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The role which gauge transformations of noninteger winding numbers might play in non-Abelian gauge theories is studied. The phase factor acquired by the semiclassical physical states in an arbitrary background gauge field when they undergo a gauge transformation of an arbitrary real winding number is calculated in the path integral formalism assuming that a {theta}{ital F{tilde F}} term added to the Lagrangian plays the same role as in the case of integer winding numbers. Requiring that these states provide a representation of the group of {open_quote}{open_quote}large{close_quote}{close_quote} gauge transformations, a condition on the allowed backgrounds is obtained. It is shown that this representability condition is only satisfied in the monopole sector of a spontaneously broken gauge theory, but not in the vacuum sector of an unbroken or a spontaneously broken non-Abelian gauge theory. It is further shown that the recent proof of the vanishing of the {theta} parameter when gauge transformations of arbitrary fractional winding numbers are allowed breaks down in precisely those cases where the representability condition is obeyed because certain gauge transformations needed for the proof, and whose existence is assumed, are either spontaneously broken or cannot be globally defined as a result of a topological obstruction. {copyright} {italmore » 1996 The American Physical Society.}« less

On the diagram level, we describe the mechanism of compensations in Ward identities for non-Abelian gauge theories. The description of this mechanism allows one to analyze the non-Lagrangian regularizations. We propose a regularization (different from dimensional regularization and the higher covariant derivative method) from the class of “universal” regularizations. Our regularization keeps the gauge invariance of the Yang-Mills theory in the one-loop approximation, and in two-loop approximation at least for the polarization operator.

A class of field configurations, those that satisfy (∂ · D)(∂ · A)=0 with ∂ · A≠0, and satisfying spherical symmetry is shown to be not transformable to the Coulomb gauge in the context of L2 fields. This is shown by considering all kinds of gauge transformations — infinitesimal, finite, and large gauge transformations.

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 are concerned with the covariant description of the Hamiltonian formalism for constrained field systems. The relation with the Lagrangian formalism is considered and applications to gauge theories are given. Both formalisms are developed on the same space, namely the momentum space. The equivalence of solutions is shown to hold for affine and quadratic Lagrangians. The Yang-Mills equations are put into a Hamiltonian form by means of acomplete family of Hamiltonians. This completeness property appears in a nice way as agauge-type condition connected with the Hamilton equations and generalizing the notion of gauge condition usually dealt with in gauge theory.

For a time-dependent quantum system, the instantaneous stationary Schrodinger equation is not covariant under gauge transformation. Therefore, the formal definition of the Berry phase may depend on the choice of mutually gauge-equivalent Hamiltonians. It is shown that the Lewis-Riesenfeld phase is manifestly gauge independent. The generalized time-dependent harmonic oscillator is used as an example to demonstrate this result.

We give a method by which we can construct solutions to the self-dualSU(N) gauge field equations, some of which can be chosen as seed solutions.

A review is made of the basic properties of the Hamiltonian description of classical Yang-Mills theory with fermions described by Grassmann-valued Dirac spinors, in the case of a trivial principal bundle with a compact, semisimple, connected, simply connected Lie structure group over Minkowski space-time. The Poincaré group is assumed to be globally implementable and only the field configurations producing finite Poincaré generators are considered. A detailed study of the Hamiltonian group of gauge transformations is made, trying to elucidate the meaning of the global gauge transformations (connected with the non-Abelian charges and with the center of the gauge group), of the winding number (connected with the large gauge transformations and with the topological charge) and of the small gauge transformations generated by the first class constraints. This leads to the identification of boundary conditions on the gauge potentials and their conjugate momenta suitable for the Hamiltonian description and allowing covariance of the non-Abelian charges. Finally, a review is made of the problem of the Gribov ambiguity, whose basis is connected with the existence of stability subgroups of gauge transformations for certain gauge potentials (gauge symmetries) and/or certain field strengths (gauge copies) in generic Sobolev spaces.

In this paper we study the differential structure of a spinor bundle in spaces where the metric tensor gμν(x, ξ, \gx) of the base manifold depends on the position x as well as on the spinor variables ξ and \gx. Notions such as: gauge covariant derivatives of tensors, connections, curvatures, torsions and Bianchi identities are presented in the context of a gauge approach, different than the one proposed in [11, 13], due to the introduction of a Poincare group and the use of d-connections [6, 8] in the spinor bundle S(2) M. The introduction of basic 1-form fields ϱμ and spinors ζa, \gza with values in the Lie algebra of the Poincare group is also essential in our study. The gauge field equations are derived by the authors [12]. Finally, we give the Yang-Mills and the Yang-Mills-Higgs equations in a form sufficiently generalized for our approach.

Locally anisotropic gauge theories for semisimple and nonsemisimple groups are examined. A gauge approach to generalized Lagrange gravity based on local linear and affine structural groups is proposed.

A recently proposed version of the chiral Schwinger model is studied in detail in this paper. It is shown that a suitable Pauli-Villars regularization can be devised to reproduce the bosonized form of the effective action that was earlier written down. It is then shown how this anomalous gauge theory can be made gauge invariant by the introduction of a Wess-Zumino field. The equations of motion of this theory are explicitly solved in Lorentz covariant gauges. Finally, the operator solution of the fermionic form of the theory is constructed.

It is shown that there exists a close connection between Sogami's method in the standard model and our previous formulation of non-commutative differential geometry. This connection emerges from an identification of an extra one-form basis in the latter with an ordinary one-form valued in Dirac algebra. Application of the method to the standard model with (or without) singlet va's gives a new intepretation of SU(5) relations among gauge coupling constants in the model without SU(5) symmetry. In a recent paper, 1 > Sogami proposed a new method of deriving the bosonic sector of the standard model lagrangian from the fermionic one in an ingenious way. This is based on the observation that the generalized covariant derivative, which acts on a set of chiral fermion fields (total fermion field) interacting with both gauge and Higgs fields, determines generalized field strengths by taking a commutator, which then fixes the bosonic lagrangian. This procedure is applied 2 > to both gauge and non-gauge theories by performing the usual chiral decomposition of the Dirac spinor so as to make the generalized covariant derivative a 2 X 2 matrix in chiral space. This matrix structure resembles that of non-commutative geometry (NCG) 3 > as for­

An asymptotically flat spacetime (AFST) at spatial infinity is studied. The technique of the field formulation of general relativity developed earlier is used. The properties of the latter are similar to those of an ordinary gauge field theory in a fixed background spacetime. We give a definition of an AFST. The role of a background is played by Minkowski space. Integrals of motion are defined with the use of a stress-energy tensor of the gravitational field together with its sources and Killing vectors of the background spacetime. For each of the following AFST characteristics, namely the definition and the values of the integrals of motion and the action integral, we have found the weakest asymptotics for gauge transformations such that this characteristic does not change under the gauge transformations. These three kinds of falloff conditions are compared. The suggested approach is discussed and compared with some known results and methods.

A general covariant conservation law of energy-momentum in complex general relativity is obtained by way of general displacement transformation in terms of Ashtekar's new variables. The energy is exactly the adm Hamiltonian on the constraint surface on condition that an appropriate time function is chosen. The energy-momentum is gauge covariant and commutes with all the constraints whence they are physical observables. Furthermore, the Poisson brackets of the momentum and the internalSU(2) charges form a 3-Poincare algebra.

The anti-BRST transformation for an arbitrary irreducible gauge invariant action is implemented in the usual Batalin-Vilkovisky (BV) approach. This is done without duplicating the gauge symmetries, but rather by duplicating all fields and antifields of the theory. In this way the Sp(2)-covariant quantization can be accomplished in the standard BV approach and its equivalence with the Batalin-Lavrov-Tyutin (BLT) approach is proved by a special gauge-fixing procedure.

The concept of covariant derivatives for quark and lepton fields is generalized on algebras of internal symmetries and the Dirac matrices. Commutators of the covariant derivatives define field strengths for both the gauge fields and the Higgs field. The bosonic part of the Lagrangian is determined using a symmetric combination of quadratic invariants of the gauge and Higgs field strengths. The new version of the standard model thus formulated predicts the Higgs boson mass in terms of the top quark mass as mu"'=f2m, at low energies and the Weinberg angle sin 2 In this article, we formulate a new unified theory of the gauge and Higgs fields without changing the notion of the ordinary Minkowski spacetime by generalizing the covariant derivative Dp. (rather than the Dirac operator l/J) on algebras of internal symmetries and the Dirac matrices in a Lorentz and gauge covariant way. Taking commutators of the covariant derivatives, field strengths are defined for both the gauge fields and the Higgs field. The bosonic part of the Lagrangian is determined by a symmetric combination of quadratic invariants of the gauge and Higgs field strengths. Essential results of the Connes theory are reproducible in this version of the standard model, where a strong parallelism is imposed between the gauge and Higgs field strengths. Three generations of leptons are represented by the chiral fields of the electr­ oweak doublets and singlets as t/Jlj=rh j (j=1, 2, 3). Similarly, those of quarks are described by the colored chiral fields of electroweak doublets t/Jqj= tPL (qJj, and the singlets t/Jw=t/JR j and t/Jdi=t/JR i. With these chiral fields, the fermionic part of the Lagrangian density of the standard model is given by ..( ~ "' ;r. ~'(·a + A ca>a 1 1 + A c2>a 1 + A co 1 y)-'· .

We describe Sogami's method of generating the bosonic sector of the standard model lagrangian from the generalized covariant derivative acting on chiral fermion fields in a simpler setting using well-known field theory models with either global or local symmetries. It is shown that Sogami's method is applicable not only to gauge theories but also to non-gauge theories. It is also shown that the present method offers a different interpretation of Sogami's high energy limit with SU(5) relations among gauge coupling constants in the standard model. Recently, SogamP) proposed an interesting method of generating the bosonic sector of the standard model lagrangian from the generalized covariant derivative operator acting on a set of chiral fermion fields (total fermion field). Its spirit is similar to Cannes' approach based on the non-commutative geometry, 2 ), 3 ), 4 ) but its mathematical nature is quite different, and much easier to understand, using more familiar tools. The purpose of this note is to show that Sogami's method is appli­ cable to both gauge and non-gauge theories in greater generality. We first explain it in terms of simpler field theory models with either global or local symmetries, and then re-study Sogami's reformulation of the standard model in a different setting described in this note. A gauge theory coupled to massive fermions of mass m is described by the lagrangian

In terms of the Dirac's finite contact transformation, we derive the principle on the choice of gauge conditions in the general singular Lagrangian system. It is applied successfully to Cawley's first counter-example of Dirac's conjecture. The number of gauge conditions or gauge freedoms is given and it is shown to be different from the accustomed conclusion which is always equal to the number of all the first-class constraints.

The correspondence between dual thermal transformations of the thermo field dynamics (TFD) and gauge fields is studied both for abelian and non-abelian theories. It is found that the action for the TFD representation of a Dirac's lagrangian remains invariant under local transformations in k -space. The conserved charge coincides with the TFD vacuum-generator G introduced by Takahashi and Umezawa. The relationship between the rules of the TFD and the principle of gauge invariance in a thermal subspace is discussed in the context of a thermal symmetry breaking.

For a distance dependent potential ø, Newtonian dynamics is characterized by conservation of energy, linear momentum, and angular momentum. In addition, the basic dynamical equations are covariant, that is, invariant under fundamental coordinate transformations. In this paper, it is shown that related N-body problems can be solved numerically in such a fashion that, independently of the time step, the identical conservation laws continue to be valid for the approximating difference equations. In addition, the numerical method is covariant.

The massless Rarita–Schwinger field propagator used in perturbative interacting field theories involving this field is derived in covariant and the Coulomb-like gauges for the first time by explicitly invoking the gauge constraints as it was for a massless vector field over the years.

It is shown that the congruent transference introduced by Weyl in 1921 defines a non-Abelian gauge field. The simplest gauge-invariant equations are proposed for this field. Its relation with the Riemann–Cartan geometry is also discussed.

A method is proposed for constructing infinitesimal gauge transformations for an arbitrary degenerate Lagrangian without restrictions on the constraint algebra. It is thereby shown that in the general case degeneracy of a theory is due to its gauge invariance.

Much of calculus deals with the interplay between differentiation and integration. The antiquated term “antidifferentiation” emphasizes the fact that differentiation and integration are inverses of one another. We will take it for granted that readers are acquainted with the mechanics of integration. The current chapter develops just enough integration theory to make our development of differentiation in Chap. 4 and the calculus of variations in Chap. 17 respectable. It is only fair to warn readers that in other chapters a few applications to probability and statistics will assume familiarity with properties of the expectation operator not covered here.

Models with higher order derivative terms in the kinetic energy appear not only as effective theories, they can be considered as elementary, renormalizable models in their own right. The extension of Higgs mechanism is discussed for Yang-Mills-Higgs models where the kinetic energy of the scalar field contains higher order derivatives. The connection term of the covariant derivatives generates a local potential for the gauge field in the phase with spontaneously broken global gauge symmetry which may break Lorentz symmetry in the vacuum. Relativistic, massless gauge bosons spectrum is found for small fluctuations where the only remnant of the breakdown of the Lorentz symmetry is a shift of the energy induced by the expectation value of the temporal component of the gauge field, reminiscent of a chemical potential., 15 pages, no figures

In order to generate mass for an abelian spin-1 vector field while preserving gauge invariance we couple it to a symmetric tensor. The derivative coupling includes up to three derivatives. We show that unitarity, causality and absence of Stueckelberg (compensating) fields single out a unique model up to trivial field redefinitions. The model contains one massive and one massless spin-1 particle. It is shown by means of a master action to be dual to the direct sum of a Maxwell plus a Maxwell–Proca theory.

In this paper, the $SIM(2)$ superspace formulation of the supersymmetric Yang-Mills gauge theory minimally coupled to chiral superfields is discussed. The super-Poincare invariant supersymmetric Yang-Mills theory is rewritten to $SIM(2)$ superspace formalism and the effects of $SIM(2)$ invariant but Lorentz breaking terms are discussed. Two approaches are investigated. The first is based on the gauge chiral representation of the supersymmetric gauge theory and the second is based on the covariant representation of the supersymmetric gauge theory.

It is argued by the author that the canonical form of the quark energy-momentum tensor with a partial derivative instead of the covariant derivative is the correct definition for the quark momentum and angular momentum fraction of the nucleon in covariant quantization. Although it is not manifestly gauge invariant, its matrix elements in the nucleon will be non-vanishing and are gauge invariant. We test this idea in the path-integral quantization by calculating correlation functions on the lattice with a gauge-invariant nucleon interpolation field and replacing the gauge link in the quark lattice momentum operator with unity, which corresponds to the partial derivative in the continuum. We find that the ratios of three-point to two-point functions are zero within errors for both the u and d quarks, contrary to the case without setting the gauge links to unity.

Given a divergence operator δ on a probability space such that the law of δ ( h ) is infinitely divisible with characteristic exponent (0.1) h ↦ − 1 2 ∫ 0 ∞ h t 2 d t , or ∫ 0 ∞ ( e i h ( t ) − i h ( t ) − 1 ) d t , h ∈ L 2 ( R + ) , we derive a family of Laplace transform identities for the derivative ∂ E [ e λ δ ( u ) ] / ∂ λ when u is a non-necessarily adapted process. These expressions are based on intrinsic geometric tools such as the Carleman–Fredholm determinant of a covariant derivative operator and the characteristic exponent (0.1) , in a general framework that includes the Wiener space, the path space over a Lie group, and the Poisson space. We use these expressions for measure characterization and to prove the invariance of transformations having a quasi-nilpotent covariant derivative, for Gaussian and other infinitely divisible distributions.

In this paper we introduce and study some mathematical structures on top of transitive Lie algebroids in order to formulate gauge theories in terms of generalized connections and their curvature: metrics, Hodge star operator and integration along the algebraic part of the transitive Lie algebroid (its kernel). Explicit action functionals are given in terms of global objects and in terms of their local description as well. We investigate applications of these constructions to Atiyah Lie algebroids and to derivations on a vector bundle. The obtained gauge theories are discussed with respect to ordinary and to similar non-commutative gauge theories., Comment: 30 pages. Final version. arXiv admin note: substantial text overlap with arXiv:1109.4282

We reexamine the 10-dimensional type II and heterotic superstring theories using the bosonic language. The aim of this bosonic formulation is the construction of the covariant fermion vertex operators, which involves a proper treatment of the \((\beta ,\gamma )\)ghost system, This will in turn lead to the introduction of the so-called covariant lattices.

The quantum charged rigid membrane model, which is a higher derivative theory has been considered to explore its gauge symmetries using a recently developed first order formalism \cite{BMP}. Hamiltonian analysis has been performed and the gauge symmetry of the model is identified as reparametrisation symmetry. First class constraints are shown to have a truncated Virasoro algebraic structure. An exact correspondence between the higher derivative theory and the first order formalism has been shown from the point of view of equations of motion., Comment: Latex, 13 pages, minor changes in introduction, references added

We describe a systematic way of the generalization, to models with non-linear duality, of the space-time covariant and duality-invariant formulation of duality-symmetric theories in which the covariance of the action is ensured by the presence of a single auxiliary scalar field. It is shown that the duality-symmetric action should be invariant under the two local symmetries characteristic of this approach, which impose constraints on the form of the action similar to those of Gaillard and Zumino and in the non-covariant formalism. We show that the (twisted) self-duality condition obtained from this action upon integrating its equations of motion can always be recast in a manifestly covariant form which is independent of the auxiliary scalar and thus corresponds to the conventional on-shell duality-symmetric covariant description of the same model. Supersymmetrization of this construction is briefly discussed., 1+21 pages, v2: typos corrected, references added

The one-loop quantum corrections of Chern-Simons spinor electrodynamics in the light-cone gauge has been investigated. We have calculated the vacuum polarization tensor, fermionic self-energy and on-shell vertex correction with a hybrid regularization consisting of a higher covariant derivative regularization and dimensional continuation. The Mandelstam-Leibbrandt prescription is used to handle the spurious light-cone singularity in the gauge field propagator. We then perform the finite renormalization to define the quantum theory. The generation of the parity-even Maxwell term and the arising of anomalous magnetic moment from quantum corrections are reproduced as in the case of a covariant gauge choice. The Ward identities in the light-cone gauge are verified to satisfy explicitly. Further, we have found the light-cone vector dependent sector of local quantum effective action for the fermion is explicitly gauge invariant, and hence the covariance of $S$-matrix elements of the theory can be achieved., Comment: 29 pages, no fugures, RevTex

In this paper we give a summary of the geometrical background of the idea of spontaneous symmetry breaking. For this purpose, we set out to discuss Yang–Mills–Higgs gauge theories from the perspective of reducible bundles. From this viewpoint, “elementary particles” are identified with vector bundles, and sections are considered to geometrically represent the states of the corresponding particle. Some physical background on the notion of “mass” is given in the introduction. Since the geometrical interpretation of a gauge boson is that of a connection, we proceed to discuss how, from a geometrical point of view, the Higgs boson can also be considered a connection. We start out with Connes’ algebraic approach, where the “shifted Higgs boson” is considered a gauge potential on a non-commutative space. We summarize how a specific generalization of the notion of a Dirac operator can be used in order to define a generalization of de Rham’s algebra. This generalization is used to define the non-commutative equivalent of the Yang–Mills action where its minima spontaneously break the gauge symmetry. In the last section we summarize how the Higgs boson can be considered a connection on a Clifford module bundle.

In previous work [1, 2], we studied the linearized gravity with covariant gauge β = 2/3 and α = 5/3. It was found that the sum of the source and initial contributions reproduces the correct field configuration over the whole de Sitter spacetime. In this paper, we extend this work to generalizing the linearized gravitational field in an arbitrary value of the gauge parameter α but the gauge parameter β remains the same.

The gauge transformations for ( n +1)×( n +1) eigenvalue problems are introduced. In particular, extended forms of Ablowitz-Kaup-Newell-Segur (AKNS) system and Kaup-Newell (KN) system are found to be related by the gauge transformation. As an application of this theory, the interrelations between the coupled nonlinear Schrodinger equation and the coupled derivative nonlinear Schrodinger equation are explicitly shown.

The tetradic Lorentz-gauge invariant formulation of the SU(2) × U(1) theory in S3 × R space-time is presented and the general gauge covariant Dirac-Klein-Gordon-Maxwell-Yang-Mills equations are derived. A direct comparison of these equations to those of the SU(2) × U(1) gauge theory on Minkowskian background points out major differences effectively induced by the minimally coupling to S3 × R gravity.

The manifold Γ defined by the equations of motion (EM) of the gauge and ghost fields w.r.t. the gauge‐fixed action is regarded as a supercanonical line bundle over the manifold Γ defined by the EM of the gauge fields only w.r.t. the classical action. In this language the BRST operator Q is a local section of the bundle Γ. Using this we give a local expression for Q as being, on the other hand, the nilpotent exterior derivative on Γ with the ghost field Ψ as its generator. This fiber bundle setup allows us to prove that any ‘‘second level’’ gauge condition, i.e., a gauge condition on Γ is equivalent to a gauge on the base manifold Γ and thus does not break the BRST symmetry of the quantized theory.

The correspondence between dual thermal transformations and gauge fields is discussed in the context of Thermo Field Dynamics (TFD). The equivalence between the rules of TFD and the principle of gauge invariance in a thermal subspace is shown starting from the invariance of TFD action under local thermal transformations in k -space.

A representation of the algebra Open image in new window (3)=t(3) ⊕ S0(3, ℝ) by differential Schaefer's operators is proposed, and an external algebra of Open image in new window (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the Open image in new window (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.

A universal covariant form proposed for the equations of motion of the second kind can be used to study the dynamics of various systems of bodies interacting with each other through arbitrary gauge fields and combinations of such fields. The correspondence between the equations obtained and the analogous equations of motion derived by other authors is considered as the properties of the dynamic system become simpler.

The algebraic structure of covariant anomalies and covariant Schwinger terms in an anomalous Yang-Mills theory is investigated. A cohomological characterization is formulated and geometrically interpreted. A new method of determining covariant anomalies and covariant Schwinger terms based on the BRS-anti-BRS complex is presented.