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S18e. Are there indefinite Hilbert spaces?
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There are no indefinite Hilbert spaces. There are, however,
vector spaces with a distinguished indefinite inner product;
these are called Krein spaces. Their structure is much weaker than
that of Hilbert spaces; there is no natural topology, no completeness,
nothing resembling a Hilbert space except the inner product.
Since there are physical situations where indefinite inner products
arise naturally, some people show their lack of knowledge of the
literature by referring to Krein spaces as indefinite Hilbert spaces.
But if a few people do so, it doesn't mean that the terminology is
justified.
For example, quant-ph/0211048 uses this poor terminology.
The ghosts referred to in this paper are nonphysical vectors in a
Krein 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 points in general relativity)
are 4-dimensional spaces with the ordinary Euclidean topology
(although the metric is non-Euclidean).