Clifford-Valued Clifforms: A Geometric Language for Dirac Equations

Vargas, Jose G.; Torr, D. G.

doi:10.1007/978-1-4612-1368-0_9

Cited by 10 publications

(8 citation statements)

References 9 publications

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“…In spite of the correspondence between (8) and (11), the right hand side of (8) may give the impression that something is wrong with it, given the absence of the commutator. This is not a real absence; it simply happens that the −Y X term is zero in the y coordinate system, where the components of X are (0, 0, .…”

Section: The Cartan-kähler View Of Basic Concepts Of Geometry and Thementioning

confidence: 90%

“…Commutation relations still emerge in any case from (9), and thus from (8) in the last instance. Indeed, formula (9) becomes…”

Section: The Cartan-kähler View Of Basic Concepts Of Geometry and Thementioning

confidence: 92%

“…Kähler-Atiyah algebra) of inhomogeneous cochains, the homogeneous r-cochains being functions of r-surfaces. As we shall explain later in the paper, we shall use the term differential forms for these cochains, in accordance with Kähler, since they, rather than the antisymmetric multilinear functions of vectors, are of the essence of his calculus [8].…”

Section: Introductionmentioning

confidence: 99%

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The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

Vargas¹

2008

Found Phys

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In 1960In -1962. Kähler enriched É. Cartan's exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His "Kähler-Dirac" (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons.The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler's tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron's and positron's large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in Kähler's work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent.A new but as yet modest new interpretation of QM starts to emerge from that calculus' peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of "wave functions" and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.

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Section: The Cartan-kähler View Of Basic Concepts Of Geometry and Thementioning

confidence: 90%

“…Commutation relations still emerge in any case from (9), and thus from (8) in the last instance. Indeed, formula (9) becomes…”

Section: The Cartan-kähler View Of Basic Concepts Of Geometry and Thementioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

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The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

Vargas¹

2008

Found Phys

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“…and how to interpret it [19]. To complete the argument, we use ds 2 to name the right hand side of Eq.…”

Section: A Kaluza-klein Space For a "Kählerian Approach" To The Equatmentioning

confidence: 99%

“…We proceed to review different options for the KD equation when the valuedness of the input differential form is not scalar. We proposed elsewhere [19] a KD equation of the type ∂Ψ = a(∨, ∨)Ψ.…”

Section: A Kähler-dirac Equation For a Kählerian Approach To Klein-camentioning

confidence: 99%

The Kähler-Dirac Equation with Non-Scalar-Valued Differential Form

Vargas¹,

Torr²

2008

AACA

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The original, matrix-free Kähler-Dirac (KD) equation of 1960 has an unpleasant feature when the input differential form is not scalar-valued. Possibly to deal with that feature, Kähler proposed in 1962 a highly cumbersome alternative equation involving matrices. It coincides with the one from 1960 when the input differential form is scalar-valued.In the 5th Clifford Conference, we proposed our own alternative, where the tensor (1960) and matrix (1962) products of the valuedness factors were replaced with the Clifford product. We also suggested a sort of Kaluza-Klein (KK) space canonically determined by a subset of the fundamental invariants that define the spacetime manifold. In this paper we show that our KK space seems to accommodate the physics well for appropriate choice of the torsion. We further show that one can choose an input differential form highly appropriate for when the aforementioned torsion is assumed to be the output differential form of a geometrized KD equation. This leads to equations of structure for the torsion that have a more quantum-mechanical and less moving frame theory flavor than the standard ones.

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From the Cosmological Term to the Planck Constant

Vargas

Torr

2002

Gravitation and Cosmology: From the Hubble Radius to the Planck Scale

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Clifford-Valued Clifforms: A Geometric Language for Dirac Equations

Cited by 10 publications

References 9 publications

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

The Kähler-Dirac Equation with Non-Scalar-Valued Differential Form

From the Cosmological Term to the Planck Constant

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