Fwd: Mirror Image Frame



Subject: Mirror Image Frame
Date: Thu, 2 Apr 2009 04:50:11 EDT

You improve the understanding of the Dirac equation and its implications here Myron. The introduction of ideas such as “Dirac Sea”, “holes in the Dirac Sea”, antimatter etc had once more taken us away from a proper understanding (and physical reality).

Your description of the positron (and anti-particles in general – “particles in a frame of opposite handedness or chirality”) is simple, comes out of the geometry, has a sounder physical basis, and is more easily realised and accepted. “Anti-matter”, like the positron, may be more difficult to detect but it doesn’t imply the existence of some sort of mysterious new matter and world.

Best, Gareth

Date: Wed, 1 Apr 2009 14:08:22 -0400
Subject: Fwd: The Fallacy of Negative Energy in the Dirac Equation

The Dirac sea depends on E = – mc squared and on negative frequency, which is very difficult to accept. There are other and simpler ways of interpreting the Dirac spinor, for example:

1) It is the positive energy spinor for the electron of spin half and the electron of spin minus 1/2. These are distinct particles by the Stern Gerlach experiment. The parity operator changes the +1/2 spin to the -1/2 spin for the electron. Charge conjugation symmetry is obeyed

2) It is made up of the positive energy Pauli spinor of the positron superimposed on the positive energy Pauli spinor of the positron with spin -1/2. Parity symmetry is obeyed. Chareg conjugation symmetry is obeyed

4) The positron can be developed with the mirror image Pauli matrix cyclical relations, with energy always positive.

What introduces spin into the Dirac equation? The answer is the Dirac matrices, made up of the Pauli matrices. In the Einstein energy equation there is no spin at all, and so the Klein Gordon equation is for a scalar particle, also with no spin. The Dirac gamma5 matrix on the other hand describes helicity, related to chirality. So I intend to develop these ideas in paper 129 without any use of negative energy. Antiparticles can be viewed as particles in a frame of opposite handedness or chirality. I think that the end result will be to make the Dirac equation even more powerful, but without indeterminacy or the Dirac sea.

Charge does not appear in the Dirac equation, so the idea of a positron as a hole in the Dirac sea introduces a positively charged antiparticle arbitrarily. I should seek a better explanation of why an antiparticle is oppositely charged to a particle. C, P, T, CP, CT, PT an CPT must be conserved if there is no violation of symmetry as observed in electroweak theory and some parts of elementary particle theory.

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Subject: RE: The Fallacy of Negative Energy in the Dirac Equation
Date: Wed, 1 Apr 2009 15:11:20 +0000

So a description of positrons as “holes in the Dirac Sea” is nonsensical.

Best, Gareth

Date: Wed, 1 Apr 2009 10:33:07 -0400
Subject: Fwd: The Fallacy of Negative Energy in the Dirac Equation

The positron has the same positive mass as the electron, but opposite charge and opposite helicity (often vaguely called “spin”). The right and left handedness of the Pauli spinors in the Dirac spinor give the electron and positron. Their basic origin is geometry. Unless one asserts E = – m c squared there is no Dirac sea of virtual particles. The Pauli spinors in a moving particle (finite momentum) are generated from the Pauli spinors for the rest particle. See notes 129(1) and 129(2) for further details. The particle zoo comes out of an SU(3) analysis basically, instead of the SU(2) analysis of Dirac. Spinors were inferred by Cartan in 1913, and tetrads by Cartan in the early twenties.

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Subject: RE: The Fallacy of Negative Energy in the Dirac Equation
Date: Wed, 1 Apr 2009 14:26:40 +0000

Interesting – so, what does this tell us about the positron Myron?

Best, Gareth

Date: Wed, 1 Apr 2009 10:04:51 -0400
Subject: The Fallacy of Negative Energy in the Dirac Equation

The rest energy in the Dirac equation is taken directly from the classical rest mass equation, the famous:

E0 = m c squared

which quantizes to

E0 psi = gamma0 m c squared psi

where psi is the Dirac spinor and gamma0 is the zero’th Dirac matrix. Note carefully that teh unit 4 x 4 matrix is implied in this notation to multiply the left hand side term. Written in terms of the Pauli spinors phiR and phiL, and using:

E0 = i h bar partial / partial t

the E = m c squared equation becomes

i partial / partial t [ phiR ] = (mc squared / h bar) [phiL ] [ phiL ] [phiR ]

For a particle at rest there is no helicity, so

phiR(0) = phiL (0)

Note that it is nowhere indicated by these equations that m can be negative. Ryder asserts at the foot of his page 44 of the second edition of “Quantum Field Theory” (Cambridge 1996) that gamma0 has four eigenvalues, +1, +1, -1, -1, two indicating positive energy, two indicating negative energy. This obscure statement only makes sense if the negative energy comes fro the starting classical equation:

E = – m c squared = – h bar omega

So I think that the negative energy problem of the Dirac equation is pure mathematics with no physical meaning. The usual de Broglie equation is

E = m c squared = h bar omega

and angular frequency is radians per seconds, it does not make sense in physics to use a negative angular frequency. So there is no Dirac sea, and the vacuum is not filled with virtual particles. This has been the source of endless confusion throughout the twentieth century, leading to a great deal of nonsense physics. The Dirac equation in other respects is a triumph. ECE shows that it is the result of the tetrad postulate of Cartan geometry. The ECE vacuum is filled simply with the voltage density cA(0).

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I will work on this in new notes for paper 129. The mirror image frame in Cartesian coordinates is

i x j = –

and so on instead of i x j = k and so on. So there are going to be mirror image Pauli matrices and Dirac matrices, and antiparticles. The Dirac gamma5 matrix is the handedness or chirality matrix. The lack of complete detail in textbooks on the Dirac equation has caused a lot of confusion down the years, so I wish to give all the details in these notes. The thing that has to be explained is why there is a great preponderance of electrons over positrons in nature. I cannot promise to give the answer to that one, but I think that positrons do not come from negative energy. On the other hand, in molecules, enantiomers occur in equal proportions in the racemic mixture. The other thing is why are there two signs of electric charge, and one sign of mass. I have made some attempts to explain that in earlier papers.


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