------------------------------------------ S18e. Are there indefinite Hilbert spaces? ------------------------------------------ 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).