Local Gauge Theory Research Articles

Gauge unification of relativistic and unitary spin

We consider the problem of unifying relativistic and unitary spin within the framework of local gauge theory. The difference between Einstein and Yang-Mills gauge models, in being constrained spin-2 and unconstrained spin-1 models, respectively, is traced to the assumption of Poincare invariance and the unification difficulty is thereby related to the Coleman-O’Raifeartaigh theorem. First, we consider a constrained, Poincare-invariantSL2,C⊗U1 gauge model which employs complex tetrads and modifies the Einstein-Maxwell theory for strong fields. Secondly, we consider an unconstrained general covariantSL2n,C⊃ ⊃SL2,C⊗SUn Yang-Mills-type gauge model for two spin-1 fields coupled to energy and spin, respectively. ItsSL2,C classical weak curvature limit is identical to the classical limit of the Weyl-Fock-Ivanenko spin-2 formulation of general relativity, and we conclude that quantum gravity is not necessarily a spin-2 theory. For a spin coupling constant of magnitude α, Poincare invariance breaks down to homogeneous Lorentz invariance and relativistic and unitary spin are effectively unified for radii of curvature smaller than α-½ × Planck length. For these curvatures gravity becomes repulsive.

Local gauge invariant QED with fundamental length

A local gauge theory of electromagnetic interactions with the fundamental length l as a new universal scale is worked out. The lagrangian contains new extra terms in which the coupling constant is proportional to the fundamental length. The theory has an elegant geometrical basis: in momentum representation one faces de Sitter momentum space with curvature radius 1 l .

Variational calculation of vacua in the Z(2) gauge theory

We apply a variational technique which is a generalization of mean field theory of a local gauge theory in 2 + 1 dimensions. We reproduce the correct features of the gauge and matter Z(2) theory within the matter-screening phase. Using big enough clusters of links, we calculate the behaviour of a Wilson loop. Applying the method to the pure gauge theory, we obtain two phases characterized by order and disorder of local surface order parameters. The clusters calculation leads to a first-order transition for the pure gauge theory and to a non-leading minimum for the gauge screening phase of the gauge with matter theory. Yet it can be employed to calculate the physical features inside all the phases of these theories.

Variational approach to Hamiltonian lattice theories

A variational calculation of the vacuum energy of a Hamiltonian lattice theory is formulated in terms of a finite box Hamiltonian for a cluster of points. The box Hamiltonian contains surface terms which are proportional to order parameters of the system. It is tested on the Ising model and applied to $Z(N)$ spin models in 1 + 1 and 2 + 1 dimensions. $Z(3)$ is found to have an exceptional phase-transition structure. The application of the method to local gauge theories is discussed.

GRADED ALGEBRA SU (l|m|n) AND GAUGE MODEL

We have suggested that the extension of SU (m|n) is to SU (l|m|n). We develop a local gauge theory of the graded algebra SU (l|m|n). We find the restrictions on the gauge potentials and matter fields in order that a physically suitable gauge invariant lag-rangian exists. It is natural to extend the graded algebra SU (2|1) electro weak model to include quarks in a superunified model.

Unified affine gauge theory of gravity and strong interactions with finite and infinite [formula omitted] spinor fields

A theory in which the Linear Group in four dimensions GL(4, R) and its Affine Extension GA(4, R) bear a direct relationship to the Physics of hadrons, and indirectly to that of the leptons is outlined. The Poincare group is embedded in a Global GA(4, R) Symmetry. Hadrons correspond to (infinite-dimensional) unitary representations of GA(4, R) , as defined by their GL(3, R) or SL(3, R) bandor content. These SL(3, R) bandors are embedded in infinite-component (poly-) fields, representing SL(4, R) bandors. All multiplicity free unitary irreducible representations are explicitly constructed, and the formulas for the scalar products of the representations' Hilbert spaces as well as the matrix elements for all noncompact (shear) operators are given. Leptons are described by nonlinear representations of GL(4, R) , realized through a Metric tensor field (i.e. Gravity). Nonlinear representations of GL(4, R) are presented in detail. The anholonomic description in terms of tetrad deformations is used. GA(4, R) is then postulated as a local Gauge Theory. It generates an Einstein-like Gravitational Interaction, plus a Confining Strong Interaction which applies only to the linear representations, i.e. to hadrons. Local GA(4, R) symmetry is spontaneously broken to local Poincare invariance for leptons (Einstein-Cartan Gravity). The variations of all the fields in the theory are presented and the Noether currents are derived. The choice of a Lagrangian that could explain hadron confinement is discussed.

Gauging [formula omitted

We develop a local gauge theory of the graded algebra su( n m ) . We find the restrictions on the gauge potentials and matter fields in order that a physically suitable gauge invariant lagrangian exists. These conditions are: triviality of fields in extended dimensions, non-extended gauge components only lie in su( n) × su( m), and suitable restrictions satisfied by the extended gauge components. For matter the grading has to be given by chirality.

A map between corner and link operators in lattice gauge theories

A completely local gauge-invariant lattice gauge theory is formulated in terms of a new set of variables introduced earlier in the continuum. This theory uses local “corner” variables defined on lattice sites only, as opposed to the conventional “link” variables. It is shown via a map that the new formulation gives identical results to the usual lattice gauge theory. The properties of the quantum commutators in the continuum limit is also discussed and contrasted for the two lattice approaches. In terms of the corner operators the quantized lattice theory is seen to be closely related to continuum QCD.

The vanishing norm of Faddeev-Popov ghost fields

It is shown that the charge conjugation properties of relativistic, local gauge theory require that the Faddeev-Popov ghost fields have vanishing norm.

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA μ ( x) = ∂ μ ,…, with □ Λ = 0, and to the operator gauge transformation δA μ ( x) = ∂ μ φ( x), …, δφ( x) = α −1 ∂· A( x), with □ φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations δA μ(x) ∼ ▪ μC(x) , …, introduced by Becchi, Rouet and Stora, where ▪ μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation δA μ(x) = (Z 1/Z 3)gA μ(x) × Λ(x)+∂ μ Λ(x) , …, is never a symmetry transformation and is never finite in perturbation theory. Only for Λ(x) = (Z 3/Z 1)L with L finite constants or for Λ(x) = Ω z ̃ 3C(x) with Ω a finite constant does it become a finite symmetry transformation, where z ̃ 3 is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.

Local gauge theory for chiral SU(3) × SU(3) symmetry breaking

The spontaneous breakdown of local (gauge of the second kind) chiral ${\mathrm{SU}(3)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{R}$ symmetry is analyzed. The pattern of strong symmetry breaking from ${\mathrm{SU}(3)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(3)}_{R} \mathrm{to} {\mathrm{SU}(2)}_{L}\ifmmode\times\else\texttimes\fi{}{\mathrm{SU}(2)}_{R} or \mathrm{to} \mathrm{SU}(3)$ is demonstrated. A possible (but not necessary) consequence of this approach is the existence of few vector mesons with low mass.

General relativity with spin and torsion: Foundations and prospects

A generalization of Einstein's gravitational theory is discussed in which the spin of matter as well as its mass plays a dynamical role. The spin of matter couples to a non-Riemannian structure in space-time, Cartan's torsion tensor. The theory which emerges from taking this coupling into account, the ${U}_{4}$ theory of gravitation, predicts, in addition to the usual infinite-range gravitational interaction medicated by the metric field, a new, very weak, spin contact interaction of gravitational origin. We summarize here all the available theoretical evidence that argues for admitting spin and torsion into a relativistic gravitational theory. Not least among this evidence is the demonstration that the ${U}_{4}$ theory arises as a local gauge theory for the Poincar\'e group in space-time. The deviations of the ${U}_{4}$ theory from standard general relativity are estimated, and the prospects for further theoretical development are assessed.

Canonical quantization of gauge theories

A canonical quantization scheme in a Hilbert space with positive-definite metric is proposed for local gauge theories, based on Dirac's general theory of singular dynamical systems. A theorem on the subsidiary conditions, permitting a perturbation treatment, is stated and proved. Unitarity in the subspace of allowed states is demonstrated. The method is applied to the case of free electrodynamics in a nonlinear gauge and Yang-Mills theory in the covariant Feynman gauge. The description does not require introduction of ghost particles. The rules for calculating graphs are shown to be equivalent to those in a Lagrangian approach with a ghost. (AIP)

Geometrical approach to local gauge and supergauge invariance: Local gauge theories and supersymmetric strings

Geometrical approach to local gauge and supergauge invariance: Local gauge theories and supersymmetric strings

Unification of gauge and gravitation theories

A new approach to local Gauge theories is presented. The theory is developed in a curved space–time, and therefore gravitation is not neglected. Besides the Yang–Mills Aαμ vector bosons associated with a symmetry group, scalar bosons gαβ appear just as naturally. The Lagrangian describing the interaction of these fields is the Ricci scalar for an extended Riemannian geometry.

Gauge variation of proper vertices in local gauge theories

Gauge variation of proper vertices in local gauge theories

Calculability and naturalness in gauge theories

Calculability conditions are discussed for local gauge theories with Higgs-type symmetry breaking. We focus on the naturalness of $\ensuremath{\mu}e$ universality, the naturalness of the Cabibbo angle $\ensuremath{\theta}$, the naturalness of $\mathrm{CP}$-violating phases, and the naturalness of the nonleptonic $\ensuremath{\Delta}I=\frac{1}{2}$ rule. In this context we examine many published gauge models and construct others to illuminate the questions at hand. We note that naturalness of $\ensuremath{\mu}e$ universality for charged currents does not necessarily imply universality for neutral currents (natural "restricted" universality), and we emphasize the need for ${\ensuremath{\nu}}_{e}$-beam experiments. For SU(2) \ifmmode\times\else\texttimes\fi{} U(1) and SU(2) \ifmmode\times\else\texttimes\fi{} U(1) \ifmmode\times\else\texttimes\fi{} U(1) we give first examples of how a nontrivial natural $\ensuremath{\theta}$ can appear. Models with $\mathrm{CP}$ violation are classified as to whether their $\mathrm{CP}$-violating phases are natural or not. For O(4) \ifmmode\times\else\texttimes\fi{} U(1) we give a first example in which all the above naturalness criteria can be implemented. Here the natural $\ensuremath{\mu}e$ universality is necessarily restricted. The principal tool used in these investigations is the strict renormalizability relative to a gauge group enlarged by discrete symmetries, and the union of representations reducible under the gauge group to irreducible ones under the enlarged group. To implement this program, it is sometimes necessary to introduce Higgs couplings involving right-handed neutrinos; here the zero neutrino mass is associated with a discrete symmetry which remains unbroken upon spontaneous breakdown. We also find that strict renormalizability can lead to mass relations between fermions. In O(4) \ifmmode\times\else\texttimes\fi{} U(1) models, such mass relations as well as right-handed neutrinos are necessary ingredients. Furthermore, for these models the spontaneity of $\mathrm{CP}$ violation acquired an operational significance, namely, as a discrete symmetry necessary (but not sufficient) to give a $\mathrm{CP}$-violating phase a natural value (90\ifmmode^\circ\else\textdegree\fi{}). While the models we discuss are rather cumbersome, particularly due to the complexity of the symmetry-breaking mechanism, we expect that the tools we have developed may well have wider applicability.

Asymmetry in the asymptotic identification

The compatibility of the notion of an internal symmetry group with the asymptotic identification for an electromagnetic part of the non-Abelian gauge field is investigated on the classical level. It is shown that the asymptotic identification would restrict, without the necessity of violating local gauge invariance, internal symmetry transformations to those which commute with the generator of electromagnetic gauge transformations. Local gauge transformations would be restricted in the same way in the region of definability of the gauge field’s electromagnetic part. Based upon these restrictions, a model for the structure of the gauge field’s sources and interactions is conjectured in order to illustrate how the knownSU 2 andSU 3 dynamical symmetry and internal particle labeling schemes might arise from a local gauge theory employing an asymptotic identification.

Gauge Fields with Positive-Definite Energy Density

The relationship between the couplings of the non-Abelian gauge field and the internal holonomy group is investigated on the classical level, under the requirements of full local gauge invariance and positive definiteness of the gauge field's energy density. For the free gauge field it is found that each solution to the gauge field equation with self-coupling is associated with a simple compact internal holonomy group. When the gauge field is coupled to other multiplet fields, an internal symmetry group is present in addition to the holonomy group H. In this case the internal holonomy group must also be simple and compact. In addition, homogeneity and isotropy of event space for closed systems of fields require that H be identical in group structure with either the symmetry group or a subgroup of the symmetry group and that all the field equations can be decoupled into sets of equations acting on multiplet fields transforming under irreducible representations of H. Thus, in local gauge theory, multiplet fields can be classified with respect to simple and compact internal holonomy groups. Some comments are offered on the relationship between internal holonomy and symmetry groups.

Internal Holonomy Groups of Yang-Mills Fields

Classical Yang-Mills potentials define a Lie group, the internal holonomy group, which is analogous to the ordinary holonomy group defined by the Christoffel symbols in general relativity. The internal holonomy group is an important tool for the dissection and study of the Yang-Mills equations and of local gauge theory in general. A number of theorems on internal holonomy groups is presented, together with some applications.