Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics

Abstract

We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group $Sp(2n)$ with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.

Revised and expanded version, includes an application to symplectic transformations and groups, accepted for publication in J. Math. Phys

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• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
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• Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics
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• Higher winding number in a non-unitary photonic quantum walk
• Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
• On Hamiltonians Generating Optimal-Speed Evolutions
• Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials
• Nonlinear Pseudo-Supersymmetry in the Framework of N-fold Supersymmetry
• A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models
• Exactly solvable time-dependent pseudo-Hermitian $su$(1,1) Hamiltonian models
• Hidden-symmetry-enforced nexus points of nodal lines in layer-stacked dielectric photonic crystals
• Some results on the dynamics and transition probabilities for non self-adjoint hamiltonians
• Nonlinear Non-Hermitian Landau-Zener-Stückelberg-Majorana interferometry
• Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries
• Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms
• Pseudo-Hermitian Hamiltonians Generating Waveguide Mode Evolution
• Holonomic Gates in Pseudo-Hermitian Quantum Systems
• Fate of Topological Edge States in Disordered Periodically-driven Nonlinear Systems
• Bosonic pair creation and the Schiff-Snyder-Weinberg effect
• Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation
• Integrable Local and Non-local Vector Non-linear Schrodinger Equation with Balanced loss and Gain
• $\mathcal{P}\mathcal{T}$ Symmetric Hamiltonian Model and Dirac Equation in 1+1 dimensions
• Solvable Limits of a class of generalized Vector Nonlocal Nonlinear Schrödinger equation with balanced loss-gain
• Lower bound of minimal time evolution in quantum mechanics
• Pseudounitary Floquet scattering matrix for wave-front shaping in time-periodic photonic media
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• Infinite-dimensional representations of the rotation group and Dirac's monopole problem
• Sufficient Conditions and Constraints for Reversing General Quantum Errors
• Self-adjoint Elements in the Pseudo-unitary Group $\mathbf{U}\left(p,p\right)$
• Non-semisimple Levin-Wen Models and Hermitian TQFTs from quantum (super)groups
• Quantum implementation of non-unitary operations with biorthogonal representations
• Unitarization of Pseudo-Unitary Quantum Circuits in the S-matrix Framework
• Reciprocity Theorem and Fundamental Transfer Matrix
• Classical Hamiltonian Systems with Balanced loss and gain
• Quantum backreaction effect in optical solitons
• Balancing gain and loss in symmetrised multi-well potentials
• Fixed points and normal automorphisms of the unit ball of bounded operators on ${\mathbb{C}}^n$
• Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter
• On Automorphism Groups of Hardy Algebras