Mixed symmetry tensors in the worldline formalism

TL;DR

This paper develops a worldline formalism for quantum fields with non-Abelian gauge symmetry, enabling projection onto arbitrary tensor representations using auxiliary variables and Chern-Simons terms.

Contribution

It introduces a method to represent and project matter fields onto any irreducible tensor representation within the worldline formalism using auxiliary variables and gauged flavor symmetry.

Findings

Method to project onto arbitrary irreducible representations

Verification of the representation counting via degrees of freedom

Potential application to one-loop scattering amplitude calculations

Abstract

We consider the first quantised approach to quantum field theory coupled to a non-Abelian gauge field. Representing the colour degrees of freedom with a single family of auxiliary variables the matter field transforms in a reducible representation of the gauge group which - by adding a suitable Chern-Simons term to the particle action - can be projected onto a chosen fully (anti-)symmetric representation. By considering F families of auxiliary variables, we describe how to extend the model to arbitrary tensor products of F reducible representations, which realises a U(F) "flavour" symmetry on the worldline particle model. Gauging this symmetry allows the introduction of constraints on the Hilbert space of the colour fields which can be used to project onto an arbitrary irreducible representation, specified by a certain Young Tableau. In particular the occupation numbers of the wavefunction - i.e. the lengths of the columns (rows) of the Young Tableau - are fixed through the introduction of Chern-Simons terms. We verify this projection by calculating the number of colour degrees of freedom associated to the matter field. We suggest that, using the worldline approach to quantum field theory, this mechanism will allow the calculation of one-loop scattering amplitudes with the virtual particle in an arbitrary representation of the gauge group.