In this paper, we regularize the Kepler problem on κ-spacetime in several different ways. First, we perform a Moser-type regularization and then we proceed for the Ligon-Schaaf regularization to our problem. In particular, generalizing Heckman and de Laat [J. Symplectic Geom. 10, 463-473 (2012)] in the noncommutative context, we show that the Ligon-Schaaf regularization map following from an adaptation of the Moser regularization can be generalized to the Kepler problem on κ-spacetime.
REFERENCES
1.
B.
Cordani
, “The Kepler problem
,” in Progress in Mathematical Physics
(Birkhäuser
, Basel
, 2003
), Vol. 29
.2.
V.
Guillemin
and S.
Sternberg
, Variations on a Theme by Kepler
(AMS Colloquium Publications
, 1990
), Vol. 42
.3.
T.
Levi-Civita
, “Sur la regularisation du probleme des trois corps
,” Acta Math.
42
, 99
(1920
).4.
J.
Moser
, “Regularization of Kepler’s problem and the averaging method on a manifold
,” Commun. Pure Appl. Math.
23
, 609
(1970
).5.
J.
Milnor
, “On the geometry of the Kepler problem
,” Am. Math. Mon.
90
, 353
(1983
).6.
T.
Ligon
and M.
Schaaf
, “On the global symmetry of the classical Kepler problem
,” Rep. Math. Phys.
9
, 281
(1976
).7.
R.
Cushman
and L.
Bates
, Global Aspects of Classical Integrable Systems
(Birkhauser
, Basel
, 1997
).8.
R.
Cushman
and J. J.
Duistermaat
, “A characterization of the Ligon-Schaaf regularization map
,” Commun. Pure Appl. Math.
50
, 773
(1997
).9.
C. M.
Marle
, “A property of conformally Hamiltonian vector fields; application to the Kepler problem
,” J. Geom. Mech.
4
, 181
(2012
).10.
S.
Hu
and M.
Santoprete
, “Regularization of the Kepler problem on the sphere
,” Canad. J. Math.
66
, 760
(2014
).11.
G.
Heckman
and T.
de Laat
, “On the regularization of the Kepler problem
,” J. Symplectic Geom.
10
, 463
(2012
).12.
C.
Rovelli
, Quantum Gravity
(Cambridge University Press
, Cambridge UK
, 2004
).13.
14.
A.
Ashtekar
and J.
Stachel
, Conceptual Problems of Quantum Gravity
(Birkhauser
, Boston, Basel, Berlin
, 1991
).15.
N.
Seiberg
and E.
Witten
, “String theory and noncommutative geometry
,” J. High Energy Phys.
9909
, 032
(1999
).16.
S.
Doplicher
, K.
Fredenhagen
, and J.
Roberts
, “The quantum structure of space-time at the Planck scale and quantum fields
,” Commun. Math. Phys.
172
, 187
(1995
).17.
H.
Snyder
, “Quantized space-time
,” Phys. Rev.
71
, 38
(1947
).18.
V. O.
Rivelles
, “A review of noncommutative field theories
,” J. Phys.: Conf. Ser.
287
, 012012
(2011
).19.
V. O.
Rivelles
, “Noncommutative field theories and gravity
,” Phys. Lett. B
558
, 191
(2003
).20.
G.
Amelino-Camelia
, “Testable scenario for relativity with minimum length
,” Phys. Lett. B
510
, 255
(2001
).21.
J.
Kowalski-Glikman
, “Observer independent quantum of mass
,” Phys. Lett. A
286
, 391
(2001
).22.
G.
Amelino-Camelia
, D.
Benedetti
, F.
D’Andrea
, and A.
Procaccini
, “Comparison of relativity theories with observer independent scales of both velocity and length/mass
,” Classical Quantum Gravity
20
, 5353
(2003
).23.
J.
Lukierski
and A.
Nowicki
, “Doubly special relativity versus kappa deformation of relativistic kinematics
,” Int. J. Mod. Phys. A
18
, 7
(2003
).24.
J.
Kowalski-Glikman
and S.
Nowak
, “Noncommutative space-time of doubly special relativity theories
,” Int. J. Mod. Phys. D
12
, 299
(2003
).25.
S.
Meljanac
, A.
Samsarov
, M.
Stojic
, and K. S.
Gupta
, “Kappa-Minkowski space-time and the star product realizations
,” Eur. Phys. J. C
53
, 295
(2008
).26.
E.
Harikumar
and A. K.
Kapoor
, “Newton’s equation on the kappa space-time and the Kepler problem
,” Mod. Phys. Lett. A
25
, 2991
(2010
).27.
P.
Guha
, E.
Harikumar
, and N. S.
Zuhair
, “MICZ-Kepler systems in noncommutative space and duality of force laws
,” Int. J. Mod. Phys. A
29
, 1450187
(2014
).28.
P.
Guha
, E.
Harikumar
, and N. S.
Zuhair
, “Fradkin-Bacry-Ruegg-Souriau vector in kappa-deformed space-time
,” Eur. Phys. J. Plus
130
, 205
(2015
).29.
M. R.
Douglas
and N. A.
Nekrasov
, “Noncommutative field theory
,” Rev. Mod. Phys.
73
, 977
(2001
).30.
J.
Lukierski
, H.
Ruegg
, and W.
Zakrzewski
, “Classical and quantum mechanics of free κ-relativistic systems
,” Ann. Phys.
243
, 90
(1995
).31.
S.
Kresic-Juric
, S.
Meljanac
, and M.
Stojic
, “Covariant realizations of kappa-deformed space
,” Eur. Phys. J. C
51
, 229
(2007
).32.
E.
Harikumar
, T.
Juric
, and S.
Meljanac
, “Geodesic equation in κ-Minkowski spacetime
,” Phys. Rev. D
86
, 045002
(2012
).33.
G.
Amelino-Camelia
and M.
Arzano
, “Coproduct and star product in field theories on Lie-algebra non-commutative space-times
,” Phys. Rev. D
65
, 084044
(2002
).34.
A.
Borowiec
, Kumar S.
Gupta
, S.
Meljanac
, and A.
Pachol
, “Constraints on the quantum gravity scale from κ-Minkowski spacetime
,” Europhys. Lett.
92
, 20006
(2010
).35.
K. S.
Gupta
, E.
Harikumar
, T.
Juric
, S.
Meljanac
, and A.
Samsarov
, “Effects of noncommutativity on the black hole entropy
,” Adv. High Energy Phys.
2014
, 139172
.© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
