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Fuzzy Sphere This is an extension to the sphere S 2 of the quantization of the unit circle It is a nice illustration of noncommutative geometry (approached in a rather pedestrian way) We show explicitly how the coherent state quantization of the ordinary sphere leads to its fuzzy geometry The continuous limit at in nite spins restores commutativity Fuzzy Hyperboloid We then describe the construction of the two-dimensional fuzzy de Sitter hyperboloids by using a coherent state quantization 18 Conclusion and Outlook In this last chapter we give some nal remarks and suggestions for future developments of the formalism presented
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2 The Standard Coherent States: the Basics
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21 Schr dinger De nition
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The coherent states, as they were found by Schr dinger [1, 17], are denoted by |z in Dirac ket notation, where z = |z| e i is a complex parameter They are states for which the mean values are the classical sinusoidal solutions of a one-dimensional harmonic oscillator with mass m and frequency : z|Q(t)|z = 2l c |z| cos ( t ) The various symbols that are involved in this de nition are as follows: the characteristic length l c =
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the Hilbert space H of quantum states for an object which classically would be viewed as a point particle of mass m, moving on the real line, and subjected to a harmonic potential with constant k = m 2 , H=
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+ 1 m 2 Q 2 is the Hamiltonian, 2
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operators position Q and momentum P are self-adjoint in the Hilbert space H of quantum states, their commutation rule is canonical, that is, [Q, P ] = i I d , the time evolution of the position operator is de ned as Q(t) = e
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In the sequel, we present the different ways to construct these speci c states and their basic properties We also explain the raison d tre of the adjective coherent
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22 Four Representations of Quantum States
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The formalism of quantum mechanics allows different representations of quantum states: position, momentum, energy or number or Fock representation, and phase space or analytical or Fock Bargmann representation
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Coherent States in Quantum Physics Jean-Pierre Gazeau Copyright 2009 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40709-5
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2 The Standard Coherent States: the Basics
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221 Position Representation
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The original Schr dinger approach was carried out in the position representation Operator Q is a multiplication operator acting in the space H of wave functions (x, t) of the quantum entity Q (x, t) = x (x, t) , P (x, t) = i (x, t) x (23)
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The quantity P(S) = S | (x, t)|2 dx is interpreted as the probability that, at the instant t, the object considered lies within the set S R, in the sense that a classical localization experiment would nd it in S with probability P(S) Consistently, we have the normalization 1 = R | (x, t)|2 dx < , and so, at a given time t, ~ H = L2 (R) Time evolution of the wave function is ruled by the Schr dinger equation H (x, t) = i (x, t) t
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or equivalently (x, t) = e H(t t 0 ) (x, t 0 ) i Stationary solutions read as (x, t) = e E n t n (x), where the energy eigenvalues are equally distributed on the positive line, E n = n + 1 , n = 0, 1, 2, To 2 each eigenvalue corresponds the normalized eigenstate n , H n = E n n , n (x) = n
x2 1 1 e 4l c Hn 2 n n! 2 l c 2 +
x 2l c
, (25)
| n (x)| dx = 1
Here, Hn denotes the Hermite polynomial of degree n [18], with n nodes The functions { n , n N} form an orthonormal basis of the Hilbert space H = L2 (R): mn = m | n = H, =