### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 079, 23 pages      arXiv:1403.3038      https://doi.org/10.3842/SIGMA.2014.079
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

### Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

Michele Arzano a, Danilo Latini b and Matteo Lotito c
a) Dipartimento di Fisica and INFN, ''Sapienza'' University of Rome, P.le A. Moro 2, 00185 Roma, Italy
b) Dipartimento di Fisica and INFN, Università  di Roma Tre, Via Vasca Navale 84, I-00146 Roma, Italy
c) Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA

Received March 13, 2014, in final form July 15, 2014; Published online July 24, 2014

Abstract
We present an in-depth investigation of the ${\rm SL}(2,\mathbb{R})$ momentum space describing point particles coupled to Einstein gravity in three space-time dimensions. We introduce different sets of coordinates on the group manifold and discuss their properties under Lorentz transformations. In particular we show how a certain set of coordinates exhibits an upper bound on the energy under deformed Lorentz boosts which saturate at the Planck energy. We discuss how this deformed symmetry framework is generally described by a quantum deformation of the Poincaré group: the quantum double of ${\rm SL}(2,\mathbb{R})$. We then illustrate how the space of functions on the group manifold momentum space has a dual representation on a non-commutative space of coordinates via a (quantum) group Fourier transform. In this context we explore the connection between Weyl maps and different notions of (quantum) group Fourier transform appeared in the literature in the past years and establish relations between them.

Key words: $2+1$ gravity; Lie group momentum space; deformed symmetries; Hopf algebra.

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References

1. Agostini A., Amelino-Camelia G., Arzano M., D'Andrea F., A cyclic integral on $\kappa$-Minkowski noncommutative space-time, Internat. J. Modern Phys. A 21 (2006), 3133-3150.
2. Agostini A., Amelino-Camelia G., D'Andrea F., Hopf-algebra description of noncommutative-space-time symmetries, Internat. J. Modern Phys. A 19 (2004), 5187-5219, hep-th/0306013.
3. Agostini A., Lizzi F., Zampini A., Generalized Weyl systems and $\kappa$-Minkowski space, Modern Phys. Lett. A 17 (2002), 2105-2126, hep-th/0209174.
4. Alesci E., Arzano M., Anomalous dimension in three-dimensional semiclassical gravity, Phys. Lett. B 707 (2012), 272-277, arXiv:1108.1507.
5. Amelino-Camelia G., Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale, Internat. J. Modern Phys. D 11 (2002), 35-59, gr-qc/0012051.
6. Amelino-Camelia G., Doubly-special relativity: facts, myths and some key open issues, Symmetry 2 (2010), 230-271, arXiv:1003.3942.
7. Amelino-Camelia G., Arzano M., Bianco S., Buonocore R.J., The DSR-deformed relativistic symmetries and the relative locality of 3D quantum gravity, Classical Quantum Gravity 30 (2013), 065012, 17 pages, arXiv:1210.7834.
8. Arzano M., Anatomy of a deformed symmetry: field quantization on curved momentum space, Phys. Rev. D 83 (2011), 025025, 12 pages, arXiv:1009.1097.
9. Arzano M., Kowalski-Glikman J., Trześniewski T., Beyond Fock space in three-dimensional semiclassical gravity, Classical Quantum Gravity 31 (2014), 035013, 13 pages, arXiv:1305.6220.
10. Baez J.C., Wise D.K., Crans A.S., Exotic statistics for strings in 4D BF theory, Adv. Theor. Math. Phys. 11 (2007), 707-749, gr-qc/0603085.
11. Bais F.A., Muller N.M., Topological field theory and the quantum double of ${\rm SU}(2)$, Nuclear Phys. B 530 (1998), 349-400, hep-th/9804130.
12. Bais F.A., Muller N.M., Schroers B.J., Quantum group symmetry and particle scattering in $(2+1)$-dimensional quantum gravity, Nuclear Phys. B 640 (2002), 3-45, hep-th/0205021.
13. Baratin A., Dittrich B., Oriti D., Tambornino J., Non-commutative flux representation for loop quantum gravity, Classical Quantum Gravity 28 (2011), 175011, 19 pages, arXiv:1004.3450.
14. Baratin A., Oriti D., Group field theory with noncommutative metric variables, Phys. Rev. Lett. 105 (2010), 221302, 4 pages, arXiv:1002.4723.
15. Deser S., Jackiw R., 't Hooft G., Three-dimensional Einstein gravity: dynamics of flat space, Ann. Physics 152 (1984), 220-235.
16. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
17. Freidel L., Majid S., Noncommutative harmonic analysis, sampling theory and the Duflo map in $2+1$ quantum gravity, Classical Quantum Gravity 25 (2008), 045006, 37 pages, hep-th/0601004.
18. Fuchs J., Schweigert C., Symmetries, Lie algebras and representations. A graduate course for physicists, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997.
19. Guedes C., Oriti D., Raasakka M., Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups, J. Math. Phys. 54 (2013), 083508, 31 pages, arXiv:1301.7750.
20. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
21. Koornwinder T.H., Muller N.M., The quantum double of a (locally) compact group, J. Lie Theory 7 (1997), 101-120, q-alg/9605044.
22. Majid S., Schroers B.J., $q$-deformation and semidualization in 3D quantum gravity, J. Phys. A: Math. Theor. 42 (2009), 425402, 40 pages, arXiv:0806.2587.
23. Matschull H.-J., Welling M., Quantum mechanics of a point particle in $(2+1)$-dimensional gravity, Classical Quantum Gravity 15 (1998), 2981-3030, gr-qc/9708054.
24. Meljanac S., Samsarov A., Stojić M., Gupta K.S., $\kappa$-Minkowski spacetime and the star product realizations, Eur. Phys. J. C 53 (2008), 295-309, arXiv:0705.2471.
25. Meljanac S., Škoda Z., Svrtan D., Exponential formulas and Lie algebra type star products, SIGMA 8 (2012), 013, 15 pages, arXiv:1006.0478.
26. Noui K., Three-dimensional loop quantum gravity: towards a self-gravitating quantum field theory, Classical Quantum Gravity 24 (2007), 329-360, gr-qc/0612145.
27. Sasai Y., Sasakura N., The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime, J. High Energy Phys. 2009 (2009), no. 6, 013, 22 pages, arXiv:0902.3050.
28. Sasai Y., Sasakura N., Massive particles coupled with $2+1$ dimensional gravity and noncommutative field theory, arXiv:0902.3502.
29. Schroers B.J., Lessons from $(2+1)$-dimensional quantum gravity, PoS Proc. Sci. (2007), PoS(QG-PH), 035, 15 pages, arXiv:0710.5844.
30. Schroers B.J., Wilhelm M., Towards non-commutative deformations of relativistic wave equations in $2+1$ dimensions, SIGMA 10 (2014), 053, 23 pages, arXiv:1402.7039.
31. Staruszkiewicz A., Gravitation theory in three-dimensional space, Acta Phys. Polon. 24 (1963), 735-740.
32. 't Hooft G., Quantization of point particles in $(2+1)$-dimensional gravity and spacetime discreteness, Classical Quantum Gravity 13 (1996), 1023-1039, gr-qc/9601014.
33. Welling M., Two-particle quantum mechanics in $2+1$ gravity using non-commuting coordinates, Classical Quantum Gravity 14 (1997), 3313-3326, gr-qc/9703058.