New Perspectives on the Kähler Calculus and Wave Functions

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

doi:10.1007/s00006-008-0112-9

Cited by 3 publications

(4 citation statements)

References 4 publications

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“…The approach of this author consists in virtually reading the interior derivative from the first equation of structure [21]. It makes bases of 1-cochains (which live on the base manifold) correspond to fields of bases of tangent vectors (which live on the tangent bundle) [22]. We shall not discuss at this point to which of those two options our approach pertains.…”

Section: Further Developments Of Kähler's Calculusmentioning

confidence: 99%

“…5. One could think in principle of a more sophisticated version of the primordial field where protons and neutrons would be Kähler particles whose spinorial components, like in (52), would represent quarks [22]. These terms would not, however, have the symmetries that would allow them to become free particles.…”

Section: From the Bottom Up Extension Of Kähler's Quantum Mechanicsmentioning

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: Further Developments Of Kähler's Calculusmentioning

confidence: 99%

Section: From the Bottom Up Extension Of Kähler's Quantum Mechanicsmentioning

confidence: 99%

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

Vargas¹

2008

Found Phys

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“…(The term differentiation is being used here in the wide sense of Kähler, which is explained in the accompanying paper [20]). It is then to be expected that, for a KD equation to enter and thus enrich the equations of structure of a (Klein-Cartan) geometry, it must involve differential invariants more explicitly.…”

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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“…It seems to me that you are the guy who will go down in history as having shown that SU (3) is space-time physics, a most difficult piece of the required breakthrough for real progress in physics: the realization that U (1)xSU (2)xSU (3) is a space-time group". I felt this, but later Jose's work on the Kähler equation and "A Different Line of Evolution of Geometry on Manifolds Endowed with Pseudo-Riemannian Metrics of Lorentz Signature" was the key to the continuation of a common work [9] (Vargas 2005b . What I like in these papers that these authors are on the way to see the one-sided ideals (in Clifford algebra) almost basis-free, almost basis-free language is more transparent, seems to me.…”

Section: Prologuementioning

confidence: 99%

The Surabale of Space-Time and Its Algebra Signature Split

Schmeikal¹

2006

AACA

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A "surabale" is a category given by a special manifold of geometric algebra frames. It is a bale brought on by a surjective map the equivalence classes of which can constitute base elements of the associative algebra. It is also a stranded braid of idempotents based on a sheaf of base unipotents. The stranded braid of idempotents which are thought to represent particles or fields consists of fibers strictly separated by mutual annihilation throughout the bundle. The surabale can be defined on the Clifford algebra of spacetime. Then it constitutes a bundle of frames which -though covering all dimensions of the geometric algebra -turn out as isomorphic to the ground space generating the algebra. Because of this, the mass shell and the DiracHestenes equation can be defined on the whole surabale. As a result the equation is preserved when acted on by the symmetries of the transformation group of the standard model. The Kähler-equation simply turns into a DiracHestenes equation on the inhomogenous surabale, yet with the same simple differential 1-forms of the linear equation. This shows very beautifully that the equation of motion as well as the invariance of the surabale under the standard model symmetry can be formulated base free. The Clifford bases -instead of Graßmann -just brings in the riches of representation, that is, the emergence of the standard model. But its grading, in a way, is an illusion. Studying the dimension of the space-time-split in quadratic Clifford algebras, it turns out that the dimension of the positive space-like component reproduces their period-8 properties. Considering as an example the Minkowski space-time in the Lorentz metric rather than in (+ − − −) we can see that physicists found the electroweak symmetries in the negative part of the geometry, here denoted as N − , but did not realize the strong force symmetries in the positive part since those depend on the graded motion in the graded subspace P + . It is comparatively difficult to find the generators of the group capable to represent the classic SU(3) with its root space A2. Though the approach put forward gives satisfying answers to some classical problems of relativistic quantum mechanics, it does not solve the most important riddle which has been variously pointed out by Professor Oziewicz, namely, mechanics is not governed by the Lorentz-or Poincaré-group. The simplest argument to be held against it, is that the Lorentz/Poincaré group by definition is the sym-

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New Perspectives on the Kähler Calculus and Wave Functions

Cited by 3 publications

References 4 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

The Surabale of Space-Time and Its Algebra Signature Split

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