**Subject:** Summary by Dr Gareth J. Evans

**Date:** Wed, 1 Apr 2009 04:18:13 EDT

I think that this is a very good summary by GJE, who was Mansel Davies’s penultimate Ph. D. student at EDCL. The Dirac equation for example is completely non-intuitive and springs from the development of the d’Alembertian operator in four dimensions in terms of 4 x 4 Dirac matrices. Dirac used geometry to derive his equation initially, and talked about this to Prof. John B. Hart.

I agree with you Myron – intuition can take you so far but, as you have shown, physics is geometry and geometry is a set of mathematical representations. A theory devoid of mathematical representations is like music without notes – it may sound nice, it may feel right but there there is no structure (and logic in the strict sense).

Your great contribution Myron, has been in choosing the right geometry, accurately developing and interpreting it, and demonstrating that the physical world we live in can be explained in four dimensions (with time as one of the dimensions). There is no need for higher dimensions at our current level of understanding. Representaions derived within this geometry reduce to all the well known laws of physics (accounting for all the well known experiments of physics).

Best, Gareth

**Date:** Wed, 1 Apr 2009 03:35:29 -0400 **Subject:** Fwd: Just What is 4-d torsion

It is better not to try to use intuition in relativity, the mathematics must be used. Popular books in words are of very little use. The torsion tensor is the difference of two gamma connections, and by hypothesis is proportional to angular energy momentum density. The theory of angular momentum has been highly developed in many ways – see for example the articles and book on the Omnia Opera section of http://www.aias.us. A good book on angular momentum in quantum mechanics is P. W. Atkins, “Molecular Quantum Mechanics” (Oxford University Press, many editions). To see how ot get angular momentum form angular momentum density see L. H. Ryder, “Quantum Field Theory” (Cambridge University Press, second edition, 1996), chapter three. In quantum mechanics the theory of angular momentum has been very highly developed, for example the orbitals of hydrogen. Dirac theory has also been very highly developed. So before starting to read GCUFT one needs courses like this. I taught Atkins at graduate level at UNCC. when Dirac was asked if the general public could understand relativistic quantum mechanics he said “no”. Inded few chemists ever go into the Dirac equation, they use L + 2S. To understand even L + 2S one needs an undergraduate degree in chemistry. So GCUFT is an advanced level monograph. The best way to understadn it is to code it up and produce computer generated animations.

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**Subject:** Just What is 4-d torsion **Date:** Tue, 31 Mar 2009 17:39:37 -0400

Myron,

The problem I have been struggling with lately is “just what is 4-d torsion?” I have I think an intuitive understanding of 2-d and maybe 3-d torsion, but my intuition draws a blank trying to think about 4-d torsion. Have you written an article about that or is there an article you can point me to?

Thanks, Dave Feustel

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