  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.
