The disappearing $Q$ operator

Abstract

In the Schroedinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\eta \equiv e^{-Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.

14 pages, no figures

References in corpus (13)

• 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
• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• On Pseudo-Hermitian Hamiltonians and Their Hermitian Counterparts
• 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
• PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
• On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian
• Analysis of the Strong Coupling Limit of the Richardson Hamiltonian using the Dyson Mapping

Cited by in corpus (22)

• Pseudo-Hermitian Representation of Quantum Mechanics
• Scattering theory with localized non-Hermiticities
• PT-symmetric quantum field theory in D dimensions
• 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
• Which Green Functions Does the Path Integral for Quasi-Hermitian Hamiltonians Represent?
• Possible treatment of the Ghost states in the Lee-Wick Standard Model
• PT symmetric fermionic field theories with axions: Renormalization and dynamical mass generation
• Towards perturbative renormalization of ${\varphi }^2(i\varphi {)}^{\epsilon }$ quantum field theory
• On the foundations of partially quenched chiral perturbation theory
• Path-Integral Formulation of Pseudo-Hermitian Quantum Mechanics and the Role of the Metric Operator
• Quantum and Classical Statistical Mechanics of a Class of non-Hermitian Hamiltonians
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• Dual oscillators and Quantum Pendulums: spectrum and correlators
• Bicoherent-State Path Integral Quantization of a non-Hermitian Hamiltonian
• Effective Field calculations of the Energy Spectrum of the $\mathcal{P}\mathcal{T}$-Symmetric ($-x^4$) Potential
• Schwinger-Dyson equations and mass generation for an axion theory with a PT symmetric Yukawa fermion interaction
• Scattering in pseudo-Hermitian quantum field theory and causality violation
• Original study of the $g{\varphi }^2(i\varphi {)}^{\epsilon }$ theory. Analysis of all orders in $\epsilon$ and resummations
• Unitarity through PT symmetry in Quantum Quadratic Gravity
• PT-Symmetry Quantum Electrodynamics--PTQED
• Vacuum Stability of the $\mathcal{P}\mathcal{T}$-Symmetric $\left(-{\varphi }^4\right)$ Scalar Field Theory