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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Paul Dirac’s Original Thinking

Feed: Dr. Myron Evans
Posted on: Sunday, January 16, 2011 6:39 AM
Author: metric345
Subject: Paul Dirac’s Original Thinking

This thinking was described by Paul Dirac to John B. Hart as in Kerry’s biography, with a photogaph of Dirac during a conference organized by Hart. It was based on Clifford algebra. For a change, there is a good wikipedia article on the Dirac equation as originally formulated. In my opinion negative energy and the Dirac sea is a non starter, and as the wiki article shows, was actually rejected by Dirac and contemporaries in the thirties. The chiral rep is the correct one to use, it eliminates negative energy, so I have been using it for some years. As can be seen on the foot of page 2 of the wiki article it was thought originally that one could use E = c root (p squared + (mc) squared) of the Einstein equation and use the usual quantum mechanical operator equivalents for E and p, expand in a series and iterate to a solution. Clearly, the negative root of E was never considered as meaningful, and it obviously is not, because negative energy or electron cascade have never been observed. Essentially, what Dirac did was to factorize the d’alembertian in terms of the 4 x 4 Dirac matrices in standard rep. That meant automatically that the wavefunction must be a column vector with four entries, all four of which pertain to the fermion. The Dirac matrices in chiral rep are different, but nonetheless give all that the standard rep gives, but without the artificial problem of negative energy. The Dirac matrices in standard rep are given in the wiki article on page 4. In my books and articles they are given in the modern chiral rep. I also derived the Dirac equation from geometry, but a more powerful one – Cartan geometry. I realized that thr Dirac spinor can be rearranged into a 2 x 2 matrix from a 1 x 4 column vector, and vice versa. The 2 x 2 matrix is a tetrad in SU(2) rep space. As described in page 8 of the wiki article the Dirac sea has been replaced by the Bogoliubov transformation and from the forties onwards by the method of QED (albeit to me, AND Feynman himself, very unsatisfactory). About a year or more ago I discovered (UFT 129 and 130) that the Dirac equation can be written as one chiral rep equation in 2 x 2 matrices in which negative energy is not a problem. I intend to develop this in UFT 172 giving complete details. Some of these have already been posted as notes for UFT 171. Chemists are taught the Dirac equation very vaguely as L + 2S and that’s that.

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Some Comments on Electric Power from Spacetime

Feed: Dr. Myron Evans
Posted on: Saturday, January 01, 2011 4:10 AM
Author: metric345
Subject: Some Comments on Electric Power from Spacetime

The Poynting theorem for this process is eq. (5) of note 170(2). The spacetime power is always present in any circuit, that is the key new inference. This is simply because the vacuum electric field strength is always present – the Lamb shift is always present, and interacts with the sources within the circuit, notably an electric current within the circuit. So the task has shifted to discerning the EXTRA effect of the vacuum electric field strength in volts per metre and amplifying that EXTRA effect. One very simple example is to use a very thin wire of low conductivity or high resistance. The latter maximizes the heat produced. The spacetime power in the wire is :

P = I (spacetime) squared / (area of wire multiplied by conductivity of wire)

Here I (spacetime) is fixed by the value of E (spacetime), commonly denoted E (vacuum), the quantity responsible for the Lamb shift. It should be easy now for electrical engineers to build on this key inference, the inference of the Poynting theorem due to the vacuum electric field strength. I am always prepared to do consultancy work for any corporation interested in this problem.

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The Role of the Gravitomagnetic Field Strength (h) in Counter Gravitation

Feed: Dr. Myron Evans
Posted on: Thursday, December 30, 2010 5:03 AM
Author: metric345
Subject: The Role of the Gravitomagnetic Field Strength (h) in Counter Gravitation

The curl h term might be important in counter gravitation, the general equation for the mass current density being:

curl h – partial d / partial t = J sub M

so in general:

g dot (curl h – partial d / partial t) = g dot J sub M

and

E dot J = g dot J sub M

These are completely new ideas so must be tested carefully by the engineers at each stage, e..g Northrop Grumman, Lockheed Martin, BAE, European Space Agency, various NASA laboratories, etc. All have been following www.aias.us and ECE sites for years. Even a very small counter gravitational effect in a spacecraft would have a cumulative effect. I advocate working into the circuit design a conventional resonance device, so the change of g may be amplified by resonance.

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Geometrical Origin of D, P, H and M

Feed: Dr. Myron Evans
Posted on: Saturday, December 04, 2010 6:15 AM
Author: metric345
Subject: Geometrical Origin of D, P, H and M

This will be developed and discussed in the next note, which I will label 167(1). This note follows on from 165(9) and 165(10). The origin is found in the fact that the correctly antisymmetric connection has a Hodge dual, which is defined using the metric. So D, P, H, and M can be expressed in terms of a potential and spin connection. Finally conditions for resonance can be looked for while maintaining antisymmetry. If the error is perpetuated of arbitrary asserting a symmetric connection, then it can have no Hodge dual. The latter is defined only for an antisymmetric tensor. We have, in S.I. units:

E = (D – P) / eps0 ; B = mu0 (H + M)

so we can take the combinations D – P and H + M to be defined by geometry. Here D is displacement, P is polarization, H is magnetic field strength, M is magnetization, E is electric field strength and B is magnetic flux density. eps0 amd mu0 are the vacuum permittivity and permeability. There is plenty of scope here for spin connection resonance while maintaining antisymmetry.

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New Resonance Solutions

Feed: Dr. Myron Evans
Posted on: Friday, December 03, 2010 11:54 PM
Author: metric345
Subject: New Resonance Solutions

I will look in to new resonance structures in UFT 167, using the inhomogeneous field equations rather than the homogeneous ones. These are, for each a:

del D = rho; curl H – partial D / partial t = J

The displacement D and magnetic field strength H are related to E and B by

D = eps0 E + P ; B = mu0 (H + M)

and the E and B fields are related to the potentials A and rho with the spin connection included. There are therefore many possibilities for resonance. Choose an antisymmetry condition that allows resonance. Here J is electric current density, P is polarization, M is magnetization, eps0 and mu0 are the vacuum permittivity and permeability respectively. The inhomogeneous equations are obtained from the attached geometry, page two, second column. In the inhomogeneous approach there is no problem with the driving force, it is derived form the charge current density. Magnetic charge current density is perennially controversial. Finally the interaction of gravitation and electromagnetism is best approached through the quadratic term in the kinetic energy due to the minimal prescription applied to p. This was done in recent papers.

a165thpapernotes9.pdf

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