有限维李群的群Fourier变换与相空间路径积分
Group Fourier transform and the phase space path integral for finite dimensional Lie groups
摘要：
我们提出了有限维李群的群傅里叶变换的概念。该变换提供了从群上的平方可积函数到非交换对偶空间上的平方可积函数的酉映射。然后，我们推导出一阶相空间路径积分的量子力学的群通过使用一个非交换的对偶空间表示通过变换得到。该形式的优点包括：（1）通过李群的表示理论和特殊函数的使用，该变换提供了谱分解的替代方案。(2)非对易对偶变量在物理上更直观，因为尽管它们是非对易的，但它们与相应的经典变量类似。这项工作预计，在其他可能的应用中，允许在量子引力的背景下的洛伦兹自旋泡沫模型的度量表示。
英文摘要：
We formulate a notion of group Fourier transform for a finite dimensional Lie group. The transform provides a unitary map from square integrable functions on the group to square integrable functions on a non-commutative dual space. We then derive the first order phase space path integral for quantum mechanics on the group by using a non-commutative dual space representation obtained through the transform. Possible advantages of the formalism include: (1) The transform provides an alternative to the spectral decomposition via representation theory of Lie groups and the use of special functions. (2) The non-commutative dual variables are physically more intuitive, since despite the non-commutativity they are analogous to the corresponding classical variables. The work is expected, among other possible applications, to allow for the metric representation of Lorentzian spin foam models in the context of quantum gravity.
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