Convergence of Migdal-Kadanoff iterations: A simple and general proof

A short proof is presented showing convergence of Migdal-Kadanoff iterations for a large class of functions on compact connected Lie groups. From our estimate of the rate of convergence, we deduce lower bounds for string tension and mass gap in hierarchical four-dimensional lattice gauge and twodimensional spin models.     

Stochastic models for lattice gauge theories

Stochastic equations are derived which describe the (Euclidean) time evolution of lattice field configurations, with and without fermions, on a three-dimensional space lattice. It is indicated how the drifts and transition functions may be obtained as asymptotic solutions of a differential equation or from a ground state ansatz. For non-Abelian gauge fields (without fermions) a ground state is constructed which is an exact eigenstate of a Hamiltonian with the same (naive) continuum limit as the Kogut-Susskind Hamiltonian. It is described how Euclidean correlations (like the Wilson loop) are obtained from the stochastic equations and how mass gaps may be obtained from the technique of exit times.     

Generalised actions for lattice gauge models

We consider simple modifications of the conventional Wilson action for lattice gauge theory. An SU(2) action is defined on “plaquettes” of 2×1 links. It is found to possess phase transitions in three- and four-dimensional realisations of the model. A similar model with gauge group Z(2) is also studied, and found to have two phases in three and four dimensions. We discuss the phase structure of Z(N) gauge models in four dimensions with several coupling constants and present phase diagrams for Z(4), Z(5) and Z(6).     

Chiral bosonization in non-abelian gauge theories

The chiral bosonization in non-abelian gauge theories is described starting directly from the QCD functional. For a given mass scale Λ, QCD may be equivalently represented by colour chiral fields, gauge fields and high energy fermions. The effective action for colour chiral fields may admit the existence of a colour skyrmion-boson with baryon number 2/3.     

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.     

Electric-magnetic duality in non-abelian gauge theories

The duality transformation of the vacuum expectation value of the operator which creates magnetic vortices (the 't Hooft loop operator in the Higgs phase), is performed in the radial gauge (x_uA_u^a(x) = 0). It is found that in the weak coupling region (small g) of a pure Yang-Mills theory the dual operator creates electric vortices whose strength is $\text{1}\text{g}$. The theory is self-dual in this region, and the effective coupling of the dual Lagrangian is $\text{1}\text{g}$. (It is self-dual also in the extreme strong coupling region.) Thus the above duality transformation reduces to electric-magnetic duality where the electric field in the 't Hooft loop operators transforms into a magnetic field in the dual operator. In a spontaneously broken gauge theory these results are valid only within the region where the vortices (or the monopoles) are concentrated, or in directions of the algebra space of unbroken symmetry, as self-duality holds only for this subset of fields. Noting that the 't Hooft loop operator project into the subspace of these field configurations we find that it is an electric-magnetic duality for the spontaneously broken theory as well. In the strong coupling region a strong coupling expansion in powers $\text{1}\text{g}$ is suggested.     

Infra-red behaviour in non-abelian gauge theories

The leading infra-red divergent corrections to massive quark scattering in quantum chromodynamics are very simple to fourth order. A simple dependence on the effective coupling constant is conjectured to hold to all orders.     

Charge definition in non-abelian gauge theories

Conserved gauge-invariant electric and magnetic charges are defined for non-abelian gauge theories in terms of the asymptotic symmetries of the field configurations. They are expressed as flux integrals. Illustrations include the magnetic charge of the 't Hooft-Polyakov monopole and the electric and magnetic charges of the Julia-Zee dyon.     

Convergence of strong-coupling expansion and linear potential in lattice gauge theories

We give a numerical lower bound on the radius of convergence of the strong coupling expansion in lattice Yang-Mills theories. From this we infer that the static potential rises linearly starting at ≈1.4 fm.     

The radiation gauge static potential for non-abelian gauge fields

The static potential between a fermion and an anti-fermion in a group singlet state is calculated, through two loops, in the radiation gauge first order formalism. The results of this calculation imply that the Coulomb propagator is not sufficient to determine the static potential: a new function of the coupling constant α_s(?t) is also required.     

The Schwinger mechanisms in non-abelian gauge theories

An alternative to the Higgs-Kibble mechanism for the formulation of spontaneously broken gauge theories has been recently proposed. It is shown how the new procedures can be generalised to the case when non-Abelian gauge groups are present.     

Colliding plane waves in non-abelian gauge theory

A new explicit solution of the classical SU(2) Yang-Mills equations is given. It describes two colliding wave packets, which asymptotically are abelian plane waves localized in the direction of propagation. Although the interaction is nonlinear, the wave packets retain their identity.     

Forbidden ghost states in non-abelian gauge theories

A two-particle state consisting of isotriplets of scalar and zero-helicity gluons and of Faddeev—Popov ghosts is identified as being exterior to the physical subspace for a Yang—Mills theory in a manifestly covariant gauge.     

Operator quantization of non-abelian gauge theories in a completely fixed axial gauge

A consistent quantization of chromodynamics in a completely fixed axial gauge is carried out by using the Dirac bracket quantization procedure. The main results are: The translation of Dirac brackets into equal-time commutators is possible, without ambiguities, because of the absence of ordering problems. All equal-time commutators are compatible with constraints and gauge conditions holding as strong operator relations. All equal-time commutators are compatible with chromoelectric, chromomagnetic, and fermionic fields vanishing at spatial infinity. The colored gauge potentials A^0,a, A^1,a, and A^2,a are seen to develop a physically significant, although pure gauge, behavior at x³ = ± ∞, as required by the presence of a nontrivial topological content. Poincaré invariance is satisfied without introducing in the Hamiltonian “extra” quantum mechanical potentials. The determinant of the Faddeev-Popov matrix does not depend upon the field variables.     

Commutator of gauge generators in non-abelian chiral theory

Commutators among non-abelian fermion currents are calculated using the BJL limit. The relation between the covariant seagull and the gauge dependence of the fermion current is derived for a canonical non-abelian theory using the path integral formulation. We observe that in a non-abelian theory with coupling to chiral fermions this relation is violated and this produces a non-trivial commutator of gauge group generators.     

Renormalization of non-abelian gauge theories in curved space-time

We use indirect, renormalization group arguments to calculate the gravitational counterterms needed to renormalize an interacting non-abelian gauge theory in curved space-time. This method makes it straightforward to calculate terms in the trace anomaly which first appear at high order in the coupling constant, some of which would need a 4-loop calculation to find directly. The role of gauge invariance in the theory is considered, and we discuss briefly the effect of using coordinate-dependent gauge-fixing terms. We conclude by suggesting possible applications of this work to models of the very early universe.