Talk:Indefinite inner product space

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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)

[edit] bad/wrong defintion

The very first sentance 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 defition 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 tat 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)

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 , where and .

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)