13 Coherent State or Frame Quantization

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Their (anti-)commutator reads as

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In the purely fermionic case, k = 2, we recover the canonical anticommutation rule {A , A } = I 2

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137 Final Comments

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Let us propose in this conclusion, on an elementary level, some hints for understanding better the probabilistic duality lying at the heart of the coherent state quantization procedure Let X be an observation set equipped with a measure and let a real-valued function X x a(x) have the status of observable This means that there exists an experimental device, a , giving access to a set or spectrum of numerical outcomes a = {a j , j J } R, commonly interpreted as the set of all possible measured values of a(x) To the set a are attached two probability distributions de ned by the set of functions a = { p j (x), j J }: 1 A family of probability distributions on the set J , J j p j (x), p j (x) = 1, indexed by the observation set X This probability enj J codes what is precisely expected for this pair (observable a(x), device a ) 2 A family of probability distributions on the measure space (X , ), X x p j (x), X p j (x) (dx) = 1, indexed by the set J Now, the exclusive character of the possible outcomes a j of the measurement of a(x) implies the existence of a set of conjugate functions X x j (x), j J , playing the role of phases, and making the set of complex-valued functions j (x) =

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an orthonormal set in the Hilbert space L2 (X , ), X j (x) j (x) (dx) = j j K There follows the existence of the family of coherent states or frame in the Hilbertian closure K of the linear span of the j s: X x |x =

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(1349)

Consistency conditions have to be satis ed together with this material: they follow from the quantization scheme resulting from the frame (1349)

137 Final Comments

Condition 132 (Frame consistency conditions) (i) Spectral condition The frame quantization of the observable a(x) produces an essentially self-adjoint operator Aa in K that is diagonal in the basis { j , j J }, with spectrum precisely equal to a : Aa =

def X

a(x) |x x| (dx) =

a j | j j |

(1350)

(ii) Classical limit condition The frame {|x } depends on a parameter [0, ), |x = |x, such that the limit

def Aa (x, ) = x, |Aa ( )|x, a(x) 0

(1351)

holds for a certain topology Tcl assigned to the set of classical observables Once these conditions have been veri ed, one can start the frame quantization of other observables, for instance, the quantization of the conjugate observables j (x), and check whether the observational or experimental consequences or constraints due to this mathematical formalism are effectively in agreement with our reality Many examples will be presented in the next chapters For some of them, we have in view possible connections with objects of noncommutative geometry (such as fuzzy spheres or pseudospheres) These examples show the extreme freedom we have in analyzing a set X of data or possibilities just equipped with a measure, by just following our coherent state quantization procedure The crucial step lies in the choice of a countable orthonormal subset in L2 (X , ) obeying the nitude condition (138) A CN (or l 2 if N = ) unitary transform of this original subset would actually lead to the same speci c quantization, and the latter could also be obtained by using unitarily equivalent continuous orthonormal distributions de ned within the framework of some Gel fand triplet Of course, further structure such as a symplectic manifold combined with spectral constraints imposed on some speci c observables will considerably restrict that freedom and should lead, hopefully, to a unique solution, such as Weyl quantization, deformation quantization, and geometrical quantization are able to achieve in speci c situations Nevertheless, we believe that the generalization of the Berezin Klauder Toeplitz quantization that has been described here, and that goes far beyond the context of classical and quantum mechanics, not only sheds light on the speci c nature of the latter, but will also help to solve in a simpler way some quantization problems