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where, at given j = 1, 2, , J, the integer k assumes its values in the range 0 u k u 2 j 1 The total number of elements of this orthonormal system is N = 2 J+1 N 2 J+1 The normalization function is equal to N (x) = , and this n=1 = | n (x)| = 2 clearly diverges at the limit J Then, it is remarkable if not expected that spectral values as well as average values of the position operator are given by x 0 |Ax |x 0 = (2k + 1)/2 J+1 for k/2 J u x 0 u (k + 1)/2 J where 0 u k u 2 J 1 Hence, our quantization scheme yields a dyadic discretization of the localization in the unit interval

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Now we choose another orthonormal system, namely, the rst two elements of the trigonometric Fourier basis, (1432) 1 (x) = 1(x), 2 (x) = 2 sin 2 x Then we have

N (x) =

(1433)

(1434)

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To any integrable function f (x) on the interval there corresponds the linear operator A f on R2 or C2 (in its matrix form): Af = dx f (x) 1 2 0 dx f (x) sin 2 x

(1435)

This is not a great image builder, being a traditional Dutch product that you have with french fries rather than one you expect to see on an Italian s table Also, unlike the spread, the product and pack lack any real added value versus the competition and as a result mayonnaise is small in sales terms Support is being reduced and it may eventually be delisted

Like in the previous case, with the choice f = 1 we recover the identity, whereas for f = 2 we get A 2 = 1 , the rst Pauli matrix We now have to deal with a Jordan algebra of operators A f , like in the N = 3 real case in the previous section It is generated by the identity matrix and the two real Pauli matrices 1 and 3 In this context, the position operator is given by Ax =

1 2 1 2

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1 2

with eigenvalues 1 1 Note its average values as a function of the coherent state 2 2 parameter x 0 [0, 1]: x 0 |Ax |x 0 = 1 2 sin 2 x 0 2 1 + 2 sin2 2 x 0

In Figure 141 we give the curve of x 0 |Ax |x 0 as a function of x 0 It is interesting to compare with the two-dimensional Haar quantization presented in the previous subsection

We now come back to the unit circle We gave an example of quantization in Section 135, resulting from the orthonormal system {cos , sin } in L2 (S 1 , d / ) Here, we explore the coherent state quantization based on an arbitrary number of elements of the Fourier exponential basis [187] As an interesting byproduct of this fuzzy circle, we give an expression for the phase or angle operator, and we discuss its relevance in comparison with various phase operators proposed by other authors

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In classical physics, angle and action are conjugated canonical variables It was hoped to get the exact correspondence for their quantum counterpart after some quantization procedure Taking for granted (true for the harmonic oscillator) the correspondence I N , N being the number operator, and , we should get

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