Waldyr A. Rodrigues

The bundles of algebraic and Dirac-Hestenes spinor fields

Our main objective in this paper is to clarify the ontology of Dirac–Hestenes spinor fields (DHSF) and its relationship with even multivector fields, on a Riemann–Cartan spacetime (RCST) M=(M,g,∇,τg,↑) admitting a spin structure, and to give a mathematically rigorous derivation of the so-called Dirac–Hestenes equation (DHE) in the case where M is a Lorentzian spacetime (the general case when M is a RCST will be discussed in another publication). To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (ClSpin1,3el(M)) and right (ClSpin1,3er(M)) spin-Clifford bundles on the spin manifold (M,g). The relation between left ideal algebraic spinor fields (LIASF) and Dirac–Hestenes spinor fields (both fields are sections of ClSpin1,3el(M)) is clarified. We study in detail the theory of covariant derivatives of Clifford fields as well as that of left and right spin-Clifford fields. A consistent Dirac equation for a DHSF Ψ∈sec ClSpin1,3el(M) (denoted DECll) on a Lorentzian spacetim...

On the existence of undistorted progressive waves (UPWs) of arbitrary speeds 0≤ϑ<∞ in nature

We present the theory, the experimental evidence and fundamental physical consequences concerning the existence of families of undistorted progressive waves (UPWs) of arbitrary speeds 0≤ϑ<∞, which are solutions of the homogeneuous wave equation, the Maxwell equations, and Dirac, Weyl, and Klein-Gordon equations.

Nonlocality in quantum physics

Preface. 1. Introduction. 2. Fundamentals. 3. How Collapses. 4. Bells Inequalities. 5. Copenhagen Interpretation. 6. Many Worlds Interpretation (MWI). 7. Quantum Logic (QL). 8. Macroscopic realizations of QL. 9. Hidden Variables Theories (HVT). 10. De Broglie-Bohm Non Relativistic HVT. 11. De Broglie-Bohm Relativistic HVT. 12. Statistical Interpretation. 13. Non-Unitary Evolution? 14. Histories Approach. 15. Quantum Miracles and Cryptography. 16. Where are we now? Appendices: A: Set Theory and Lattices. B: Hilbert Spaces. Index.

Covariant, algebraic, and operator spinors

We deal with three different definitions for spinors: (I) thecovariant definition, where a particular kind ofcovariant spinor (c-spinor) is a set of complex variables defined by its transformations under a particular spin group; (II) theideal definition, where a particular kind of algebraic spinor (e-spinor) is defined as an element of a lateral ideal defined by the idempotente in an appropriated real Clifford algebra ℝp,q (whene is primitive we writea-spinor instead ofe-spinor); (III) the operator definition where a particular kind of operator spinor (o-spinor) is a Clifford number in an appropriate Clifford algebra ℝp,q determining a set of tensors by bilinear mappings. By introducing the concept of “spinorial metric” in the space of minimal ideals ofa-spinors, we prove that forp+q≤5 there exists an equivalence from the group-theoretic point of view among covariant and algebraic spinors. We also study in which senseo-spinors are equivalent toc-spinors. Our approach contain the following important physical cases: Pauli, Dirac, Majorana, dotted, and undotted two-component spinors (Weyl spinors). Moreover, the explicit representation of thesec-spinors asa-spinors permits us to obtain a new approach for the spinor structure of space-time and to represent Dirac and Maxwell equations in the Clifford and spin-Clifford bundles over space-time.

Spin and electron structure

The recent literature shows a renewed interest, with various independent approaches, in the classical theories for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such theories, the ordinary electron is in general associated to the mean motion of a point–like “constituent” Q, whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac–like equation, such that —when averaging over an internal cycle (which corresponds to linearization)— it transforms into the ordinary Dirac equation (valid for the electron as a whole). PACS nos.: 03.70.+k ; 11.10.Qr ; 14.60.Cd . 0 † Work partially supported by INFN, CNR, MURST; by CAPES, CNPq, FAPESP; and by the Slovenian Ministry of Science and Technology. 0∗ On leave from the J.Stefan Institute; University of Ljubljana; 61111–Ljubljana; Slovenia. 0∗∗ Also: Facolta di Ingegneria, Universita statale di Bergamo, 24044 Dalmine (BG), Italy; and C.C.S., State University at Campinas, 13083-970 Campinas, S.P.; Brazil. 0∗∗∗ On leave from Departamento de Matematica Aplicada — Imecc; UNICAMP; 13084–Campinas, S.P.; Brazil.Abstract The recent literature shows a renewed interest, with various independent approaches, in the classical models for spin. Considering the possible interest of those results, at least for the electron case, we purpose in this paper to explore their physical and mathematical meaning, by the natural and powerful language of Clifford algebras (which, incidentally, will allow us to unify those different approaches). In such models, the ordinary electron is in general associated to the mean motion of a point-like “constituent” Q , whose trajectory is a cylindrical helix. We find, in particular, that the object Q obeys a new, non-linear Dirac-like equation, such that — when averaging over an internal cycle (which corresponds to linearization) — it transforms into the ordinary Dirac equation (valid, of course, for the electron as a whole).

Antiparticles from special Relativity with ortho-chronous and antichronous Lorentz transformations

Special Relativity can be based on the whole proper group of both ortho- and antichronous Lorentz transformations, and a clear physical meaning can be given also to antichronous (i.e., nonorthochronous) Lorentz transformations. From the active point of view, the latter requires existence, for any particle, of its antiparticle within a purely relativistic, classical context. From the passive point of view, they give rise to frames “dual” to the ordinary ones, whose properties—here briefly discussed—are linked with the fact that in relativity it is impossible to teach another, far observer (by transmitting only instructions, and no physical objects) our own conventions about the choices right/left, matter/antimatter, and positive/negative time direction. Interesting considerations follow, in particular, by considering—as it is the case—theCPT operation as an actual (even if antichronous) Lorentz transformation.

The Clifford bundle and the nature of the gravitational field

In this paper we formulate Einsteins gravitational theory with the Clifford bundle formalism. The formalism suggests interpreting the gravitational field in the sense of Faraday, i.e., with the field residing in Minkowski spacetime. We succeeded in discovering the condition for this interpretation to hold. For the variables that play the role of the gravitational field in our theory, the Lagrangian density turns out to be of the Yang-Mills type (with an auto-interaction plus gauge-fixing terms). We give a brief comparison of our theory with other field theories of the gravitational field in the flat Minkowski spacetime.

Dirac and Maxwell equations in the Clifford and spin-Clifford bundles

We show how to write the Dirac and the generalized Maxwell equations (including monopoles) in the Clifford and spin-Clifford bundles (of differential forms) over space-time (either of signaturep=1,q=3 orp=3,q=1). In our approach Dirac and Maxwell fields are represented by objects of the same mathematical nature and the Dirac and Maxwell equations can then be directly compared. We show also that all presentations of the Maxwell equations in (matrix) Dirac-like “spinor” form appearing in the literature can be obtained by choosing particular global idempotents in the bundles referred to above. We investigate also the transformation laws under the action of the Lorentz group of Dirac and Maxwell fields (defined as algebraic spinor sections of the Clifford or spin-Clifford bundles), clearing up several misunderstandings and misconceptions found in the literature. Among the many new results, we exhibit a factorization of the Maxwell field into two-component spinor fields (Weyl spinors), which is important.

Causal explanation for observed superluminal behavior of microwave propagation in free space

Abstract In this Letter we present a theoretical analysis of an experiment by Mugnai and collaborators where superluminal behavior was observed in the propagation of microwaves. We suggest that what was observed can be well approximated by the motion of a superluminal X wave. Furthermore, the experimental results are also explained by the so called scissor effect which occurs with the convergence of pairs of signals coming from opposite points of an annular region of the mirror and forming an interference peak on the intersection axis traveling at superluminal speed. We clarify some misunderstandings concerning this kind of electromagnetic wave propagation in vacuum.

Quaternionic Electron Theory: Dirac's Equation

We perform a one-dimensional complexifiedquaternionic version of the Dirac equation based oni-complex geometry. The problem of the missing complexparameters in quaternionic quantum mechanics withi-complex geometry is overcome by a nice“trick” which allows us to avoid the Diracalgebra constraints in formulating our relativisticequation. A brief comparison with other quaternionicformulations is also presented.