The ECE Fermion Field

Feed: Dr. Myron Evans
Posted on: Saturday, January 29, 2011 12:56 AM
Author: metric345
Subject: The ECE Fermion Field

UFT 173 will develop the fermion field equation as distinct from the single particle fermion equation, in itself a powerful new equation of physics. In a standard text such as “Quantum Field Theory” by Lewis Ryder (Cambridge Univ. Press, 1996, 2nd ed.), pp. 137 ff. the modern way of dealing with antiparticles is described. The fermion equation has the great advantage of removing negative energy. What has to be done now is to recognize that the ECE fermion field is a quantum field and is an operator of quantum mechanics. This is the procedure known as “second quantization”. It has been applied to the Dirac field for many years, but suddenly we know that the Dirac field is obsolete, it uses 4 x 4 matrices whereas the 2 x 2 Pauli matrices of the fermion equation are sufficient and preferred by Ockham’s Razor (the simpler theory is always chosen in physics and the natural sciences, and indeed the whole of philosophy). One questions now whether the Dirac equation had any advantages, because the spin half of the electron can be described by Pauli matrices; the Lande (g = 2) factor by the Schroedinger Pauli equation, and the Thomas factor by other methods. Dirac also made the major error of using the wrong gamma matrices, producing a “negative energy” that has caused endless confusion. Our lecturers at undergraduate certainly had no idea what the Dirac sea was and I suspect that no one ever did, least of all Paul Dirac himself. Having written this I suppose that the advantage of Dirac is its ability to pull things together, but now the ECE fermion equation can do all that in a simpler and much more powerful way. This is why a monograph on the fermion equation is timely. First I give the open source material as usual and then order it for a book at a later stage.

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Mirror Image Pauli Matrices of the Fermion Equation

Feed: Dr. Myron Evans
Posted on: Wednesday, January 26, 2011 5:49 AM
Author: metric345
Subject: Mirror Image Pauli Matrices of the Fermion Equation

These are used in the fermion equation. Their equivalent in Cartesian rep is explained as follows. The ordinary rep is i x j = k, k x i = j, j x k = i. Now apply a mirror in the ik plane to obtain the same rep again but with j replaced by – j in the three equations. These are known as cyclic permutations. The replacement of j by -j has the effect of making a mirror image Cartesian coordinate system, this is a chiral transformation, left hand out of right hand. Both reps are equally valid. The fermion equation is the most fundamental first order equation known to date. Its sigma sup 2 Pauli matrix is the opposite in sign to the sigma sup 2 Pauli matrix of the now obsolete Dirac equation. This is one of the many interesting mathematical properties of the fermion equation. All this also applies to the anti fermion equation, and many other particle equations based on the old Dirac equation, e.g. quark equations, electroweak equations and so on. A fermion type equation can also be derived for the strong field using the Gell-Mann matrices and SU(3) rep space instead of the Pauli matrices and SU(2) rep space. I have already done some work in that direction in UFT 135 and 136.

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Probability Current of the Fermion Equation

Feed: Dr. Myron Evans
Posted on: Monday, January 24, 2011 7:11 AM
Author: metric345
Subject: Probability Current of the Fermion Equation

This is the next stage of the notes for UFT 172 and the short lecture notes type monograph on the Fermion Equation I plan for CISP. The problem with the Klein Gordon wave equation of 1928 was that the probability could be negative. The Dirac equation is traditionally said to produce a rigorously non negative probability with the use of the Dirac spinor. I think that this result will remain true, but I note that Ryder used the chiral rep to obtain this current in his book, not the standard rep as usually used by Dirac. The ECE fermion equation should be capable of development in quantum field theory: second quantization and quantum electrodynamics, but without the use of renormalization. So to summarize the results to date.

1) Negative energy is not meaningful and the ECE fermion equation produces positive energy.
2) The fermion equation produces the Lande factor of the electron (g = 2) more simply than the Dirac equation and without negative energy.
3) Half integral spin is intrinsic to the fermion equation.
4) The anti fermion equation is produced by parity inversion.
5) The wave equation with ECE spinor is an example of an ECE wave equation in the general spacetime, and part of a generally covariant unified field theory.
6) The complete details in the notes show that all calculations are correct

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Derivation of g = 2 without the Dirac Equation

Feed: Dr. Myron Evans
Posted on: Saturday, January 22, 2011 10:51 AM
Author: metric345
Subject: Derivation of g = 2 without the Dirac Equation

This derivation can be derived entirely without the Dirac equation using the Schroedinger Pauli equation as in M. W. Evans and L. B. Crowell, “Classical and Quantum Electrodynamics and the B(3) Field” . So Dirac did not predict the g = 2 factor of the electron nor did he predict antiparticles. So all that remains is the Thomas factor, which can also be predicted in other ways. These things need to be brought out in to the open. The Dirac equation’s strength is that it brings all these together, but it went badly wrong with negative energy.

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172(6): Derivation of the g Factor of the Electron from ECE Fermion Eq.

Feed: Dr. Myron Evans
Posted on: Saturday, January 22, 2011 5:21 AM
Author: metric345
Subject: 172(6): Derivation of the g Factor of the Electron from ECE Fermion Eq.

This note gives the first correct derivation of the g = 2 factor of the electron, using the ECE fermion equation equivalent to the chiral rep of the Dirac equation. In this derivation eienstates of energy are always positive as required by experimental observation. For some reason Ryder switches to the standard Dirac rep for his derivation of the g factor of the electron in “Quantum Field Theory”, having initially used the chiral rep. The derivation of the g = 2 electron factor given in this note is entirely original and simpler than the one using the standard rep (which incorrectly introduces negative energy from an incorrect choice of gamma matrices). I would say that Dirac just made the wrong choice of gamma matrices and incorrectly introduced negative energy eigenstates. He then tried to cure this problem with the Dirac sea, which was abandoned shortly after (nineteen thirties). Dirac did not in fact predict anti particles. The latter are correctly understood only with ECE theory and Cartan geometry. Unfortunately a mythology and iconology has grown up around Dirac, so it takes some time and effort to clear away all the cobwebs. The truth is that Dirac himself was always uncertain about the Dirac sea and no contemporary accepted the concept.

a172ndpapernotes6.pdf

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172(7): The Lande or g = 2 Factor from the Schroedinger Pauli Equation

Feed: Dr. Myron Evans
Posted on: Sunday, January 23, 2011 5:19 AM
Author: metric345
Subject: 172(7): The Lande or g = 2 Factor from the Schroedinger Pauli Equation

This note shows that the g = 2 factor of the electron and also its spin half factor, can be obtained on the non-relativistic level from the Schroedinger Pauli equation. This was first shown in:

M. W. Evans and L. B. Crowell, “Classical and Quantum Electrodynamics and the B(3) Field” (World Scientific, 2001) (see Omnia Opera of www.aias.us).

The value of 2 is in fact an approximation of relativistic quantum mechanics and is not exact. It is also changed by the well known radiative corrections. The next note will deal with the probability current of Dirac.

a172ndpapernotes7.pdf

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Simple Solutions of the ECE Fermion Equation

Feed: Dr. Myron Evans
Posted on: Friday, January 21, 2011 12:07 AM
Author: metric345
Subject: Simple Solutions of the ECE Fermion Equation

This will be the subject of the next note for UFT 172, considering motion in the Z axis in the Minkowski limit when R = (mc / h bar) squared. Again there are no negative energy eigenstates. After that the probability current will be discussed.

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Erratum : Eq. (16) of Note 172(1)

Feed: Dr. Myron Evans
Posted on: Monday, January 17, 2011 6:55 AM
Author: metric345
Subject: Erratum : Eq. (16) of Note 172(1)

It should be :

c (p squared) power half = c modulus p bold = (E squared – m squared c fourth) power half

which is the Einstein energy equation with POSITIVE root (no negative energy). The Einstein energy equation of special relativity is a restatement of the relativistic momentum (p = gamma mv) and is

E squared = p squared c squared + m squared c fourth

or

p sup mu p sub mu = (mc) squared

Dirac was once interviewed and asked whether the general public would understand relativistic quantum mechanics, and he said: “No”. When asked why he did not predict the positron he answered “pure cowardice” (this is another standard model myth, the positron was predicted by Oppenheimer after some remarks by Weyl. Dirac had done all the dogwork however and played it safe when he predicted the proton as the antiparticle of the electron. Understandable in view of the way some people dislike new thought and knee jerk to ad hominem attacks). Even a Lucasian Professor does not like that, and come to think of it, neither does a Civil List Scientist. My colleague Grigolini met Dirac when he was an old man in Tallahassee, Florida. More accurately he saw him studying in the library but was too shy to talk to him.

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Structure of the Dirac Equation in ECE Theory

Feed: Dr. Myron Evans
Posted on: Monday, January 17, 2011 1:06 AM
Author: metric345
Subject: Structure of the Dirac Equation in ECE Theory

The structure of the Dirac equation of the general spacetime in ECE theory is given directly from Cartan geometry in papers such as UFT 4, 129 and 130. The eigenfunction psi is a tetrad with four components in SU(2) representation space. So from the tetrad postulate the Dirac equation in second order format is obtained as a fermion wave equation:

(d’alembertian + R) psi = 0

This can also be interpreted as a Proca equation for the boson, or the Klein Gordon equation for the spinless particle, and also as a Majorana equation. The usual Dirac equation is obtained by rearranging the 2 x 2 psi in to a 1 x 4 psi and factorizing the d’alembertian with Dirac matrices. Finally take the limit of R = (mc / h bar) squared to obtain the original Dirac equation. The ECE fermion equation of UFT 129 and 130 shows that the fermion equation can be written as a first order equation with a 2 x 2 psi (see also notes for UFT 171 and the forthcoming UFT 172). The 2 x 2 psi is obtained from the definition of the tetrad as a matrix linking 2 spinors (column 2 vectors) in two different representations: 1) R and L; 2) 1 and 2. Similarly the tetrad for electromagnetism is defined as the 4 x 4 matrix linking two four vectors, one in (0), (1), (2), (3) rep, the other on 0, 1, 2, 3 rep. Similarly we can define the tetrad in any SU(n) rep space, or any rep space. This gives a generally covariant unified field theory based on geometry and the philosophy of relativity. The tetrad in ECE is more broadly defined than in the original work of Cartan, which used a tangent Minkowski spacetime labelled a at point P to a base manifold labelled mu.

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Erratum: Eq. (28) of Note 171(1)

Feed: Dr. Myron Evans
Posted on: Sunday, January 16, 2011 11:31 PM
Author: metric345
Subject: Erratum: Eq. (28) of Note 171(1)

This has been checked by computer algebra (Maxima) by Dr Horst Eckardt and it should be:

A = (omega sub 2 + omega sub 1) (omega – omega sub 2) /
(( omega squared – omega sub 0 squared) power half cos theta)

This note was not used for paper 171, in which all hand calculations have been checked for correctness by computer by co author Horst Eckardt.

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