With the appropriate choice of parameters and sufficient cooling, chargedparticles in a circular accelerator are believed to undergo a transition to ahighly-ordered crystalline state. The simplest possible crystallineconfiguration is a one-dimensional chain of particles. In this paper, we writedown the quantized version of its dynamics. We show that in a low-densitylimit, the dynamics is that of a theory of interacting phonons. There is aninfinite sequence of $n$-phonon interaction terms, of which we write down thefirst orders, which involve phonon scattering and decay processes. The quantumformulation developed here can serve as a first step towards aquantum-mechanical treatment of the system at finite temperatures.
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机译：通过适当选择参数和充分冷却，可以认为圆形促进剂中的带电粒子会转变为高度有序的晶态。最简单的结晶构型是一维颗粒链。在本文中，我们记下了其动力学的量化版本。我们表明，在低密度极限下，动力学是声子相互作用理论的动力学。 $ n $-声子相互作用项有无限个序列，我们写下其中的一阶，涉及声子散射和衰减过程。这里开发的量子配方可以作为朝着在有限温度下对该系统进行量子力学处理的第一步。
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