# Quasi-Hermitian quantum mechanics in phase space

@article{Curtright2007QuasiHermitianQM,
title={Quasi-Hermitian quantum mechanics in phase space},
author={Thomas L. Curtright and Andrzej Veitia},
journal={Journal of Mathematical Physics},
year={2007},
volume={48},
pages={102112-102112}
}
• Published 2 January 2007
• Physics
• Journal of Mathematical Physics
We investigate quasi-Hermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phase-space metrics, and explore the relationships between them.
12 Citations
Phase space formulation of density operator for non-Hermitian Hamiltonians and its application in quantum theory of decay
• Physics
International Journal of Modern Physics B
• 2018
The Wigner–Weyl transform and phase space formulation of a density matrix approach are applied to a non-Hermitian model which is quadratic in positions and momenta. We show that in the presence of a
Pseudo-Hermitian Representation of Quantum Mechanics
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
PT-symmetric quantum theory
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to
Biorthogonal quantum systems
• Physics
• 2007
Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially
COHERENT STATES AND SCHWINGER MODELS FOR PSEUDO GENERALIZATION OF THE HEISENBERG ALGEBRA
• Mathematics
• 2009
We show that the non-Hermitian Hamiltonians of the simple harmonic oscillator with $\mathcal{PT}$ and $\mathcal{C}$ symmetries involve a pseudo generalization of the Heisenberg algebra via two pairs
PT -Symmetric Interpretation of Double-Scaling
• Physics
• 2012
The conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the
Tridiagonal {\cal PT} -symmetric N-by-N Hamiltonians and a fine-tuning of their observability domains in the strongly non-Hermitian regime
A generic -symmetric Hamiltonian is assumed tridiagonalized and truncated to N < ∞ dimensions, H → H(chain model), and all its up–down symmetrized special cases with J = [N/2] real couplings are
${\mathcal{P}}{\mathcal{T}}$-symmetric interpretation of the electromagnetic self-force
• Physics
• 2014
In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a

## References

SHOWING 1-10 OF 29 REFERENCES
Moyal products -- a new perspective on quasi-hermitian quantum mechanics
• Physics
• 2006
The rationale for introducing non-Hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for
Pseudo-Hermiticity, symmetry, and the metric operator
The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction with the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and
On the regular Hilbert space representation of a Moyal quantization
It is shown that in the regular viz. phase space representation of a Moyal quantization real polynomials in the phase space variables become essentially self‐adjoint operators, and that functions in
Introduction to 𝒫𝒯-symmetric quantum theory
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability
Biorthogonal quantum systems
• Physics
• 2007
Models of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially
Isospectral Hamiltonians from Moyal products
• Physics
• 2006
Recently Scholtz and Geyer proposed a very efficient method to compute metric operators for non-Hermitian Hamiltonians from Moyal products. We develop these ideas further and suggest to use a more