3-dimensionalSU(2) lattice gauge theory in terms of gauge invariant variables

As a first step towards a duality transformation for theSU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for theO(4) symmetric Heisenberg ferromagnet Rühl's boson representation is used to derive the result.     

On the link between strong and weak coupling expansions for the SU(2) lattice gauge theory

The intermediate region between the strong and weak coupling expansions of SU(2) lattice gauge theory is analyzed by inverting the strong coupling series of the average plaquette energy E_p(J).

Taking into account the weak coupling behaviour too, we get Padé approximants for J(E_p) valid in the whole physical region. The zeros of pJ/pE_p are mapped into singularities of E_p(J); they occur for complex values of J and give rise to a peak in the specific heat in fair agreement with the Monte Carlo simulation data. The variable E_p is further used to analyze the strong coupling expansions for the roughening indicator, the string tension and the mass gap.     

The quark-antiquark potential of SU(2) lattice gauge theory in 2+1 dimensions

Within the framework of a hamiltonian formulation of SU($\text{N}$) lattice gauge theory in two spatial dimensions we develop a weak coupling expansion in the static quark-antiquark sector. For SU(2) the quark-antiquark potential is calculated to fourth order. Moreover we extend the former calculation of the energy gap in the gauge invariant sector to general SU($\text{N}$) and extrapolate to the infinite lattice limit from large lattices. Due to the infrared divergence of the weak coupling expansion on infinite lattices the quark-antiquark potential contains a term diverging logarithmically with the lattice expansion. Hence the expansion has to be interpreted as a pseudo-perturbation expansion in the sense of Symanzik.     

Convergence of theU(1)4 lattice gauge theory to its continuum limit

It is shown that in four space-time dimensions the compactU(1) lattice gauge theory with general energy function converges to a renormalized free electromagnetic field on the current sector as the lattice spacing approaches zero, provided the coupling constant is sufficiently large. For the Wilson energy function, it is possible, by judicious choice of the Gibbs state, to get convergence for arbitrary coupling strengths. Furthermore, for all but a countable number of values of the coupling constant, the limit exists and is independent of the particular state chosen to define the lattice model.This research was supported in part by an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship and in part by N.S.F. Grant Nos. DMS 84-01997 and DMS-8610730(1)     

The deconfining transition for finite-temperature U(1) lattice gauge theory in (2 + 1) dimensions

A detailed examination of the deconfining transition in finite-temperature (2 + 1)-dimensional U(1) lattice gauge theory is presented. The transition appears to be of the Kosterlitz-Thouless type, in agreement with the two-dimensional XY model behaviour expected from universality arguments. However, measurements of the critical exponent η of the Wilson line correlation function do not give the predicted XY value.     

Plaquette expansion study of compact U(1) lattice gauge theory in (2 + 1) dimensions

The plaquette expansion is applied to compact U(1) lattice gauge theory in (2 + 1) dimensions to high order: 1/N ⁸ for the ground state energy density and 1/N ⁷ for the anti-symmetric or photon mass gap, where N is defined as the number of plaquettes. Evidence of scaling in the photon mass gap is observed for low order (≤ 1/N ⁴) for inverse coupling values β = 0. 7 to 1. 25 with scaling behaviour given by the weak-coupling formula: M ²/β = exp(-5. 01β + 5. 82) in good agreement with other studies. Higher order results appear to diverge from the scaling slope past the transition point portenting the prospect that the strong coupling trial state in this region gives vise to an asymtotic series in 1/N for the photon mass gap.     

Convergence of U(1)3 lattice gauge theory to its continuum limit

It is shown that in three space-time dimensions the pure U(1) lattice gauge theory with Villain action and fixed coupling constant converges to the free electromagnetic field as the lattice spacing approaches zero. The same holds for the Wilson action on the electric sector.     

Continuum limit of a hierarchical SU(2) lattice gauge theory in 4 dimensions

We study nonperturbative renormalizability of ad=4 hierarchical SU(2) gauge model that realizes Migdal's recursion relation as an exact renormalization group transformation. A continuum limit of effective actions is shown to exist as the scaling limit, both for initial Wilson and heat kernel actions. These limit effective actions exhibit ultraviolet asymptotic freedom and provide a strictly positive string tension.     

Linked cluster expansions for U(1) lattice gauge theory in 2 + 1 and 3 + 1 dimensions

Compact U(1) lattice gauge theory is studied in 2 + 1 and 3 + 1 dimensions using strong coupling series expansions and the recently proposed exact linled cluster expansion alborithm Results for the vacuum energy, specific heat and axial string tension in 2 + 1 dimensions are in agreement with previous finite lattice estimates. In 3 + 1 dimensions, we present new strong coupling series results (order g^?40) which together with the ELCE estimates show evidence of a continuous phase transition at x = 1/g⁴ = 0.72 ± 0.08. The associated critical index for the vanishing string tension is μ = 0.65 ± 0.12. The axial string tension in D = 3 + 1 appears to undergo a non-deconfining roughening transition at smaller x (0.56 ± 0.07).     

Universality in SU(2) lattice gauge theory

A method of resumming perturbation theory is used to re-analyze previous Monte Carlo data of Bhanot and Dashen. We find no inconsistency with universality. There appears to be a relatively large region where Monte Carlo studies can reliably be done.     

A microscopic semiclassical confining field equation forU(1) lattice gauge theory in 2+1 dimensions

We present a semiclassical nonlinear field equation for the confining field in 2+1-dimensionalU(1) lattice gauge theory (compact QED). The equation is derived directly from the underlying microscopic quantum Hamiltonian by means of truncation. Its nonlinearities express the dynamic creation of magnetic monopole currents leading to the confinement of the electric field between two static electric charges. We solve the equation numerically and show that it can be interpreted as a London relation in a dual superconductor.     

U(2) four-dimensional simplicial lattice gauge theory

Monte Carlo simulations for pureU (2) gauge theory on a four-dimensional simplicial lattice with six sites in each direction are reported. Wilson loops and the string tensions for squares and triangles are presented. A first-order phase transition, similar to that found for the hypercubical lattice, is observed and found to confineSU (2) colour and deconfineU (1) charge.     

Finite temperature SU(2) lattice gauge theory

We discuss SU(2) lattice gauge theories at non-zero temperature and prove several rigorous results including i) the absence of confinement for sufficiently high temperature in the pure gauge theory, and ii) the absence of spontaneous chiral symmetry breaking for sufficiently high temperature in the theory with massless fundamental representation fermions.