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arXiv 1609.08526quant-ph
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路径积分与李群

Path Integrals and Lie Groups

Akira Inomata, Georg Junker

发表信息：
Noncompact Lie Groups and Some of Their Applications, (NATO ASI Series, Kluwer Academic Publishers, Dordrecht, 1994) p. 199
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摘要：

讨论了李群在非相对论量子力学中费曼路径积分中的作用。动力学以及几何对称性被发现有用的路径积分量子化。考虑了具有非紧李群对称性的两个例子。第一种是粒子在常负曲率空间中的自由量子运动。该系统有一个群SO（d，1）与几何结构，其中的技术的调和分析的齐次空间。作为具有非紧动力学对称性的系统的一个例子，选择d维谐振子，其除了其几何对称性SO（d）之外还具有非紧动力学群SU（1，1）。径向路径积分被看作SU（1，1）的紧群元的矩阵函数在连续基上的卷积。

英文摘要：

The roles of Lie groups in Feynman's path integrals in non-relativistic quantum mechanics are discussed. Dynamical as well as geometrical symmetries are found useful for path integral quantization. Two examples having the symmetry of a non-compact Lie group are considered. The first is the free quantum motion of a particle on a space of constant negative curvature. The system has a group SO(d,1) associated with the geometrical structure, to which the technique of harmonic analysis on a homogeneous space is applied. As an example of a system having a non-compact dynamical symmetry, the d-dimensional harmonic oscillator is chosen, which has the non-compact dynamical group SU(1,1) besides its geometrical symmetry SO(d). The radial path integral is seen as a convolution of the matrix functions of a compact group element of SU(1,1) on the continuous basis.

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