On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian

Abstract

It can be shown using operator techniques that the non-Hermitian $PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential $-g{x}^4$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naive derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.

arXiv commentLaTeX, 12 pages, no figures

References in corpus (6)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• 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
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

Cited by in corpus (20)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermitian Representation of Quantum Mechanics
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• $\mathcal{P}\mathcal{T}$-symmetric, non-Hermitian quantum many-body physics -- a methodological perspective
• The disappearing $Q$ operator
• PT Symmetry in Classical and Quantum Statistical Mechanics
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• Dual oscillators and Quantum Pendulums: spectrum and correlators
• Identification of observables in quantum toboggans
• Cryptohermitian Picture of Scattering Using Quasilocal Metric Operators
• PT symmetry and large-N models
• Quantum knots
• Isomorphic Hilbert spaces associated with different Complex Contours of the $\mathcal{P}\mathcal{T}$-Symmetric $(-x^4)$ Theory
• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory
• 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
• Vacuum Stability of the wrong sign $(-{\varphi }^6)$ Scalar Field Theory
• Effects of complex parameters on classical trajectories of Hamiltonian systems
• Hilbert space representation for quasi-Hermitian position-deformed Heisenberg algebra and Path integral formulation