non-Abelian Version Research Articles

The Lipkin-Weisberger-Peshkin difficulties in abelian and non-abelian gauge theories

The question is raised on the validity of the usual derivation of the Lipkin-Weisberger-Peshkin (LWP) difficulty. It is shown that the LWP difficulty can be derived legitimately in the framework of the Wu-Yang formalism. A non-abelian version of the LWP difficulty is presented in a Wu-Yang solution of the pure SU(2) Yang-Mills theory, showing a failure of the Jacobi identity at one point through the term − i4π( r ̂ ·T)δ 3(x) . A nonrelativistic treatment of the SU(2) doublet particle with spin 1 2 around this solution is given. Some remarks are made also on a possible LWP difficulty in the 't Hooft-Polyakov monopole.

Local observables in non-abelian gauge theories

Labelling of the physical states of a non-abelian gauge theory on a lattice in terms of local observables is considered. The labelling is in terms of local color electric field observables and (separately) local color magnetic field observables. Matter fields are also included. Non-local observables required when space is multiply connected, are specified. The non-abelian version of Stokes' theorem is considered. Relevance to the continuum theory is discussed in detail.

Exact computation of loop averages in two-dimensional Yang-Mills theory

We present an explicit algorithm that allows the exact computation of all nonlocal gauge-invariant correlation functions in two-dimensional Yang-Mills theory. Explicit expressions are given for the expectation value of entangled loop operators and their products in the $\mathrm{U}(N)$ theory with arbitrary $N$.Using these results we show that the Schwinger-Dyson equation for loops holds without singularities in the continuum, and we verify the factorizability of the correlation functions in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. A non-Abelian version of Stokes's theorem which is used in the calculations is derived for an arbitrary number of dimensions.

A superfield formulation of extended supersymmetric gauge theories

We present a superfield framework in which to discuss extended supersymmetric Yang-Mills theories in an unconstrained manner. Superfields and lagrangians are constructed for the SO(2) and SO(4) cases for both the abelian and linearised non-abelian versions and the auxiliary component fields are obtained in those cases. Extension to the interacting or non-abelian case presents severe difficulties which are discussed briefly.

CP n−1 models and their non-abelian version HP n−1 : A study of the quantum properties

A class of models with non-abelian gauge invariance called HP n−1 (they are a generalization of the abelian CP n−1 models) is considered at the quantum level, in particular in view of their relation with the O( n) non-linear σ-models, which is connected with the confinement property. Renormalization, critical behaviour and the 1 n expansion are studied in 2 or 2 + ϵ dimensions. A comparison between the abelian CP n−1 and the non-abelian HP n−1 allows us to discuss the rôle of the classical solutions in the confinement problem.

Concerning axially symmetric monopole-type solutions

We study axially symmetric static, finite-energy solutions to Yang-Mills-Higgs field equations. The Lagrangian consists of a triplet of Higgs fields and a triplet of gauge fields and has local SO(3) gauge invariance. The solutions can describe, in principle, field configurations with arbitrary monopole charge. Although we have not found analytic solutions to the coupled partial differential equations involving six unknown functions, we do find consistent solutions in the asymptotic region as well as in the region near the axis of symmetry. We investigate the cylindrically symmetric configuration which is a particular case of our axially symmetric formulation and demonstrate that a vortex-type solution exists. We thus have a non-Abelian version of the Nielsen-Olesen model. We discuss the relevance of this vortex-type solution to our axially symmetric system and present plausibility arguments concerning the emergence of a non-Abelian stringlike configuration.