[GiNaC-devel] Improved dummy index renaming -> clifford exam fails

[Chris Dams]

Dear Vladimir,

>    CD> Here one
>    CD> clearly sees that neither contractiong g~mu~mu nor g.mu.mu makes
>    CD> any sense. 
>
>	The mentioned expression indexed(squared_metric, alpha, alpha) did
>  not stand for a contractions, it was simply an abbreviation for
>  pow(e.alpha,4)=pow(g.alpha.alpha,2). However I will rewrite 
>  clifford.cpp without this sort of "optimisation". 

Yes, the this interpretation was never in agreement with other code that
has been in GiNaC for ages. The question has to be asked how you can
guarantee that this alpha is not to be summed over? What if somewhere else
in the expression there is a v~alpha? Then we would have an Einstein
summation convention involving three indices, however, GiNaC always had
the convention that dummy indices were to be contracted over in pairs.  
Also not that B~mu~mu + 1 has yielded an exception as long as I can
remember when, for instance, asked what its free indices are.

If you decide to go with the idea of interpreting a matrix that is given
as metric to clifford objects as really being the metric in the sense as
is common in physics, I have an even better optimisation for you. And it
doesn't even depend on the metric being anti-commuting or not. Realizing
that in that case the metric with up-indices is the inverse of the metric
with down-indices, we arive at the identity

e~mu e~alpha e.mu = (2 - dim) * e~alpha,

where dim is the number of row/columns of the metric.

>    CD> This is why I was wondering in an
>    CD> earlier email whether we should even allow matrices to carry
>    CD> indices with a variance. If g is the metric tensor g with up
>    CD> indices should be the inverse of g with down indices.
> 
> Why not use the agreement from the second sentence to make sense for
> the first one.

That certainly is a possibility. However, If I create M = [[2, 0], [0, 2]]
then one of M.1.1 and M~1~1 should evaluate into 2 and the other into 1/2.  
Which of the two? Seems a bit arbitrary and confusing to me. Besides that,
what I just wrote is not even true. It is only true if M is considered to
be the metric. Not all matrices are the metric... And who says what matrix
is the metric and what matrix is not... IMO it would be much less
problematic to simply disallow varidxes as indices of matrices. Or maybe
GiNaC should have the possibility to globally set what the metric is
considered to be. Not that I would be very much in favour of that. It 
makes one think "global variable... blech...".

>    CD> If g is the metric tensor, it automatically obeys g.i~j =
>    CD> delta.i~j.  Actually, this simplification is done automatically
>    CD> in GiNaC. It is in the function ex
>    CD> tensmetric::eval_indexed(const basic & i) const.
>
>	So I only need take care on it if metric is given by a matrix?

Well, if you are really set on giving clifford units with a matrix as 
metric varidxes, I suppose the answer is yes.

Best wishes,
Chris