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We study initial value problems of the Chern–Simons–Dirac equations. With the Lorentz gauge condition they are formulated in the second-order hyperbolic equations. Under the Coulomb gauge condition Dirac equation is coupled with the elliptic equations which show some smoothing properties of the gauge field. With the temporal gauge condition divergence-curl decomposition and elliptic estimates will be used.

In lattice gauge theories, the renormalization transformation and its properties are formally defined and formally proved by making use of Dirac's δ function and its properties. In this Letter, we shall give a mathematically rigorous definition of a renormalization transformation for lattice pure gauge field theories and show the required properties, which are use to show ultraviolet stability of lattice gauge theories.

When studying gauge theories (e.g. with finite energy conditions), attention is traditionally restricted to the subset of irreducible connections, which is open and dense in the full space of connections. We point out that generally the residual set of reducible connections contains critical points of the gauge functionals which, moreover, are the only ones common to all theories with a given symmetry, i.e. those determined by the symmetry and geometry of the problem alone, and not by the specific choice of functional.

We present a straightforward model of the U(1) gauge equations of Dirac and Maxwell, as well as the U(n) Yang–Mills equations where all fields and gauge transformations take values in a Clifford algebra. When expressed in terms of the Clifford components of the fields, the equations display various gauge symmetries which we intestigate for all Clifford algebras. In particular, for the Pauli algebra, the Dirace CA equations possess the SU(2) × U(1)-symmetry.

We present some classical solutions to a gauge theory based on quadratically nonlinear Lie algebras without a central term. We observe that instanton-like and meron-like solutions force the internal fields to behave like solitons.

We reduce the functional integral of quantum electrodynamics to an integral containing only local gauge invariant quantities. The set of (bosonic) invariants contains bilinear combinations of the spinor field and a real-valued covector field.

An algebraic technique of separation of gauge modes in Abelian gauge theories on homogeneous spaces is proposed. An effective potential for the Maxwell-Chern-Simons theory on S3 is calculated. A generalization of the Chern-Simons action is suggested and analyzed with the example of SU(3)/U(1) X U(1).

Within a differential-geometrical framework, the notion of a generalized connection form on a principal fibre bundle is applied to a generalized model for the unification of two (or more) gauge interactions. An example with gauge groups SU(2) and U(1) is considered.

We discuss the problem of the choice of test functions in gauge quantum field theories. Analysis of explicitly soluble models suggests that the test function spaces which are suitable for local and covariant formulation of gauge theories are the Gelfand and Shilov spaces ℒ α β , α+β>1. We also discuss a possible generalization of the spectral condition.

Using the method of infrared bounds and partial-integration formulas, we prove that there is a chiral phase transition in four-dimensional strongly coupled lattice gauge theory with gauge group U(N) and staggered fermions for all N≤5.

The three-dimensional analog of singleton gauge theory turns out to be related to the topological gauge theory of Schwartz and Witten. It is a fully-fledged gauge theory, though it involves only a single scalar field. Real, physical degrees of freedom propagate in 3-space, but they are ‘confined’ in the sense that they cannot be detected locally. The physical Hamiltonian density is not zero, but it is concentrated on the boundary at spatial infinity. This boundary surface, a torus, supports a two-dimensional conformal field theory.

A geometrical treatment of the gauge coupling constant is proposed in terms of a generalized connection form using fibre-bundle language. This extends the notion of the coupling constant to a notion of a field. The reduction of a curvature form for the generalized connection form is described in the case of a reduction of a structure group G to a subgroup H (broken gauge symmetry), and a coupling constant for the gauge group H is constructed from the corresponding one for the gauge group G.

We show that the sequence of Jordan algebras M inf3 sup1 , M inf3 sup2 , M inf3 sup4 , and M inf3 sup8 , whose elements are in the 3×3 Hermitean matrices over ℝ, ℂ, ℚ, and O, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.

The gauge fixing problem in the conformal (spinor and scalar) QED is examined. For the analysis, we generalize Dirac's manifestly conformal-covariant formalism. It is shown that the (vector and matter) fields must obey a certain mixed (conformal and gauge) type of transformation law in order to fix the local gauge symmetry preserving the conformal invariance in the Lagrangian.

A complete set of gauge invariant quantities for generic configurations of the theory of an SU(3) gauge field interacting with a Higgs field in the fundamental representation is constructed. Among the invariants, there appear quantities of a nontensorial type reflecting the nontrivial topological structure of the gauge orbit space. In the terminology introduced by Mandelstam, our description of classes of gauge invariant configurations corresponds to the superconductivity phase of this model containing magnetic monopoles which are confined by magnetic vortices.

An elementary derivation, using Witten's Ansatz, is given of the elliptic meron-antimeron solution of the (Minkowski) SU(2) gauge theory in the W 0=0 gauge.

In this note it is shown that for every dynamic vierbein field there is a gauge transformation such that the transformed field satisfies the algebraic gauge condition of Deser and Van Nieuwenhuizen. In this sense there is no Gribov phenomenon for the above hauge condition.

Quantisation of the Abelian gauge field in some classes of the noncovariant infrared-soft gauges can be carried out consistently based on the classical field equations and the basic principles of stochastic mechanics.

In this paper we find some connections between simple (Riemann) waves and nonlinear representations of local gauge groups originating from orthogonal and pseudo-orthogonal groups. These representations are obtained by introducing fibre bundles for Riemann waves and are symmetries of the original equation.

We find two classes of supersymmetry theories such that after Fermi-field integration and Bose-field transformation the graded partition function reduces to that of a free theory. We study theories with or without gauge symmetries and relate these results with the possibility of dynamical breaking of supersymmetry. Within these theories are N=2 and N=4 Super Yang-Mills.

A new theory of gravity in Bose space is extended to a local gauge group of arbitrary coordinate transformations in superspace. We find that global supersymmetry can be recovered from the curved non-Riemannian superspace theory.

The gauge potential for the 2-monopole solution is shown to have an infinite line of singularity running from −∞ to +∞ in any fixed direction in the unitary gauge. The string structures for higher monopole solutions in this gauge are also speculated.

Conformal gauge fixing on simply connected parts of world sheets in Minkowski space is examined in detail. It is proved that conformal gauge fixing can be imposed, including the usual boundary conditions.

A formulation of composite electrodynamics, with scalar singleton constituents, is here extended to a supersymmetric theory of composite photons and neutrinos, in which the (massless) neutrinos are composed of one each of the scalar and spinorial singletons. These particles interact with an arbitrary scalar matter superfield, with massive, charged components. The feasibility of an interaction that is gauge invariant (in the sense of singleton gauge invariance) arises from the use of unconventional quantization rules which, in turn, are available because singletons are ‘kinematically confined’.

The process of the reduction of a principal fibre bundle P(M, G) to a reduced subbundle Q(M, H) is analysed in view of applications to the description of symmetries of gauge theories. The main idea is to relate symmetries of the nonuniqueness of the reduction by gauging the different ways in which for each fiber the unbroken group H is imbedded into the large group G. Gauge theories encompassing gravitation are especially taken into account.

For a lattice gauge theory a necessary and sufficient condition on the gauge group is stated, which assures that the linear span of products of Wilson loop observables is dense in the space of continuous, gauge invariant functions on the configuration space. Some groups which fulfill this condition are exhibited, among themU(N) andSU(N),N=1, 2, 3, ... Finally we prove that generically it is fulfilled for all connected, compact Lie groups.

Three-dimensional supersymmetric Higgs models with an additional U(N) ‘flavor’ symmetry are considered within the 1/N expansion. Explicit expressions for the renormalization group functions are obtained in the large N limit which exhibit logarithmic dependence on the gauge coupling constants.

The correspondence principle is used to extend nonabelian gauge theories with chiral fermions up to anomaly-free theories by introducing unitary scalar fields. Resulting theories can be treated as a low-energy approximation to left-right theories. This approach fixes the Weinberg angle of the electroweak theory at θw=30°.

Starting with the determination of the two-points function in the Landau gauge given by Wightman and Strocchi, we build a more economic realization of this gauge, following the basic principles they have proposed.

A description is presented of a charged particle in an external gauge field with the help of a previously introduced group of paths. This description exploits the group representation induced by known contour factors (ordered exponents, holonomy group). Then some novel class of representations is defined by integration of covariantly constant 2-forms. The representation from this class connected with the dual form *F for the strength of field F is shown to describe analogues of test magnetic monopoles and dyons in the gauge theory. The quarks are supposed to be gauge dyons, whose ‘magnetic’ degrees of freedom are connected with a colour.

We formulate a path integral of chiral gauge theories by means of the canonical quantization of fermions in time-dependent background gauge fields. The expression of the path integral is composed of two parts. One is due to the nontrivial holonomy of the fermionic Fock vacua and the other is the conventional form which is used in the perturbation theory. The nontrivial holonomy part is expected to be a nonlocal counter term. We show a possibility of the perturbative calculation

The symmetry of the stationary vacuum Einstein equations under the group SL(2, ℝ) is extended to a symmetry of SL(2, ℂ) self-dual equations of gauge theory. For a dual parametrization of Yang's R gauge, three independent infinitesimal changes under an Ehlers-type transformation are derived.

We give a C*-algebra description of the quantisation of the electromagnetic field in Landau gauge and discuss the gauge transformation which relates it to the Lorentz gauge.

Any gauge theory is proved to arise from spontaneous breakdown of symmetry under certain infinite parameter group, the corresponding gauge field being the Goldstone field by which this breakdown is accompanied.

Parity-invariant three-dimensional gauge theories with N=2 extended supersymmetry are studied by the heat kernel method. The parity-anomalous part of the one-loop effective action is exactly found. It is expressed in terms of the N=2 supersymmetric Chern-Simons term and is identified as a N=2 superspace Atiyah-Patodi-Singer eta-invariant.

A conformal nonabelian gauge theory with a five-component gauge potential is considered. In this theory the conformal-invariant two-point function has a nonzero transverse part and a Lagrangian with a conformal-invariant gauge-fixing term is found. The corresponding local effective Lagrangian, where the dimensionless Faddeev-Popov ‘ghost’ field is included, obeys a global supersymmetry of the Becchi-Rouet-Stora type. In the gauge-invariant sector it is shown that this theory is equivalent to the ordinary Yang-Mills theory.

The nonlocal gauge group of unitary integral operators Un (∞) is considered. Gluons in the nonlocal version of QCD are described by the bilocal gauge fields (BGF). The Higgs effect for BGF are discussed. It is shown that the local SU(n) gauge theory can be treated as a local limit of spontaneously broken Un(∞)/SU(n) BGF theory.

It is shown that a class of linearized solutions of the Poincare gauge theory also solves the exact field equations of the same theory. Utilizing this property, some new solutions are given.

For a classical Yang-Mills field which is periodic in the longitudinal light-cone coordinate: (a) a gauge condition is formulated, (b) the presence of field singularities in this gauge is shown, and (c) the relevance of these singularities to the topological charge is demonstrated.

The equations of motion (for N=3, 4) and the constraint equations (N=1, 2) for supersymmetric Yang-Mills theories are shown to be the compatibility conditions of some system of linear equations with a parameter.

The geometrical and mechanical aspects of a particle interacting with a Poincare gauge field are considered and the relation with a gravitational interaction is studied.