Mixed symmetry Wilson-loop interactions in the worldline formalism

Using the worldline formalism of the Dirac field with a non-Abelian gauge
symmetry we show how to describe the matter field transforming in an arbitrary
representation of the gauge group. Colour degrees of freedom are carried on the worldline by auxiliary fields, responsible for providing path ordering and the
Wilson-loop coupling. The Hilbert space of these fields is reducible but we
make use of recent work in order to project onto a single, arbitrary, irreducible representation. By functionally quantising the resulting theory we show that this procedure correctly generates the Wilson-loop interaction between the gauge field and the matter field taken to transform in a chosen representation. This work has direct application to physical observables such as scattering amplitudes in the presence of such a matter multiplet and lifts the restriction on the type of matter that has previously featured in worldline calculations.