**Subject:** Fwd: AW: 130(5) : New Discovery in the Mathematics of the Fermion

**Date:** Sat, 18 Apr 2009 10:20:06 EDT

Many thanks to Dr. Horst Eckardt for these comments, incisive as always. He is a leading ECE scholar. In eq. (18) the unfactorized solution (7) occurs and phi1R appears on both sides, and in eq. (17) the factorized solutions (15) and (16) occur, and this has the effect that phiR appears on the LHS of eq. (17) and phiL on the RHS. Of course, eqs. (17) and (18) are also the results form the two forms of the Dirac equation, eq. (18) being the wave form (Klein Gordon form). This is directly analogous ot the Dirac method, except that 2 x 2 matrices are used throughout. There is no longer any need for 4 x 4 matrices. The key point is encapsulated in eqs. (13) and (14), which are pure 2 x 2 matrix properties. This whole note 130(5) is for the particle. For the antiparticle, the same phase and 2 x 2 matrix algebra is used, the only difference is that sigma 3 becomes – sigma 3. This note 130(5) is just for the rest particle. As Dr Eckardt mentions, a great deal of development is possible in many directions. Up to now it was thought to be impossible to solve the Dirac equation without 4 x 4 matrices. I will dialogue some more on Dr Eckardt’s comments later. This is just a first response to his interesting remarks.

This is an interesting result. Eqs. (17) and (18) show that the Dirac equation leads to a coupling of right and left handed components for equal spin. The wave equation contains the components in decoupled form (as already discussed for the tetrad elements). The question is why both forms are compatible.

From LDA calculations it is known that the right-handed (or upper) components are by orders of magnitude bigger than the left-handed components, one speaks of major and minor components. The minor components are often neglected to reduce computational complexity. If we interpret both kinds of handedness as particle and antiparticle, there must be a significant difference, since experimentally we have one sort of particle only. This is in accordance with the new interpretation. We can add the philosophical statement that each particle contains a certain amount of its antiparticle in itself.

Another point is the interpretation of spin in the Dirac equation. Spin cannot be separated from handedness, i.e. one sort of spin always has both kinds of handedness present. This is a problem in relativistic DFT calculations when the so-called scalar-relativistic equation is used. This is the squared Dirac equation which does no more contain spin components. Spin can be re-introduced non-relativistically, but this is an approximation a priori.

Perhaps your result can give hints how to interpret spin in presence of spin-orbit coupling. The latter is often handled non-relativistically as an additional term. So far we have only considered kinetic energy and rest mass energy. For a comparison with DFT or quantum chemical methods, the external potential needs to be introduced into the Dirac equation. Can this be done ab initio as the derivation of the Dirac equation from geometry?

Horst

—–Urspr?ngliche Nachricht—– Von: EMyrone] at [aol.com [mailto:EMyrone] at [aol.com] Gesendet: Samstag, 18. April 2009 12:29 An: fdamador] at [comcast.net; sean] at [somewhere.ws; HorstEck] at [aol.com; dfeustel] at [mindspring.com; rob] at [rfmicrosystems.co.uk; dlindstrom] at [shaw.ca; kp.phys] at [btinternet.com; thenarmis] at [yahoo.com; raydela] at [sbcglobal.net; scullinan] at [uwic.ac.uk; garethjohnevans] at [hotmail.co.uk; ccefalas] at [eie.gr; avanderm] at [du.edu; spomenka.kobe] at [ijs.si; burleigh.personal] at [gmail.com; DWLindstrom] at [gmail.com; karel.jelinek] at [gmail.com Betreff: 130(5) : New Discovery in the Mathematics of the Fermion

Subject to the opinion of other AIAS / TGA scholars and other objective scholars and academicians unaffected by standard dogma, this looks like a major discovery in the mathematics of the fermion, removing the need for Dirac matrices in mathematics and physics, and giving great insight to the physics of the fermion and antifermion. The discovery is summarized in eqs. (15) and (16) and eqs. (21) and (22). It is that the wavefunction of the fermion can be expressed in terms of 2 x 2 (Pauli) matrices. Up to now this was thought to be impossible (e.g. a text such as Ryder’s), which is why Dirac introduced 4 x 4 matrices. In note 129(1) the Pauli matrices were shown to be tetrad elements as defined in Cartan geometry. So the origin of the fermion and antifermion wavefunction is geometry. It cannot be “indeterministic”. The antifermion is generated from the fermion by reversing the sign of a given Pauli matrix such as sigma sup 3. Thsi dicovery works its way through into all areas of particle field physics and also into any area where the fermion is considered.