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 SIGMA, 2009, Volume 5, 053, 22 pages (Mi sigma399)

An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais–Uhlenbeck Oscillator

A. Déctora, H. A. Morales-Técotlab, L. F. Urrutiaa, J. D. Vergaraa

a Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A. Postal 70-543, México D. F., México
b Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, CP 09340, México D. F., México

Abstract: Our purpose in this paper is to analyze the Pais–Uhlenbeck (PU) oscillator using complex canonical transformations. We show that starting from a Lagrangian approach we obtain a transformation that makes the extended PU oscillator, with unequal frequencies, to be equivalent to two standard second order oscillators which have the original number of degrees of freedom. Such extension is provided by adding a total time derivative to the PU Lagrangian together with a complexification of the original variables further subjected to reality conditions in order to maintain the required number of degrees of freedom. The analysis is accomplished at both the classical and quantum levels. Remarkably, at the quantum level the negative norm states are eliminated, as well as the problems of unbounded below energy and non-unitary time evolution. We illustrate the idea of our approach by eliminating the negative norm states in a complex oscillator. Next, we extend the procedure to the Pais–Uhlenbeck oscillator. The corresponding quantum propagators are calculated using Schwinger's quantum action principle. We also discuss the equal frequency case at the classical level.

Keywords: quantum canonical transformations; higher order derivative models

DOI: https://doi.org/10.3842/SIGMA.2009.053

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ArXiv: 0905.0589
MSC: 70H15; 70H50; 81S10
Received: November 14, 2008; in final form April 22, 2009; Published online May 5, 2009
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Citation: A. Déctor, H. A. Morales-Técotl, L. F. Urrutia, J. D. Vergara, “An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais–Uhlenbeck Oscillator”, SIGMA, 5 (2009), 053, 22 pp.

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Martinez S.A., Montemayor R., Urrutia L.F., “Perturbative Hamiltonian Constraints for Higher-Order Theories”, Internat J Modern Phys A, 26:26 (2011), 4661–4686
2. Laemmerzahl C., Rademaker P., “Higher Order Equations of Motion and Gravity”, Phys. Rev. D, 86:12 (2012), 124017
3. Gallardo A., “Time Boundary Terms and Dirac Constraints”, Int. J. Mod. Phys. A, 27:11 (2012), 1250058
4. Margalli C.A., David Vergara J., “Complex Higher Order Derivative Theories”, Ix Workshop of the Gravitation and Mathematical Physics Division of the Mexican Physical Society, AIP Conference Proceedings, 1473, eds. UrenaLopez L., BecerrilBarcenas R., LinaresRomero R., Amer Inst Physics, 2012, 255–259
5. Lopez-Sarrion J., Reyes C.M., “Myers-Pospelov Model as an Ensemble of Pais-Uhlenbeck Oscillators: Unitarity and Lorentz Invariance Violation”, Eur. Phys. J. C, 73:4 (2013), 2391
6. Rami, E.-N.A., Soulati, T., Rezazadeh, H., “Non-standard complex Lagrangian dynamics”, Journal of Advanced Research in Dynamical and Control Systems, 5:1 (2013), 50–62
7. Masterov I., “An Alternative Hamiltonian Formulation For the Pais-Uhlenbeck Oscillator”, Nucl. Phys. B, 902 (2016), 95–114
8. Masterov I., “The odd-order Pais–Uhlenbeck oscillator”, Nucl. Phys. B, 907 (2016), 495–508
9. Cumsille P. Reyes C.M. Ossandon S. Reyes C., “Polymer quantization, stability and higher-order time derivative terms”, Int. J. Mod. Phys. A, 31:9 (2016), 1650040
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