Variational method in hamiltonian lattice gauge theories

A general expression for the expectation value of the hamiltonian of a d + 1 dimensional lattice gauge theory as a function of the norm of the variational state (that itself has the form of a partition function of a d-dimensional lattice gauge theory) is given. Applications include U(1), SU(2), U(2) and U(N) gauge theories for large N in d = 2 + 1 dimensions. It is also demonstrated that the deconfining phase transition is of first order in every dimension above the critical one, provided it is of first or second order at the critical dimension.     

Analytic approximations to hamiltonian lattice field theories: (I). Periodic gaussian model for U(1) theories

Periodic gaussian models are introduced for local and global U(1) invariant hamiltonian lattice field theories. The models coincide with standard lattice theories at weak coupling, but the leading non-perturbative contributions to wave functions and physical quantities are exactly calculable. Electric charges are confined and the mass gap is finite if correlations of an integer-valued magnetic field are of infinite range (d = 2 + 1 gauge model). Otherwise, for short-range correlations, the mass gap and the string tension vanish at weak coupling (QED, XY model, etc.)     

Large dimensional lattice gauge theories: (I). Z2 pure gauge system

The transition pattern of lattice gauge theories can be stydied by a variational method based on strong coupling series. For large space-time dimension d, this leads to a 1/d expansion when the parameter βd is kept fixed. The first-order phase transition of the Z₂ pure gauge system is studied here.     

How to do mean field theory in Feynman gauge and doing it for U(1) with corrections to fourth order

It is demonstrated how mean field theory with corrections from fluctuations may be applied to lattice gauge theories in covariant gauges. By fixing the gauge at tree level the importance of fluctuations is decreased. This is understood as inclusion of terms of next-to-leading-order in d in the definition is the mean field tree approximation, d being the dimension of the lattice. The gauge group U(1) and Wilson's action are used as testing ground. Tree and one-loop results comparable to those previously obtained in axial gauge are obtained for d = 4. The next three correction terms to the free and plaquette energies are evaluated in Feynman gauge. The truncated asymptotic series thus obtained is compared to that of the ordinary weak coupling expansion. The mean field series gives, to those orders studied, a much better approximation. The location of phase transitions in 4d and 5d are predicted with 1% error bars.     

Universal properties of U(1) gauge theories

The previously known analogies between four-dimensional compact U(1) lattice gauge theories and the two-dimensional planar model are extended to a number of other results. We show that the monopoles in the gauge theory renormalize the coupling constant α by an amount proportional to the susceptibility of the monopole gas. Confinement occurs when this susceptibility diverges. We argue that α is analogous to the critical exponent η of the planar model, and that the transition occurs at a universal critical value α_c.We also define an analogue of the superfluid density for the gauge theory, in terms of the dependence of the free energy on the boundary conditions, and show that it is universally related to α. Finally, we show that the same physics emerges from a continuum U(1) theory with real magnetic monopoles.     

Spectrum of lattice gauge theories with fermions from a 1/d expansion at strong coupling

We study the strong coupling limit of U(N) or SU(N) gauge theories with fermions on a lattice. The integration over the gauge and fermion degrees of freedom is performed by analytic methods, leading to a partition function in terms of localized meson and baryon fields. A method for deriving a systematic expansion in the inverse of the space-time dimension of the corresponding Green functions is developed. It is applied to the study of spontaneous breakdown of chiral symmetry, which occurs for any U(N) or SU(N) theory with fermions in the fundamental representation. Meson and baryon spectra are then computed, and found to be in close agreement with those obtained by numerical methods at finite coupling. The pion decay constant is estimated.     

Phase transition analysis in Z2 and U(1) lattice gauge theories

The transition region of Z₂ lattice gauge theory is investigated by inverting the strong coupling series of the average plaquette energy E_P(J). We find a clear evidence for a first-order transition and the existence of a metastable phase. In the U(1) case we confirm a second-order phase transition even if there is a little discrepancy on the critical point position as indicated by Monte Carlo simulations.     

Grand unification of effective gauge theories

An effective non-renormalizable SU(3)×SU(2)×U(1) invariant gauge theory results at ordinary energies when superheavy fields are integrated out from a grand unified theory based on a simple gauge group G. The solutions of the second-order renormalization-group equations for the gauge coupling constants of the effective theory are examined. General formulae for the superheavy vector boson mass and for sin²θ near M_W are given in this approach to grand unification. The superheavy vector boson mass is plotted against the QCD scale parameter Λ for a certain set of grand unified models. Corrections to the prediction when the set of models is enlarged are discussed, and illustrated with examples from G≡SU(5) and O(10).     

Kosterlitz-Thouless transition for the finite-temperatured=2+1,U(1) Hamiltonian lattice gauge theory

We prove that in thed=2+1,U(1) Hamiltonian (continuous time) lattice gauge theory the confining potential between two static external charges grows logarithmically with their distance, at sufficiently high temperatures. As it is known that for zero or low temperatures and large coupling constant the model confines linearly, we have therefore established the existence of a Kosterlitz-Thouless transition. Our results are based on a Mermin-Wagner type of argument combined with correlation inequalities and known results for the two-dimensional (spin) Villain model.     

Migdal renormalization group approach to deconfinement phase transition

The Migdal renormalization group approach is applied to a finite temperature lattice gauge theory. Imposing the periodic boundary condition in the timelike orientation, the phase structure of the finite temperature lattice gauge system with a gauge groupG in (d+1)-dimensional space is determined by two kinds of recursion equations, describing spacelike and timelike correlations, respectively. One is the recursion equation for ad-dimensional gauge system with the gauge groupG, and the other corresponds to ad-dimensional spin system for which the effective theory is described by the nearest neighbor interaction of the Wilson lines. Detailed phase structure is investigated for theSU(2) gauge theory in (3+1)-dimensional space. Deconfinement phase transition is obtained. Using the recursion equation for the three dimensional spin system of the Wilson lines, it is shown that the flow of the renormalization group trajectories leads to a phase transition of the three dimensional Ising model.     

On the connection between spin glasses and gauge field theories

A simple connection between Ising spin glasses and the Z₂ lattice gauge theory, at negative plaquette temperatures, is presented. It is first shown that annealed models give useful lower bounds on the free energy and ground-state energy of spin glasses. However, they have unphysical low temperature properties (e.g. a negative entropy), which are related to a temperature dependence of the frustration. A restricted annealing scheme is presented which remedies this deficiency through the introduction of a pure gauge coupling counterterm. The possible phase diagrams of the lattice gauge system and their relevance to spin glass transitions are discussed.     

Renormalization group equations in lattice gauge theories

New recursion equations for renormalization group transformations of the Migdal-Kadanoff type are obtained for gauge systems including fermion variables on a d-dimensional Euclidean space-time lattice. It is shown that in the weak gauge coupling region these equations have β-functions similar to those of continuum field theories in the case of U(1), SU(2) gauge groups (QED, QCD). On the other hand in the strong-coupling limit there is an infrared attractive fixed point corresponding to a color-confining effective system in both groups. A possible entire trajectory of the non-Abelian system is briefly conjectured.     

Variational approach for the N-state spin and gauge Potts model

A hamiltonian variational treatment is applied both to the spin Potts model and to its gauge version for any number of states N and spatial dimensions d?2. Regarding the former we reproduce the correct critical coupling and latent heat for not too low N and d. For the latter, our approach turns the gauge theory into an equivalent d-dimensional classical spin model, which evaluated for d + 1 = 4 gives results in agreement with 1/N expansions.     

Anomaly-free supergravity theories remain anomaly-free after dimensional reduction

We show that the anomaly after compactification of a supergravity theory coupled to Yang-Mills matter is usually given by an integral of the original anomaly over the compact space, as long as there are no isometries for the compact space. This means that a supergravity theory, whose anomaly vanishes identically (i.e., without the addition of local counter terms to the action), will remain anomaly-free after compactification to any lower dimension, subject to some restrictions on self-dual antisymmetric tensors. We next consider the case where the original anomaly cancels by the Green-Schwarz mechanism. In this case, again subject to the restrictions on self-dual antisymmetric tensors, the anomaly will still cancel after compactification to any lower dimension D > 2, provided that: (1) There are no U(1) gauge groups after compactification. (2) There exists a three-form field strength H such that dH = (TrR₀² + kTrF₀²), or that the compact space is chosen such that (TrR₀² + kTrF₀²) = 0.     

Strong coupling expansions for the mass gap in lattice gauge theories

We derive strong coupling expansions for the mass gap in euclidean lattice gauge theories in any space-time dimension. For gauge groups SU(2), SU(3), Z₂ and Z₃ the series are calculated up to order g^?16. They are used to get rough estimates for the lowest glueball mass in continuum SU(2) and SU(3) gauge theories, assuming a sudden crossover from strong to weak coupling behaviour in the lattice theory.     

Alternative scaling hypothesis forSU(2) andSU(3) lattice gauge theories

High precision data from a variety of sources forSU(2) andSU(3) Wilson action lattice gauge theory are analyzed with respect to the hypothesis of the possible existence of a zero temperature deconfining phase transition, in analogy with theU(1) theory. The internal energy, specific heat, string tension, and Wilson line, fit well to correlation length scaling laws associated with a finite order transition occurring at the weak coupling end of the crossover region for both theories. TheSU(2) theory is consistent with a correlation length exponent ν=2/3 and critical pointβ _c ≈2.47. ForSU(3) the data fit well to ν=1 andβ _c ≈6.69. Additional indirect evidence for the existence of such phase transitions is discussed, as is the possible crucial role of light dynamical fermions in the confinement mechanism.     

Strong coupling expansion of meson propagator in finite temperature lattice gauge theories

We consider Susskind fermions on a (d+1)-dimensional lattice interacting with aU(n) gauge field at finite temperature. We calculate the meson propagator in an expansion in 1/g ² and 1/d and determine the meson masses. To the order considered the results are identical to those obtained at zero temperature.     

Compactification of Einstein-Yang-Mills theories on CPn

We study compactification of Einstein-Yang-Mills theories in 2n + 4 dimensions on the manifold CP_n, with a classical gauge field that is equal to the spin connection. The complete boson fluctuation spectrum is calculated and no tachyons, ghosts or massless scalars are found for the minimal Yang-Mills group SU(n) × U(1). For larger groups, tachyons or massless scalars may appear.     

General relativity and gauge theories

The theory of general relativity is presented in the form of a gauge field theory by use of the group SL(2,C). The following topics are discussed: (1)Spinor representation of the group SL(2,C); (2)Connection between spinors and tensors; (3)Maxwell, Weyl and Riemann Spinors; (4)Classification of Maxwell spinor; (5)Classification of Weyl spinor; (6)Isotopic spin and gauge fields; (7)Lorentz invariance and the gravitational field; (8)SL(2,C) invariance and the gravitational field; (9)Gravitational field equations.     

CPN−1 coupled to gauge fields: Mean field theory and monte carlo results

We define a two parameter lattice field theory which interpolates between the O (2N) Heisenberg model, pure U(1) gauge theory, and a lattice version of the CP^N?1 model. The phase diagram in space-time dimension d=4 is obtained by Monte Carlo simulation on a 4⁴ lattice, and the nature of the phases is discussed in mean field approximation.