7 research outputs found
An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
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
Time boundary terms and Dirac constraints
Time boundary terms usually added to action principles are systematically
handled in the framework of Dirac's canonical analysis. The procedure begins
with the introduction of the boundary term into the integral Hamiltonian action
and then the resulting action is interpreted as a Lagrangian one to which
Dirac's method is applied. Once the general theory is developed, the current
procedure is implemented and illustrated in various examples which are
originally endowed with different types of constraints.Comment: 12 page
An Alternative Canonical Approach to the Ghost Problem in a Complexified Extension of the Pais-Uhlenbeck Oscillator
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
Perturbative Hamiltonian constraints for higher order theories
We present a method for constructing a consistent low energy canonical
formalism for higher order time-derivative theories, extending the Dirac method
to include perturbative Hamiltonian constraints. We apply it to two
paradigmatic examples: the Pais-Uhlenbeck oscillator and the Bernard-Duncan
scalar field. We also compare the results, both at the classical and quantum
level, with the ones corresponding to a direct perturbative construction
applied to the exact higher order theory. This comparison highligths the
soundness of the present formalism.Comment: 26 pages, 4 figures; review section shortened and appendices change