Dynamical Similarity in Multisymplectic Field Theory

TL;DR

The work develops a rigorous geometric framework for dynamical similarity in covariant multisymplectic field theory and extends it to multicontact geometry to handle dissipative, action-dependent dynamics. It shows that a scaling symmetry can reduce both Lagrangian and Hamiltonian descriptions to a lower-dimensional, action-dependent theory whose reduced space carries a multicontact structure, with the eliminated scale appearing as a friction-like component that preserves observable dynamics. Through explicit analyses of free and interacting scalar fields, the authors illustrate how the reduced theory decouples reduced field dynamics from the dissipation while revealing how interactions alter this separation. The paper also provides a classification scheme for field theories based on gauge coverage and metric dynamics, discusses the interplay with General Relativity, and outlines future work toward gauge fixing, quantization, and extensions to fermionic degrees of freedom.

Abstract

Symmetry under a particular class of non-strictly canonical transformations may be used to identify, and subsequently excise degrees of freedom which do not contribute to the closure of the algebra of dynamical observables. In this article, we present a mathematical framework which extends this symmetry reduction procedure to classical field theories described within the covariant De-Donder Weyl formalism. Both the Lagrangian and Hamiltonian formalisms are discussed; in the former case, the Lagrangian is introduced as a bundle morphism $\mathcal{L}:J^1E\rightarrow\wedge^mM$, whilst the Hamiltonian description takes as its multiphase space the restricted multimomentum bundle $J^1E^*$. The contact reduction process is implemented in a number of particular cases, and it is found that the geometry of the reduced space admits an elegant physical interpretation.