Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation

TL;DR

This paper explores the asymptotic symmetry group of SU(N)-Yang-Mills theory in Hamiltonian formalism, revealing that under standard conditions, no non-trivial symmetries or charges exist, hinting at classical color confinement.

Contribution

It extends the Hamiltonian approach used in gravity and electrodynamics to non-abelian Yang-Mills fields, analyzing conditions for non-trivial asymptotic symmetries and charges.

Findings

Standard fall-off conditions admit no non-trivial asymptotic symmetries.

Relaxing conditions does not yield non-trivial symmetries without violating Hamiltonian formalism.

Classical color confinement may be linked to the absence of asymptotic charges in non-abelian gauge theories.

Abstract

We investigate the asymptotic symmetry group of the free SU(N)-Yang-Mills theory using the Hamiltonian formalism. We closely follow the strategy of Henneaux and Troessaert who successfully applied the Hamiltonian formalism to the case of gravity and electrodynamics, thereby deriving the respective asymptotic symmetry groups of these theories from clear-cut first principles. These principles include the minimal assumptions that are necessary to ensure the existence of Hamiltonian structures (phase space, symplectic form, differentiable Hamiltonian) and, in case of Poincar\'e invariant theories, a canonical action of the Poincar\'e group. In the first part of the paper we show how these requirements can be met in the non-abelian SU(N)-Yang-Mills case by imposing suitable fall-off and parity conditions on the fields. We observe that these conditions admit neither non-trivial asymptotic symmetries nor non-zero global charges. In the second part of the paper we discuss possible gradual relaxations of these conditions by following the same strategy that Henneaux and Troessaert had employed to remedy a similar situation in the electromagnetic case. Contrary to our expectation and the findings of Henneaux and Troessaert for the abelian case, there seems to be no relaxation that meets the requirements of a Hamiltonian formalism and allows for non-trivial asymptotic symmetries and charges. Non-trivial asymptotic symmetries and charges are only possible if either the Poincar\'e group fails to act canonically or if the formal expression for the symplectic form diverges, i.e. the form does not exist. This seems to hint at a kind of colour-confinement built into the classical Hamiltonian formulation of non-abelian gauge theories.