Der innere Differentialkalkül

Kahler, Erich

doi:10.1007/bf02992927

Cited by 22 publications

(27 citation statements)

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“…The long history of the Dirac and related equations has been reviewed some years ago by Esposito [6]. In the present context, we just mention that ideas similar than ours were discussed long time ago in the paper titled "Fermions without spinors" [7], whereby the authors used the classical Kähler tensor field equation [8], which was extended by them and then applied to fermions.…”

Section: Introductionmentioning

confidence: 77%

Dirac equation based on the vector representation of the Lorentz group

Marsch

Narita

2020

Eur. Phys. J. Plus

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In this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU(2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.

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Section: Introductionmentioning

confidence: 77%

Dirac equation based on the vector representation of the Lorentz group

Marsch

Narita

2020

Eur. Phys. J. Plus

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“…We compare our results with some others that appear in the literature for the covariant Dirac Spinor field Hehl and Datta, 1971) and also for Dirac-Kähler fields (Kähler, 1962;Graf, 1978;Ivanenko and Obukhov, 1985).…”

Section: Introductionmentioning

confidence: 56%

“…It is clear from our discussion of the Fierz identities that are fundamental for the physical interpretation of Dirac theory that these fields cannot be used in a physical theory. The same holds true for the so-called Dirac-Kähler fields (Kähler, 1962;Graf, 1978;Becher, 1981;Hehl and Datta, 1971) which are sections of Cℓ(M). These fields do not have the appropriate transformation law under a Lorentz rotation of the local tetrad field.…”

Section: A Comment On Amorphous Spinor Fieldsmentioning

confidence: 69%

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Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

et al. 1996

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Abstract. In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named Spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to the famous Geroch's theorem concerning to the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all the three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of a RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.

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“…If the vector space has dimension n, its Clifford algebra has dimension 2 n which is the same dimension as the exterior algebras Λ(V ) and Λ(V * ). Therein it is possible to establish a one-to-one 18]). Exploiting this correspondence we realize a Clifford algebra on Λ(V * ) and Λ(V ) by defining the following product…”

Section: Clifford Algebrasmentioning

confidence: 99%