Canonical structure and boundary conditions in Yang-Mills theory

Abstract

The canonical structure of pure Yang-Mills theory is analysed in the case when Gauss' law is satisfied identically by construction. It is shown that the theory has a canonical structure in this case, provided one uses a special gauge condition, which is a natural generalisation of the Coulomb gauge condition of electrodynamics. The emergence of a canonical structure depends critically also on the boundary conditions used for the relevant field variables. Possible boundary conditions are analysed in detail. A comparison of the present formulation in the generalised Coulomb gauge with the well known Weyl gauge ($A_{0} = 0$) formulation is made. It appears that the Hamiltonians in these two formulations differ from one another in a non-trivial way. It is still an open question whether these differences give rise to truly different structures upon quantisation. An extension of the formalism to include coupling to fermionic fields is briefly discussed.