Quantum Theories and Geometry
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MATHEMATICAL PHYSICS STUDIES A SUPPLEMENTARY SERIES TO LETTERS IN MATHEMATICAL PHYSICS
Editors: M. FLATO, Universite de Dijon, France E. H. LIEB, Princeton University, U.SA. W. THIRRING, Institut fur Theoretische Physik der Universiliit Wien, Austria A. TRAUTMAN, Institute of Theoretical Physics, Warsaw, Poland
Editorial Board: H. ARAKI, Kyoto University, Japan M. CAHEN, Universite Libre de Bruxelles, Belgium A. CONNES, /.lIES., France L. FADDEEV, Steklov Institute of Mathematics, Leningrad, U.S.s.R. B. NAGEL, K.T.ll., Stockholm, Sweden R. RACZKA,lnstitut Badan Jadrowych, Warsaw, Poland A. SALAM, International Centre for Theoretical Physics, Trieste, Italy W. SCHMID, Ilarvard University, U.s.A. J. SIMON, Universite de Dijon, France D. STERNHEIMER, College de France, France I. T.TODOROV, Institute of Nuclear Research, Sofia, Bulgaria J. WOLF, University of Cal ifomia, Berkeley, USA.
VOLUME 10
Quantum Theories and Geometry Edited by
M. Cahen Department o/Malhematics. Free Unil'ersil.l' oj' Bru.Hels. Bl'/gium
and
M. Flato U/liI'enit.l' oj' Dijo/l. 1-i"1Incl'
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
library of
Congres~
Catalogin, in Publication Data
Ou.ntu. th,grlU and g.o . . trV I ,little! by H. C,h,n ,nd H. Fino. p. c a . -- IHnh,lItlul phYSiCS stUd ies) BneO on lecturn glVln.1 . . . .,Ing held ,I thl FonOI1\on LIS Tr.llln, Harch 23-27, 1987 . Inc ludu Ind ... IS8N·13: 9r8-9-I-Ol0·7874-t1 1 . Qu.ntu . th ,ory--Congr""s. 2. Geo.et ry--Congrl$s,l . 11. Flalo, H. I HgShi l , 1937I. C.hen. H. IMlch e ll , 1935-
HI. S,ron.
OC173 . 96.0818 1966 53D . 1 '2--c1C 19
68- 23251
ISBN· U: 978-')4.(111)..7814-0 e-ISBN·\J: 97S.
S being defined by
the 2-form
G 1n
S(X-,Y-)
< E;,[X,Y] >.
I~
In fact ife
restrict ourselves to that case. Let us suppose now we have a covariant
*
product on
W>
that means that
v X, Y € l.i We suppose finally that
*
u
*u * vd~(E;)
v
fu
v
V u, v € A
1S a scalar product on A
Then, as In the first part :
Definition 1.
[X *
G with
If there exists an unitary representation of
as infinitesimal generator, we shall say that representation is l.i
generated by
(=
{X, X € l.i})
and denote it by
E(x), x € G, espe-
cially :
= Exp *
E(exp tX)
tx
m.
V x E Ii, t €
Conditions for existence of such a representation was given by Nelson
[17]
and Flato, Simon, Snellman and Sternheimer [18].
If
G
IS nilpotent and
W a coadjoint orbit, by induction on
dim l.i, we have
Proposition 1. such that
S =
~
dp.
J
E; A
There exists variables -+
dq.) J
(p.,q.) J
J
and:
(p.,q.), j = 1, ... ,k J
J
is a global canonical chart
(W
Ri
on
m2k ,
W
38
D.ARN,
v
k
~, x(~)
X E
a.(q)p. + a (q) J J 0
~
j=1
Hi th these very peculiar form, we push back the Moyal product from
W, using this chart;
to
h = 1,
we put
*
JR2
is thus covariant.
Moreover i f :
J
e
-i ( ) u p,q
dp I (2;r)k 2
then
k
A(P,q)3 p .(fp u) +
~
j=1
x X
k ~
j=1
A.d
J Pj
J
k ~
j=1
B.(P,q)3 J
qj
(fpu)
k
+
L B.d j=1 J qj
is a complete vector fields on
and
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