论文标题
SL（2，R）轨道上的路径积分
Path Integrals on sl(2,R) Orbits
论文作者
论文摘要
我们量化了隔离组对SL（2，r）元素的元素作用的轨道。沿椭圆形切片的路径积分类似于紧凑型谎言组的共同连接轨道量化，椭圆形元素的特征的计算沿与紧凑型组相同的线进行。双曲线元件元件的痕迹以对角线的计算以及以双曲线为基础的整个组作用计算需要更多的技术。我们确定PSL（2，r）对伴随轨道的双曲线单参数亚组的作用，并讨论适应坐标系的选择中的全局微妙。使用轨道的双曲线切片，我们在双曲线基础上描述了不可还原SL（2，R）表示的量子力学，并将基础与Mellin积分转换的数学联系起来。此外，我们讨论了PSL（2，r）的双层覆盖SL（2，r）的表示理论以及其通用覆盖物的表示理论。计算这些组的这些组表示的轨迹。最后，我们通过指示应用来激励对这个基本量化问题的治疗。
We quantise orbits of the adjoint group action on elements of the sl(2,R) Lie algebra. The path integration along elliptic slices is akin to the coadjoint orbit quantization of compact Lie groups, and the calculation of the characters of elliptic group elements proceeds along the same lines as in compact groups. The computation of the trace of hyperbolic group elements in a diagonal basis as well as the calculation of the full group action on a hyperbolic basis requires considerably more technique. We determine the action of hyperbolic one-parameter subgroups of PSL(2,R) on the adjoint orbits and discuss global subtleties in choices of adapted coordinate systems. Using the hyperbolic slicing of orbits, we describe the quantum mechanics of an irreducible sl(2,R) representation in a hyperbolic basis and relate the basis to the mathematics of the Mellin integral transform. We moreover discuss the representation theory of the double cover SL(2,R) of PSL(2,R) as well as that of its universal cover. Traces in the representations of these groups for both elliptic and hyperbolic elements are computed. Finally, we motivate our treatment of this elementary quantisation problem by indicating applications.
