Its lower symbol reads as m |[N , A ]| m = 2 NM

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(1476)

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144 Coherent State Quantization of the Unit Circle and the Quantum Phase Operator

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Its asymptotic behavior at large M > N , > m |[N , A ]| m W i N

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(1477)

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shows that we have, to some extent, improved the situation

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1444 A Phase Operator from the Interplay Between Finite and In nite Dimensions

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We show now that there is no need to discretize the angle variable as in [193] to recover a suitable commutation relation We adopt instead the Hilbert space of square-integrable functions on the circle as the natural framework for de ning an appropriate phase operator in a nite-dimensional subspace So we take as an observation set X the unit circle S 1 provided with the measure (d ) = d /2 The Hilbert space is L2 (X , ) = L2 (S 1 , d /2 ) and has the inner product

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(1478)

In this space we choose as an orthonormal set the N rst Fourier exponentials with negative frequencies:

N 1

n ( ) = e in ,

with N ( ) =

| n ( )|2 = N

(1479)

The phase states are now de ned as the corresponding coherent states : 1 | ) = N

N 1

e in | n ,

(1480)

where the kets | n can be directly identi ed with the number states |n , and the round bracket denotes the continuous labeling of this family We have, by construc~ tion, normalization and resolution of the unity in HN = CN :

( | ) = 1,

| )( | N (d ) = I N

(1481)

Unlike in (1445), the states (1480) are not orthogonal but overlap as ( | ) = ei

N 1 2 (

N 2 ( ) sin 1 ( ) 2

(1482)

14 Coherent State Quantization of Finite Set, Unit Interval, and Circle

Note that for N large enough these states contain all the Pegg Barnett phase states and besides they form a continuous family labeled by the points of the circle The coherent state quantization of a particular function f ( ) with respect to the continuous set (1480) yields the operator A f de ned by f ( )

f ( )| )( | N (d ) = A f

(1483)

An analogous procedure has been used within the framework of positive-operator-valued measures [194, 195]: the phase states are expanded over an in nite orthogonal basis with the known drawback of de ning the convergence of the series | = n e in |n from the Hilbertian arena and the related questions concerning operator domains When it is expressed in terms of the number states, the operator (1483) takes the form

N 1

Af =

n,n =0

c n n ( f )|n n | ,

(1484)

where c n ( f ) is the Fourier coef cient of the function f ( ),

cn( f ) =

f ( )e in

d 2

Therefore, the existence of the quantum version of f is ruled by the existence of its Fourier transform Note that A f is self-adjoint only when f ( ) is real-valued In particular, a self-adjoint phase operator of the Toeplitz matrix type is readily obtained by choosing f ( ) = :

N 1

A = i

n= /n n,n =0

1 |n n |, n n

(1485)

an expression that has to be compared with (1213) Its lower symbol or expectation value in a coherent state is given by ( |A | ) = i N

N 1

n,n =0 n= /n

e i(n n ) n n

(1486)

Owing to the continuous nature of the set of | ), all operators produced by this quantization are different from the Pegg Barnett operators As a matter of fact, the commutator [N , A ] expressed in terms of the number basis reads as

N 1

[N , A ] = i

n,n =0 n= /n

|n n | = iI N + ( i)IN ,

(1487)

144 Coherent State Quantization of the Unit Circle and the Quantum Phase Operator

and has all diagonal elements equal to 0 Here IN = n,n =0 |n n | is the N ~ N matrix with all entries equal to 1 The spectrum of this matrix is reduced to the values 0 (degenerate N 1 times) and N, that is, (1/N ) IN is an orthogonal projector on one vector More precisely, the normalized eigenvector corresponding to the eigenvalue N is along the diagonal in the rst quadrant in CN : 1 |v N = | = 0) = N

N 1

N 1