Gauge Invariant Decomposition of Yang-Mills Potentials

Abstract

The Yang‐Mills (YM) potentials are decomposed into an isovector part and a part which transforms nonhomogeneously under local gauge transformations. Two decompositions are shown; one of them is based on a gauge‐invariant version of the transversality condition, and the other arises from a gauge‐invariant modification of the Lorentz condition. The latter is Lorentz as well as gauge invariant. The gauge invariance of the decompositions is obtained at the expense of locality since the separate parts of the decomposed potential are functionals of the full YM potential. The transverse‐longitudinal decomposition is used to throw the YM sourceless field equations into a gauge‐invariant Hamiltonian form. Static fields in the Hamiltonian formulation are discussed. The decompositions are used to construct massive, gauge‐invariant but nonlocal Lagrangians. A Lorentz and gauge‐invariant nonlocal interaction of the YM field with a spinor‐isospinor field is formed. The transverse‐longitudinal decomposition is used to investigate the geometric structure of a configuration space Ω of YM potentials. The nonexistence of submanifolds of Ω orthogonal to the gauge‐invariant manifoldsX ∈ Ω is proved in contradistinction to the electromagnetic case. A Green's functional for the Yang‐Mills field is represented explicitly by an infinite power series of functionals and is shown to be self‐adjoint.