Extended Einstein–Cartan theory à la Diakonov: The field equations

Diakonov formulated a model of a primordial Dirac spinor field interacting gravitationally within the geometric framework of the Poincaré gauge theory (PGT). Thus, the gravitational field variables are the orthonormal coframe (tetrad) and the Lorentz connection. A simple gravitational gauge Lagrangian is the Einstein–Cartan choice proportional to the curvature scalar plus a cosmological term. In Diakonov?s model the coframe is eliminated by expressing it in terms of the primordial spinor. We derive the corresponding field equations for the first time. We extend the Diakonov model by additionally eliminating the Lorentz connection, but keeping local Lorentz covariance intact. Then, if we drop the Einstein–Cartan term in the Lagrangian, a nonlinear Heisenberg type spinor equation is recovered in the lowest approximation.     

Derivation of the Lorentz Boost from the Evans Wave Equation

The Lorentz boost is derived from the Evans wave equation of generally covariant unified field theory by constructing the Dirac spinor from the tetrad in the SU(2) representation space of non-Euclidean spacetime. The Dirac equation in its wave formulation is then deduced as a well-defined limit of the Evans wave equation. By factorizing the dAlembertian operator into Dirac matrices, the Dirac equation in its original first differential form is obtained from the Evans wave equation. Finally, the Lorentz boost is deduced from the Dirac equation using geometrical arguments. A self-consistency check of the Evans wave equation is therefore forged by deducing therefrom the Lorentz boost in the appropriate limit. This procedure demonstrates that the Evans wave equation governs the properties of matter and anti-matter in general relativity and unified field theory and leads both to Fermi-Dirac and Bose-Einstein statistics in general relativity.     

Hamiltonian formulation of the theory of interacting gravitational and electron fields

The action which describes the interaction of gravitational and electron fields is expressed in canonical form. In addition to general covariance, it exhibits the local Lorentz invariance associated with four-dimensional rotations of the local orthonormal frames. The corresponding Hamiltonian constraints are derived and their (Dirac) bracket relations given. The derivative coupling of the gravitational tetrad and spinor fields is not present in the Hamiltonian, but rather in the unusual bracket relations of the field variables in the theory. If the timelike leg of the tetrad field is fixed to be normal to the x^o = constant hyper-surfaces (“time gauge”) the derivative coupling drops from the theory in the sense that the relation between the gravitational velocities and momenta is the same as when the spinor fields are absent.     

Spinor theory of general relativity without elementary gravitons

We propose a generally covariant and locally Lorentz invariant theory of a Majorana spinor field ψ_μ^α. Our theory has no elementary spin-2 quanta, but does reproduce Einstein's general relativity as a classical solution. We compare this situation to the possibility of finding classical monopoles in a gauge theory, even though no such elementary object is introduced at the outset.     

Determination of Phase Space Spectra of the Generalized Chiral Schwinger model with the Lorentz invariant Mass like Term for the Choice a − r2 = 0

We have considered the generalized version of chiral schwinger model with the Lorentz covariant masslike term for gauge field with the choice a ? r² =?0. We carry out the quantization by the canonical Dirac method of both the gauge-invariant and non-invariant version of this model to determine the phase space structure. Therefore we have shown that the gauge invariant theory has the same physical spectrum as that of the original gauge noninvariant formulation.

     

Covariant extensions and the nonsymmetric unified field

The problem of generally covariant extension of Lorentz invariant field equations, by means of covariant derivatives extracted from the nonsymmetric unified field, is considered. It is shown that the contracted curvature tensor can be expressed in terms of a covariant gauge derivative which contains the gauge derivative corresponding to minimal coupling, if the universal constantp, characterizing the nonsymmetric theory, is fixed in terms of Planck's constant and the elementary quantum of charge. By this choice the spinor representation of the linear connection becomes closely related to the spinor affinity used by Infeld and Van Der Waerden in their generally covariant formulation of Dirac's equation.     

Construction of a covariant spinor field theory

A covariant theory is constructed of a spinor field in a space which is represented by the local topological product of a space X_n and a space of values of a geometrical object η. The covariant nonlinear spinor field theory constructed preserves the principles of the theory of the unified field and is compatible with the theory of gauge fields.     

The role of singular spinor fields in a torsional gravity,Lorentz-violating,framework

In this work, we consider a generalization of quantum electrodynamics including Lorentz violation and torsional-gravity, in the context of general spinor fields as classified in the Lounesto scheme. Singular spinor fields will be shown to be less sensitive to the Lorentz violation, as far as couplings between the spinor bilinear covariants and torsion are regarded. In addition, we prove that flagpole spinor fields do not admit minimal coupling to the torsion. In general, mass dimension four couplings are deeply affected when singular—flagpoles—spinors are considered, instead of the usual Dirac spinors. We also construct a mapping between spinors in the covariant framework and spinors in Lorentz symmetry breaking scenarios, showing how one may transliterate spinors of different classes between the two cases. Specific examples concerning the mapping of Dirac spinor fields in Lorentz violating scenarios into flagpole and flag-dipole spinors with full Lorentz invariance (including the cases of Weyl and Majorana spinors) are worked out.     

Classical radiation reaction off-shell corrections to the covariant Lorentz force

We show that the problem of radiation reaction may be formulated in a space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg–Feynman–Schwinger covariant mechanics (the zero mode fields of the 0,1,2,3 components correspond to the Maxwell fields). The particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five-dimensional field equations provides the classical Abraham–Lorentz–Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution as well as an autonomous nonlinear equation of third order for the off-shell mass. The theory does not admit radiation if the particle does not move off-shell. The structure of the equations implies that the mass-shell deviation is bounded when the external field is removed.     

Gauge Covariant Fermion Propagator in the Presence of Arbitrary External Gauge Field and Its Schwinger--Dyson Equation

Gauge covariance for Green＇s functions of a gauge theory through a fermion propagator in the presence of arbitrary external gauge field is proven and a formalism of gauge and Lorentz covariant Schwinger-Dyson equation for the fermion propagator with external gauge field is built up within ladder approximation.     

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.Research supported by Ministero della Pubblica Istruzione and CNR-GNAFA     

Solution of the Dirac equation by separation of variables in spherical coordinates for a large class of non-central electromagnetic potentials

A systematic and intuitive approach for the separation of variables of the three-dimensional Dirac equation in spherical coordinates is presented. Using this approach, we consider coupling of the Dirac spinor to electromagnetic four-vector potential that satisfies the Lorentz gauge. The space components of the potential have angular (non-central) dependence such that the Dirac equation becomes separable in all coordinates. We obtain exact solutions for a class of three-parameter static electromagnetic potential whose time component is the Coulomb potential. The relativistic energy spectrum and corresponding spinor wave functions are obtained. The Aharonov–Bohm and magnetic monopole potentials are included in these solutions.     

Radiation Reaction of the Classical Off-Shell Relativistic Charged Particle

It has been shown by Gupta and Padmanabhan that the radiation reaction force of the Abraham–Lorentz–Dirac equation can be obtained by a coordinate transformation from the inertial frame of an accelerating charged particle to that of the laboratory. We show that the problem may be formulated in a flat space of five dimensions, with five corresponding gauge fields in the framework of the classical version of a fully gauge covariant form of the Stueckelberg–Feynman–Schwinger covariant mechanics (the zero mode fields of the 0, 1, 2, 3 components correspond to the Maxwell fields). Without additional constraints, the particles and fields are not confined to their mass shells. We show that in the mass-shell limit, the generalized Lorentz force obtained by means of the retarded Green's functions for the five dimensional field equations provides the classical Abraham–Lorentz–Dirac radiation reaction terms (with renormalized mass and charge). We also obtain general coupled equations for the orbit and the off-shell dynamical mass during the evolution as well as an autonomous non-linear equation of third order for the off-shell mass. The theory does not admit radiation if the particle does not move off-shell. The structure of the equations implies that mass-shell deviation is bounded when the external field is removed.     

Derivation of an effective Lagrangian from a generalized scalar curvature

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral SU(n) × SU(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections.The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.     

Equivalence of Dirac and Maxwell equations and quantum mechanics

In this paper we present an analysis of the possible equivalence of Dirac and Maxwell equations using the Clifford bundle formalism and compare it with Campolattaro's approach, which uses the traditional tensor calculus and the standard Dirac covariant spinor field. We show that Campolattaro's intricate calculations can be proved in few lines in our formalism. We briefly discuss the implications of our findings for the interpretation of quantum mechanics.     

Quantum gauge equivalence in QED

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb, and spatial axial gauges to what we call “common form,” in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.     

Massless spinors in more than four dimensions

We consider fermions in theories of higher dimensional gravity where the four-dimensional gauge group is embedded in the invariance group of d dimensional (d>4) Lorentz and general co-ordinate transformations. It is a necessary condition for obtaining massless chiral fermions from dimensional reduction that the d dimensional spinor does not admit a mass term consistent with Lorentz and general co-ordinate transformations. This is the case for a Weyl spinor for d = 6 8 mod 8, a Majorana spinor for d = 9 mod 8 or a Majorana-Weyl spinor for d = 2 mod 8.     

Heavy fermions in gauge theories: (I). Analysis of a general spinor loop

A general one-loop spinor diagram is analyzed in the coordinate space with an arbitrary number of external scalar, pseudoscalar, vector and axial vector legs. We identify chiral anomalies, and an unambiguous definition of a renormalized spinor loop amplitude in gauge theories is given by studying its symmetry properties. We then study the case when some of fermions carry very large masses compared to external momentum scales. Using a new calculational technique based on Schwinger's proper-time method, we provide the explicit forms of dominant effective local vertices induced by virtual heavy fermions in general spontaneously broken gauge theories. In the sequel to the present paper, these results will be applied to various interesting field theory models.