Multivector functions of a multivector variable

Journal of Mathematical Physics

2004

122

Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF ) and Dirac-Hestenes spinor fields (DHSF ) on Minkowski spacetime as some equivalence classes of pairs (Ξu, ψ Ξu ), where Ξu is a spinorial frame field and ψ Ξu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the 'bilinear covariants' (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary RiemannCartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of

Section: A6 Some Useful Properties Of the Real Clifford Algebrasmentioning

confidence: 99%

Algebraic and Dirac–Hestenes spinors and spinor fields

Journal of Mathematical Physics

2004

122

“…To end we would like to observe that eventually some readers will not fell happy for seeing us to introduce an universal medium filling all spacetime, because it eventually looks like the revival of ether of the XIX th century. To those people we ask to study the arguments for the existence of such a medium in, e.g., [52,82,83,84,97] and also analyze references [95,96] which show that if we interpret particles as defects in the medium 59 , then some characteristic relativistic and quantum field theory properties of those objects emerge in a natural way and also there is hope to also describe the electromagnetic field as some kind of distortion of the medium. We will come back to those issues in a forthcoming paper.…”

Section: Discussionmentioning

confidence: 99%

“…The MECM has its origin, by the best of our knowledge in some mathematical ideas first proposed by Hestenes and Sobczyc in their book "Clifford Algebra to Geometrical Calculus" [43]. Those ideas have been investigated in [25,56,86] and attained maturity in [23,26,27,57,58,59,60,28,29,30,31]. In particular in [60] the theory of derivatives of functionals of extensor fields necessary for the developments of the present paper has been introduced in a rigorous way.…”

Section: Killing Vector Fields Symmetries and Conservation Lawsmentioning

confidence: 97%

See 1 more Smart Citation

Gravitation as Plastic Distortion of the Lorentz Vacuum

Fernández

Gravitation as a Plastic Distortion of the Lorentz Vacuum

2010

In this paper we present a theory of the gravitational field where this field, represented by a (1, 1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a real substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor η generating what may be interpreted as an effective Lorentzian metric extensor g = h † ηh and also it permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may have non null curvature and/or torsion and/or non metricity tensors. We thus have different possible effective geometries which may be associated to the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we developed with enough details the theory of multiform functions and multiform functionals that permitted us to successfully write a Lagrangian for h and to obtain its equations of motion, that results equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R 4 ). However, in our theory, differently from the case of General Relativity, trustful energy-momentum and angular momentum conservation laws exist. We express also the results of our theory in terms of the gravitational potentials g µ = h † (ϑ µ ) where {ϑ µ } is an orthonormal basis of Minkowski spacetime in order to have results which may be easily expressed with the * Some (odd) misprints and typos have been corrected, some sentences have been changed for better intelligibility and Appendix F has new important remarks which result from discussions that W.A .R. had with A. Lasenby at ICCA10 (Tartu) in August 2014.

“…In that way the correspondence between Eqs. (27), (30) and (31) 12 We denote the space of sections of p-vectors by sec p (M ).…”

Section: Dirac Equation For Covariant Dirac Fieldsmentioning

confidence: 99%

The bundles of algebraic and Dirac–Hestenes spinor fields

Mosna

Journal of Mathematical Physics

2004

204

The main objective of 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 (Cℓ l We also obtain a representation of the DE Cℓ l in the Clifford bundle Cℓ(M, g). It is such equation that we call the DHE and it is satisfied by Clifford fields ψΞ ∈ sec Cℓ(M, g). This means that to each DHSF Ψ ∈ sec Cℓ l Spin e 1,3 (M ) and to each spin frame Ξ ∈ sec P Spin e 1,3 (M ), there is a well-defined sum of even multivector fields ψΞ ∈ sec Cℓ(M, g) (EMFS ) associated with Ψ. Such an EMFS is called a representative of the DHSF on the given spin frame. And, of course, such a EMFS (the representative of the DHSF ) is not a spinor field. With this crucial distinction between a DHSF and its representatives on the Clifford bundle, we provide a consistent theory for the covariant derivatives of Clifford and spinor fields of all kinds. We emphasize that the DE Cℓ l and the DHE, although related, are equations of different mathematical natures. We study also the local Lorentz invariance and the electromagnetic gauge invariance and show that only for the DHE such transformations are of the same mathematical nature, thus suggesting a possible link between them. *