Giving up the ghost

Abstract

The Pais-Uhlenbeck model is a quantum theory described by a higher-derivative field equation. It has been believed for many years that this model possesses ghost states (quantum states of negative norm) and therefore that this model is a physically unacceptable quantum theory. The existence of such ghost states was believed to be attributable to the field equation having more than two derivatives. This paper shows that the Pais-Uhlenbeck model does not possess any ghost states at all and that it is a perfectly acceptable quantum theory. The supposed ghost states in this model arise if the Hamiltonian of the model is (incorrectly) treated as being Dirac Hermitian (invariant under combined matrix transposition and complex conjugation). However, the Hamiltonian is {\it not} Dirac Hermitian, but rather it is PT symmetric. When it is quantized correctly according to the rules of PT quantum mechanics, the energy spectrum is real and bounded below and all of the quantum states have positive norm.

8 pages, o figures; talk given by the first author at the 5th International Symposium on Quantum Theory and Symmetries (QTS5) held in Vallodolid, Spain, July 2007; published in a special issue of the Journal of Physics A, Ed. by M. Gadella, J. M. Izquierdo, S. Kuru, J. Negro, M. del Olmo

References in corpus (18)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• 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
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Introduction to PT-Symmetric Quantum Theory
• Exact PT-Symmetry Is Equivalent to Hermiticity
• Must a Hamiltonian be Hermitian?
• Supersymmetry and the spontaneous breakdown of PT symmetry
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Solution to the ghost problem in fourth order derivative theories
• Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator
• 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
• Calculation of the Hidden Symmetry Operator for a $\cP\cT$-Symmetric Square Well

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• Pais-Uhlenbeck Oscillator and Negative Energies
• Tackling Higher Derivative Ghosts with the Euclidean Path Integral
• Testing symmetries in effective models of higher derivative field theories
• Quartic propagators, negative norms and the physical spectrum
• Conformal Supergravity Tree Amplitudes from Open Twistor String Theory
• From PT-symmetric quantum mechanics to conformal field theory
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• Gauge Theory of Gravity with de Sitter Symmetry as a Solution to the Cosmological Constant Problem and the Dark Energy Puzzle
• Quasinormal modes of black holes in Weyl gravity: Electromagnetic and gravitational perturbations
• Astrophysical Gravitational Waves in Conformal Gravity
• Foundations of ghost stability
• Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians
• PT symmetry in relativistic quantum mechanics
• Can cosmic acceleration be caused by exotic massless particles?
• Resonance and stability of higher derivative theories of derived type
• Conservation Laws and Stability of Field Theories of Derived Type
• Gravitational Waves and Degrees of Freedom in Higher Derivative Gravity
• Astrophysical Evidence for the Non-Hermitian but $PT$-symmetric Hamiltonian of Conformal Gravity
• Gravitational waves from inspiralling compact binaries in conformal gravity
• Polymer quantization, stability and higher-order time derivative terms
• Schwarzschild and linear potentials in Mannheim's model of conformal gravity
• The initial value problem for gravitational waves in conformal gravity