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.     

Remarks on the use of a microcanonical ensemble to study phase transitions in lattice gauge theory

The phase transitions of the Z₂ and U₁ lattice gauge theories are described by sampling a microcanonical ensemble, in which the gauge field is in thermal equilibrium with a system of auxiliary variables called demons. It is shown that the microcanonical ensemble yields a more complete picture of a first-order transition than that obtained by sampling the Boltzmann distribution.     

Electric dipole moment for supersymmetric monopoles

The electric dipole moment for the monopoles that can be present in N = 2 and N = 4 supersymmetric SU(2) gauge theories, spontaneously broken by imposing a non-zero expectation value of a scalar field at infinity, is determined by considering the response to a weak external electric field. The magnetic g factor g_M = 2 which is in accord with the duality conjecture of Montonen and Olive.     

Abelian gauge glasses

Gauge glasses are lattice gauge theories with quenched random couplings; in this paper, the two simplest abelian models, having Z₂ and U(1) gauge symmetries respectively, are constructed. An important extension of gauge invariance is defined and the disorder invariant under this symmetry, the frustration, is identified. Simple energetic properties of frustrations are derived using duality arguments. The question of the existence of a weakly coupled glassy phase is raised, and then addressed using replica mean field theory and real-space renormalisation group techniques, both in the context of the Z₂ model. A phase transition is found for dimension six and above. The implications for random dynamics are discussed.     

Radiative corrections in compactified superstring models

One-loop corrections to the effective potential in models obtained from compactification of ten-dimensional superstring theories are calculated. It is found that no masses are generated for gauge non-singlet scalars even in the presence of supersymmetry breaking terms induced by gauge and gaugino condensation, but that the gravitino mass is determined at one loop. The scales of grand unification, supersymmetry breaking and condensation are fixed by the gauge singlet scalars and are found to be close to Planck scale. Requiring M_GUT<M_Planck restricts the other parameters of the theory. The one-loop effective potential at scales between the condensate and compactification scales is also discussed, with possible implications for the allowed particle content of the effective theory.     

Mean fields and self consistent normal ordering of lattice spin and gauge field theories

Classical Heisenberg spin models on lattices possess mean field theories that are well defined real field theories on finite lattices. These mean field theories can be self consistently normal ordered. This leads to a considerable improvement over standard mean field theory. This concept is carried over to lattice gauge theories. We construct first an appropriate real mean field theory. The equations determining the Gaussian kernel necessary for self-consistent normal ordering of this mean field theory are derived.     

A New Class of Inhomogeneous Cosmological Models with Electromagnetic Field in Normal Gauge for Lyra’s Manifold

A new class of exact solutions of Einstein’s modified field equations in inhomogeneous space-time for perfect fluid distribution with electromagnetic field is obtained in the context of normal gauge for Lyra’s manifold. We have obtained solutions by considering the time dependent displacement field. The source of the magnetic field is due to an electric current produced along the z-axis. Only F ₁₂ is a non-vanishing component of the electromagnetic field tensor. It has been found that the displacement vector β(t) behaves like the cosmological constant Λ in the normal gauge treatment and the solutions are consistent with the recent observations of Type Ia supernovae. Physical and geometric aspects of the models are also discussed in the presence of magnetic field.     

Continuum limit of A Z2 lattice gauge theory

Phase diagrams of lattice gauge theories have in several cases lines of first-order transitions ending at points at which continuous (second-order) transitions take place. In the vicinity of this critical point, a continuum field theory may be defined. We have analyzed here a Z₂ gauge plus matter model (which has no formal continuum limit) and identified the critical point with a usual Ø⁴, globally Z₂ invariant, field theory. The analysis relies on a mean field functional formalism and on a loop-wise expansion around it, which is reviewed.     

Large-N lattice models with mixed actions

U(N) lattice gauge theories and spin systems with mixed fundamental and adjoint representation actions are studied. Exact results are found for two-dimensional gauge theories and one-dimensional spin chains. Phase diagrams for higher-dimensional systems are found using mean field theory. Various implications of this work are discussed.     

Renormalization constants in asymptotically free field theories

Renormalization constants Z_i for asymptotically free field theories can be computed via renormalization group techniques from perturbation theory. We show that there exists a subclass of these theories in which, by virtue of a new eigenvalue condition on the gauge parameter, the Z_i are asymptotically gauge independent, and hence can vanish in all gauges.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

Gaussian fluctuations to U(N) and SU(N) lattice gauge theories in the mean field approximation

The mean field can be considered as a classical solution of an appropriately reformulated version of lattice gauge theories. Axial gauge fixing renders it stable. The quadratic forms for the fluctuations in the gaussian approximation are analyzed. The gaussian correction to the mean field free energy is expressed for all U(N) and SU(N) in terms of structure functions that are explicitly calculated for U(N), SU(∞, and SU(∞) numerical calculations are performed for the phase transition point, its latent heat, and some correlation lengths that are characteristic for this kind of mean field approach.     

CRAVITATION AND SPINOR GAUGE FIELD

Basing on the Lorentz covariance and SO (4, 2) symmetry of Dirac theory, anobvious covariant theory of spinor gauge field is obtained by expanding the Lorentztransformation to general coordinate tranformation and making the SO (4, 2) to belocalized. We have proved that, by the gauge independence, the symmetrygroup is reduced to the localized rotation of Lorentz group in Riemann space automa-tically. So our theory is the natural generalization of Dirac theory in curved space.We have also proved that, the spinor gauge field can not appear in flat space, thenthe existence of spinor gauge field is closely related to the curvature. The differencesbetween our theory and Utiyama and Kibble theories are also discussed, and it is poin-ted out that the so-called scalar property of Dirac wave function in general relativity isa misunderstanding caused by the unobvious covariance of those theories, even inthose theories We can not distinguish what is the genuine gauge. field and what is theeffect of the structure of space. In obvious covariant theory this paradox disappears.     

On the energy-momentum tensor in gauge field theories

The energy-momentum tensor in spontaneously broken non-Abelian gauge field theories is studied. The motivation is to show that recent results on the finiteness and gauge independence of S-matrix elements in gauge theories extends to observable amplitudes for transitions in a gravitational field. Path integral methods and dimensional regularization are used throughout. Green's functions Γ_μν^(j)(q; p₁,…,p_j) involving the energy-momentum tensor and j particle fields are proved finite to all orders in perturbation theory to zero and first order in q, and finite to one loop order for general q. Amputated Green's functions of the energy momentum tensor are proved to be gauge independent on mass shell.     

Field theories on a lattice

Lattice field theories are described as a way to regularize continuum quantum field theories. They are obtained by replacing ordinary space time by a lattice, space time derivatives by suitable differences and Minkowski by Euclidean space. The connection between a quantum field theory isd space dimension and classical statistical mechanics in (d+1) dimensions is brought outvia elementary examples. The problem of regaining the continuum limit and of handling nonabelian gauge theories are briefly discussed.     

Twisted Gauge Theories

Gauge theories on a space-time that is deformed by the Moyal–Weyl product are constructed by twisting the coproduct for gauge transformations. This way a deformed Leibniz rule is obtained, which is used to construct gauge invariant quantities. The connection will be enveloping algebra valued in a particular representation of the Lie algebra. This gives rise to additional fields, which couple only weakly via the deformation parameter θ and reduce in the commutative limit to free fields. Consistent field equations that lead to conservation laws are derived and some properties of such theories are discussed.     

Chemical potentials in gauge theories

One-loop calculations of the thermodynamic potential Ω are presented for temperature gauge and non-gauge theories. Prototypical formulae are derived which give Ω as a function of both (i) boson and/or fermion chemical potential, and in the case of gauge theories (ii) the thermal vacuum parameter A₀=const (A_μ is the euclidean gauge potential). From these basic abelian gauge theory formulae, the one-loop contribution to Ω can readily be constructed for Yang-Mills theories, and also for non-gauge theories.     

Classifying bases for 6D F-theory models

We classify six-dimensional F-theory compactifications in terms of simple features of the divisor structure of the base surface of the elliptic fibration. This structure controls the minimal spectrum of the theory. We determine all irreducible configurations of divisors (??clusters??) that are required to carry nonabelian gauge group factors based on the intersections of the divisors with one another and with the canonical class of the base. All 6D F-theory models are built from combinations of these irreducible configurations. Physically, this geometric structure characterizes the gauge algebra and matter that can remain in a 6D theory after maximal Higgsing. These results suggest that all 6D supergravity theories realized in F-theory have a maximally Higgsed phase in which the gauge algebra is built out of summands of the types su(3), so(8), f₄, e₆, e₈, e₈, (g₂ ?? su(2)); and su(2) ?? so(7) ?? su(2), with minimal matter content charged only under the last three types of summands, corresponding to the non-Higgsable cluster types identified through F-theory geometry. Although we have identified all such geometric clusters, we have not proven that there cannot be an obstruction to Higgsing to the minimal gauge and matter configuration for any possible F-theory model. We also identify bounds on the number of tensor fields allowed in a theory with any fixed gauge algebra; we use this to bound the size of the gauge group (or algebra) in a simple class of F-theory bases.     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Scalar lattice gauge theory

Scalar lattice gauge theories are models for scalar fields with local gauge symmetries. No fundamental gauge fields, or link variables in a lattice regularization, are introduced. The latter rather emerge as collective excitations composed from scalars. For suitable parameters scalar lattice gauge theories lead to confinement, with all continuum observables identical to usual lattice gauge theories. These models or their fermionic counterpart may be helpful for a realization of gauge theories by ultracold atoms. We conclude that the gauge bosons of the standard model of particle physics can arise as collective fields within models formulated for other “fundamental” degrees of freedom.