Re: tensorial question

From: Bjoern Feuerbacher (feuerbac_at_thphys.uni-heidelberg.de)
Date: 07/23/04 

Date: Fri, 23 Jul 2004 11:45:19 +0200

Edward Green wrote:
> The space of all linear functions on a vector space forms a second
> vector space, known as a dual space. An example of a vector space and
> its dual is given by the space of Kets and Bras in any quantum
> mechanical set-up.
>
> To what extent do the spaces of covariant and contravariant vectors at
> any point on a manifold enjoy this relation?

It's a while ago that I learned differential geometry, but aren't
the covariant vectors *defined* to be the linear functions of the
contravariant vectors? (or was it the other way round?)

> Some possible simularities:
>
> We can form inner products between one from column A, one from column
> B.
>
> Some possible dissimularities:
>
> To form the inner products in the second case, we need the help of the
> metric.

Well, inner products aren't limited to the standard scalar product.
I don't think that the fact that one has to use the metric here is
a big problem.

> (As if we always had to stick and operator inside the Bra and Ket)

Or as if we had defined the scalar product of Bra and Ket in a slightly
different way.

> We can interconvert between types, again with help of the metric.
>
> (Though we can do something similar in QM, since we implicitly know
> how to make a Bra from a Ket?)

Again, good point...

> We can have objects with an indefinite number of each type of "index",
> not just vectors.

In QM, it is also possible to construct tensors, e.g. by using a
tensor product of the state vectors.

> (Could we extend the parallel, say, by learning how to write operators
> in "Bra,Bra" form, which would act on a pair of Kets, for example?)

Such things indeed exist. Think e.g. of the direct product of
two spinors.

> I'm just groping towards grokking, which often excites griping,
> particularly among those with personality disorders, mollusc fetishes
> and francophone names.

Good luck. ;-)

Bye,
Bjoern