中微子台球的半经典量化,Physical Review E

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中微子台球的半经典量化
Physical Review E ( IF 2.2 ) Pub Date : 2020-10-16 , DOI: 10.1103/physreve.102.042214
Barbara Dietz , Zi-Yuan Li

现在，基于半经典方法可以很好地理解经典动力学对非相对论量子台球（QBs）特征值谱中波动特性的影响。

ρ^{luc} （ ķ ）

用迹线公式表示的频谱密度，即其经典对应物的经典周期性轨道上的总和，简称为CB。量子系统的特征值谱与经典周期轨道之间的这种联系在

ρ^{luc} （ ķ ）

从本征波数

ķ

长度，在周期轨道的长度上出现峰值。与其相对论的狄拉克光谱有关的石墨烯性质的兴趣上升暗示了对相对论中微子台球（NBs）的深入研究的出现，该研究包括由Dirac方程控制并限于有界平面域的自旋1/2粒子。与QB不同，NB没有明确定义的经典限制。然而，它们的长度光谱的比较显示，对于无质量自旋1/2粒子，NB的那些粒子在对应于周期性轨道长度的位置处出现峰，并且在与相应QB相关的CB边界处具有偶数个反射。为了理解从相对论向非相对论的过渡，我们得出大量NB的精确量化条件，并使用它获得跟踪公式。该跟踪公式通过关联的CB的周期性轨道，在NB的频谱密度与相应QB的经典动力学之间提供了直接联系。

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Semiclassical quantization of neutrino billiards

The impact of the classical dynamic on the fluctuation properties in the eigenvalue spectrum of nonrelativistic quantum billiards (QBs) are now well understood based on the semiclassical approach which provides an approximation for the fluctuating part

ρ^{fluc} (k)

of the spectral density in terms of a trace formula, that is, a sum over classical periodic orbits of its classical counterpart, abbreviated as CB. This connection between the eigenvalue spectrum of a quantum system and the classical periodic orbits is discernible in the Fourier transform of

ρ^{fluc} (k)

from eigenwave number

k

to length, which exhibits peaks at the lengths of the periodic orbits. The uprise of interest in properties of graphene related to their relativistic Dirac spectrum implicated the emergence of intensive studies of relativistic neutrino billiards (NBs), consisting of a spin-1/2 particle governed by the Dirac equation and confined to a bounded planar domain. In distinction to QBs, NBs do not have a well-defined classical limit. Yet comparison of their length spectra showed that for massless spin-1/2 particles those of the NB exhibit peaks at positions corresponding to the lengths of periodic orbits with an even number of reflections at the boundary of the CB associated with the corresponding QB. In order to understand the transition from the relativistic to the nonrelativistic regime, we derive an exact quantization condition for massive NBs and use it to obtain a trace formula. This trace formula provides a direct link between the spectral density of a NB and the classical dynamic of the corresponding QB through the periodic orbits of the associated CB.

更新日期：2020-10-17

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