论文标题
关于完全可解决的高衍生系统
On exactly solvable higher-derivative systems
论文作者
论文摘要
我们讨论涉及运动积分具有更高力量的动力的确切可解决的系统。如果为哈密顿人选择了其中一个积分，我们将获得涉及鬼魂的更高衍生系统，即，一个既没有从下方也不是从上方界定的哈密顿量的系统。但是，这些鬼魂是良性的：没有崩溃，没有违反一野性。 例如，我们考虑三颗粒TODA周期链，我选择了用于哈密顿式的立方体不变。经典的轨迹表现出规则的振荡，量子量，哈密顿量的光谱是从负到加上无穷大的离散运行。我们还与Hamiltonian H = I + V讨论了扰动系统的经典动力学，其中V是振荡器的电位。这样的系统并非完全可以解决，但是其经典轨迹并不规律，但仍然没有崩溃的良性行为。这意味着相应的量子问题也被很好地定义。 对于涉及无限数量的保护法律的准确解决（1+1）的维度（1+1） - 维度野外理论可以进行相同的观察：可以为哈密顿人选择任何一个。我们为正弦和KDV模型说明了这一点。在后一种情况下，拉格朗日和标准运动的积分涉及较高的空间衍生物，而不是时间衍生物。但是，人们总是可以互换X和T，之后我们获得了具有良性幽灵的系统。
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.
