not fix a specific configuration value (e.g., requiring only that velocities vanish at the boundary).

Methodology
The author employs the geometric framework of GiMmsy, which utilizes a 3+1 decomposition of spacetime $M$ into Cauchy surfaces $\Sigma_t$ . The methodology proceeds as follows:

Framework Extension: The instantaneous configuration space $Q_t = \Gamma(Y_t)$ and its tangent bundle $TQ_t$ are defined via the jet decomposition map $\beta_\zeta$ , which relates covariant jets to instantaneous spatial jets and velocities.
Boundary Condition Analysis: The paper distinguishes between two types of boundary conditions:
- Constant Dirichlet: The field configuration is fixed to a specific value $b_t$ at the boundary, which remains constant under the slicing flow.
- Velocity-Vanishing: The field configuration is allowed to vary at the boundary, but its time derivative (velocity) is required to vanish ( $\dot{\phi}|_{\partial \Sigma} = 0$ ).
Geometric Characterization: Using the properties of the boundary restriction map $r_t: Q_t \to \Gamma(Y_t|_{\partial \Sigma_t})$ , the author analyzes the structure of the admissible state space. For constant Dirichlet conditions, the state space is shown to be a single tangent bundle. For velocity-vanishing conditions, the state space is shown to be a disjoint union of tangent bundles, indexed by the boundary configurations.
Legendre Transform: The instantaneous Legendre transform is defined on this "sectored" state space (an admissible velocity bundle $D$ ) using the fiber derivative. The resulting phase space is analyzed for its symplectic and Poisson structures.
Application to Yang-Mills: The formalism is applied to Yang-Mills theory on a compactified spatial slice with asymptotic boundary conditions. The gauge group action is analyzed to determine which transformations are Hamiltonian (possessing a momentum map) and which are not.

Key Contributions and Results

Structure of Instantaneous State Space:
- Under a constant Dirichlet boundary condition, the instantaneous state space is isomorphic to the tangent bundle $TQ_b$ of the configuration space satisfying that specific boundary condition (Proposition 3.5).
- Under the condition that only the velocity vanishes at the boundary (without fixing the boundary value), the instantaneous state space is not a single tangent bundle but a disjoint union (a "sector structure") of tangent bundles: $D = \bigsqcup_{b} TQ_b$ (Corollary 3.6). Each sector corresponds to a fixed boundary configuration $b$ .
Legendre Transform and Phase Space Structure:
- The Legendre transform is rigorously defined on the admissible velocity bundle $D = \ker(Tr)$ . The resulting primary phase space $P$ is a disjoint union of phase spaces $P_b \subset T^*Q_b$ .
- The total phase space $P$ does not carry a global symplectic structure because there are no conjugate momenta for the boundary configurations. Instead, it possesses a leafwise canonical Poisson structure, where the symplectic leaves are the individual sectors $T^*Q_b$ (Remark 4.5).
Gauge Transformations and Momentum:
- In Yang-Mills theory, gauge transformations that change the boundary configuration (sector-moving) are shown to be non-Hamiltonian with respect to the Poisson structure on $P$ . They map between different symplectic leaves and lack a classical momentum map because the boundary momentum is undefined in this formulation.
- Gauge transformations that preserve the boundary sector (stabilizer subgroup $G_b$ ) are Hamiltonian. Their momentum map includes a boundary term corresponding to the electric flux.
Definition of Physical Symmetry Group:
- The paper proposes a novel definition for the physical boundary symmetry group: $G_{phys} := G_{Ham} / G_{triv}$ , where $G_{Ham}$ are the boundary-preserving Hamiltonian transformations and $G_{triv}$ are the trivial (redundant) ones.
- Electromagnetism ( $U(1)$ ): The physical symmetry group across all sectors is found to be isomorphic to the global gauge group $U(1)$ , as constant transformations preserve all sectors and carry non-zero flux momentum.
- Non-Abelian Yang-Mills: In a boundary trivialization, the physical symmetry group across all sectors is trivial (the center of the Lie algebra) unless specific restrictions (like $Ad(G)$-invariant boundary configurations) are imposed, in which case the global group $G$ re-emerges.

Significance and Claims
The paper claims to provide a rigorous geometric foundation for understanding how boundaries interact with the instantaneous formulation of field theories, specifically clarifying the relationship between 3+1 decompositions, the Legendre transform, and boundary fluxes.

The primary significance lies in demonstrating that when boundary values are not fixed, the state space naturally decomposes into sectors, leading to a phase space with a leafwise Poisson structure rather than a global symplectic one. This structural insight explains why "sector-moving" gauge transformations (those changing the boundary connection) cannot be generated by Hamiltonian functions in this formulation. Consequently, the paper argues that physical boundary symmetries should be restricted to those transformations that are Hamiltonian (i.e., those that preserve the sector and carry a momentum map). This redefines the understanding of asymptotic symmetries, distinguishing them from transformations that merely shift between superselection sectors without a corresponding physical observable (momentum).

The results are presented as a generalization of previous work (specifically [9]) to the full generality of velocity-vanishing boundary conditions, offering a clearer geometric picture of flux superselection sectors and the role of edge modes in the absence of boundary conjugate variables.