Fundamental length hypothesis and new concept of gauge vector field

Through an analysis of quantum field theory with “fundamental length” l[1–10], a new concept of gauge vector field is determined. The electromagnetic field is considered in detail. The new electromagnetic potential turns out to be a 5-vector associated with the De Sitter group SO(4,1). The extra fifth component, called τ-photon, similar to the scalar and longitudinal photons, does not correspond to an independent dynamical degree of freedom. Gauge-invariant equations of motion for all components of the electromagnetic 5-potential are found. Though the new gauge group remains Abelian, it is nevertheless larger than the conventional gauge group. In particular, the new gauge transformations intrinsically depend on the fundamental length l. Therefore one can consider them as a base for modification of QED at small distances (?l) in a profound way. The underlying physics becomes much richer due to the appearance of new interactions mediated by the τ-photons [14].     

A gravitating SO(3, 1) gauge field

In this article, we postulate SO(3, 1) as a local symmetry of any relativistic theory. This is equivalent to assuming the existence of a gauge field associated with this noncompact group. This SO(3, 1) gauge field is the spinorial affinity which usually appears when we deal with weighting spinors, which, as is well known, cannot be coupled to the metric tensor field. Furthermore, according to the integral approach to gauge fields proposed by Yang, it is also recognized that in order to obtain models of gravity we have to introduce ordinary affinities as the gauge field associated with GL(4) (the local symmetry determined by the parallel transport). Thus if we assume both L(4) and SO(3, 1) as local independent symmetries we are led to analyze the dynamical gauge system constituted by the Einstein field interacting with the SO(3, 1) Weyl-Yang gauge field. We think this system is a possible model of strong gravity. Once we give the first-order action for this Einstein-Weyl-Yang system we study whether the SO(3, 1) gauge field could have a tetrad associated with it. It is also shown that both fields propagate along a unique characteristic cone. Algebraic and differential constraints are solved when the system evolves along a null coordinate. The unconstrained expression for the action of the system is found working in the Bondi gauge. That allows us to exhibit an explicit expression of the dynamical generator of the system. Its signature turns out to be nondefinite, due to the nondefinite contribution of the Weyl-Yang field, which has the typical spinorial behavior. A conjecture is made that such an unpleasant feature could be overcome in the quantized version of this model.     

Local aspects of superselection rules. II

In a theory where the local observables are determined by local field algebras as the fixed points under a (a priori noncommutative) group of gauge transformations of the first kind, we show that, if the field algebras possess intermediate type I factors, we can construct observables having the meaning of local charge measurements, and local current algebras in the field algebras.     

Reissner--Nordström-de--Sitter-type Solution by a Gauge Theory of Gravity

We use the theory based on a gravitational gauge group （Wu＇s model） to obtain a spherical symmetric solution of the field equations for the gravitational potential on a Minkowski spacetime. The gauge group, the gauge covariant derivative, the strength tensor of the gauge feld, the gauge invariant Lagrangean with the cosmological constant, the field equations of the gauge potentiaIs with a gravitational energy-momentum tensor as well as with a tensor of the field of a point like source are determined. Finally, a Reissner-Nordstrom-de Sitter-type metric on the gauge group space is obtained.     

The general caloron correspondence

We outline in detail the general caloron correspondence for the group of automorphisms of an arbitrary principal G-bundle Q over a manifold X, including the case of the gauge group of Q. These results are used to define characteristic classes of gauge group bundles. Explicit but complicated differential form representatives are computed in terms of a connection and Higgs field.     

Does a generic connection depend continuously on its curvature?

For a principal bundle with semi-simple structure group over a smooth four-dimensional base manifold, the set of connections (gauge potentials)A which are uniquely determined by their curvature (field or field strength)F is generic in the set of all potentials, endowed with the WhitneyC ^∞ topology. However, the operator taking each such fieldF to its potentialA is not continuous. Partial negative results are given concerning the existence of a smaller generic set on which this operator is continuous.     

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.     

The statistical quantum number and gauge groups in parafermi field theory

Properties of a system consisting of a single parafermi field of order p are studied mainly in connection with gauge groups. Following the theory of Drühl, Haag and Roberts, the algebra of observables is classified into four cases according to the types of gauge groups, i.e., SO(p), O(p), U(p), and SU(p). A detailed study is made of irreducible representations of these gauge groups that are realized in the state-vector space of the parafermi field. Superselection operators which give rise to the corresponding superselection rules related to the gauge groups are studied, and their explicit expressions given. The statistical quantum number which we introduced before is found to be nothing other than the eigenvalues of a superselection operator for the gauge group O(p).     

Mean field calculations and defect gases in lattice gauge theory

We investigate the relationship between local defects and the mean field method in lattice gauge theory. In particular we clarify the role of defects in establishing the equivalence between mean field calculations with and without gauge fixing. In two dimensions we derive the area law for the Wilson loop by a mean field calculation incorporating defects. We also establish a general rule about mean field variables which are appropriate to handle defects induced by an action that almost possesses a local symmetry group, and we apply it to theZ(2) Higgs model and to the mixedSU(2)-SO(3) model.     

Canonical formulation of the spherically symmetric einstein-Yang-Mills-Higgs system for a general gauge group

The dynamics of the spherically symmetric system of gravitation interacting with scalar and Yang-Mills fields is presented in the context of the canonical formalism. The gauge group considered is a general (compact and semisimple) N parameter group. The scalar (Higgs) field transforms according to an unspecified M-dimensional orthogonal representation of the gauge group. The canonical formalism is based on Dirac's techniques for dealing with constrained hamiltonian systems. First the condition that the scalar and Yang-Mills fields and their conjugate momenta be spherically symmetric up to a gauge is formulated and solved for global gauge transformations, finding, in a general gauge, the explicit angular dependence of the fields and conjugate momenta. It is shown that if the gauge group does not admit a subgroup (locally) isomorphic to the rotation group, then the dynamical variables can only be manifestly spherically symmetric. If the opposite is the case, then the number of allowed degrees of freedom is connected to the angular momentum content of the adjoint representation of the gauge group. Once the suitable variables with explicit angular dependence have been obtained, a reduced action is derived by integrating away the angular coordinates. The canonical formulation of the problem is now based on dynamical variables depending only on an arbitrary radial coordinate r and an arbitrary time coordinate t. Besides the gravitational variables, the formalism now contains two pairs of N-vector variables (R, π_r), (Θ, π_Θ), corresponding to the allowed Yang-Mills degrees of freedom and one pair of M-vector variables, (h, π_h), associated with the original scalar field. The reduced Hamiltonian is invariant under a group of r-dependent gauge transformations such that R plays the role of the gauge field (transforming in the typically inhomogeneous way) and in terms of which the gauge covariant derivatives of Θ and h naturally appear. No derivatives of R appear in the Hamiltonian and the gauge freedom allows us to define a gauge in which R is zero. Also the r and t coordinates are fixed in a way consistent with the equations of motion. Some nontrivial static solutions are found. One of these solutions is given in closed form; it is singular and corresponds to a generalization of the singular solution found in the literature with different degrees of generality and the geometry is described by the Reissner-Nordström metric. The other solution is defined through its asymptotic behavior. It generalizes to curved space the finite energy solution discyssed by Julia and Zee in flat space.     

Gauge Gravitational Field in a Fractal Space-Time

Considering the fractal structure of space-time, the scale relativity theory in the topological dimension D_T=2 is built. In such a conjecture, the geodesics of this space-time imply the hydrodynamic model of the quantum mechanics. Subsequently, the gauge gravitational field on a fractal space-time is given. Then, the gauge group, the gauge-covariant derivative, the strength tensor of the gauge field, the gauge-invariant Lagrangean, the field equations of the gauge potentials and the gauge energy-momentum tensor are determined. Finally, using this model, a Reissner-Nordström type metric is obtained.     

Reformulation of general relativity as a gauge theory

The starting point is a spinor affine space-time. At each point, two-component spinors and a basis in spinor space, called “spin frame,” are introduced. Spinor affine connections are assumed to exist, but their values need not be known. A metric tensor is not introduced. Global and local gauge transformations of spin frames are defined with GL(2) as the gauge group. Gauge potentials B_μ are introduced and corresponding fields F_μν are defined in analogy with the Yang-Mills case. Gravitational field equations are derived from an action principle. Incases of physical interest SL(2, C) is taken as the gauge group, instead of GL(2). In the special case of metric space-times the theory is identical with general relativity in the Newman-Penrose formalism. Linear combinations of B_μ are generalized spin coefficients, and linear combinations of F_μν are generalized Weyl and Ricci tensors and Ricci scalar. The present approach is compared with other formulations of gravitation as a gauge field.     

Susy-breaking without sliding vev

In the absence of gauge interactions the spontaneous breakdown of supersymmetry (Susy) is accompanied by a scalar field whose vacuum expectation value is not determined by the tree potential. Inclusion of the gauge interactions can fix this vev at the tree level. The situation is exemplified by a simpleSU(2)-model yielding nonvanishingF-andD-vevs.     

Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism

Spontaneous symmetry breakings in indefinite metric quantum field theories are analyzed and a generalization of the Goldstone theorem is proved. The case of local gauge quantum field theories is discussed in detail and a characterization is given of the occurrence of the Higgs mechanism versus the Goldstone mechanism. The Higgs phenomenon is explained on general grounds without the introduction of the so-called Higgs fields. The basic property is the relation between the local internal symmetry group and the local group of gauge transformations of the second kind. Spontaneous symmetry breaking ofc-number gauge transformations of the second kind is shown to always occur if there are charged local fields. The implications about the absence of mass gap in the Wightman functions and the occurrence of massless particles associated with the unbroken generators in the Higgs phenomenon 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.     

Gauge invariance for extended objects

Just as the vector potential (one-form) couples to charged point-particles, antisymmetric tensor fields of higher rank (p-forms) couple to elementary objects of higher dimensionality (strings, membranes, …). It is shown that the only possible gauge invariant interaction of such an extended object with a gauge field in spacetime is based on the abelian group U(1). This is unlike the situation for particles where Yang-Mills actions based on any gauge group may be written down. The properties of the abelian theory are explored. It is pointed out that a compact object is analogous to a particle-antiparticle pair and its quantum rate of production in a constant external field is calculated semiclassically. The analysis is performed keeping generic both the dimension of the object and that of spacetime.     

Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism

We consider noncommutative gauge theory defined by means of Seiberg–Witten maps for an arbitrary semisimple gauge group. We compute the one-loop UV divergent matter contributions to the gauge field effective action to all orders in the noncommutative parameters θ. We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group. We use path-integral methods in the framework of dimensional regularisation and consider arbitrary invertible Seiberg–Witten maps that are linear in the matter fields. Surprisingly, it turns out that the UV divergent parts of the matter contributions are proportional to the noncommutative Yang–Mills action where traces are taken over the representation of the matter fields; this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.