Nonunique C operator in PT Quantum Mechanics

Abstract

The three simultaneous algebraic equations, $C^2=1$, $[C,PT]=0$, $[C,H]=0$, which determine the $C$ operator for a non-Hermitian $PT$-symmetric Hamiltonian $H$, are shown to have a nonunique solution. Specifically, the $C$ operator for the Hamiltonian $H=1/2p^2+1/2{\mu }^2{q}^2+i\epsilon q^3$ is determined perturbatively to first order in $\epsilon$ and it is demonstrated that the $C$ operator contains an infinite number of arbitrary parameters. For each different $C$ operator, the corresponding equivalent isospectral Dirac-Hermitian Hamiltonian $h$ is calculated.

10 pages, 1 figure

References in corpus (22)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Must a Hamiltonian be Hermitian?
• Supersymmetry and the spontaneous breakdown of PT symmetry
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Scalar Quantum Field Theory with Cubic Interaction
• Dual PT-Symmetric Quantum Field Theories
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Ghost Busting: PT-Symmetric Interpretation of the Lee Model
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• The C Operator in PT-Symmetric Quantum Theories
• PT-Symmetric Quantum Electrodynamics
• Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics
• Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well
• The C Operator in PT-Symmetric Quantum Field Theory Transforms as a Lorentz Scalar
• Anomalies in PT-Symmetric Quantum Field Theory
• Dual oscillators and Quantum Pendulums: spectrum and correlators

Cited by in corpus (14)

• A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field
• Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories
• Fundamental length in quantum theories with PT-symmetric Hamiltonians
• Complete Set of Inner Products for a Discrete PT-symmetric Square-well Hamiltonian
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• Unbounded C-symmetries and their nonuniqueness
• Small-$\epsilon$ behavior of the Non-Hermitian PT-Symmetric Hamiltonian $H=p^2+x^2(ix{)}^{\epsilon }$
• On $J$-self-adjoint extensions of the Phillips symmetric operator
• Nonuniqueness of the C operator in PT-symmetric quantum mechanics
• Matrix representation of the time operator
• Three perspectives on entropy dynamics in a non-Hermitian two-state system
• Non-Hermitian CP-Symmetric Dirac Hamiltonians with Real Energy Eigenvalues
• Comment on ``New ansatz for metric operator calculation in pseudo-Hermitian field theory''
• Identification of the metric for diagonalizable (anti-)pseudo-Hermitian Hamilton operators represented by two-dimensional matrices