It is then easily veri ed that z is given by the M bius action (829): z = g z in Visual Studio .NET

Encoder QR Code in Visual Studio .NET It is then easily veri ed that z is given by the M bius action (829): z = g z
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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)
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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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87 Group-Theoretical Content of the Coherent States
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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)
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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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This countable set of representations constitutes the almost complete holomorphic discrete series of representations of SU (1, 1) [97 99] It is almost complete because the lowest one, which corresponds to the value = 1/2, requires a special treatment owing to the nonexistence of the inner product (837) in this case: there is no Fock Bargmann realization in that case We will come back to this important question in the last section Had we considered the continuous set [1/2, + ), we would have been led to involving the universal covering of SU (1, 1) [100, 101] The matrix elements of the operator U ( g ) with respect to the orthonormal basis (838) are given (see, eg, [102] or Appendix A in [103]) in terms of hypergeometrical polynomials by U ( g ) = p n |U ( g )| p n = nn ~ where ( , ) = n> = n n> = n
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n > ! (2 + n > ) n < ! (2 + n < )
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2 n > n < | |2 | |2 , (840)
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( ( , ))n > n < 2F 1 (n > n < )!
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n < , n > + 2 ; n > n < + 1 ;
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n> =
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max (n, n ) v 0 min
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| | 1 Owing to the relation | |2 = 1 | |2 , this expression is alternatively given in terms of Jacobi polynomials as follows: 1/2
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U (g) = nn
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n < ! (2 + n > ) n > ! (2 + n < ) ~
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2 n > n < ( ( , ))n > n < (n > n < )! P n <>
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(n n < , 2 1)
1 | |2 1 + | |2
(841)
8 SU (1,1) or SL(2, R) Coherent States
873 Lie Algebra Aspects
Any element g SU (1, 1) can also be factorized, in a nonunique way, in terms of three one-parameter subgroup elements: g = h( ) s(u) l(v ) Besides the sign , the rst factor was already encountered in the Cartan decomposition (835), whereas the others are of noncompact hyperbolic type and are given by s(u) = cosh u sinh u sinh u cosh u , l(v ) = cosh v i sinh v i sinh v cosh v , u, v R (842) The rst subgroup is isomorphic to U (1), whereas the two others are isomorphic to R Their respective generators, N , = 0, 1, 2, are de ned by h( ) = e N 0 , s(u) = e u N 1 , h( ) = e v N 2 , (843)
and are given in terms of the Pauli matrices by N0 = i 3 , 2 N1 = 1 1 , 2 1 N 2 = 2 2 (844)
They form a basis of the Lie algebra su(1, 1) and obey the commutation relations [N 0 , N 1 ] = N 2 , [N 0 , N 2 ] = N 1 , [N 1 , N 2 ] = N 0 (845)
Their respective self-adjoint representatives under the unitary irreducible representation (839), de ned generically as i / t U ( g (t)), are the following differential operators on the Fock Bargmann space F B d + , dz i d + i z , i N 1 K 1 = (1 z 2 ) 2 dz 1 d + z , i N 2 K 2 = (1 + z 2 ) 2 dz i N0 K0 = z and obey the commutation rules [K 0 , K 1 ] = i K 2 , [K 0 , K 2 ] = i K 1 , [K 1 , K 2 ] = i K 0 (847) (846a) (846b) (846c)
We may check that the elements of the orthonormal basis (838) are eigenvectors of the compact generator K 0 with equally spaced eigenvalues: K 0 | p n = ( + n) | p n (848)
The particular element | p 0 of the basis is a lowest weight or vacuum state for the representations U Indeed, let us introduce the two operators with their commutation relation: K = i (K 1 iK 2 ) = K 2 iK 1 , [K + , K ] = 2K 0 (849)