2020/08/31 by Andrei Smilga, A. V. Smilga · 1 citation
Mathematics · Physics and Astronomy · #Bounded function #Classical mechanics #Conservation law #Hamiltonian (control theory) #Hamiltonian system #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Physics #Quantum #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Quantum mechanics #Quantum system #Unitarity #hep-th #math-ph #math.MP
paper · pdf · doi:10.1016/j.physleta.2020.127104
Final version published in Physics Letters A. The title is modified compared to the first version
openalex created_date 2020/09/08 · openalex publication_date 2020/12/17 · arxiv created 2021/06/04 · arxiv updated 2021/06/07 · openalex updated_date 2026/08/05
We discuss exactly solvable systems involving integrals of motion with higher powers of momenta. If one of these integrals is chosen for the Hamiltonian, we obtain a higher-derivative system involving ghosts, i.e. a system whose Hamiltonian is not bounded neither from below, nor from above. However, these ghosts are benign: there is no collapse and unitarity is not violated. As an example, we consider the 3-particle Toda periodic chain, with the cubic invariant I chosen for the Hamiltonian. The classical trajectories exhibit regular oscillations, and the spectrum of the quantum Hamiltonian is discrete running from minus to plus infinity. We also discuss the classical dynamics of a perturbed system with the Hamiltonian H = I + v, where v is an oscillator potential. Such a system is not exactly solvable, but its classical trajectories exhibit not regular, but still benign behaviour without collapse. This means that also the corresponding quantum problem is well defined. The same observation can be made for exactly solvable (1+1)-dimensional field theories involving an infinite number of conservation laws: any of them can be chosen for the Hamiltonian. We illustrate this for the Sine-Gordon and KdV models. In the latter case, the Lagrangian and standard integrals of motion involve higher spatial rather than temporal derivatives. But one can always interchange x and t, after which we obtain a system with benign ghosts.