Gauge Symmetries, Contact Reduction, and Singular Field Theories

The reduction of dynamical systems which are invariant under changes of global scale is well-understood, for classical theories of particles, and fields. The excision of the superfluous degree of freedom describing such a scale leads to a dynamically-equivalent theory, which is frictional in nature. In this article, we extend the formalism to physical models, of both particles and fields, described by singular Lagrangians. Our treatment of classical field theory is based on the manifestly covariant Hamilton De-Donder Weyl formalism, in which the Lagrangian density is introduced as a bundle morphism on the pre-multisymplectic velocity phase space $J^1E$. The results obtained are subsequently applied to a number of physically-motivated examples, as well as a discussion presented on the implications of our work for classical General Relativity.

💡 Research Summary

The paper develops a systematic framework for reducing classical mechanical and field theories that possess a global scale invariance, extending the well‑known procedure for regular (non‑singular) systems to those described by singular Lagrangians. The authors begin by reviewing the pre‑symplectic geometry underlying ordinary Lagrangian mechanics, emphasizing that singularity manifests as a non‑maximal rank of the Hessian with respect to velocities, which in turn makes the Legendre map a surjection onto a primary constraint submanifold rather than a diffeomorphism. For almost‑regular systems they introduce the primary constraint manifold (M_{0}=F_{L}(TQ)) equipped with the induced pre‑symplectic form (\omega_{0}) and Hamiltonian (H_{0}).

A detailed pre‑symplectic constraint algorithm is then presented. Starting from the equation (\iota_{X_{H}}\omega_{0}=dH_{0}), the algorithm iteratively isolates points where the Hamiltonian differential lies in the image of the bundle map (\flat_{0}). At each step secondary constraints are generated by demanding tangency of the Hamiltonian vector field to the previously obtained constraint surface. The process terminates when a final constraint submanifold (M_{f}) is reached, which is decomposed into first‑class (gauge) and second‑class constraints. Second‑class constraints are imposed strongly using Dirac brackets, while first‑class constraints generate gauge orbits; quotienting by these orbits yields the physical phase space equipped with a genuine symplectic form.

The novel contribution of the work is to combine this pre‑symplectic reduction with contact geometry, which naturally appears when the redundant scale degree of freedom is eliminated. By extending the configuration space to (TQ\times\mathbb{R}) and defining the contact 1‑form (\eta_{L}=dz-\theta_{L}), the authors obtain a (pre‑)contact Lagrangian system ((TQ\times\mathbb{R},\eta_{L},E_{L})). In the hyper‑regular case the Legendre map becomes a diffeomorphism onto (T^{*}Q\times\mathbb{R}) with canonical contact form (\eta=dz-p_{i}dq^{i}). The associated contact Hamiltonian (H) satisfies the contact Hamilton’s equations (\bar\flat(X_{H})=dH-(R(H)+H)\eta), where (R) is the Reeb vector field. These equations describe dissipative (non‑conservative) dynamics, showing that the removal of the scale variable converts a conservative system into a frictional one.

To treat field theories, the authors adopt the manifestly covariant De‑Donder‑Weyl multisymplectic formalism. The Lagrangian density is regarded as a bundle morphism on the first jet bundle (J^{1}E), providing a pre‑multisymplectic structure on the velocity phase space. The pre‑symplectic constraint algorithm is lifted to this infinite‑dimensional setting, and contact reduction is shown to be compatible with the multisymplectic geometry, yielding a “multicontact” reduced space when the field theory is action‑dependent.

Three illustrative examples are worked out in detail. First, a simple particle model with a scaling symmetry is reduced in two different orders—first excising the scale variable, then applying the constraint algorithm, and vice‑versa—demonstrating that the final frictional dynamics is independent of the order. Second, a string‑inspired low‑energy non‑Abelian gauge theory is examined; the dilaton field, which controls the string coupling, plays the role of the redundant scale degree of freedom. Contact reduction removes the dilaton while preserving the gauge structure, producing a dissipative gauge dynamics without fixing the coupling constant a priori. Third, the Einstein–Hilbert action of General Relativity is decomposed into a conformal factor and a unit‑determinant metric, exposing a hidden scaling symmetry. Applying the contact reduction yields a non‑conservative version of gravity, suggesting new avenues for exploring dissipative effects in classical GR.

In the concluding section the authors argue that their combined pre‑symplectic/contact reduction framework offers a unified treatment of singular theories, bridging the gap between constraint analysis and non‑conservative dynamics. They point out that the approach is readily adaptable to quantisation schemes, especially path‑integral formulations of dissipative field theories, and that the multisymplectic‑contact synthesis may illuminate the structure of quantum gravity candidates where scale invariance and gauge redundancy play central roles. Future work is suggested on extending the method to higher‑order theories, incorporating fermionic degrees of freedom, and exploring the physical interpretation of the emergent friction terms in both particle and field contexts.