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 world-line 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.
Highlights
To guarantee the gauge invariance) has been lifted, making the perturbative expansion more manageable
Which we define as the product of their eigenvalues for functions on S1 with anti-periodic boundary conditions (APB)
We examined the Hilbert space of the auxiliary worldline fields which carry the colour information of the matter field and described how to isolate wavefunction components transforming in a desired irreducible representation
Summary
First quantised approaches to quantum field theory begin with a quantum mechanical description of the dynamical and spin degrees of freedom of the field. We will take the generators of this algebra, {T a}, to be Hermitian and choose them in the fundamental representation of the symmetry group (we limit attention to the gauge group SU(N ) for physical reasons), so Aμ = AaμT a These additional details can be incorporated into the worldline action at a classical level by introducing additional Grassmann variables which carry the colour degrees of freedom that upon quantisation will create the associated Hilbert space [19, 20]. In this work we generalise the worldline action to overcome this inadequacy and provide the means to project onto any, arbitrarily chosen representation of the symmetry group This provides a complete framework in which the worldline approach can be applied to any form of fermionic matter field by merely introducing some additional degrees of freedom to take part in quantisation whose role is to ensure that the contribution from undesired wavefunction components are excluded from physical results
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