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

SIGMA 5 (2009), 053, 22 pages      arXiv:0905.0589      https://doi.org/10.3842/SIGMA.2009.053
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

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

A. Déctor a, H.A. Morales-Técotl a, b, L.F. Urrutia a and J.D. Vergara a
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

Received November 14, 2008, in final form April 22, 2009; Published online May 05, 2009

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.

Key words: quantum canonical transformations; higher order derivative models.

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