[Paper Review] Comment on "Dirac Quantization of Pais-Uhlenbeck Fourth Order Oscillator"

Katarzyna Bolonek-Lasoń, Kosinski, Piotr|ArXiv.org|Dec 1, 2006

Quantum Mechanics and Non-Hermitian Physics5 references3 citations

TL;DR

This paper critically examines the Dirac quantization of the Pais-Uhlenbeck fourth-order oscillator in the equal-frequency limit, demonstrating that the standard limiting procedure leads to a theory with an unphysical indefinite metric and zero-norm eigenvectors. Despite attempts to redefine the metric and conjugation rules, the resulting Hamiltonian cannot be diagonalized in a positive-definite Hilbert space, rendering the limiting theory unsatisfactory for physical interpretation.

ABSTRACT

The structure of Pais-Uhlenbeck oscillator in the equal-frequency limit has been recently studied by Mannheim and Davidson [Phys.Rev. A71 (2005), 042110]. It appears that taking this limit, as presented in the above paper, is quite subtle and the resulting structure of space of states - involved. In order to clarify the situation we present here the proper way of taking the equal-frequency limit, first under the assumption that the scalar product in the space of states is positive defined. We discuss also the case of indefinite metric space of states. We show that, irrespective of the way the limit is defined, the limiting theory can be hardly viewed as satisfactory.

Motivation & Objective

• To clarify the proper mathematical procedure for taking the equal-frequency limit of the Pais-Uhlenbeck fourth-order oscillator.
• To assess whether the limiting theory can yield a consistent quantum theory with positive-definite metric and bounded Hamiltonian.
• To investigate the structure of the state space and the behavior of the Hamiltonian under different definitions of hermitian conjugation in the degenerate limit.
• To determine whether the limiting theory can be made physically viable through metric redefinition or alternative conjugation rules.

Proposed method

• Performs a canonical transformation to diagonalize the Hamiltonian in the non-degenerate case, expressing it in terms of two independent harmonic oscillators with opposite signs in energy.
• Analyzes the singular behavior of the transformation in the equal-frequency limit, showing the breakdown of standard quantization procedures.
• Introduces a metric operator η = (-1)^{N₁} to define an indefinite inner product, enabling a positive-definite norm for the physical states.
• Constructs a new conjugation rule using τ = e^{iπA₂⁺A₂} to restore standard commutation relations in a Fock space with indefinite metric.
• Examines the Jordan structure of the Hamiltonian in subspaces of fixed total number operator N, showing it forms single Jordan blocks.
• Computes the norm of energy eigenvectors, proving they vanish identically, indicating a breakdown of physical state interpretation.

Experimental results

Research questions

• RQ1 Can the equal-frequency limit of the Pais-Uhlenbeck oscillator be consistently defined in a way that preserves a positive-definite Hilbert space?
• RQ2 What is the structure of the state space and the spectrum of the Hamiltonian in the degenerate limit?
• RQ3 Does the use of a metric operator and redefined conjugation rules lead to a physically acceptable quantum theory?
• RQ4 Why does the limiting Hamiltonian fail to be diagonalizable, and what are the consequences for the physical interpretation of the theory?
• RQ5 Can the eigenvectors of the Hamiltonian be assigned a non-zero norm in the limiting case?

Key findings

• The equal-frequency limit of the Pais-Uhlenbeck oscillator leads to a Hamiltonian that is not diagonalizable and forms single Jordan blocks in each subspace of fixed total number operator N.
• All energy eigenvectors of the Hamiltonian have zero norm, as shown by ⟨n|n⟩ = 0, rendering the physical state space unphysical.
• The Hamiltonian H does not commute with its adjoint (H, H⁺] ≠ 0, confirming it is not normal and cannot be diagonalized in the standard sense.
• Despite redefining the metric and conjugation rules, the resulting theory still suffers from zero-norm physical states and an unbounded spectrum.
• The standard limiting procedure breaks down because the canonical transformation becomes singular, and the conjugation rules diverge in the limit.
• The paper concludes that no consistent positive-energy quantization of the degenerate Pais-Uhlenbeck oscillator exists via the limiting procedure.