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d 2 z f (z, z ) (z, z ) ,
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(1219)
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and the notation is the same for all tempered distributions T In view of (1218), we can extend (1219) to locally integrable functions f (z, z ) that increase like 2 e |z| p(z, z ) for some < 1 and some polynomial p, and to all distributions that are derivatives (in the distributional sense) of such functions We recall here that partial derivatives of distributions are given by r s T, z r z s = ( 1)r+s T, r s z r z s (1220)
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We also recall that the multiplication of distributions T by smooth functions (z, z ) C (R2 ) is understood through S(R2 ) T , = T ,
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Now, equipped with the above distributional material, we consider as acceptable observables those distributions in D (R2 ) that obey the following condition
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De nition 122 (Coherent state quantizable observable) A distribution T D (R2 ) is a coherent state quantizable classical observable if there exists < 1 such that the 2 product e |z| T S (R2 ), that is, is a tempered distribution
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125 Quantization of Distributions: Dirac and Others
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Of course, all compactly supported distributions such as the Dirac distribution and its derivatives are tempered and so are coherent state quantizable classical observable The Dirac distribution supported by the origin of the complex plane is denoted as usual by (and abusively in the present context by (z, z )): C (R2 ) , = d 2 z (z, z ) (z, z ) = (0, 0)
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C def
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(1222)
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Let us now coherent state quantize the Dirac distribution using the recipe provided by (123) and (128): 1 (z, z ) |z z| d 2 z =
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1 n!n !
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d 2 z |z|2 n n z z (z, z ) |n n | e
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1 1 = |z = 0 z = 0| = 00 (1223) We thus nd that the ground state (as a projector) is the quantized version of the Dirac distribution supported by the origin of the phase space The obtaining of all possible diagonal projectors nn = |n n| or even all possible operators nn = |n n | is based on the quantization of partial derivatives of the distribution First, let us quantize the various derivatives of the Dirac distribution: U a,b = = b a (z, z ) |z z| d 2 z z b z a ( 1)n+a
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b! a! 1 n b,n a nn (b n)! n!n !
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Once this quantity U a,b is at hand, one can invert the formula to get the operator r+s,r = |r + s r| as
r+s,r =
r!(r + s)!( 1)s
1 U p, p+s , p!(s + p)!(r p)!
(1225)
and so its upper symbol is given by the distribution supported by the origin:
f r+s,r (z, z ) =
r!(r + s)!( 1)s
1 p!(s + p)!(r p)!
p+s p (z, z ) z p+s z p (1226)
Note that this distribution, as is well known, can be approached, in the sense of the topology on D (R2 ), by smooth functions, such as linear combinations of derivatives of Gaussians The projectors r,r are then obtained trivially by setting s = 0 in (1225) to get
r,r =
1 p!
U p, p
(1227)
12 Standard Coherent State Quantization: the Klauder Berezin Approach
This phase space formulation of quantum mechanics enables us to mimic at the level of functions and distributions the algebraic manipulations on operators within the quantum context By carrying out the coherent state quantization of Cartesian powers of planes, we could obtain an interesting functional portrait in terms of a star product on distributions for the quantum logic based on manipulations of tensor products of quantum states
126 Finite-Dimensional Canonical Case
The idea of exploring various aspects of quantum mechanics by restricting the Hilbertian framework to nite-dimensional space is not new, and has been intensively used in the last decade, mainly in the context of quantum optics [165 167], but also in the perspective of noncommutative geometry and fuzzy geometrical objects [168], or in matrix model approaches in problems such as the quantum Hall effect [169] For quantum optics, a comprehensive review (mainly devoted to the Wigner function) is provided in [170] In [166, 167], the authors de ned normalized nite-dimensional coherent states by truncating the Fock expansion of the standard coherent states Let us see through the approach presented in this chapter how we recover their coherent states [171] We just restrict the choice of the orthonormal set { n } to a nite subset of it, more precisely to the rst N elements: |z|2 n z n (z) = e 2 , n!