Towards a quantization of gauge fields on de Sitter group by functional integral method

A formulation of the de Sitter symmetry as a purely inner symmetry defined on a fixed Minkowski space-time is presented. We define the generators of the de Sitter group and write the structure equations using a constant deformation parameter λ. The conserved gauge currents are calculated, and their physical meaning is given. Local gauge transformations and the corresponding covariant derivative depending on the gauge fields are also obtained. We study the behavior of gauge fields, the torsion and curvature tensors and give a regularization technique in terms of the ζ function.     

A METHOD ON DERIVATION OF GAUGE FIELDS

Usually the study of gauge field is based on the wave function. By discussing thebehaviour of Dirac particles in gravitation, one has a famous difficulty, that is, thewave functions appear as scalars under general coordinate transformations. In thispaper, a method is suggested to constitute the gauge fields directly from algebraicstructures, Lie algebra and Jordan algebra. We introduce a concept called represen-tation group of algebras, the transformations, of wave function are connected with therepresentation group. The global and local representation groups are connected withglobal and local transformations of wave function respectively. According to thismethod we find that it is equivalent to the usual one for all of the problems concernedwith internal freedom as Yang-Mills field etc. For spinors, one can introduce gravi-tation by changing the algebraic structure, one find that the vierbein is unneccessaryand the wave functions transform as spinors corresponding to Dirac theory. Somerelated problems are also discussed.     

Supergravity as a gauge theory of supersymmetry

In a new approach to supergravity we consider the gauge theory of the 14-dimensional supersymmetry group. The theory is constructed from 14×4 gauge fields, 4 gauge fields being associated with each of the 14 generators of supersymmetry. The gauge fields corresponding to the 10 generators of the Poincaré subgroup are those normally associated with general relativity, and the gauge fields corresponding to the 4 generators of supersymmetry transformations are identified with a Rarita-Schwinger spinor. The transformation laws of the gauge fields and the Lagrangian of lowest degree are uniquely constructed from the supersymmetry algebra. The resulting action is shown to be invariant under these gauge transformations if the translation associated field strength vanishes. It is shown that the second-order form of the action, which is the same as that previously proposed, is invariant without constraint.     

Gravitation and unified covariant formalism of classical gauge fields in equiaffine space

In this paper, we construct a unified covariant formalism for the classical gauge fields in an equiaffine space. The gauge transformation groups are the Lie groups, induced according to the third Lie theorem by the structure constants. As a result of the gauge transformations, one set of geometric objects is replaced by another. It is confirmed that the differential conservation laws in the equiaffine spaces are a result of the equations of the gauge fields. The particular case when the gauge transformation group is a four-parameter group and is abelian is distinguished. This group corresponds to gauge fields that are induced by an energy-momentum tensor and, which, as a result, are called gravitational fields. As a particular case of the equations of the given gravitational fields, we obtain Einstein's equations with the help of a Lagrangian, which is quadratic with respect to the gravitational field intensities. In concluding, we note the possibility of describing gauge fields, corresponding to nongravitational interactions of vector mesons with nonzero rest mass, without invoking the scalar Higgs mesons. This possibility appears both as a result of the generalization of the Yang-Mills covariant derivative and as a result of including gravitational interactions in the general gauge field formalism.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 12, 47–51, December, 1981.     

A gauge field theory of spacetime based on the de Sitter group

A new theory of spacetime is proposed in which translations are considered as a part of the de Sitter gauge group. The theory is built along the general principles of classical gauge field theories, which are outlined. Applications of gauge principles to linear and affine connections are also given in order to make the presentation self-sufficient. A de Sitter invariant Lagrangian is constructed, which yields approximately Einstein's vacuum equations when it is subjected to variation with respect to gauge potentials and the result expressed in a specific gauge class. As a difference from the usual use of de Sitter groups, the radius of its translations must be small in the present approach, which probably has the meaning of an elementary subatomic length. The solution of the equations describing flat spacetime is not the trivial zero-curvature connection of the conventional approach.     

Generalizations of gravitational theory based on group covariance

The mathematical structure, the field equations, and fundamentals of the kinematics of generalizations of general relativity based on semisimple invariance groups are presented. The structure is that of a generalized Kaluza-Klein theory with a subgroup as the gauge group. The group manifold with its Cartan-Killing metric forms the source-free solution. The gauge fields do not vanish even in this case and give rise to additional modes of free motion. The case of the de Sitter groups is presented as an example where the gauge field is tentatively assumed to mediate a spin interaction and give rise to spin motion. Generalization to the conformal group and a theory yielding features of Dirac's large-number hypothesis are discussed. The possibility of further generalizations to include fermions are pointed out. The Kaluza-Klein theory is formulated in terms of principal fibre bundles which need not to be trivial.     

On nonlinear higher spin curvature

We present the first nonlinear term of the higher spin curvature which is covariant with respect to deformed gauge transformations that are linear in the field. We consider the case of spin 3 after presenting spin 2 as an example, and then construct the general spin s quadratic term of the de Wit–Freedman curvature.     

On the gauge variance of action functions under transformations on space-time

The problem of the gauge variance or invariance of action functions in classical mechanics is discussed from a group and path-theoretic viewpoint. By using the elementary theory of the cohomology of groups, criteria are introduced which enable one to decide when action functions gauge variant under a kinematical group are equivalent to action functions invariant under the transformations of the group. The criteria are applied to action functions gauge variant under Lorentz and Galilei transformations, where we deduce that any action function gauge variant under the Lorentz group is equivalent to an action function invariant under Lorentz transformations, whilst action functions gauge variant under the Galilei group are not necessarily equivalent to Galilei-invariant action functions. It is also shown that any action function gauge variant in a more restricted fashion which we define in the text, is necessarily equivalent to a kinetic-energy action.     

Invariance operator groups for a charged particle in an external electromagnetic field

In this paper a general group theoretical approach is given for the problem of a charged particle moving in an external electromagnetic field F. From a knowledge of the symmetry transformations of the field (Galilean or Poincaré), it is possible to explicitly construct groups of operators which commute with the operators of the equations of motion (classical, quantum mechanical, Klein-Gordon or Dirac) using the concept of compensating gauge transformations together with a uniquely chosen map π: F → A fixing the gauge of the potential A. Other choices of gauges give rise to isomorphic operator groups. The general structure of the possible symmetry groups of the fields is discussed and the corresponding invariance operator groups are explicitly given for (almost) arbitrary fields. The structure of these groups is then investigated and it is shown in particular that a large class of fields give rise to non-Type I groups, i.e. to groups which have (unitary continuous) representations whose corresponding von Neumann algebras have non-discrete factors. A general criterion for these pathological cases is given. As an application, we study the problem of a Bloch electron in arbitrary constant uniform electric and magnetic fields.     

Unusual representations of local groups

By means of duality transformations we obtain representations of Yang-Mills groups which are neither tensors nor gauge fields, and fields which under general coordinate transformations are neither tensors nor tensor densities.     

Bäcklund-calogero group and general form of integrable equations for the two-dimensional Gelfand-Dikij-Zakharov-Shabat problem bilocal approach

The general form of the integrable equations and their Bäcklund transformations connected with the general two-dimensional Gelfand-Dikij-Zakharov-Shabat spectral problem is found within the framework of the generalized AKNS method. The bilocal tensor product of the solutions of the spectral problem is used successively, which essentially simplifies the calculations of recursion operators. The transformation properties of the integrable equations and Bäcklund transformations under the gauge group are discussed.     

Renormalization of Wilson operators in gauge theories

The operators in a Wilson expansion are not in general multiplicatively renormalized in non-Abelian gauge theories. This is because of the renormalization of the gauge transformations themselves. Renormalized fields may be defined, which have the old gauge transformations. Alternatively, a special choice of gauge may be made, in which the gauge transformations are unchanged on renormalization. In any case, one gauge invariant factor appears in the renormalization of the Wilson operators.     

A unified approach to conservation laws in general relativity,gauge theories,and elementary particle physics

A unified treatment of conservation laws in general relativity, gauge theories, and elementary particle physics is formulated in the setting of principal fiber bundles. The group AUT(P) is introduced as the general gauge transformation group that covers space-time coordinate transformations. A set of master equations is exhibited for any Lagrangian density generally covariant with respect to AUT(P). The symmetry group for elementary particle theory is shown to be the structure group of the bundle only in the special case when the gauge potential is flat and the space-time is simply connected. In the general case, the symmetry group is reduced to the symmetry group of the gauge potential. This natural mechanism for a reduction of the symmetry group is speculated on as a model for spontaneous symmetry breaking.This essay received an honorable mention from the Gravity Research Foundation for the year 1981-Ed.Partially supported by a grant from the National Science Foundation.     

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.     

Chiral symmetry breaking and vacuum structure in QCD

The solution of the axial U(1) problem, the role of the topology of the gauge group in forcing the breaking of axial symmetry in any irreducible representation of the observable algebra and the θ vacua structure are revisited in the temporal gauge with attention to the mathematical consistency of the derivations. Both realizations with strong and weak Gauss law are discussed; the control of the general mechanisms and structures is obtained on the basis of the localization of the (large) gauge transformations and the local generation of the chiral symmetry. The Schwinger model in the temporal gauge exactly reproduces the general results.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

On Gauge Invariant Theories of Gravitation

Theories of gravitation are called gauge invariant if the invariance of the gravitational field lagrangian with respect to gauge transformations of the gravitational field variables is independend of the invariance of this lagrangian with respect to the Einstein group of general coordinate transformations. They are bimetric theories because the coordinate covariance is ensured by constructing scalar densities relative to a globally flat background metric. Such a theory is represented by the PAUL-FIERZ equations for massless spin 2 particles. But this theory is inconsistent if nongravitational matter is enclosed as a source. All attempts to overcome this inconsistancy preserving gauge invariance lead to Einstein's GRT. We review this problem and compare the situation with a theory proposed by LOGUNOV showing that he overcomes the inconsistency of linear Einstein's equations by replacing the field variables by a gauge invariant combination of new ones, which turns out to be the first order form of v. FREUD'S superpotential.     

Cosmological term in the general theory of relativity and localization of the de Sitter and Einstein groups

The theory of a gauge gravitational field with localization of the de Sitter group is formulated. Starting from the tetradic components of the de Sitter universe, a relationship is established between the Riemannian metric and the de Sitter gauge field. It is shown that the general theory of relativity with the cosmological term is the simplest variant of the de Sitter gauge theory of gravitation, which transforms in the limit of an infinite radius of curvature of the de Sitter universe into the Poincaré-invariant GTR without the cosmological term. A theory of a gauge gravitational field with localization of Einstein's group of motions of the uniform static universe (the Einstein group R × S0 (4)) is formulated in an analogous manner.Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 8, pp. 86–90, August, 1984.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.