论文标题
循环量子宇宙学的星级产品方法
Star product approach for Loop Quantum Cosmology
论文作者
论文摘要
在循环量子宇宙学(LQC)形式主义内实施变形量化程序的最新发展的指导下,在本文中,我们介绍了LQC星星产品的积分和差异表示。为此,我们考虑了在实物的Bohr紧凑型上定义的圆柱函数的Weyl量化图。积分表示包含所有表征恒星产物的共同特性,在此处研究的情况下,该恒星代表了圆柱函数的通常点乘积的变形。我们的构造还承认与Moyal产品的积分表示直接比较,这可以通过明智地替代识别这种表示形式的适当特征来从我们的配方中复制。此外,我们介绍了合适的恒星换向器,该换向器既正确又重现了LQC的固体 - 浮温代数的量子表示,并且在适当的极限下,在宇宙学设置中出现了全能频率经典泊松代数。最后,我们提出了一种自然的方法,以使用离散有限差异来获取圆柱函数的恒星换向器以及恒星产品的差异表示,以获取LQC中的量子动力学演化。我们希望我们的发现可能有助于更好地理解LQC计划中引起的某些问题,特别是与半经典限制和量子状态的动态演变有关的问题。
Guided by recent developments towards the implementation of the deformation quantization program within the Loop Quantum Cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.