Abstract
In the worldline approach to non-Abelian field theory the colour degrees of freedom of the coupling to the gauge potential can be incorporated using worldline “colour” fields. The colour fields generate Wilson loop interactions whilst Chern-Simons terms project onto an irreducible representation of the gauge group. We analyse this augmented worldline theory in phase space focusing on its supersymmetry and constraint algebra, arriving at a locally supersymmetric theory in superspace. We demonstrate canonical quantisation and the path integral on S1for simple representations of SU(N).
Highlights
Non-Abelian symmetry groups are a crucial aspect of theoretical physics, with focus on groups such as SU(2), SU(3) of the standard model and SU(5) or SO(10) that appear in attempts at unification [1, 2]
We carried out a phase space analysis of the extended worldline action describing a Dirac spinor transforming in an arbitrary irreducible representation of SU(N)
The action is augmented by auxiliary colour fields that generate the gauge degrees of freedom
Summary
Non-Abelian symmetry groups are a crucial aspect of theoretical physics, with focus on groups such as SU(2), SU(3) of the standard model and SU(5) or SO(10) that appear in attempts at unification [1, 2]. Its origins can be traced to Feynman [6], with a revival of interest following Bern and Kosower’s Master Formulae for field theory amplitudes derived from string theory [7, 8] It is recently finding application in a wide range of physical problems, including multi-loop effective actions and scattering amplitudes [3, 9], constant field QED and the Schwinger effect [10, 11], gravitational interactions [12, 13], higher spin fields [14, 15], quantum fields in non-commutative space-time [16, 17] and the form factor decomposition of the four gluon vertex reported by Schubert elsewhere in these proceedings. Any matter multiplet can be described with a single worldline theory without the need for a manual path ordering prescription, that is instead generated automatically by the path integral over the colour fields This procedure leads to compact expressions in perturbation theory that combine many Feynman diagrams through preservation of permutation symmetry of external gluon legs
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