Talk:Indefinite inner product space

Needs intro
This article needs a introduction, a bit of style, links to relevant subjects, and a spellcheck. That just to start. :) Oleg Alexandrov (talk) 01:28, 16 June 2006 (UTC)


 * I have put it in a new version. C. Trunk 12:14, 18 June 2006 (UTC)

bad/wrong definition
The very first sentence is wrong:
 * A Krein space is a Hilbert space which has an additional structure: An inner product

But Hilbert spaces have an inner product, that's part of the definition of a Hilbert space. Soo, what was actually meant ?? Reading further into the article, it seems that the correct definition would be:


 * A Krein space is a topological vector space endowed with an inner product, such that the inner product is a semi-norm on the vector space.

Err, I see that it has to be a complex vector space. So then:


 * A Krein space is a complex vector space endowed with a non-positive-definite Hermitian form.

This is my guess; can someone correct this please? linas 00:18, 23 June 2006 (UTC)


 * Well, you are right. In the language of Krein spaces, people uses inner product for a hermitian sesquilinear form (which is in general indefinite). However, here, as I checked it, an inner product is positive definite by definition. I changed now the intoduction and I hope it will now fit better. Just to explain: A Krein space has two hermitian forms, one is an inner product which turns the space into a Hilbert space, but the other is an indefinte one. C. Trunk 17:04, 26 June 2006 (UTC)

Null directions
I am no expert on operator theory, but I have been reading some papers into which Krein spaces enter, and have concluded that possible remaining null directions in the "Hilbert" inner product needed more delicate handling. I am not 100% convinced I have this straight yet and will do some more homework before I do another editing pass. In the meantime, comments and fixes are of course welcome. Michael K. Edwards 11:49, 3 September 2006 (UTC)


 * In his lectures, Heinz Langer (who is cited in this article and who was a student of Krein himself) defined a Krein space to be a pair $$(K, [.,.])$$ where K is the vector space direct sum of two Hilbert spaces H+ and H-, they have inner products $$(.,.)_+ $$ and $$(.,.)_-$$, respectively, and $$[.,.]$$ is an indefinite inner product (this terminology is O.K., also in Minkowski-space one calls it an indefinite inner product, for instance) given by $$[\hat{x},\hat{y}] := (x_+,y_+)_+ - (x_-,y_-)_-$$, where $$\hat{x} = (x_+,x_-)$$ and $$\hat{y} = (y_+,y_-)$$. A. Slateff, 128.131.37.74 20:04, 3 September 2006 (UTC)


 * Do you think it is appropriate to continue to use the term "Krein space" for the situation I describe, in which there is a third sector that is null in both inner products and must be quotiented out in order to obtain a Hilbert space? I am coming from the context of Horuzhy and Voronin, Commun. Math. Phys. 123, 677-685 (1989).  The physical space of states of the Hamiltonian formulation of a BRST theory resembles a standard Krein space in having indefinite and "Hilbert" inner products related by $$J$$.  (See notes in BRST Quantization, which is still in draft.)  However, if one tries to go over to the quotient space (asymptotic "physical" states containing no quanta of the ghost/anti-ghost/longitudinal gauge fields) too soon, some of the operators in the theory become non-local.  Perhaps there is another term of which I am ignorant for the spaces I am trying to describe.  Michael K. Edwards 23:18, 3 September 2006 (UTC)