[Top][All Lists]

[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

Re: [Axiom-developer] [quantum] Geometric Algebra

From: Arthur Ralfs
Subject: Re: [Axiom-developer] [quantum] Geometric Algebra
Date: Mon, 14 Nov 2016 10:11:41 -0800
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:45.0) Gecko/20100101 Thunderbird/45.4.0


Yours and Tim's private response has put my mind at ease, I guess. My concern was based on a book, Geometric Algebra by Dorst, Fontijne, and Mann, recommended by Martin Baker, a few years ago. When I got to section 13.8 it was mentioned that some of the work was patented and not freely available for commercial use.


On 11/14/2016 09:43 AM, Bertfried Fauser wrote:
Hi Arthur,

  there were rumors about this when Rafal and I developed our Clifford
package (mid 90ies), so we registered it as original work with the US
Library of Congress (Rafal knows how he did that) and that makes it a
publicly available piece of software (and unpatentable). Though it
needs (alas) Maple to run, the software as it stands is open source
freely available and can be used (modified rewritten). Anyhow I will
patent natural numbers and all derived concepts, hey why not sets
(after Bourbaki all math is derivable from that :) )

  Martin Baker has written a new Grassmann and Clifford package for
FriCAS, which goes beyond what was done in AXIOM before. The
implementation of Clifford algebras in AXIOM is at best a proof of
concept. Rafal's and mine Clifford package have algorithms which in
certain situations can be proven to be optimal, and also special
algorithms for diagonal (in suitable bases) bilinear (polar) forms (of
quadratic forms). Without such fast algorithms computations in
Clifford algebras of dimension 5 or higher are not feasible, even with
fast algorithms, the computation of a Clifford algebra multiplication
is not possible for dimension 8/9 upwards. Rafal and I investigated
how to parallelise the Clifford product, as Maple has some coarse
grained parallelism. We got 11 times faster code on a two core machine
(amazing isn't it?) and it showed to us that a much better design of
data structures brought more gain in speed that doing parallel
computations (2 versus 5.5 at best). Robotics people do computations
in 9 to 11 (base space) dimenional Clifford algebras and would also be
interested in fast software for doing that.
  While Hestenes was influential in the Geometric Algebra camp, he was
not the person who did this first. There is a continuous literature
going back to Hamilton, Grassmann and Clifford. Though Geometric
Algebra people do tend not to cite these papers.
  In the mid 90ies people tried to do quantum computing stuff using
Clifford algebras in the Hestenes (operator spinor) style. However
they ran into problems:
  When you tensor matrices not spinors, you pick up additional
dimensions, the proper tensor product is an amalgamated (central
product) one. People then used 'quantum correlators' (aka projection
operators) to fix this. However, in the end they just reconstructed
what people did with (spin-tensor) matrices anyhow. Unless there is
more insight or more abstraction (basis free Clifford algebras are
still a challenge to be done in a CAS) it might be not that fruitful?
Anyhow, a search for literature might be helpful. If I remember rightly
there was a special issue for Hestenes' 60ies birthday in Foundatons
of Physics, and in that you will find some of the work mentioned above
by Doran, Lasenby et al.

Unfortunately I do not have much time at the moment to help out, but if
question arise regarding algorithms related to Grassmann and Clifford
algebras etc I try to help.

Kind regards

reply via email to

[Prev in Thread] Current Thread [Next in Thread]