Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 017, 13 pages      arXiv:0808.0139      https://doi.org/10.3842/SIGMA.2009.017
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Comments on the Dynamics of the Pais-Uhlenbeck Oscillator

Andrei V. Smilga
SUBATECH, Université de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307, France

Received November 24, 2008, in final form February 05, 2009; Published online February 12, 2009

Abstract
We discuss the quantum dynamics of the PU oscillator, i.e. the system with the Lagrangian
      L = ½ [ ¨q2 - (Ω12 + Ω22) ·q2 + Ω12Ω22q ]     (+ nonlinear terms).
When Ω1 ≠ Ω2, the free PU oscillator has a pure point spectrum that is dense everywhere. When Ω1 = Ω2, the spectrum is continuous, E Î {–∞, ∞}. The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary at Ω1 ≠ Ω2, but unitarity is broken in the equal frequencies limit.

Key words: higher derivatives; ghosts; unitarity.

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References

  1. Smilga A.V., 6D superconformal theory as the theory of everything, in Gribov Memorial Volume, Editors Yu.L. Dokshitzer, P. Levai and J. Nyiri, World Scientific, 2006, 443-459, hep-th/0509022.
  2. Smilga A.V., Benign versus malicious ghosts in higher-derivative theories, Nuclear Phys. B 706 (2005), 598-614, hep-th/0407231.
  3. Pais A., Uhlenbeck G.E., On field theories with nonlocalized action, Phys. Rev. 79 (1950), 145-165.
  4. Robert D., Smilga A.V., Supersymmetry versus ghosts, J. Math. Phys. 49 (2008), 042104, 20 pages, math-ph/0611023.
  5. Smilga A.V., Ghost-free higher-derivative theory, Phys. Lett. B 632 (2006), 433-438, hep-th/0503213.
  6. Bolonek K., Kosinski P., Comments on "Dirac quantization of Pais-Uhlenbeck fourth order oscillator", quant-ph/0612009.
  7. Mannheim P.D., Davidson A., Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Phys. Rev. A 71 (2005), 042110, 9 pages, hep-th/0408104.
  8. Bender C.M., Mannheim P.D., No ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model, Phys. Rev. Lett. 100 (2008), 110402, 4 pages, arXiv:0706.0207.
  9. Bender C.M., Mannheim P.D., Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart, Phys. Rev. D 78 (2008), 025022, 20 pages, arXiv:0804.4190.
  10. Ostrogradsky M., Mémoires sur les équations différentielles relatives au problème des isopérimètres, Mem. Acad. St. Petersbourg, VI 4 (1850), 385-517.
  11. Heiss W.D., Exceptional points of non-Hermitian operators, J. Phys. A: Math. Gen. 37 (2004) 2455-2464, quant-ph/0304152.
  12. Anderson A., Canonical transformations in quantum mechanics, Ann. Phys. 232 (1994) 292-331, hep-th/9305054.
  13. Case K.M., Singular potentials, Phys. Rev. 80 (1950), 797-806.
    Meetz K., Singular potentials in non-relativistic quantum mechanics, Nuovo Cimento 34 (1964), 690-708.
    Perelomov A.M., Popov V.S., Collapse onto scattering center in quantum mechanics, Teor. Mat. Fiz. 4 (1970), 48-65.
  14. Bronzan J.B., Shapiro J.A., Sugar R.L., Reggeon field theory in zero transverse dimensions, Phys. Rev. D 14 (1976), 618-631.
    Amati D., Le Bellac M., Marchesini G., Ciafaloni M., Reggeon field theory for α(0) > 1, Nuclear Phys. B 112 (1976), 107-149.
    Gasymov M.G., Spectral analysis of a class of second-order non-self-adjoint differential operators, Funktsional. Anal. i Prilozhen. 14 (1980), 14-19.
    Caliceti E., Graffi S., Maioli M., Perturbation theory of odd anharmonic oscillators, Comm. Math. Phys. 75 (1980), 51-66.
    Scholtz F.G., Geyer H.B., Hahne F.J.W., Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Physics 213 (1992), 74-101.
    Bender C.M., Boettcher S., Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998), 5243-5246, physics/9712001.
    Mostafazadeh A., PT-symmetric cubic anharmonic oscillator as a physical model, J. Phys. A: Math. Gen. 38 (2005), 6557-6570, quant-ph/0411137.

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