A global construction of a family of symplectic connections on a cotangent bundle compatible with some linear symmetric connection on the base space is proposed. Examples of the compatible symplectic connections are given. A detailed analysis of an Abelian connection and of flat sections for this kind of symplectic connection in the Fedosov algorithm are presented. Some properties of the *–product determined by the compatible symplectic connection are shown.
REFERENCES
1.
J. E.
Moyal
, Proc. Cambridge Philos. Soc.
45
, 99
(1949
).2.
reprinted (
Dover
, New York
, 1950
).3.
E. P.
Wigner
, Phys. Rev.
40
, 749
(1932
).4.
H. J.
Groenewold
, Physica
12
, 405
(1946
).5.
F.
Bayen
, M.
Flato
, C.
Fronsdal
, A.
Lichnerowicz
, and D.
Sternheimer
, Lett. Math. Phys.
1
, 521
(1977
).6.
F.
Bayen
, M.
Flato
, C.
Fronsdal
, A.
Lichnerowicz
, and D.
Sternheimer
, Ann. Phys. NY
111
, 61
(1978
);F.
Bayen
, M.
Flato
, C.
Fronsdal
, A.
Lichnerowicz
, and D.
Sternheimer
, Ann. Phys. (N.Y.)
111
, 111
(1978
).7.
8.
B.
Fedosov
, Deformation Quantization and Index Theory
, (Akademie Verlag
, Berlin
, 1996
).9.
P.
Bieliavsky
, M.
Cahen
, S.
Gutt
, J.
Rawnsley
, and L.
Schwachhöfer
, Int. J. Geom. Methods Mod. Phys.
3
, 375
(2006
).10.
11.
J.
Tosiek
, J. Phys.: Conf. Ser.
128
, 012024
(2008
).12.
F.
Bourgeois
and M.
Cahen
, J. Geom. Phys.
30
, 233
(1999
).13.
M.
Cahen
and S.
Gutt
, Lett. Math. Phys.
6
, 395
(1982
).14.
M.
Bordemann
, N.
Neumaier
, and S.
Waldmann
, J. Geom. Phys.
29
, 199
(1999
).15.
B.
Mielnik
, private communication, 18 May 1998
.16.
17.
I.
Gelfand
, V.
Retakh
, and M.
Shubin
, Adv. Math.
136
, 104
(1998
).18.
S.
Kobayashi
and K.
Nomizu
, Foundations of Differential Geometry
(Interscience
, New York
, 1963
).19.
K.
Yano
and S.
Ishihara
, Tangent and Cotangent Bundles
(Marcel Dekker
, New York
, 1973
).20.
S.
Gutt
and J.
Rawnsley
, Lett. Math. Phys.
66
, 123
(2003
).21.
O.
Kravchenko
, e-print arXiv:math.SG/0008157.22.
I.
Vaisman
, J. Math. Phys.
43
, 283
(2002
).© 2011 American Institute of Physics.
2011
American Institute of Physics
You do not currently have access to this content.
