Multi-Hamiltonian formulations and stability of higher-derivative extensions of $3d$ Chern-Simons

TL;DR

This paper explores the stability and Hamiltonian structures of third-order 3D gauge theories, revealing conditions for classical stability and multiple inequivalent Hamiltonian formulations, including implications for quantization.

Contribution

It introduces multi-Hamiltonian formulations for higher-derivative 3D gauge theories and analyzes their stability and quantization properties.

Findings

Certain parameter ranges yield bounded conserved quantities indicating classical stability.

The theory admits multiple, inequivalent Hamiltonian formulations not related by canonical transformations.

Interacting theories with spinor fields can remain stable and Hamiltonian if bounded conserved quantities exist.

Abstract

Most general third-order $3d$ linear gauge vector field theory is considered. The field equations involve, besides the mass, two dimensionless constant parameters. The theory admits two-parameter series of conserved tensors with the canonical energy-momentum being a particular representative of the series. For a certain range of the model parameters, the series of conserved tensors include bounded quantities. This makes the dynamics classically stable, though the canonical energy is unbounded in all the instances. The free third-order equations are shown to admit constrained multi-Hamiltonian form with the zero-zero components of conserved tensors playing the roles of corresponding Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski's one, which is unbounded. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. This means, the theory admits inequivalent quantizations at the free level. Covariant interactions are included with spinor fields such that the higher-derivative dynamics remains stable at interacting level if the bounded conserved quantity exists in the free theory. In the first-order formalism, the interacting theory remains Hamiltonian and therefore it admits quantization, though the vertices are not necessarily Lagrangian in the third-order field equations.