Dynamical dimensional reduction in multi-valued Hamiltonians

TL;DR

This paper explores how multi-valued Hamiltonians in complex physical systems lead to a form of dynamical dimensional reduction through degeneracies, revealing new features and interpretations in their symplectic structures.

Contribution

It introduces a framework for describing the evolution of systems with multi-valued Hamiltonians as degenerate dynamical systems, linking multi-valuedness to gauge symmetries and dimensional reduction.

Findings

Multi-valued Hamiltonians cause degeneracies in symplectic structures.

Degeneracies lead to some degrees of freedom becoming gauge symmetries.

The approach applies to various physical systems like higher-dimensional gravity and fluids.

Abstract

Several interesting physical systems, such as the Lovelock extension of General Relativity in higher dimensions, classical time crystals, k-essence fields, Horndeski theories, compressible fluids, and nonlinear electrodynamics, have apparent ill defined sympletic structures, due to the fact that their Hamiltonians are multi-valued functions of the momenta. In this paper, the dynamical evolution generated by such Hamiltonians is described as a degenerate dynamical system, whose sympletic form does not have a constant rank, allowing novel features and interpretations not present in previous investigations. In particular, it is shown how the multi-valuedness is associated with a dynamical mechanism of dimensional reduction, as some degrees of freedom turn into gauge symmetries when the system degenerates.