Transposition in Clifford Algebra: SU(3) from Reorientation Invariance

Schmeikal, Bernd

doi:10.1007/978-1-4612-2044-2_23

Cited by 5 publications

(4 citation statements)

References 18 publications

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“…These terms would not, however, have the symmetries that would allow them to become free particles. One cannot explain at this point why three of them would be quarks and the fourth one would not be so (Schmeikal has conjectured that it would be a neutrino [33]). The gluon field would just be the name for the primordial field, and protons and neutrons would be just solutions of the Kähler equation that it satisfies.…”

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

confidence: 94%

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

confidence: 94%

“…For completeness purposes, we consider nucleons as per Schmeikal's representation of SU(3) with Clifford algebra [33]. We combine it with the intrinsic entanglement discussed in Sect.…”

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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“…Early attempts at relating SU(3) to Clifford algebra are due to Chisholm and Farwell [15] and [16]. But it was Schmeikal's relating of idempotents to quarks that inspired the present author's connecting of Kähler's solutions of exterior systems with quarks [7]. Schmeikal exploits his virtuosity in algebra.…”

Section: Schmeikal's Quarksmentioning

confidence: 95%

“…Apart from the work of Kähler on idempotent solutions of his "Kähler-Dirac equation" [4] -to which he referred simply as Dirac equation in spite of its paradigm changing character-a major inspiration for our work is a paper on algebraic quarks by Schmeikal [7]. The part of his work that occupies us in subsection 3.1 is of algebraic nature.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Quarks from the Tangent Bundle: Methodology

Vargas

2015

Preprint

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In a previous paper, we developed a table of components of algebraic solutions of a system of equations generated by an inhomogeneous proper-value equation involving Kähler's total angular momentum. This table looks as if it were a representation of real life quarks. We did not consider all options for solutions of the system of equations that gave rise to it. We shall not, therefore, claim that the present distribution of those components as a well ordered table has strict physical relevance. It, however, is of great interest for the purpose of developing methodology, which may then be used for other solutions.We insert into our present table concepts that parallel those of the phenomenology of high energy physics (HEP): generations, color, flavor, isospin, etc. Breaking then loose from that distribution, we consider simpler alternatives for algebraic "quarks" of primary color (The mathematics speaks of each generation having its own primary color). We use them to show how stoichiometric argument allows one to reach what appear to be esthetically appealing idempotent representation of particles for other than electrons and positrons (Kähler already provided these half a century ago with idempotents similar to our hypothetical quarks). We then use neutron decay to obtain formulas for also hypothetical algebraic neutrinos and the Z 0 , and use pair annihilation to obtain formulas for gamma particles.We finally go back to the aforementioned system of equations and start to develop an alternative option. We solve the system of equations for this new option but stop short of studying it along the lines of the present paper. This would thus be an easy entry point to this theory by HEP physicists. Their knowledge of the phenomenology will allow them to go faster and further.

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New Perspectives on the Kähler Calculus and Wave Functions

Vargas¹,

Torr²

2008

AACA

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In 1960In -1962, Kähler produced a generalization of Cartan's calculus of differential forms which he claimed to be a common language for relativity and quantum mechanics. It revolves around the "Kähler-Dirac (KD) equation", which plays a role in this calculus comparable to that of the equations of structure in geometry. Its reach is such that the input differential form needed to solve the relativistic hydrogen atom is just a very simple one among the possible inputs.We show how Cartan's methods allow one to simplify or ignore many of Kähler's cumbersome formulas. We interpret the three series of indices in the components of this calculus and show a promising connection between the structure of its solutions to problems with symmetries and Schmeikal's algebraic representation of low energy quarks.

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Transposition in Clifford Algebra: SU(3) from Reorientation Invariance

Cited by 5 publications

References 18 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

Algebraic Quarks from the Tangent Bundle: Methodology

New Perspectives on the Kähler Calculus and Wave Functions

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