DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2024): 0.9

SJR (2024): 0.193

SNIP (2024): 0.586

Index Copernicus (2024): 119.29

H-Index: 6

Discussiones Mathematicae - General Algebra and Applications

Article in volume

Authors:

J. Bilski

Jakub Bilski

Institute of Mathematics, University of Zielona Góra, 4a Szafrana, 65-516 Zielona Góra, Poland, EU

email: j.bilski@im.uz.zgora.pl

0000-0001-5438-6496

Title:

Nonlocal holonomy representations for Lie algebra-valued Ashtekar-Barbero connection

PDF

Source:

Discussiones Mathematicae - General Algebra and Applications 45(2) (2025) 507-520

Received: 2024-12-30 , Revised: 2025-02-09 , Accepted: 2025-02-10 , Available online: 2025-10-01 , https://doi.org/10.7151/dmgaa.1494

Abstract:

Holonomies of the Ashtekar-Barbero connection can be considered as abstract elements of a Lie group, exponentially mapped from their algebra representation. This idea allows for the definition of the states in loop quantum gravity, which preserve the group symmetry that is equivalent to the Ashtekar-Barbero connection symmetry of Lie algebra. The equivalence of the symmetries requires either the quadratic- or linear-order precision in the expansion of group elements either around a finite value of the expansion parameter or by taking the limit as this parameter approaches zero, respectively. These conditions put different constraints on the holonomy regularization method in loop quantum gravity, where holonomies are expanded around finite values of the related paths' lengths. This article investigates the possibility of increasing the linear-order precision, postulated in canonical loop quantum gravity, into the quadratic order. It demonstrates that the regularization method can be defined more accurately.

Primary keywords:

gauge theory, holonomy, Lie algebra, Lie group, loop quantum gravity, nonlocality

Secondary keywords:

Wilson loops, quantization, lattice gravity

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