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Bi-Hamiltonian Structures and Equivalent Representations of the Pais-Uhlenbeck Model

Bethan Turner

TL;DR

The paper investigates whether the Pais-Uhlenbeck oscillator—the prototypical higher time derivative system—can be represented in two dimensions with a positive-definite Hamiltonian. By exploiting dynamical Lie symmetries, it constructs a Bi-Hamiltonian structure that yields a second conserved charge $H_2$ and a second Poisson tensor, allowing a positive-definite $H'$ in the free theory under suitable constraints. It then analyzes all two-dimensional embeddings and shows that preserving the PU dynamics while maintaining the original Poisson structure generally leads to unbounded Hamiltonians, while the Lie-symmetry approach provides a systematic path to well-behaved formulations in the absence of interactions. However, introducing interactions breaks the Bi-Hamiltonian framework (the only compatible Poisson tensor is $J_1$), causing the system to regain instability beyond a coupling threshold. The results motivate exploring nonlinear or higher-order Bi-Hamiltonian generalizations as potential routes to mitigate ghost-like instabilities in higher-derivative theories.

Abstract

We provide a complete classification of all the ways the Pais-Uhlenbeck osicllator might be embedded in two dimensional space. We discuss the Bi-Hamiltonian structures of this model, and examine how alternative Hamiltonian structures might be generated from the dynamical Lie symmetries of the theory. We then examine how the Bi-Hamiltonian strucutre may be exploited to evade the problem of unbounded Hamiltonians that is usually associated with Higher Time Derivative Theories. The effect of interactions on this Bi-Hamiltonian structure is also considered.

Bi-Hamiltonian Structures and Equivalent Representations of the Pais-Uhlenbeck Model

TL;DR

The paper investigates whether the Pais-Uhlenbeck oscillator—the prototypical higher time derivative system—can be represented in two dimensions with a positive-definite Hamiltonian. By exploiting dynamical Lie symmetries, it constructs a Bi-Hamiltonian structure that yields a second conserved charge and a second Poisson tensor, allowing a positive-definite in the free theory under suitable constraints. It then analyzes all two-dimensional embeddings and shows that preserving the PU dynamics while maintaining the original Poisson structure generally leads to unbounded Hamiltonians, while the Lie-symmetry approach provides a systematic path to well-behaved formulations in the absence of interactions. However, introducing interactions breaks the Bi-Hamiltonian framework (the only compatible Poisson tensor is ), causing the system to regain instability beyond a coupling threshold. The results motivate exploring nonlinear or higher-order Bi-Hamiltonian generalizations as potential routes to mitigate ghost-like instabilities in higher-derivative theories.

Abstract

We provide a complete classification of all the ways the Pais-Uhlenbeck osicllator might be embedded in two dimensional space. We discuss the Bi-Hamiltonian structures of this model, and examine how alternative Hamiltonian structures might be generated from the dynamical Lie symmetries of the theory. We then examine how the Bi-Hamiltonian strucutre may be exploited to evade the problem of unbounded Hamiltonians that is usually associated with Higher Time Derivative Theories. The effect of interactions on this Bi-Hamiltonian structure is also considered.

Paper Structure

This paper contains 6 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Pais-Uhlenbeck Oscillator, Classical Dynamics and Quantisation
  3. Two Dimensional Representations
  4. Lie Symmetires and Bi-Hamiltonian Structures
  5. Interactions
  6. Conclusion

Figures (1)

  • Figure 1: Phase space plots for Pais-Uhlenbeck oscillator with quartic potential $W\left(q \right) = \frac{1}{4} \lambda q^4$. The coordinate $x_i$ and $p_i$ are the coordinates given in (\ref{['PUphase']}). The frequencies are $\omega_1 =1$, $\omega_2=2$. The intial conditions are $x_1\left(0\right)=x_2\left(0\right)=0$, $p_1\left(0\right)=-p_2\left( 0\right)=0.5$.