Unified Field Theory from Enlarged Transformation Group. The Consistent Hamiltonian

A theory has been presented previously in which the geometrical structure of a real four-dimensional space time manifold is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group. The group enlargement was accomplished by including those transformations to anholonomic coordinates under which conservation laws are covariant statements. Field equations have been obtained from a variational principle which is invariant under the larger group. These field equations imply the validity of the Einstein equations of general relativity with a stress-energy tensor that is just what one expects for the electroweak field and associated currents. In this paper, as a first step toward quantization, a consistent Hamiltonian for the theory is obtained. Some concluding remarks are given concerning the need for further development of the theory. These remarks include discussion of a possible method for extending the theory to include the strong interaction.     

Gravitational and electroweak unification by replacing diffeomorphisms with larger group

The covariance group for general relativity, the diffeomorphisms, is replaced by a group of coordinate transformations which contains the diffeomorphisms as a proper subgroup. The larger group is defined by the assumption that all observers will agree whether any given quantity is conserved. Alternatively, and equivalently, it is defined by the assumption that all observers will agree that the general relativistic wave equation describes the propagation of light. Thus, the group replacement is analogous to the replacement of the Lorentz group by the diffeomorphisms that led Einstein from special relativity to general relativity, and is also consistent with the assumption of constant light velocity that led him to special relativity. The enlarged covariance group leads to a non-commutative geometry based not on a manifold, but on a nonlocal space in which paths, rather than points, are the most primitive invariant entities. This yields a theory which unifies the gravitational and electroweak interactions. The theory contains no adjustable parameters, such as those that are chosen arbitrarily in the standard model.     

Unified Field Theory From Enlarged Transformation Group. The Covariant Derivative for Conservative Coordinate Transformations and Local Frame Transformations

Pandres has developed a theory in which the geometrical structure of a real four-dimensional space-time is expressed by a real orthonormal tetrad, and the group of diffeomorphisms is replaced by a larger group called the conservation group. This paper extends the geometrical foundation for Pandres’ theory by developing an appropriate covariant derivative which is covariant under all local Lorentz (frame) transformations, including complex Lorentz transformations, as well as conservative transformations. After defining this extended covariant derivative, an appropriate Lagrangian and its resulting field equations are derived. As in Pandres’ theory, these field equations result in a stress-energy tensor that has terms which may automatically represent the electroweak field. Finally, the theory is extended to include 2-spinors and 4-spinors.     

Space-time theories and symmetry groups

This paper addresses the significance of the general class of diffeomorphisms in the theory of general relativity as opposed to the Poincaré group in a special relativistic theory. Using Anderson's concept of an absolute object for a theory, with suitable revisions, it is shown that the general group of local diffeomorphisms is associated with the theory of general relativity as its local dynamical symmetry group, while the Poincaré group is associated with a special relativistic theory as both its global dynamical symmetry group and its geometrical symmetry group. It is argued that the two groups are of equal significance as symmetry groups of their associated theories.     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Gravitation theory as theory of non-Lorentzian transformations of the systems of reference

According to the programme of Einstein as discussed with Abraham, gravitation can be described by the bending of the systems of inertia in special relativity. This bending means non-Lorentzian transformations of the systems of reference, depending on the point in space-time. Einstein's equations for the metric imply equations for the transformation matrix, which are also of the same structure.The non-Lorentzian transformations of the reference systems of a manifold can lead to a general map of the set of metrics into the set of vacuum metrics on the same manifold. Resulting new aspects in problems of gravitation theory are discussed.     

Lie Symmetries of Einstein’s Vacuum Equations in N Dimensions

Abstract

We investigate Lie symmetries of Einstein’s vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein’s equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein’s equations), we set to zero the coefficients of derivatives that do not appear in Einstein’s equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein’s equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.     

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.     

Electromagnetic solutions of the field equations in presence of zeromass mesons

The field equations of general relativity with electromagnetic stress tensor and zeromass scalar meson field are investigated. The metric coefficients are assumed to be functions of three variables only. It is then shown that, if one assumes a functional relation between some one of the metric coefficients and the electromagnetic potentials, that one can find a solution of the coupled Einstein-Maxwell equations in terms of a solution of the Einstein equations with zeromass scalar meson field as source.     

Non-local Effects of Conformal Anomaly

It is shown that the nonlocal anomalous effective actions corresponding to the quantum breaking of the conformal symmetry can lead to observable modifications of Einstein’s equations. The fact that Einstein’s general relativity is in perfect agreement with all observations including cosmological or recently observed gravitational waves imposes strong restrictions on the field content of possible extensions of Einstein’s theory: all viable theories should have vanishing conformal anomalies. It is shown that a complete cancellation of conformal anomalies in \(D=4\) for both the \(C^2\) invariant and the Euler (Gauss–Bonnet) invariant can only be achieved for N-extended supergravity multiplets with \(N \ge 5\).     

Finslerian Multi-Dimensionality,Associated Gauge Fields,and Stochastic Space-Time

The theory is constructed which elucidates all the gauge fields associated with fibration of the tangent vectors and of the vectors of higher degrees of tangency. The approach synthesizes intrinsically the Einstein equations with the Yang-Mills equations and gives an adequate framework for extending the physical field equations. Then, by considering the statistical behaviour of internal variables, it is possible to get a deeper insight in understanding the origin of quantum laws. Actually, we construct what may be called the general relativity in vector fibrations over space-time manifold.     

Gauge Dependence in the Theory of Non-Linear Spacetime Perturbations

Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we introduce the notion of knight diffeomorphism, that generalises the usual concept of flow, and prove a Taylor's formula for the action of a knight on a general tensor field. Then, we show that any one-parameter family of diffeomorphisms can be approximated by a family of suitable knights. Since in perturbation theory the gauge freedom is given by a one-parameter family of diffeomorphisms, the expansion of knights is used to derive our transformation formula. The problem of gauge dependence is a purely kinematical one, therefore our treatment is valid not only in general relativity, but in any spacetime theory. Received: 21 November 1996 / Accepted: 20 August 1997     

Von Neumann’s quantization of general relativity

Von Neumann’s procedure is applied to quantizing general relativity. Initial data for dynamical variables in the Planck epoch, where the Hubble parameter value coincided with the Planck mass are quantized. These initial data are defined in terms of the Fock orthogonal simplex in the tangent Minkowski spacetime and the Dirac conformal interval. The Einstein cosmological principle is used to average the logarithm of the determinant of the spatial metric over the spatial volume of the visible Universe. The splitting of general coordinate transformations into diffeomorphisms and transformations of the initial data is introduced. In accordance with von Neumann’s procedure, the vacuum state is treated is a quantum ensemble that is degenerate in quantum numbers of nonvacuum states. The distribution of the vacuum state leads to the Casimir effect in gravidynamics in just the same way as in electrodynamics. The generating functional for perturbation theory in gravidynamics is found by solving the quantum energy constraint. The applicability range of gravidynamics is discussed along with the possibility of employing this theory to interpret modern observational data.     

Gödel type metrics in Einstein–aether theory

Aether theory is introduced to implement the violation of the Lorentz invariance in general relativity. For this purpose a unit timelike vector field is introduced to the theory in addition to the metric tensor. Aether theory contains four free parameters which satisfy some inequalities in order that the theory to be consistent with the observations. We show that the Gödel type of metrics of general relativity are also exact solutions of the Einstein–aether theory. The only field equations are the 3D Maxwell field equations and the parameters are left free except c ₁ ? c ₃ = 1.     

Gauge Fields in General Parametrical Approach and Their Finslerian Reduction

The covariance principle of general relativity is extended to internal space. Associated gauge fields and tensors are systematically described, whereupon the variational principle is set up for all gauge fields by applying a Palatini-type method, thereby giving rise to an attractive self-contained theory in which the Einstein equations are intrinsically synthesized with the generalized Yang-Mills equations. General gauge-covariant physical field equations are formulated, showing that currents, external + internal spin tensors, and energy-momentum tensors can be introduced unambiguously under these general conditions and that the associated conservation laws can be derived. The electromagnetic field finds its gauge-geometric origin as the gauge field related to internal densities. To be operative with the tensor indices of external and internal types, this general theory must be bimetric. The assumptions that the gauge-covariant derivatives of metric tensors should vanish simplify the theory to the level of a Finslerian gauge approach.     

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.     

Derivation of the Geometrical Phase from the Evans Phase Law of Generally Covariant Unified Field Theory

The phase law of generally covariant electrodynamics is used to explain straightforwardly the origin of the geometrical and Berry phase effects, exemplified by the Tomita-Chiao effect. Both effects are described by a phase factor that is constructed from the generally covariant Stokes formula of differential geometry, a phase factor in which the contour integral over the potential field A ⁽³⁾ is equated to the area integral over the gauge invariant field B ⁽³⁾, the Evans-Vigier field. The latter is the fundamental spin Casimir invariant of the Einstein group of general relativity applied to electrodynamics. General relativity as extended in the Evans unified field theory is needed for a correct understanding of all phase effects in physics, an understanding that is forged through the Evans phase law, the origin both of the Berry phase and the geometrical phase of electrodynamics observed in the Sagnac and Tomita-Chiao effects.     

Counting rule of Nambu–Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

When continuous symmetry is spontaneously broken, there appear Nambu–Goldstone modes (NGMs) with linear or quadratic dispersion relation, which is called type-I or type-II, respectively. We propose a framework to count these modes including the coefficients of the dispersion relations by applying the standard Gross–Pitaevskii–Bogoliubov theory. Our method is mainly based on (i) zero-mode solutions of the Bogoliubov equation originated from spontaneous symmetry breaking and (ii) their generalized orthogonal relations, which naturally arise from well-known Bogoliubov transformations and are referred to as “σ$\sigma$-orthogonality” in this paper. Unlike previous works, our framework is applicable without any modification to the cases where there are additional zero modes, which do not have a symmetry origin, such as quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the presence of a topological soliton or a vortex. As a by-product of the formulation, we also give a compact summary for mathematics of bosonic Bogoliubov equations and Bogoliubov transformations, which becomes a foundation for any problem of Bogoliubov quasiparticles. The general results are illustrated by various examples in spinor Bose–Einstein condensates (BECs). In particular, the result on the spin-3 BECs includes new findings such as a type-I–type-II transition and an increase of the type-II dispersion coefficient caused by the presence of a linearly-independent pair of zero modes.     

THE PROBLEM OF MOTION IN THE KALUZA THEORY

It is shown that the equations of motion for a charged massive particle are consequences of the field equations in Kaluza unification theory of gravitation and electromagnetism, i.e., the equations of motion for the particle can be deduced from Kaluza field equations, just as that in Einstein's theory of motion of general relativity the equations of motion for a massive particle are consequences of the Einstein equations. Furthermore, the Lorentz equations for a particle maving in the Maxwell electromagnetic field on the Minkowskian space-time can also be obtained from the Maxwell equations by means of the Kaluze mechanism of the Maxwell theory.     

New infinite-dimensional hidden symmetries for the stationary axisymmetric Einstein－Maxwell equations with multiple Abelian gauge fields

By proposing a so-called extended hyperbolic complex (EHC) function method, an Ernst-like (p+2)×(p+2) matrix EHC potential is introduced for the stationary axisymmetric (SAS) Einstein－Maxwell theory with p Abelian gauge fields (EM-p theory, for short), then the field equations of the SAS EM-p theory are written as a so-called Hauser－Ernst-like self-dual relation for the EHC matrix potential. Two Hauser－Ernst-type EHC linear systems are established, based on which some new parametrized symmetry transformations for the SAS EM-p theory are explicitly constructed. These hidden symmetries are found to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac－Moody algebra su(p+1,1)\otimes R(t,t^{-1}) and Virasoro algebra (without centre charges). All of the SAS EM-p theories for p=0,1,2,… are treated in a unified formulation, p=0 and p=1 correspond, respectively, to the vacuum gravity and the Einstein－Maxwell cases.