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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Antisymmetry of Connection

Feed: Dr. Myron Evans
Posted on: Friday, December 03, 2010 11:14 PM
Author: metric345
Subject: Antisymmetry of Connection

To Dr Douglas Lindstrom:

The precise interpretation is that the commutator of covariant derivatives acting on any tensor produces the torsion and curvature tensors in any space of any dimension. The connection is not a tensor because it does not transform as a tensor under the general coordinate transformation, but is always antisymmetric in its lower two indices (mu and nu). The curvature tensor is always antisymmetric in mu and nu, the commutator is always antisymmetric in mu and nu. This means that of mu and nu are switched to nu and mu, the tensor changes sign. If mu is the same as nu the tensor is zero (all elements of it are zero). In general a non zero tensor may contain non-zero and zero elements. This is true in Riemann geometry itself and there exists a Riemannian torsion which is always non-zero., i.e. it is a non-zero tensor in general. Cartan’s geometry reduces to Riemann’s geometry, and in Cartan’s geometry the torsion is a vector valued two-form. A two-form is antisymmetric in mu and nu. The Cartan curvature is a tensor valued two-form. There can be other geometries too, but ECE uses Cartan’s elegant geometry because it is simple and profound. There seems to be no need for another geometry in natural philosophy. In the 166 ECE papers to date there are many proofs of all details of Cartan geometry, many of them check themselves. At first, tensor analysis and form analysis is difficult, looking like a blizzard of subscripts and superscripts, but I developed a basic notation which reveals its basic simplicity. This is

T = D ^ q; R = D ^ omega;
D ^ T := R ^ q

as given by Cartan, and my own identity (proven precisely in UFT 137):

D ^ T tilde := R tilde ^ q

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Curvatures for atomic H transition

Feed: Dr. Myron Evans
Posted on: Saturday, November 27, 2010 11:54 PM
Author: metric345
Subject: Curvatures for atomic H transition

These are very interesting results again from Dr Horst Eckardt. They show that every transition in atomic H will have its own R pattern which can be used for spectral analysis. The same is true for all atomic and molecular spectra. So this is the first description of atoms and molecules in terms of general relativity as corrected by ECE theory. The basic equation is very simple and is the absorption equation itself:

E2 – E1 = h bar omega

but with considerations of conservation of linear momentum included for the first time. The concepts are simple and clear. Compared with this flood of new results from ECE theory, something like CERN is a complete waste of time and about as exciting as growing grass. It may have produced technological spin offs, but there are cheaper ways of doing that. It is important to criticise all these white elephants because they seriously inhibit real science. The US Government has pointed towards this conclusion.

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165(7) : R Theory of Group Velocity, Superluminal Signalling

Feed: Dr. Myron Evans
Posted on: Friday, November 26, 2010 7:51 AM
Author: metric345
Subject: 165(7) : R Theory of Group Velocity, Superluminal Signalling

This theory accounts for why the group velocity has recently been observed to be zero, negative or greater than c, and the de Broglie equation v sup p v sub g = c squared is developed for use in ECE theory and general relativity. If the numerator in eq. (5) is greater than omega, then v sub g is greater than c. Unlike the standard model this may allow for superluminal signalling, the signal velocity v sub g is greater than c. There are many reports by now of superluminal signalling. Superluminal effects may occur in matter waves.

a165thpapernotes7.pdf

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