2020/12/18 by Jakub Bilski, Bilski, Jakub
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Lattice (hep-lat) #High Energy Physics - Theory (hep-th) #Noncommutative and Quantum Gravity Theories
paper · pdf · doi:10.48550/arxiv.2012.10465
openalex publication_date 2020/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.