论文标题
Ashtekar-Barbero连接操作员的Lie代数
Lie algebra of Ashtekar-Barbero connection operators
论文作者
论文摘要
Ashtekar-Barbero连接的全能人物可以被视为谎言群体的抽象元素，该元素是根据其连接表示形式呈指数映射的。这个想法提供了比较这些对象的几何和代数属性的可能性。结果允许在单位的几何和代数扩展中识别次阶术语。这种识别导致对相关希尔伯特空间公式的验证。如果各州是尸体对称群体的表示，则根据Wigner的定理，它们可以保留规格的变换。因此，循环量子重力中的自旋网络满足该定理。此外，对不同扩展的考虑识别确保了Ashtekar连接的现实。只有真正的联系的全能才能导致满足Wigner定理的状态。
Holonomies of the Ashtekar-Barbero connection can be considered as abstract elements of a Lie group exponentially mapped from their connections representation. This idea provides a possibility to compare the geometric and algebraic properties of these objects. The result allows to identify the next-to-the-leading-order terms in the geometric and algebraic expansion of a holonomy. This identification leads to the verification of the related Hilbert space formulation. If states are the representations of the holonomy's symmetry group, they preserve gauge transformations according to Wigner's theorem. Thus, the spin network in loop quantum gravity satisfies this theorem. Moreover, the considered identification of the different expansions ensures the reality of the Ashtekar connection. Only the holonomies of real connections lead to the formulation of states that satisfy Wigner's theorem.
