Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry

Abstract

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by $i$ and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a $\mathcal{P}\mathcal{T}$-invariant supersymmetric model with $N$ subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic $N=2$ supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the $\mathcal{P}\mathcal{T}$-invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the "exotic rotational invariant harmonic oscillator" system governed by a dynamical parameter $\gamma$. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of $\gamma$.

arXiv comment38 pages. The relation of the construction with the Swanson model and Dyson map is clarified, and the relationship of our basic PT-symmetric Hamiltonian with the Berry-Keating Hamiltonian and Hilbert-Polya conjecture is indicated

References in corpus (61)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Supersymmetry and Quantum Mechanics
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• PT-Symmetric Quantum Mechanics
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Pseudo-Hermitian Representation of Quantum Mechanics
• Rapidly Rotating Atomic Gases
• Observation of Parity-Time Symmetry in Optically Induced Atomic Lattices
• Making the Case for Conformal Gravity
• SUSY in the sky
• Theory of Non-Hermitian Fermionic Superfluidity with a Complex-Valued Interaction
• Black holes, hidden symmetries, and complete integrability
• Parity-time-symmetric topological superconductor
• Self-isospectrality, special supersymmetry, and their effect on the band structure
• Hamiltonian for the zeros of the Riemann zeta function
• Time evolution of non-Hermitian Hamiltonian systems
• Hidden Symmetries of Dynamics in Classical and Quantum Physics
• Hidden supersymmetry in quantum bosonic systems
• Nonlinear Supersymmetry, Quantum Anomaly and Quasi-Exactly Solvable Systems
• Theory of superconductivity with non-Hermitian and parity-time reversal symmetric cooper pairing symmetry
• Integrable nonlocal Hirota equations
• PT-symmetric quantum field theory in D dimensions
• Landau levels and Riemann zeros
• $PT$-symmetric invisible defects and confluent Darboux-Crum transformations
• Non-Hermitian Hamiltonians of Lie algebraic type
• Spontaneously Breaking Non-Abelian Gauge Symmetry in Non-Hermitian Field Theories
• The origin of the hidden supersymmetry
• Pseudo-Hermitian approach to Goldstone's theorem in non-Abelian non-Hermitian quantum field theories
• Self-isospectrality, mirror symmetry, and exotic nonlinear supersymmetry
• Regularized degenerate multi-solitons
• PT-Symmetric Non-Hermitian Quantum Field Theories with Supersymmetry
• 't Hooft-Polyakov monopoles in non-Hermitian quantum field theory
• Conformal bridge between asymptotic freedom and confinement
• Perfectly invisible $\mathcal{P}\mathcal{T}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry
• Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator
• Degenerate multi-solitons in the sine-Gordon equation
• Complex solitons with real energies
• Towards perturbative renormalization of ${\varphi }^2(i\varphi {)}^{\epsilon }$ quantum field theory
• Pseudo supersymmetric partners for the generalized Swanson model
• Time-delay and reality conditions for complex solitons
• Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry
• Hidden symmetries of rationally deformed superconformal mechanics
• Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions
• The Riemann zeros as spectrum and the Riemann hypothesis
• Klein four-group and Darboux duality in conformal mechanics
• Geometry of Lax pairs: particle motion and Killing-Yano tensors
• Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background
• Hidden superconformal symmetry: Where does it come from?
• Conformal bridge transformation and PT symmetry
• Hidden superconformal symmetry of spinless Aharonov-Bohm system
• Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons
• PT -symmetric rational Calogero model with balanced loss and gain
• Conformal generation of an exotic rotationally invariant harmonic oscillator
• Astrophysical Evidence for the Non-Hermitian but $PT$-symmetric Hamiltonian of Conformal Gravity
• Hidden symmetry and (super)conformal mechanics in a monopole background
• $\mathcal{P}\mathcal{T}$ deformation of angular Calogero models
• Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation
• $\mathcal{P}\mathcal{T}$ deformation of Calogero-Sutherland models
• Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian