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 Poincaré invariant theories, a canonical action of the Poincaré 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 Poincaré 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.
Highlights
Asymptotic symmetries are those symmetries that appear in theories with long-ranging fields, such as gravity and electrodynamics
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
Summary
Asymptotic symmetries are those symmetries that appear in theories with long-ranging fields, such as gravity and electrodynamics. Based on the results obtained in the study of the asymptotic symmetries of Yang-Mills fields at null infinity [34, 35] and of the results obtained in the Hamiltonian approach of other gauge theories, such as electrodynamics [21] and general relativity [20], one expects to find a well-defined Hamiltonian formulation of the non-abelian Yang-Mills theory, which features a canonical action of a non-trivial group of asymptotic symmetries. These parity conditions seem too strong in that they exclude the possibility of a non-trivial asymptotic Lie-algebra of symmetries and in preventing us to have a non-zero total colour charge. Examples of such dual-Lie-algebra-valued functions that we will encounter and identify with their corresponding Liealgebra-valued functions are the conjugated momenta πα and the Gauss constraint G
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