Re: [quantum] Geometric Algebra

Bertfried,

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.

Arthur

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 :) )
>
> @Tim:
>   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
> (fast)
> 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
> table
> 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
> BF.
>
>