Restricted Multimomentum Bundle in Field Theory

• Restricted multimomentum bundle is a finite-dimensional geometric structure that generalizes canonical phase space for covariant multisymplectic field theories.
• It employs canonical forms and the De Donder–Weyl Hamiltonian formalism to describe field evolution using spacetime covariant derivatives, avoiding a split between space and time.
• Symmetry reduction techniques applied to this bundle yield contact structures that isolate observables from redundant degrees of freedom in classical field theories.

A restricted multimomentum bundle is a finite-dimensional geometric structure central to covariant multisymplectic formulations of classical field theory, generalizing canonical phase space to accommodate field-theoretic degrees of freedom and their conjugate momenta. It serves as the multiphase space in the De Donder–Weyl Hamiltonian formalism, with its geometry supporting covariant Hamiltonian dynamics, symmetry reduction procedures, and contact structural interpretations that elegantly separate observable from redundant or unphysical variables.

1. Definition and Construction of the Restricted Multimomentum Bundle

The restricted multimomentum bundle, denoted $J^1E^*$, is constructed from a fiber bundle $E \rightarrow M$ over a spacetime manifold $M$. Field configurations are identified with sections $s: M\rightarrow E$. The first jet bundle $J^1E$ encodes field variables and their first derivatives, having local coordinates $(x^\mu, y^a, y^a_\mu)$, where $x^\mu$ are spacetime coordinates, $y^a$ are field variables, and $y^a_\mu$ are their partial derivatives.

The restricted multimomentum bundle $J^1E^*$ arises as a quotient of the extended multimomentum bundle, yielding local coordinates $(x^\mu, y^a, p^\mu_a)$. Here, $p^\mu_a$ are momenta conjugate to the partial derivatives $y^a_\mu$ of the fields with respect to spacetime coordinates. The geometric structure of $J^1E^*$ is characterized by the canonical (Poincaré–Cartan) form: $$\Theta_h = p^\mu_a \, dy^a \wedge d^{d-1}x_\mu - H(x, y, p)\, d^dx$$ and the associated multisymplectic form

$\Omega_h = -d\Theta_h$

which governs the local dynamical properties of the bundle (Bell et al., 19 Sep 2025).

2. Covariant Hamiltonian Formulation and Equations of Motion

Within the De Donder–Weyl covariant Hamiltonian formalism, the restricted multimomentum bundle $J^1E^*$ replaces the infinite-dimensional phase space of canonical 3+1 formulations with a finite-dimensional ensemble suited for field theories. The covariant Hamilton–De Donder–Weyl equations

$$\frac{\partial y^a}{\partial x^\mu} = \frac{\partial H}{\partial p^\mu_a}, \qquad \frac{\partial p^\mu_a}{\partial x^\mu} = - \frac{\partial H}{\partial y^a}$$

express field evolution entirely in terms of spacetime covariant derivatives, avoiding a split between space and time. These equations are equivalent to the requirement that the multivector field $\boldsymbol{X}_h$ is decomposable and satisfies

$\iota_{\boldsymbol{X}_h}\,\Omega_h = 0$

which geometrically encodes the field dynamics on $J^1E^*$ (Bell et al., 19 Sep 2025).

For hyperregular Lagrangians, the (covariant) Legendre transformation $\mathcal{FL}: J^1E \rightarrow J^1E^*$ is a diffeomorphism, ensuring that $J^1E^*$ accurately captures the physically meaningful multiphase space for the field theory.

3. Symmetry Reduction and Dynamical Similarity

The geometry of $J^1E^*$ admits symmetry reduction procedures especially adapted for handling dynamical similarities and other non-strictly canonical symmetries (Bell et al., 19 Sep 2025). For a system exhibiting a scaling symmetry, a vector field $D$ acts on the multimomentum bundle, satisfying

$$\mathcal{L}_D \omega^\mu_h = \omega^\mu_h,\qquad \mathcal{L}_D H = \Gamma H$$

where $\omega^\mu_h$ are reduced $2$-forms associated to contracted multisymplectic forms and $\Gamma$ is the scaling degree. Such a symmetry allows for the identification of a redundant coordinate (for example, a logarithmic scale variable $\rho$), with the phase space coordinates split as $(x^\mu, \psi, p^\mu_\psi, p^\mu_\rho)$.

Reduction proceeds by quotienting $J^1E^*$ by the one-parameter group generated by $D$. The resulting quotient

$$J^1E^* / \sim \cong J^1\widetilde{E}^* \times \mathbb{R}^d$$

inherits a contact or multicontact structure. In coordinates, the reduction yields new variables

$$s^\mu = \frac{p^\mu_\rho}{e^\rho},\quad \Pi^\mu_\psi = \frac{p^\mu_\psi}{e^\rho}$$

with contact Hamiltonian

$H^c(\psi, \Pi^\mu_\psi, s^\mu) = \frac{H}{e^\rho}$

and defining one-forms

$\eta^\mu = ds^\mu - \Pi^\mu_\psi\,d\psi$

leading to the contact Hamilton–Cartan form

$$\Theta_{H^c} = \eta^\mu\wedge d^{d-1}x_\mu + H^c\,d^dx$$

This structure precisely excises the redundant scale variable, defining a reduced geometric arena where only observables and physically relevant momenta remain (Bell et al., 19 Sep 2025).

4. Geometric Interpretation and Physical Consequences

The restricted multimomentum bundle provides a manifestly covariant geometric framework for classical field theories. All spacetime coordinates are treated on equal footing, and the multisymplectic formalism enables a direct encoding of the dynamics using geometric structures. The presence of symmetry reduction yields a finite-dimensional contact structure on the reduced space, in which dissipation, conserved quantities, and the algebra of dynamical observables can be discussed.

In the process, the bundle

• Separates observable variables from redundant or gauge degrees of freedom,
• Identifies the minimal phase-space structure relevant for field evolution,
• Supports covariant quantization and geometric analysis of field-theoretic phenomena,
• Encodes both conservative and dissipative dynamics, with action-dissipation variables naturally represented as components of a Reeb distribution (Bell et al., 19 Sep 2025).

5. Connections to Bundle-Valued Multisymplectic Structures and Moduli Spaces

The mathematics of restricted multimomentum bundles is closely interconnected with contemporary developments in bundle-valued multisymplectic geometry and moduli space stratification. Recent work on bundle-valued $n$-plectic structures extends classical symplectic and multisymplectic theory by considering $(n+1)$-forms valued in vector bundles with connections. Homotopy momentum sections in this context generalize momentum maps using Lie algebroid symmetries, and reduction theorems analogous to Marsden–Weinstein–Meyer apply to vector-valued multimomentum bundles (Hirota et al., 2023). This framework unifies several classical theories into a common language of high-dimensional symplectic geometry and reduction.

Algebraic methods, including those involving Grassmannian stratification and projective construction as in the context of rational curves, provide additional tools for parametrizing and classifying families of restricted multimomentum bundles using discrete invariants and group actions (Alzati et al., 2014). The classification of associated subspaces or "vertices" and the assembly of their moduli spaces are essential for structure theorems and for understanding the interplay of geometric and algebraic properties.

6. Applications and Prospective Implications

Restricted multimomentum bundles are employed in the geometric analysis of classical field theories, including relativity, gauge theory, and systems with dynamical symmetries. Their geometry, especially the canonical and multisymplectic forms, underpins Hamiltonian and Lagrangian descriptions of field dynamics, symmetry identification, and the reduction of unobserved degrees of freedom.

Practical implications include:

• Clean separation of physical observables from non-essential variables in covariant field theories,
• Manifestly coordinate-independent descriptions,
• Natural settings for contact and multicontact structures supporting the analysis of both conservation and dissipation,
• Utility as foundational objects for geometric quantization, given the contact interpretation of the reduced space.

A plausible implication is that such bundles are likely to play a central role in future advances in multisymplectic quantization procedures and in the geometric characterization of moduli spaces of field-theoretic solutions.

7. Summary Table: Key Features of Restricted Multimomentum Bundles

Feature	Mathematical Realization	Physical/Geometric Role
Multiphase Space Structure	$J^1E^*$, local $(x^\mu, y^a, p^\mu_a)$	Arena for covariant field dynamics
Canonical Forms	$\Theta_h$, $\Omega_h = -d\Theta_h$	Multisymplectic structure, dynamics
Hamiltonian Equations	Covariant Hamilton–De Donder–Weyl equations	Evolution in spacetime, not split
Symmetry Reduction	Quotient by group generated by $D$	Excises redundant variables
Contact Structure	Contact Hamiltonian, one-forms $\eta^\mu$	Encodes observables, dissipation

Restricted multimomentum bundles unify geometric, algebraic, and physical principles, forming the backbone of covariant multisymplectic field theory and providing essential infrastructure for symmetry reduction and the identification of the true dynamical content of classical and possibly quantum field theories (Bell et al., 19 Sep 2025, Hirota et al., 2023, Alzati et al., 2014).