From: harke@acm.org
Newsgroups: sci.math.research,sci.math,sci.physics.research
Subject: Re: Geometric Algebra and/or Differential Topology
Date: 9 Apr 1998 21:07:22 -0700
In article <6fseq1$6td$1@agate.berkeley.edu>, FILTER.biggus@colorado.edu
(jeff) writes:
>Anyone know the relationship between geometric algebra and differential
>topology? I've been reading up on the former, and am learning the latter,
>but am unclear about their ultimate connection. Does one subsume the other
>in some way?
>
>Some have suggested just learning differential topology instead of
>geometric algebra, because it can do all that the other can, but this came
>from one not too familiar with geometric algebra. Any suggestions?
I cannot not say much about differential topology but I have been studying
geometric algebra and believe that is a very excellent system for
many areas of physics including gravitation. You do not mention what
works you are reading so I will go ahead and mention some refences.
First and foremost: "Clifford Algebra to Geometric Calculus" by
David Hestenes and Garret Sobczyk, D. Reidel Publishing -- The hardcover
is pricey but there is a soft cover edition. I regard this as an essential
book for the mathemtical foundations.
Also there is: "New Foundations for Classical Mechanics" by
David Hestenes, D. Reidel Publishing This book demonstrates the use
of geometric algebra in formulating classical mechanics.
>Also, does anyone know a good source for geometric algebra in gravitation
>theory? I would be very interested in reformulations of general relativity
>and post-newtonian mechanics.
You will certainly want to visit HTTP:www.mrao.cam.ac.uk/~clifford
They have quite a number of papers that can be downloaded as well as links
to other related sites. For gravitation, be sure to see:
"Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby,
Chris Doran and Stephen Gull. This paper is also available in the
Philosophical Transactions of the Royal Society, Series A, Volume 356,
Number 1737 pp 487-582 (1998 March 15)
This is a very exciting theory of gravity in that, for those predictions
of General Relativity THAT CAN BE TESTED CURRENTLY, it gives just the
same answers as GR. But it does give different results when a horizon
is present. It also provides some advantage in dealing with matter
with spin. This may also lead to testable differences. It may also
provide important insights in the direction of unifying gravitation
with quantum mechanics.
Richard Harke
==============================================================================
From: Ed Gerck
Newsgroups: sci.math.research,sci.math
Subject: Re: Geometric Algebra and/or Differential Topology
Date: Fri, 10 Apr 1998 13:58:25 -0600
In article <6gjpaq$rnu@hopper.ACM.ORG>,
harke@acm.org wrote:
>
> In article <6fseq1$6td$1@agate.berkeley.edu>, FILTER.biggus@colorado.edu
> (jeff) writes:
>
> >Anyone know the relationship between geometric algebra and differential
> >topology? I've been reading up on the former, and am learning the latter,
> >but am unclear about their ultimate connection. Does one subsume the other
> >in some way?
> >
> >Some have suggested just learning differential topology instead of
> >geometric algebra, because it can do all that the other can, but this came
> >from one not too familiar with geometric algebra. Any suggestions?
>
What has been called Grassmann, quaternionic, Dirac, Pauli, vector,
multivector, and geometric algebras are all usually considered particular
cases of a basic structure known as Clifford algebra. Their applications in
physics cover a wide range of topics from classical mechanics to general
relativity, electromagnetism, elementary particle physics and many aspects
of quantum mechanics.
One important feature of Clifford (ie, geometric) algebra is the direct sum
over different vector spaces, allowing multivectors to be defined. Another
feature is the exterior product and the eye-opening so-called "Clifford's
choices" for the elementary results. Both allow true vector division to be
defined and eliminate the problems of pseudo-vectors and pseudo-scalars found
in differential topology, while also making operations self-contained in the
algebra... unlike the vector product of two vectors in DT.
Besides the references given below, there are also some given under the recent
thread on Grassmann geometry, at sci.math. Thanks to Clifford, Grassmann`s
geometric algebra was not only better understood but also expanded.
Unfortunately, the same did not happen until today to the more geometric
aspects of Grassmann's Ausdehnungslehre ... but the algebraic aspects are
finding more and more uses in the physical sciences. More enthusiats in math
research are clearly needed.
Some additional web pointers:
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Clifford.html
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Grassmann.html
http://www.ime.unicamp.br/~walrod/
http://dvworld.nwu.edu/kleber/Grassmann/node1.html (Portuguese)
http://www.physiology.uwo.ca/TweedWeb/6MirrorAlgebra.htm#Introduction
http://www.hit.fi/~lounesto/counterexamples.htm
Cheers,
Ed Gerck
______________________________________________________________________
Dr.rer.nat. E. Gerck egerck@novaware.cps.softex.br
http://novaware.cps.softex.br
--- Meta-Certificate Group member, http://www.mcg.org.br ---
[quote of previous article deleted -- djr]
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From: saw24@hermes.cam.ac.uk (Stephen Wells)
Newsgroups: sci.math.research
Subject: Re: Geometric Algebra and/or Differential Topology
Date: Sun, 19 Apr 1998 19:54:35 +0100
Hi, I'm a student at Cambridge university, and I'm currently doing a
literature review on geometric algebra and relativity, supervised by the
guys who wrote the paper you mention below.
One of the things I've found while searching for references is that the
impact of GA on mainstream research in GR seems to be rather small- which
is a shame. Would you mind if I mention your post in my review, just as
evidence that someone outside the Cavendish lab has heard of the subject?
Yours, Stephen Wells.
In article <6gjpaq$rnu@hopper.ACM.ORG>, harke@acm.org wrote:
>
> You will certainly want to visit HTTP:www.mrao.cam.ac.uk/~clifford
>
> They have quite a number of papers that can be downloaded as well as links
> to other related sites. For gravitation, be sure to see:
> "Gravity, Gauge Theories and Geometric Algebra" by Anthony Lasenby,
> Chris Doran and Stephen Gull. This paper is also available in the
> Philosophical Transactions of the Royal Society, Series A, Volume 356,
> Number 1737 pp 487-582 (1998 March 15)
> This is a very exciting theory of gravity in that, for those predictions
> of General Relativity THAT CAN BE TESTED CURRENTLY, it gives just the
> same answers as GR. But it does give different results when a horizon
> is present. It also provides some advantage in dealing with matter
> with spin. This may also lead to testable differences. It may also
> provide important insights in the direction of unifying gravitation
> with quantum mechanics.
>
> Richard Harke
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