Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

Abstract

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\cP\cT$-symmetric wrong-sign quartic Hamiltonian $H=\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\tilde H=\half p^2+4g x^4-\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\cP\cT$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\cP\cT$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $-{\varphi }^4$ quantum field theory in higher-dimensional space-time are discussed.

arXiv comment19 pages, 7 figures, corrected references

References in corpus (19)

• 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?
• 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
• Solution of Schwinger-Dyson Equations for $\mathcal{P}\mathcal{T}$-Symmetric Quantum Field Theory
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• The C Operator in PT-Symmetric Quantum Theories
• Reflectionless Potentials and PT Symmetry
• PT-Symmetric Quantum Electrodynamics
• Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well
• Bound States of Non-Hermitian Quantum Field Theories
• PT-Symmetric Quantum Theory Defined in a Krein Space
• Kurt Symanzik - a stable fixed point beyond triviality

Cited by in corpus (52)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• The ODE/IM Correspondence
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Hamiltonian for the zeros of the Riemann zeta function
• General solution of scalar field cosmology with a (piecewise) exponential potential
• Complex Langevin Equations and Schwinger-Dyson Equations
• Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• A return to observability near exceptional points in a schematic PT-symmetric model
• Giving up the ghost
• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• $\mathcal{P}\mathcal{T}$-symmetric $-g{\phi }^4$ theory
• Conditional observability
• The disappearing $Q$ operator
• On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian
• Nonunique C operator in PT Quantum Mechanics
• The structure of supersymmetry in $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics
• Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions
• Classical and quantum cosmology of the little rip abrupt event
• PT Symmetry in Classical and Quantum Statistical Mechanics
• Pseudo-Hermitian Interactions in Dirac Theory: Examples
• Quantum and Classical Statistical Mechanics of a Class of non-Hermitian Hamiltonians
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• $\mathcal{P}T$ symmetry, pattern formation, and finite-density QCD
• On the Dirac equation with PT-symmetric potentials in the presence of position-dependent mass
• Dual oscillators and Quantum Pendulums: spectrum and correlators
• Thermodynamics of Pseudo-Hermitian Systems in Equilibrium
• PT Symmetric, Hermitian and P-Self-Adjoint Operators Related to Potentials in PT Quantum Mechanics
• Approximate Dirac solutions of complex -symmetric Pöschl-Teller potential in view of spin and pseudospin symmetries
• Exact Solutions for Non-Hermitian Dirac-Pauli Equation in an intensive magnetic field
• On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
• Phantom without phantom or how the PT symmetry saves us from the Big Rip
• Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics
• Non-Hermitian $\mathcal{P}\mathcal{T}$-Symmetric Dirac-Pauli Hamiltonians with Real Energy Eigenvalues in the Magnetic Field
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Nonuniqueness of the C operator in PT-symmetric quantum mechanics
• PT-symmetric interpretation of double-scaling
• An Algebraic PT-Symmetric Quantum Theory with a Maximal Mass
• Wilsonian approach to the interaction ${\varphi }^2(i\varphi {)}^{\epsilon }$
• PT symmetry and large-N models
• Scattering off PT-symmetric upside-down potentials
• Double-Scaling Limit of the O(N)-Symmetric Anharmonic Oscillator
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• Hypercomplex Fock States for Discrete Electromagnetic Schrödinger Operators: A Bayesian Probability Perspective
• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory
• Faster than Hermitian Time Evolution
• Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^2-g{x}^4+a/x^2$
• The Large-N Limit of PT-Symmetric O(N) Models
• Comment on: `Numerical estimates of the spectrum for anharmonic PT symmetric potentials' [Phys. Scr. \textbf{85} (2012) 065005]
• Operator based approach to PT-symmetric problems on a wedge-shaped contour
• Effects of complex parameters on classical trajectories of Hamiltonian systems