Thursday, April 10, 2014

David Hestenes of Geometric Algebra Fame to speak at Texas A&M Today

Dr. David Hestenes, (pictured to the left [2]), the original author of geometric algebra, (it was his PhD dissertation work at UCLA), will be speaking at A&M today[1].

We're learning how to do literature review matrices in our writing class, so I thought I'd try out the technique while reading Dr. Hestene's bio[2] last night.  Here are the key points I came away with






1.  Hestenes was inspired by Marcel Riesz's book "Clifford Numbers and Spinors"[3]
One day in the mathematics– engineering library I looked at a shelf of incoming new books and pulled down and some lecture notes entitled “Clifford Numbers and Spinors” by Marcel Riesz. It was about Clifford algebra as a mathematical system. I read, I think, for about 15 minutes and all of a sudden I had an epiphany. I exclaimed “Gee, differential forms and the Dirac algebra have a common algebraic structure!”…


2.  Dr. Hestenes received the Oersted medal for his work on the force concepts inventory, but delivered the accompanying lecture[4] on geometric algebra:
Yeah, except for my Oersted lecture, okay? So, I gave my Oersted lecture on elementary applications of geometric algebra instead of my educational R&D, for which the award was intended. But I related that to science education by emphasizing that what you understand about science depends critically on your facility with conceptual tools, representational tools, and mathematical tools. For example, you had to do all of your calculations with roman numerals, you wouldn’t do very well, okay?

3.  He has written a book[5] that treats advanced classical mechanics in terms of geometric algebra
I have published the first advanced book on classical mechanics worked out exclusively with geometric algebra. All equations are formulated and calculations are done without resorting to coordinates or matrices, including rotational dynamics, precessing tops, and all that. The introductory chapter is a kind of annotated history of geometric algebra


4.  The Dirac algebra is associative.  Dr. Hestenes reinterpreted the Dirac gammas in terms of vectors.
To explain since you know Dirac algebra. you know that the whole algebra is generated by the Dirac matrices, so you can understand the significance when I reinterpreted the Dirac gammas them as vectors. These vectors then generate an associative algebra, mathematically speaking, a Clifford algebra. But I developed this algebra as an encoding of geometric properties for space-time in algebraic form. I call that system space-time algebra (STA). From that viewpoint, the Pauli algebra sheds its representation by 2×2 matrices to emerge as a subalgebra of the STA. That was my second significant discovery about the Pauli algebra.

5.  Of interest to grad students with kids, (like me!).  Dr. Hestenes had four children before he was out of graduate school!
Another reason that I went to ASU and stayed there is because I was married when I was in college. I had my first child while I was in army and my second child was born on my first day in graduate school. By the time I finished my PhD I had four children. Then I went to Princeton. I have never heard of another postdoc with four children.


6.  Hestenes on spin and spinors as vectors:
The first discovery is one of the highlights of my life. And it gave me strong motivation and direction for my research. That discovery was recognition that the Pauli matrices could be reinterpreted as vectors, and their products had a geometric interpretation. I was so excited that I went and gave a little lecture about it to my father. Among other things, I said, “Look at this identity σ1 σ2 σ3 = i, which appears in all the quantum mechanics books that discuss spin. All the great quantum physicists, Pauli, Schroedinger, Heisenberg and even Dirac as well as mathematicians Weyl and von Neumann, failed to recognize its geometric meaning and the fact that it has nothings to do with spin. When you see the Pauli sigmas as vectors, then you can see the identity as expressing the simple geometric fact that three orthogonal vectors determine a unit volume. Thus there is geometric reason for the Pauli algebra, and it has nothing whatsoever to do with a spin.

7.  Dr. Hestenes thinks there are opportunities for further research in superconductivity
T: Because it just occurs to me that we need a somewhat new theory of electron probably to solve the questions of superconductivity. Do you feel a need of that sort?
H: I feel more than a need. I think I know at least one way that the theory should be changed. The standard theory of superconductivity is not as successful
as people make it out to be. When you get near the critical point they have a renormalization theory to explain what happens. But renormalization theory doesn’t get the correct result for the critical point. And not just at the critical point! The deviations of theory from the experimental data increase as you get closer
and closer to the critical point. So, what is going on there? Here is my hypothesis: the electrons have this internal Zitter motion, and as you approach the critical point there is an increase in Zitter correlations, that is, in resonances between Zitter motions of different electrons. As temperatures increase correlations are destroyed by thermal fluctuations. I submit this as a general explanation for all critical phenomena in condensed matter systems.
With a nod to the Smitten Kitchen for the new post style

Picture of the Day:
A vacant factory in Savannah, GA


References:
1.  Seminar announcement
http://calendar.tamu.edu/aero/?eventdatetime_id=20190

2.  Taşar, M., Bilici, S., Fettahlıoğlu, P., "An Interview with David Hestenes: His life and achievements", Eurasia Journal of Mathematics, Science & Technology Education, 8(2), (2012), 139-153
http://www.ejmste.com/v8n2/EURASIA_v8n2_Tasar.pdf

3. Marcel Riesz's book "Clifford Numbers and Spinors", available open access:
https://archive.org/details/cliffordnumberss00ries

4.  Dr. Hestenes Oersted Lecture
http://geocalc.clas.asu.edu/html/Oersted-ReformingTheLanguage.html

5.  Hestenes book on classical mechanics
http://books.google.com/books?id=N9ZEXX-ypSQC&printsec=frontcover&dq=david+hestenes+mechanics&hl=en&sa=X&ei=2alGU_bfKNLgsASfo4D4Cw&ved=0CD8Q6AEwAA#v=onepage&q=david%20hestenes%20mechanics&f=false

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