Gauge Symmetries, Contact Reduction, and Singular Field Theories

TL;DR

<3-5 sentence high-level summary> The article develops a comprehensive geometric framework for reducing singular (gauge) theories that possess scaling symmetries, by applying pre-symplectic and contact/pre-multicontact formalisms to both particle and field theories. It demonstrates that restriction and reduction operations commute, enabling dynamically-equivalent, frictional descriptions after excising redundant global scales. Through detailed particle-level examples and a substantive non-Abelian gauge theory with a dilaton, the work shows how scaling reductions yield dissipative dynamics while preserving the observable content. The framework has potential implications for classical General Relativity and provides a foundation for exploring reductions in covariant, manifestly relativistic field theories, with avenues toward quantum considerations noted in the outlook.

Abstract

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.

Figures (1)

• Figure 1: Commutative diagram showing how the two constraint algorithms run in parallel. Starting from the cotangent bundle, with its canonical symplectic form, the right-hand path shows how we may restrict to the primary constraint manifold $M_0$, and carry out the pre-symplectic algorithm, to reach the final space $\mathcal{P}$. This manifold is symplectic, and thus allows a contact reduction to be carried out. Alternatively, the scaling degree of freedom may be excised from the full symplectic phase space, defining the contact manifold $C$. Upon projecting the primary constraint functions that define $M_0$, $C$ is restricted to the pre-contact space $C_0$, and the left-hand path illustrates the implementation of the pre-contact algorithm. Both paths lead to the same final reduced space $C_{\mathcal{P}}$.