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Pseudo-Hermitian representation of quantum mechanics. (English) Zbl 1208.81095

In the classical quantum mechanics one fixes the physical Hilbert space of the system and develops a theory based on this preassigned Hilbert space. This Hilbert space is space of states and observables are self-adjoint operators on the space of states.
The paper under review describes an alternative approach. Namely, a diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. The author gives a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject.
In pseudo-Hermitian quantum mechanics, the physical Hilbert space is constructed using the following prescription. First one endows the vector space of state-vectors with a fixed auxiliary (positive-definite) inner product. This defines the reference Hilbert space \({\mathcal H}\) in which all the relevant operators act. Next, one chooses a Hamiltonian operator that acts in \({\mathcal H}\), is diagonalizable, has a real spectrum, but needs not be Hermitian. Finally, one determines the (positive-definite) inner products on \({\mathcal H}\) that render the Hamiltonian Hermitian. Any inner product may be defined in terms of a certain linear operator \(\eta_+\). It is this so-called metric operator that determines the kinematics of pseudo-Hermitian quantum systems. The Hamiltonian operator \(H\) that defines the dynamics is linked to the metric operator via the pseudo-Hermiticity relation, \(H^\dagger=\eta_+ H\eta_+^{-1}\). This makes the construction of \(\eta_+\) the central problem in pseudo-Hermitian quantum mechanics. There are various methods of calculating a metric operator and author examines some of the most general methods.
In other sections of the paper author explores connection of pseudo-Hermitian quantum mechanics with \({\mathcal {PT}}\)- and \({\mathcal C}\)-symmetries, the classical limit of pseudo-Hermitian quantum systems, the subtleties involving time-dependent Hamiltonians and the path-integral formulation of the theory.
The final part of the paper under review contains discussion of various known applications and manifestations of pseudo-Hermitian quantum mechanics. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.

MSC:

81Q12 Nonselfadjoint operator theory in quantum theory including creation and destruction operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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