On Fri, 2 Jun 2000, Jerzy Karczmarczuk wrote:
> I hope that this work will progress.
So it does. I started working on linear operators.
New version of the module is available for downloading.
Much still needs to be done, but the closure
formula is already there. Conceptually this seemed
to be a tough piece, but in fact the implementation
is pretty simple and consistent with all the rest.
And this is the real workhorse of the formalism and
corresponds to what one would use in matrix representation
(which we do not wish to use, as this is one of the goals
of the entire exercise).
A |x> = |y> - Abstract equation
A |k><k|x> = |y> - Theoretical closure formula
(sum over basis k).
a >< x -> y - Haskell implementation of closure
Few examples of A show how to re-label representation,
and to "rotate" the basis.
> For the moment I would only
> say, that the orthogonality requirement of Jan is a bit constraining,
> limiting the applicability of the theory to discrete (even finite?)
> spaces,
Discrete yes, finite no.
> while it would be interesting to work with, say |x>, with
> x belonging to R3...
I had to start somewhere. There is enough abstraction for
me at the moment. Hopefully it can be generalized,
but even if not than there are many discrete cases
to work with.
> Moreover, I assure you that some non-orthogonal
> bases are of extreme importance in physics, a canonical example
> being the coherent states in optics.
I am sure they are (*). Give me a break, will you? :-)
Thanks for the comments.
Jan
P.S.
I think Jerzy is talking about cases conceptually sketched below.
/ e2 |e2'
/ |
/ |
/_______ | contravariant basis
e1 \
covariant basis \
\ e1'
e1 * e2' = 0 e1*e1' = 1
e2 * e1' = 0 e2*e2' = 1
