Pseudo-Hermitian Representation of Quantum Mechanics

By Ali Mostafazadeh. 2008

716 citations recorded by the source corpus.

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. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as ${\mathcal{P}\mathcal{T}}$, the true meaning and significance of the so-called charge operators $\mathcal{C}$ and the ${\mathcal{C}\mathcal{P}\mathcal{T}}$-inner products,...

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f05f9e7a12af8572fac6def1a430c44c4642c835

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Nonlinear waves in PT -symmetric systems (2016, background): “Developments on this front are nicely covered in a series of reviews (Bender, 2005, 2007; Makris et al., 2011; Mostafazadeh, 2010) and special issues in J. Phys.”
Pseudospectra in non-Hermitian quantum mechanics (2014, background): “the reviews [14, 55], particularly on the physical relevance of PT-symmetry in the quantum-mechanical context.”
Biorthogonal quantum mechanics (2013, methodology): “for all n, where 1̂ by construction is an invertible positive Hermitian operator, which is unique and can be determined from the eigenstates [4]:”