[GiNaC-list] Question (Generic Clifford Algebra)

[Vladimir V Kisil]

Dear Javier,

At the moment Clifford algebras in GiNaC are suppossed to be 
independent unless
they are defined throught the same metric and representation label (in this
case they are exactly coinsiding). Thus if you take a product of 
e~i*E~j (for a
Clifford units e_j and E_j from different definitions) it could not be
simplified to anything shorter. If one need some relations between elements of
the basis e~i and E~j, I see two possibilities:

1. Some additional procedures which can reduce different basis to the common
ground. For example you can define something like (very schematic, sorry):

matrix transf_matrix(3, 3) = [some elements here];

basis1 = clifford_unit(mu, metric);
vector1 = lst_to_clifford(lst(a1, b1, c1), rho, e);

ex basis2 = indexed(transf_matrix, nu, mu.toggle_variance())*e;
vector2 = indexed(matrix(3,1, lst(a2,b2,c2)), nu.toggle_variance())*basis2;
...

[Sometimes the function exapand_dummy_sum() can be of use here]

2. Implementation of different Clifford units, which are linear combination of
each other, in the core GiNaC procedures. This will need some more thinking
about a careful design and should be probably moved to ginac-dev mailing list.

I will be glad to assit you in the future development.

Best wishes,
Vladimir
Quoting Javier Ros Ganuza <jros at unavarra.es>:

> Following Vladimir's  :)  instructions...
>
> I implemented it, but I've found a problem with the output, probably due
> to my lack of knowledge :(.
>
> For reference the listing is
>
> ---
> //c++ clifford.cc -o clifford -lcln -lginac
>
> #include <iostream>
> using namespace std;
>
> #include <ginac/ginac.h>
> using namespace GiNaC;
>
> int main()
> {
>
> varidx nu(symbol("nu", "\\nu"), 3), mu(symbol("mu", "\\mu"), 3), xi
> (symbol("xi", "\\xi"), 3),  rho(symbol("rho", "\\rho"),3);
>
> ex basis1 = clifford_unit(mu, diag_matrix(lst(1, 1, 1)));
> ex basis2 = clifford_unit(nu, diag_matrix(lst(1, 1, 1)));
>
> symbol a1("a1"),a2("a2"),b1("b1"),b2("b2"),c1("c1"),c2("c2");
>
> ex vector1 = lst_to_clifford(lst(a1,b1,c1), rho, basis1);
> ex vector2 = lst_to_clifford(lst(a2,b2,c2), rho, basis1);
>
> cout << 2*vector1+vector2 << endl;
>
> cout << 2*vector1*vector2 << endl;
>
> cout << lst_to_clifford(lst(1, 0, 0), rho, basis1) + lst_to_clifford(lst
> (0, 1, 0), rho, basis1) << endl;
>
> return 0;
> }
>
> ---
>
> And the output
>
> $ ./clifford
> 2*[[a1],[b1],[c1]].rho*e~rho+e~rho*[[a2],[b2],[c2]].rho
> 2*[[a1],[b1],[c1]].rho*ONE*[[a2],[b2],[c2]].rho*e~rho~rho
> [[0],[1],[0]].rho*e~rho+e~rho*[[1],[0],[0]].rho
>
>
> As it can be see the elements, [[a1],[b1],[c1]].rho, seem indivisible.
> There are not trivial groupings being made, like,
>
> [[0],[1],[0]].rho*e~rho+e~rho*[[1],[0],[0]].rho -> [[1],[1],
> [0]].rho*e~rho
>
> I tried to simplify_indexed but the result keeps the same.
>
> Also at the tutorial, when dealing with generic clifford algebras,
> refered Ginac output for elements of a Clifford Algebra are reflected as
>
> v~0 e.0 + v~1 e.1 + ... + v~n e.n
>
> so exhibiting that they are some how divisible.
>
> Also covariant and contravariant indexes are interchanged in comparison
> with the tutorial.
>
>
> Any clues???.
>
> Thanks for your help
>
> Javier Ros
>
>
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>
>



--
Vladimir V. Kisil      email: kisilv at maths.leeds.ac.uk