<sub>2</sub>-gradings of Clifford algebras and multivector structures

Mosna, Ricardo A.; Miralles, David; Vaz, Jayme

doi:10.1088/0305-4470/36/15/312

Cited by 14 publications

(18 citation statements)

References 13 publications

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“…The algebra D ⊗ Cℓ p,q can be shown not to be a Clifford algebra. Indeed, considering [24]. It follows the importance of defining and investigating algebras of type D ⊗ Cℓ p,q , allowing us completely to classify all subalgebras of a Clifford algebra.…”

Section: The Extended Grassmann Algebramentioning

confidence: 99%

Extended Grassmann and Clifford Algebras

Rocha

Vaz

2006

AACA

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This paper is intended to investigate Grassmann and Clifford algebras over Peano spaces, introducing their respective associated extended algebras, and to explore these concepts also from the counterspace viewpoint. The presented formalism explains how the concept of chirality stems from the bracket, as defined by Rota et all [1]. The exterior (regressive) algebra is shown to share the exterior (progressive) algebra in the direct sum of chiral and achiral subspaces. The duality between scalars and volume elements, respectively under the progressive and the regressive products is shown to have chirality, in the case when the dimension n of the Peano space is even. In other words, the counterspace volume element is shown to be a scalar or a pseudoscalar, depending on the dimension of the vector space to be respectively odd or even. The de Rham cochain associated with the differential operator is constituted by a sequence of exterior algebra homogeneous subspaces subsequently chiral and achiral. Thus we prove that the exterior algebra over the space and the exterior algebra constructed on the counterspace are only pseudoduals each other, if we introduce chirality. The extended Clifford algebra is introduced in the light of the periodicity theorem of Clifford algebras context, wherein the Clifford and extended Clifford algebras Cℓ p,q can be embedded in Cℓ p+1,q+1 , which is shown to be exactly the extended Clifford algebra. We present the essential character of the Rota's bracket, relating it to the formalism exposed by Conradt [25], introducing the regressive product and subsequently the counterspace. Clifford algebras are constructed over the counterspace, and the duality between progressive and regressive products is presented using the dual Hodge star operator. The differential and codifferential operators are also defined for the extended exterior algebras from the regressive product viewpoint, and it is shown they uniquely tumble right out progressive and regressive exterior products of 1-forms.

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Section: The Extended Grassmann Algebramentioning

confidence: 99%

Extended Grassmann and Clifford Algebras

Rocha

Vaz

2006

AACA

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“…In order to present proofs for the propositions above, the bundle of differential forms is embedded in the Clifford bundle 5 -T * M = 4 r=0 T r M ֒→ Cℓ(M, g), where Cℓ(M, g) is the Clifford bundle of differential forms [25] where g is the metric of the cotangent bundle. [20].…”

Section: Equationmentioning

confidence: 99%

“…Taking into account Eq. (41), the following algebraic equation 20 , relating the components A σ to the components of the energy-momentum tensor of matter and the components of the g field that is part of the original LSTS, is obtained:…”

Section: Remark 10mentioning

confidence: 99%

The Maxwell and Navier-Stokes equations that follow from Einstein equation in a spacetime containing a Killing vector field

Rodrigues¹,

Rodrigues²,

Rocha³

2012

AIP Conference Proceedings

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In this paper we are concerned to reveal that any spacetime structure M, g, D, τ g , ↑ , which is a model of a gravitational field in General Relativity generated by an energy-momentum tensor T -and which contains at least one nontrivial Killing vector field A -is such that the 2-form field F = dA (where A = g(A, )) satisfies a Maxwell like equation -with a well determined current that contains a term of the superconducting type-which follows directly from Einstein equation. Moreover, we show that the resulting Maxwell like equations, under an additional condition imposed to the Killing vector field, may be written as a Navier-Stokes like equation as well. As a result, we have a set consisting of Einstein, Maxwell and Navier-Stokes equations, that follows sequentially from the first one under precise mathematical conditions and once some identifications about field variables are evinced, as explained in details throughout the text. We compare and emulate our results with others on the same subject appearing in the literature. In Appendix A we fix our notation and recall some necessary material concerning the theory of differential forms, Lie derivatives and the Clifford bundle formalism used in this paper. Moreover, we comment in Appendix B on some analogies (and main differences) between our results to the ones obtained long ago by Bergmann and Kommar which are reviewed and briefly criticized.

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“…(This fact gives rise to a large class of multivector Dirac equations in flat spacetime, generalizing the Dirac-Hestenes equation [23,24].) In order to capture all possibilities we recall that R 1,3 can be considered as a module over itself by left (or right) multiplication.…”

Section: Left Spin-clifford Bundlementioning

confidence: 99%

The bundles of algebraic and Dirac–Hestenes spinor fields

Mosna

Rodrigues

2004

Journal of Mathematical Physics

204

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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. *

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₂-gradings of Clifford algebras and multivector structures

Cited by 14 publications

References 13 publications

Extended Grassmann and Clifford Algebras

Extended Grassmann and Clifford Algebras

The Maxwell and Navier-Stokes equations that follow from Einstein equation in a spacetime containing a Killing vector field

The bundles of algebraic and Dirac–Hestenes spinor fields

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2-gradings of Clifford algebras and multivector structures

Cited by 14 publications

References 13 publications

Extended Grassmann and Clifford Algebras

Extended Grassmann and Clifford Algebras

The Maxwell and Navier-Stokes equations that follow from Einstein equation in a spacetime containing a Killing vector field

The bundles of algebraic and Dirac–Hestenes spinor fields

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₂-gradings of Clifford algebras and multivector structures