In a way invented when AM played chess with his daughter (and
motivated by the puzzles posed by the higher-derivative theories,
e.g., of gravity), the benchmark Pais-Uhlenbeck oscillator is
considered in both the non-degenerate case (where the frequencies
differ) and in the degenerate case (where the frequencies coincide).
In the former case the non-quantum system is assigned an
alternative, amended classical Hamiltonian leading to the standard
canonical quantization recipe in which the quantum Hamiltonian
operator proves Hermitian, with the positive spectrum bounded from
below. This offers an alternative and amendment of the quantization
recipe of Ref. [5] (which was marred by certain anomalous features,
especially when the quantum - classical correspondence is
concerned). In the second half of the letter the more difficult
degenerate case is shown complex. A real description of this complex
system and certain properties of its possible quantizations are
discussed. A number of arguments is given supporting its expected
(and expectable) instability.


MR2737899 Mostafazadeh, Ali A Hamiltonian formulation of the
Pais-Uhlenbeck oscillator that yields a stable and unitary quantum
system. Phys. Lett. A 375 (2010), no. 2, 93--98. 81Q80