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By H. Araki (auth.), M. Cahen, M. Flato (eds.)
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Additional resources for Quantum Theories and Geometry
Example text
T 11;1 = Q) u Wn nh} n=l n=l by E(x)(I;) = (x + 0 ~ x·J x. p E Coo(G) defined on ~. e. if 1S the unit sphere of ~ Let ~ rr :L ~ ~* rr of G, rr irreductible and unitary, = 2rri < rr(X)z,z > by: ~(z)(X) G weight for t 1n rr. Then V z E L, V X E with Lie algebra with the Killing form, and denote by ~* X, and if 1S realized on the Hilbert J(, let us define T be a maximal torus in and rr W(rr) ! i:A. c~. ~. We identify the maximal will be the orbit of AE t* 81 is a -1 (W(rr)) ~ W(rr) principal bundle.
Q. Let us now extend and u * 0 Q. ) v for L2 (JR2k) and the space K and: u, v to some spaces of distributions. s(JR2k), proposition 1 gives a meaning to the formula: 27 THE *-EXPONENTIAL =
1 The level sets of (I,E) : T*S ... the regular values of folds of T*S. (I,E). :m 2 are compact and connected, for They are smooth 2-dimensional submani- On them, the commuting vector fields (:m 2 ,+), transitive action of HI' HE define a so these level sets are tori of the form (:m 2 ,+)/P(I,E), where P(I,E) = {(T 1 ,T 2 ) E HE :m 2 ; after time is the period lattice. tori, one gets that flow of HI after time T1 0 flow of T2 = identity on the level set of (I,E)} Using this action in order to parametrize the HI = a/at 1 , HE = a/at 2 are constant vector fields.