[GiNaC-devel] Matrices vs indexed [Was: Fixes w.r.t. indexed objects]

[Sheplyakov Alexei]

On Sun, Jul 30, 2006 at 03:26:08PM +0200, Chris Dams wrote:
> > But even GiNaC developers *do* write such things, see:
> > http://thep.physik.uni-mainz.de/pipermail/ginac-list/2005-April/000638.html
> 
> Yes, unfortonately they do :-(. Also in code that went into GiNaC :-(. See
> http://www.ginac.de/pipermail/ginac-devel/2006-July/000985.html . IIRC it
> was also Vladimir who submitted the first patch to start doing automatic
> index renaming. 

> However, if we start doing that we also should be careful
> that our indexed expressions make sense. This means that symbols in
> indices should occur once or twice in a product of indexed objects and if
> they occur twice in a varidx it should be in an up/down pair.
Unfortunatelly, the `matrix' class does not follow the same rule :(.

> For everything else there can be no guarantee whatsoever. It doesn't
> mean anyting.  We shouldn't be showing that kind of stuff on our webpage.

OK, here is better (?) variant of FAQs:

Q1: I am wondering how to convince GiNaC that the following to
  expressions are equal:
  [[-1,0],[0,1]~mu~mu * a~mu
  [[-1,0],[0,1].nu~mu * a~nu

A:
Actually, these expressions are meaningless. Indexed objects are
tensors. Thus, symbols in indices should occur once or twice in 
a product of indexed objects and if they occur twice in a varidx it
should be in an up/down pair.

Q2: I'd like to know if is it possible to unroll indexed
objects. As example, I would like to do something like this

 a_i a~i = (a_1)^2 + (a_2)^2 + (a_3)^2 + ...

A:
The indexed class (and most derived classes) is intended for tensor
manipulation without referring to a particular basis. Thus, the
indexed class is well suited for calculations involving (formally
defined) tensor algebra of non-integer-dimensional space. This
is particularly useful for evaluation of Feynman integrals in
the framework of dimensional regularization. In that framework,
unrolling a_i a^i to a_1^2 + a_2^2 + a_3^2 +... is not possible, 
since the dimension is not integer!

Q3:
I wish to mix matrices and indexed objects. How can that be done?

A:
The indexed class (and most derived classes) is intended for tensor
manipulation without referring to a particular basis. On the other hand,
the `matrix' class is _not_ treated as a tensor, so mixing matrices
with indexed objects typically gives meaningless result. See also Q2.

Best regards,
 Alexei.


P.S.:
There is no need to CC me, I read the list.

-- 
All science is either physics or stamp collecting.

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