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 (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

Article in press

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

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

References:

  1. J. Alfaro, H.A. Morales-Tecotl, M. Reyes and L.F. Urrutia, On nonAbelian holonomies, J. Phys. A 36 (2003) 12097–12107.
    https://doi.org/10.1088/0305-4470/36/48/012
  2. R.L. Arnowitt, S. Deser and C.W. Misner, Canonical variables for general relativity, Phys. Rev. 117 (1960) 1595.
    https://doi.org/10.1103/PhysRev.117.1595
  3. A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244–2247.
    https://doi.org/10.1103/PhysRevLett.57.2244
  4. A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A status report, Class. Quant. Grav. 21 (2004) #R53.
    https://doi.org/10.1088/0264-9381/21/15/R01
  5. A. Ashtekar and J. Lewandowski, Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995) (1995).
    https://doi.org/10.1063/1.531037
  6. A. Ashtekar and J. Lewandowski, Differential geometry on the space of connections via graphs and projective limits, J. Geom. Phys. 17 (1995) 191.
    https://doi.org/10.1016/0393-0440(95)00028-G
  7. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, J. Math. Phys. 36 (1995) 6456–6493.
    https://doi.org/10.1063/1.531252
  8. J.F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D 51 (1995) 5507.
    https://doi.org/10.1103/PhysRevD.51.5507
  9. J.F. Barbero G., Reality conditions and Ashtekar variables: A different perspective, Phys. Rev. D 51 (1995) 5498–5506.
    https://doi.org/10.1103/PhysRevD.51.5498
  10. J.W. Barrett, Holonomy and path structures in general relativity and Yang-Mills theory, Int. J. Theor. Phys. 30 (1991) 1171–1215.
    https://doi.org/10.1007/BF00671007
  11. J. Bilski, Continuously distributed holonomy-flux algebra, [arXiv: 2101.05295 [gr-qc]].
  12. J. Bilski, Implementation of the holonomy representation of the Ashtekar connection in loop quantum gravity, [arXiv: 2012.14441 [gr-qc]].
  13. A. Caetano and R.F. Picken, An axiomatic definition of holonomy, Int. J. Math. 5(6) (1994) 835–848.
    https://doi.org/10.1142/S0129167X94000425
  14. B.S. DeWitt, Quantum theory of gravity $1$. The canonical theory, Phys. Rev. 160 (1967) 1113.
    https://doi.org/10.1103/PhysRev.160.1113
  15. P.A.M. Dirac, The fundamental equations of quantum mechanics, Proc. Roy. Soc. Lond. A 109 (1925) 642–653.
    https://doi.org/10.1098/rspa.1925.0150
  16. F.J. Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85 (1952) 631–632.
    https://doi.org/10.1103/PhysRev.85.631
  17. A. Einstein, The foundation of the general theory of relativity, Annalen Phys. 49(7) (1916) 769–822.
    https://doi.org/10.1002/andp.200590044
  18. P.M. Fishbane, S. Gasiorowicz and P. Kaus, Stokes' theorems for nonabelian fields, Phys. Rev. D 24 (1981) 2324.
    https://doi.org/10.1103/PhysRevD.24.2324
  19. F. Frenet, Sur les courbes à double courbure, Thèse, Toulouse (1847), abstract in: J. Math. Pures Appl. 17 (1852).
  20. W. Heisenberg and W. Pauli, Zur Quantendynamik der Wellenfelder, Z. Phys. 56 (1929) 1–61.
    https://doi.org/10.1007/BF01340129
  21. S. Holst, Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action, Phys. Rev. D 53 (1996) 5966–5969.
    https://doi.org/10.1103/PhysRevD.53.5966
  22. G. Immirzi, Quantum gravity and Regge calculus, Nucl. Phys. B Proc. Suppl. 57 (1997) 65–72.
    https://doi.org/10.1016/S0920-5632(97)00354-X
  23. J.A. Serret, Sur quelques formules relatives à la théorie des courbes à double courbure, J. Math. Pures Appl. 16 (1851).
  24. T. Thiemann, Quantum spin dynamics (QSD), Class. Quant. Grav. 15 (1998) 839–873.
    https://doi.org/10.1088/0264-9381/15/4/011
  25. T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge, UK, Cambridge Univ. Pr., 2007).
    https://doi.org/10.1017/CBO9780511755682
  26. T. Thiemann, Reality conditions inducing transforms for quantum gauge field theory and quantum gravity, Class. Quant. Grav. 13 (1996) 1383–1404.
    https://doi.org/10.1088/0264-9381/13/6/012
  27. S. Weinberg, The Quantum Theory of Fields Vol. 1 Foundations (Cambridge, UK, Cambridge Univ. Pr., 1995).
    https://doi.org/10.1017/CBO9781139644167
  28. E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg+Teubner Verlag, Wiesbaden, 1931).
    https://doi.org/10.1007/978-3-663-02555-9
  29. E.P. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939) 149–204.
    https://doi.org/10.2307/1968551
  30. K.G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974) 2445.
    https://doi.org/10.1103/PhysRevD.10.2445
  31. C.N. Yang and R.L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954) 191.
    https://doi.org/10.1103/PhysRev.96.191
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