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The problem of coexistence of several non-Hermitian observables in PT-symmetric quantum mechanics
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Abstract
During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical consistency of the resulting models of stable quantum systems requires a reconstruction of an alternative, amended, physical inner product of states. We point out the less known fact that for more than one observable the task is not always feasible. The difficulty is re-analyzed and its elementary linear-algebraic interpretation and treatment are outlined.
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These reports contain an account of 2015’s meeting on noncommutative geometry. Noncommutative geometry has developed itself over the years to a completely new branch of mathematics shedding light on many other areas as number theory, differential geometry and operator algebras. A connection that was highlighted in particular in this meeting was the connection with the theory of II_1 -factors and geometric group theory.
A quasi-Hermitian operator is an operator that is similar to its adjoint in
some sense, via a metric operator, i.e., a strictly positive self-adjoint
operator. Whereas those metric operators are in general assumed to be bounded,
we analyze the structure generated by unbounded metric operators in a Hilbert
space. It turns out that such operators generate a canonical lattice of Hilbert
spaces, that is, the simplest case of a partial inner product space
(PIP-space). We introduce several generalizations of the notion of similarity
between operators, in particular, the notion of quasi-similarity, and we
explore to what extend they preserve spectral properties. Then we apply some of
the previous results to operators on a particular PIP-space, namely, a scale of
Hilbert spaces generated by a metric operator. Finally, motivated by the recent
developments of pseudo-Hermitian quantum mechanics, we reformulate the notion
of pseudo-Hermitian operators in the preceding formalism.
We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR‐theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the ‘‘Connes–Lott’’ description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincaré duality and in the unimodularity condition.
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