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present context when the unit disk, as a symplectic manifold, is given a phasespace meaning The Cartan decomposition of SU (1, 1), denoted by SU (1, 1) = P H, means that any g SU (1, 1) can be written as the product g = p h, with p P and h H It is de ned by the Cartan involution i ph : g ( g ) 1 in the sense that P is made of all p SU (1, 1) such that i ph ( p) = p 1 , that is, p = p is Hermitian, while H has all its elements unchanged under i ph , that is, h = h 1 , which means ~ that h is unitary In consequence, H = U (1) is the unitary subgroup of SU (1, 1) The decomposition reads explicitly SU (1, 1) with p(z) = and h( ) = e i /2 0 0 e i /2 , = 2 arg , 0 u < 4 (835) z z , z = 1 , = (1 |z|2 ) 1/2 (834) g= = p(z) h( ) , (833)



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The bundle section 11)D z p(z) P gives the unit disk D a symmetric space realization identi ed with the coset space SU (1, 1)/H We remark that p 2 = g g and that ( p(z)) 1 = p( z) We can exploit the Cartan factorization by making SU (1, 1) act on D through a left action on the set of matrices p(z) Explicitly, g : p(z) p(z ) de ned by g p(z) = p(z ) h (836)

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872 Discrete Series of SU(1, 1)

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We now consider a class of unitary irreducible representations of SU (1, 1), precisely indexed by the parameter appearing in the measure on the unit disk, and involved in the construction of the coherent states, to which this chapter is devoted For a given > 1, we introduce the Fock Bargmann Hilbert space F B of all analytical functions f (z) on D that are square-integrable with respect to the scalar product: f 1| f 2 = 2 1 2 f 1 (z) f 2 (z) (1 |z|2 )2 2 d 2 z (837)

11) A nontrivial ber bundle consists of four objects, (E , B , , F ), where E, B, and F are topological spaces and : E B is a continuous surjection such that E is locally (but not globally!) homeomorphic to the Cartesian product B ~ F B is called

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the base space of the bundle, E the total space, and F the ber The map is called the projection map (or bundle projection) A section (or cross section) is a continuous map, s : B E , such that (s(x)) = x for all x in B

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Note that the elements of this space are just the conjugate of the elements of K+ cleared of their nonanalytical factor (1 |z|2 )2 The orthonormal basis given by (86) is now made of powers of z suitably normalized: p n (z) = (2 )n n z n! with n N (838)

We de ne, for = 1, 3/2, 2, 5/2, the unitary irreducible representation g= U ( g )

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of SU (1, 1) on F B by F B f (z) U ( g ) f (z) = ( z + ) 2 f z z + (839)

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