Re: applications leading to a normal eigenvalue problem

ARP wrote:
There are no indefinite Hilbert spaces. If you mean an indefinite inner
product space, there is no natural topology, no completeness, nothing
resembling a Hilbert space...


One would call it a generalized linear symmetric eigenvalue problem.
It has nothing to do with normality.


Please check the references in quant-ph/0211048 (Rev.Mex.Fis. 49S2
(2003) >130-133) I thought that if we have a normal operator instead an
hermitian one, it could be taken as a generalization...

Calling the resulting structure a Hilbert space is a misnomer;
if a few people do so, it doesn't mean that the terminology is
justified. A vector space with an indefinite inner product
is commonly called a Krein space.

The ghosts referred to in quant-ph/0211048 are nonphysical vectors
in a Krein space (called there indefinite Hilbert space) which
contains a definite subspace of physical vectors whose completion
gives the physical Hilbert space. This is a natural
construction in gauge theories (Gupta-Bleuler formalism) where
the direct construction of a physical Hilbert space would
manifestly break Lorentz and/or gauge invariance, while the
nonphysical, bigger Krein space enjoys all desired invariance
properties.

The indefinite metric in relativity, also mentioned in that paper,
has nothing to do with indefinite Hilbert spaces, since the
underlying vector spaces (Minkowski space in special relativity,
the tangent spaces at space-time pointsd in general relativity)
are 4-dimensional spaces with the ordinary Euclidean topology
(although the metric is noneuclidean).

Arnold Neumaier

.

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