ap-3-10.dvi Acta Polytechnica Vol. 50 No. 3/2010 Coherent State Quantization and Moment Problem J. P. Gazeau, M. C. Baldiotti, D. M. Gitman Abstract Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion. 1 Introduction One of the most interesting properties of standard or Glauber coherent states |z〉 [1, 2, 3, 4] is the Bayesian duality [5, 6] that they encode between the discrete Poisson probability distribution, n �→ e−|z| 2 |z|2/n!, of obtaining n quantum excitations (“photons” or “quanta”) in a measurement through some counting device, and the continuous Gamma probability distri- bution measure |z|2 �→ e−|z| 2 |z|2/n! on the classical phase space. For this latter distribution, |z|2 is itself a randomvariable, denoting the averagenumber of pho- tons, given that n photons have been counted. Such a duality underlies the construction of all types of coher- ent state families, provided they satisfy a resolution of the unity condition. It turns out that this condition is equivalent to setting up a “positive operator valued measure” (POVM) [7, 4] on the phase space. Such a measure, in turn, leads to the quantization of the classical phase space, which associates to each point z ≡ (q + ip)/ √ 2 the one dimensional projection op- erator Pz, projecting onto to the subspace generated by the coherent state vector, and then for z �= z′, PzPz′ �= Pz′Pz). This “Berezin-Klauder-Töplitz” quantization (or “anti-Wick”) [1, 8, 9] turns out, in this case, to be equivalent to the canonical quantiza- tion procedure. Clearly, this non-commutative version of the complex plane is intrinsically based on the non- negative integers (appearing in the n! term). We then follow a similar path by considering sequences of non- negativenumberswhichare far ornot fromthenatural numbers [10]. The resulting quantizationswill then be looked upon as generalizations of the one yielded by the standard coherent states. We illustrate our approach with the elementary model of the 2-d motion of a charged particle in a uniformmagnetic field [11, 12]. By using a solution to a version of the Stieltjes moment problem [13, 14] we construct a family of coherent states for this model. We prove that these states form an overcomplete set that is normalized and resolves the unity. We then carry out the corresponding coherent state quantiza- tion and we examine the consequences in terms of its probabilistic, functional, and localization aspects. This article is organized as follows. In Section 2, we briefly review the standard coherent states and the way they allow painless quantization of the complex plane viewed as a phase space. The so-called non- linear coherent states built from arbitrary sequences of numbers are described in Section 3 and we show how the moment problem immediately emerges from the exigence of unity resolution. If the positive case, the corresponding quantization of the complex plane is described in Section 4. In Section 5 we apply our formalism to themotion of a chargedparticle in a uni- formmagnetic field. There exist two families of coher- ent states for suchamodel, namely theMalkin-Man’ko states [15], which are just tensor products of standard coherent states, and the Kowalski-Rembielinski states [16]. By introducing a kind of squeezing parameter q = eλ > 1 we extend the definition of the latter and solve the corresponding Stieltjes moment prob- lem. This allows us to proceed with the quantization of the physical quantities and illustrate our studywith numerical investigation. 2 Quantization with standard coherent states and A short review of standard CS Let H be a separable (complex) Hilbert space with orthonormal basis e0, e1, . . . , en ≡ |en〉, . . .. To each complex number z ∈ C there corresponds the follow- ing vector in H: |z〉= ∞∑ n=0 e− |z|2 2 zn√ n! |en〉 . (1) Selected Topics in Mathematical and Particle Physics, Prague, May 5–7, 2009 30 Acta Polytechnica Vol. 50 No. 3/2010 Such vectors are the well-known Glauber-Klauder- Schrödinger-Sudarshan coherent states or standard co- herent states. They are distinguished by many prop- erties. Here we particularly retain the following. (i) 〈z|z〉 =1 (normalization). (ii) Themap C � z �→ |z〉 is continuous (continuity). (iii) The map N ∈ n �→ |〈en|z〉|2 = e−|z| 2 |z|2n/n! is a Poisson probability distribution with aver- age number of occurrences equal to |z|2 (discrete probabilistic content). (iv) The map C � z �→ |〈en|z〉|2 = e−|z| 2 |z|2n/n! is a Gamma probability distribution (with respect to the square of the radial variable) with n as a shape parameter (continuous probabilistic content). (v) There holds resolution of the unity in H: I = ∫ C d2z π Pz , (2) where Pz = |z〉〈z| is the orthogonal projector on vector |z〉 and the integral should be under- stood in the weak sense. The proof is straight- forward and stems from the orthogonality of the Fourier exponentials and fromthe integral expres- sion of the gamma function which solves the mo- ment problem for the factorial n!∫ C d2z π Pz = ∞∑ n,n′=0 |en〉〈en′| 1√ n!n′! · ∫ C d2z π e−|z| 2 znz̄n ′ = (3) ∞∑ n=0 |en〉〈en′| = I . Berezin-Klauder-Toeplitz-“anti-Wick” quantization or “coherent state quantization” Property (v) allows to define 1. a normalized positive operator-valued measure (POVM) on the complex plane equipped with its Lebesgue measure d2z π and its σ−algebra F of Borel sets: F � Δ �→ ∫ Δ d2z π Pz ∈ L(H)+ , (4) whereL(H)+ is the cone of positive bounded op- erators on H. 2. a quantization of the complex plane, which means that to a function f(z, z̄) in the complex plane there corresponds the operator Af in H defined by f �→ Af = ∫ C d2z π f(z, z̄)Pz = ∞∑ n,n′=0 |en〉〈en′| 1 √ n!n′! · (5) ∫ C d2z π f(z, z̄)e−|z| 2 znz̄n ′ provided that weak convergence holds. For the simplest functions f(z) = z and f(z) = z̄ we obtain Az = â , â |en〉 = √ n|en−1〉 , (6) â|e0〉 = 0 , (lowering operator) Az̄ = â † , ↠|en〉 = √ n +1|en+1〉 (7) (raising operator) . These two basic operators obey the canonical com- mutation rule : [â, â†] = I. The number operator N̂ = â†â is such that its spectrum is exactly N with eigenvectors en : N̂|en〉 = n|en〉. The fact that the complex plane has become non-commutative is appar- ent from the quantization of the real and imaginary parts of z = 1 √ 2 (q + ip): Aq def = Q = 1√ 2 (â + â†) , (8) Ap def = P = 1 √ 2i (â − â†) , [Q, P ] = iI . 3 Coherent states for generic sequences Let X = {xn}n∈N be a strictly increasing sequence such that x0 = 0 and lim n→∞ xn = ∞. Then its associ- ated exponential E(t)= +∞∑ n=0 tn xn! , xn! ≡ x1x2 · · · xn , x0! = 1 , (9) has an infinite convergence radius. Associated “co- herent states” (“non-linear CS” in Quantum Optics) read as elements of H, a separable Hilbert space with orthonormal basis {|en〉 , n ∈ N}: |vz〉 = ∞∑ n=0 1√ E(|z|2) zn √ xn! |en〉 . (10) These vectors still enjoy someproperties similar to the standard ones. (i) 〈vz|vz〉 =1 (normalization). (ii) The map C � z �→ |vz〉 is continuous (continu- ity). 31 Acta Polytechnica Vol. 50 No. 3/2010 (iii) The map N ∈ n �→ |〈en|vz〉|2 = |z|2n E(|z|2)xn! is a Poisson-like distribution with average number of occurrences equal to |z|2 (discreteprobabilistic content). Consider the discrete probability distributionwith pa- rameter t ≥ 0: n �→ p(n;t)= 1 E(t) tn xn! . (11) The average of the random variable n �→ xn is 〈xn〉 = t. Contrariwise to the standard case X = N, the continuous (gammalike) distribution t �→ 1 E(t) tn xn! with parameter n is not a probabilitydistributionwith respect to the Lebesgue measure dt:∫ +∞ 0 dt E(t) tn xn! def = μn �=1 . (12) Finding the right measure amounts to solve a usually intractable moment problem. So, the map C � z �→ |〈en|vz〉|2 = |z|2n/ ( E(|z|2)xn! ) is not a (Gamma-like) probability distribution (with respect to the square of the radial variable in the complex plane)with xn+1 as a shapeparameter, and this is a serious setback for the Berezin-Toeplitz quantization program. Indeed there is no reason to get now the resolution of the unity: with Pz = |vz〉〈vz|,∫ C d2z π Pz = ∞∑ n,n′=0 |en〉〈(en′| 1√ xn!x′n! · ∫ C d2z π 1 E(|z|2) znz̄n ′ = (13) ∞∑ n=0 1 xn! I(n)|en〉〈(en| def = F . Here F is a diagonal operator determined by the se- quence of integrals I(n) = ∫ +∞ 0 tn dt E(t) . These in- tegrals form a sequence of Stieltjes moments for the measure dt E(t) . If the moment problem has a solution. Sup- pose that the Stieltjes moment problem [13, 14] has a solution for the sequence (xn!)n∈N, i.e. there exists a probability distribution t �→ w(t) on [0,+∞) with infinite support such that xn! = ∫ +∞ 0 tn w(t)dt . (14) We know that a necessary and sufficient condition for this is that the two matrices⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 1 x1! x2! . . . xn! x1! x2! x3! . . . xn+1! x2! x3! x4! . . . xn+2! ... ... ... ... ... xn! xn+1! xn+2! . . . x2n! ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ , (15) ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ x1! x2! x3! . . . xn+1! x2! x3! x4! . . . xn+2! x3! x4! x5! . . . xn+3! ... ... ... ... ... xn+1! xn+2! xn+3! . . . x2n+1! ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ have strictly positive determinants for all n. Then, a natural approach is just to modify the measure on C by including the weight w(|z|2)E(|z|2). We then ob- tain the resolution of the identity:∫ C d2z π w(|z|2)E(|z|2)Pz = ∞∑ n,n′=0 |en〉〈(en′| 1√ xn!x′n! · (16) ∫ C d2z π w(|z|2)znz̄n ′ = ∞∑ n=0 1 xn! ∫ +∞ 0 tn w(t)dt |en〉〈(en| = I . If the moment problem is solved by a measure t �→ w(t), then the vectors |vz〉 enjoy all needed properties for quantization: (iv) Themap C � z �→ |〈en|vz〉|2 = |z|2n/ ( E(|z|2)xn! ) is a (Gamma-like) probability distribution (with respect to the square of the radial variable) with xn+1 as a shape parameter and with respect to the modified measure on the complex plane ν(dz) def = w(|z|2)E(|z|2) d2z π . (17) Note that we might face with (xn!) an indetermi- nate moment sequence, which means that there are several representing measures. Then to each such a measure there corresponds a probability distribution on the classical phase space to be in- terpreted in terms of statistical mechanics. If the moment problem has no (explicit) solu- tion. See an alternative in [10]. 4 CS quantization with sequence X If the moment problem has an explicit solution, one can proceed with the corresponding CS quantization of the complex plane since the family of vectors |vz〉 solves the unity : with Pz = |vz〉〈vz|,∫ C ν(dz)Pz = ∞∑ n,n′=0 |en〉〈en′| 1 √ xn!xn′! · ∫ C d2z π w(|z|2)znz̄n ′ = (18) ∞∑ n=0 1 xn! |en〉〈en| = I , 32 Acta Polytechnica Vol. 50 No. 3/2010 Weproceedwith this quantization like in the standard case: to a function f(z, z̄) in the complex plane there corresponds the operator Af in H defined by f �→ Af = ∫ C d2z π f(z, z̄)w(|z|2)E(|z|2)Pz = ∞∑ n,n′=0 |en〉〈en′| 1√ xn!xn′! · (19) ∫ C d2z π w(|z|2)f(z, z̄)znz̄n ′ provided that weak convergence holds. For the sim- plest functions f(z, z̄)= z and f(z, z̄)= z̄ we obtain Az = â , â |en〉 = √ xn [en−1〉 , (20) â |e0〉 = 0 , (lowering operator) Az̄ = â † , ↠|en〉 = √ xn+1 |en+1〉 (21) (raising operator) . These twobasic operatorsobey the commutation rule : [â, â†] = xN+1 − xN def = ΔN. The operator xN is de- fined by xN = â †â and is such that its spectrum is ex- actly the sequence X with eigenvectors en : xN |en〉 = xn |en〉. The triple {â, â†,ΔN} equipped with the op- erator commutator [· , ·] generates (generically) an in- finite Lie algebra which replaces the Weyl-Heisenberg Lie algebra. The quantization of the real and imagi- nary parts of z = 1 √ 2 (q + ip) yields position and mo- mentum operators corresponding to the sequence X , Aq def = Q = 1 √ 2 (â + â†) , (22) Ap def = P = 1 √ 2i (â − â†) , [Q, P ] = iΔN , together with new quantum localization properties. 5 An example: charged particle in a magnetic field Consider a classicalnonrelativistic particle, charge−e, moving in the plane ( x1, x2 ) and interacting with a constant and uniform magnetic field of intensityB perpendicular to the plane, described by a vector potential A only (A0 = 0). The Hamiltonian of the particle is H(x,p) = 1 2μ [ p+ e c A(x) ]2 , x =( x1, x2 ) , p = (p1, p2). With the symmetric gauge Âi = − B 2 εij x̂ j , i, j, k = 1,2, the quantum Hamil- tonian takes the form Ĥ = 1 2μ ( P̂21 + P̂ 2 2 ) . The P̂i, i = 1,2, are components of the kinematic momentum operator, P̂i = p̂i − eB 2c εij x̂ j , [ P̂1, P̂2 ] = −ih̄ eB c , (23) where εij is the Levi-Civita symbol. Kowalski & Rembielinski coherent states KowalskiandRembielinski [16] haveproposed the con- struction of CS for a particle in a uniform magnetic field by using their coherent states for the circle [17]. The latter are constructed from the angular momen- tum operator Ĵ and the unitary operator Û that rep- resents the position of the particle on the unit circle. These operators obey the commutation relations[ Ĵ, Û ] = U , [ Ĵ, Û+ ] = −Û+ . (24) The introduction of these coherent states permits to avoid the problem of the infinite degeneracy present in the approach followed by Man’ko and Malkin [15], and, in addition, takes into in account the momentum part of the phase space. Consequently, the so obtained CS offer a better way to compare the quantumbehav- ior of the system with the classical trajectories in the phase space. Let us introduce the centre-coordinate operators x̂10 = x̂ 1 − 1 μω P̂2 , x̂ 2 0 = x̂ 2 + 1 μω P̂1 , (25) where ω = eB/μ is the cyclotron frequency. The x̂i0 are integral of motion, [H, x̂i0] = 0. Relative motion coordinate operators, r̂1 = x̂1 − x̂10 = 1 μω P̂2 , (26) r̂2 = x̂2 − x̂10 = − 1 μω P̂1 . Introduce now the operators r̂0± = x̂ 1 0 ± ix̂ 2 0 , (27) r̂± = r̂ 1 ± ir̂2 = 1 μω (P̂2 ∓ iP̂1) . They obey the commutation rules [r̂0+, r̂0−] = 2 h̄ μω , [r̂+, r̂−] = −2 h̄ μω , (28) [r̂0±, r̂±] = 0 . The “relative” angular momentum operator Ĵ is pro- portional to the Hamiltonian Ĵ = r̂1P̂2 − r̂2P̂1 = − 2 ω Ĥ = (29) μωr̂+r̂− + h̄ = μωr̂−r̂+ − h̄ . Due to the rules, [J, r̂0±] = 0 , [J, r̂±] = ±2h̄r̂± , (30) Ĵ can be viewed as the generator of rotations about the axis passing through the classical point (x10, x 2 0) 33 Acta Polytechnica Vol. 50 No. 3/2010 and perpendicular to the (x1, x2) plane. The nonuni- tary operator r̂− describes to a certain extent the an- gular position of the particle on a circle. The sym- metries and the integrability of the model can be encoded into the two independent Weyl-Heisenberg algebras, one for the center of circular orbit and the other for the relative motion. They allow one to construct the Fock space with orthonormal basis {|m, n〉 ≡ |m〉⊗|n〉, m, n ∈ Z}, as repeated actions of the raising operators r̂0− and r̂+, r̂0−|m〉 = √ 2h̄(m +1) μω |m +1〉 , (31) r̂+|n〉 = √ 2h̄(n +1) μω |n +1〉 . r̂0+|m〉 = √ 2h̄m μω |m −1〉 , (32) r̂−|n〉 = √ 2h̄n μω |n −1〉 , and the eigenvalue equation Ĵ |m, n〉 =(2n +1) h̄ |m, n〉 . (33) The K&R CS |z0, ζ〉 are constructed in the Hilbert space spanned by the orthonormal basis as solution to the eigenvalue equation: r̂0+ |z0, ζ〉 = z0 |z0, ζ〉 , (34) Ẑ |z0, ζ〉 = ζ |z0, ζ〉 , z0, ζ ∈ C , where Ẑ = e 1 2(Ĵ /h̄+1)r̂−. The projection of these CS in this Fock basis reads as 〈m, n| ζ, z0〉 = e− |z̃0| 2 2√ E(|ζ̃|2) z̃m0√ m! ζ̃n √ n! e− 1 2 n(n+1) , (35) where z̃0 = √ μω 2h̄ z0 , ζ̃ = √ μω 2h̄ ζ. The normaliza- tion factor involves the function E (t)= ∞∑ n=0 e−n(n+1) tn n! ≡ ∞∑ n=0 tn xn! , (36) where we recognize a generalized exponential with xn ≡ e2nn. Squeezing/deforming the K& R states The introduction of a “squeezing” parameter λ allows us to generalize the previous CS of a charged particle in a uniform magnetic field as an eigenvector of the commuting operators r̂0+ and Ẑλ, r̂0+ |z0, ζ〉 = z0 |z0, ζ〉 , Ẑλ |z0, ζ〉 = ζ |z0, ζ〉 , Ẑλ = exp [ λ 4 ( Ĵ/h̄ +1 )] r̂− . (37) Operator Ẑλ coincideswith theK&R Ẑ for λ =2, and with just r̂− for λ =0, i.e., the case ofMalkin-Man’ko CS, which are actually tensor products of standard CS. Operator Ẑλ controls the dispersion relations of the angular momentum Ĵ and of the “position opera- tor” r̂−. The corresponding CS read: |z0, ζ〉 = e− |z̃0| 2 2√ Eλ (∣∣∣ζ̃∣∣∣2) ∑ m,n z̃m0√ m! ζ̃n √ xn! |m, n〉 , (38) with Eλ (t)= ∞∑ n=0 tn/xn! and xn ≡ enλn. The complex numbers z0 and ζ parameterize, respectively, the posi- tion of the centre of the circle and the classical phase space state of the circular motion. Some properties of these CS make them more suitable with regard to the semi-classical behavior of a charged particle in a magnetic field, in comparisonwith theMalkin-Man’ko CS. The generalization involving λ allows one to ex- hibit better these interesting characteristics. Resolution of the moment problem The λ-CS |z0, ζ〉 are the tensor product of the states |z0〉 and |ζ〉, where the first one is a standard CS. So, in order to perform the CS quantization, we concen- trate only on the states |ζ〉. For convenience, we put μω/2h̄ = 1, and so ζ̃ = ζ. Then, in the Fock basis {|n〉}, |ζ〉 = 1√ Eλ|ζ|2) +∞∑ n=0 ζn√ xn! |n〉 , Eλ(t) = +∞∑ n=0 t xn! , xn = e λn n . They resolve the unity in the Fock space spanned by the kets |n〉, ∫ C �λ ( |ζ|2 ) d2ζ π |ζ〉〈ζ| = I . The weight function �λ solves the moment problem∫ ∞ 0 tn�λ (t) dt = n! exp { λn(n +1) 2 } ≡ xn! ,(39) λ ≥ 0 , and is given under the form of the Laplace transform, �λ (t) = e−λ/2 √ 2πλ ∫ +∞ 0 du exp ( −e−λ/2tu ) e− (ln u)2 2λ = e−λ/2 √ 2πλ L [ e− (ln u)2 2λ ] ( e−λ/2t ) . 34 Acta Polytechnica Vol. 50 No. 3/2010 Fig. 1: Error function as a function of l for λ = 2 (solid line), λ =4 (dashed line) and λ =6 (dotted line). We see that, with the λ-CS, this approximation can be improved, for |l| ≤ 1, by increasing the value of λ CS quantization The correspondingCSquantizationof functions on the complex plane is the map f ( ζ, ζ̄ ) �→ ∫ C d2ζ π �λ ( |ζ|2 ) · (40) f ( ζ, ζ̄ ) Eλ ( |ζ|2 ) |ζ〉〈ζ| def= f̂ . As expected, the CS quantization of the variables ζ and ζ̄ yields ζ �→ ζ̂ = Ẑλ , ζ̄ �→ ζ̂ = Ẑ † λ . (41) Numerical analysis One convenient criterion to evaluate the closeness of the introduced λ-CS to the classical phase space con- sists in verifying how closely the expectation value of the angular momentum operator approaches the re- spective classical quantity. This can be implemented through the evaluation of the relative error e(λ, l)= |(〈Ĵ〉ζ /h̄ − l)| l , (42) with the expectation value of the angular momentum given by 〈Ĵ〉ζ = 〈ζ|Ĵ|ζ〉 = 1 Eq(|ζ|2) +∞∑ n=0 |ζ|2n (2n +1) xn! , xn = e λ 2 n n . The parameter ζ are relatedwith the classical angular momentum l = μωr2 (where r is the classical radius) by |ζ| = √ l μω exp( λ 4 l) . Kowalski and Rembielinski observed that the approx- imate equality 〈Ĵ〉ζ � l does not hold for arbitrary small l, being really acceptable for |l| > 1 only. References [1] Klauder, I. R., Skagerstam, B. S.: Coherent States, Applications in Physics and Mathemat- ical Physics, World Scientific, Singapore, 1985, pp. 991. [2] Perelomov, A. M.: Generalized Coherent States and Their Applications, Springer-Verlag, New York 1986. [3] Malkin, I. A., Man’ko, V. I.: Dynamical Symme- tries and Coherent States of Quantum Systems, Nauka, Moscow, 1979, pp. 320. [4] Ali, S.T., Antoine, J.P.,Gazeau, J.-P.: Coherent states, wavelets and their generalizations. Grad- uate Texts in Contemporary Physics, Springer- Verlag, New York, 2000. [5] Ali, S. T. , Gazeau, J.-P., Heller, B.: J. Phys. A: Math. Theor., 41 (2008) 365302. [6] Gazeau, J.-P.: Coherent States in Quantum Physics, Wiley-VCH, Berlin, 2009. [7] Holevo, A. S.: Statistical Structure of Quantum Theory, Springer-Verlag, Berlin, 2001. [8] Berezin, F. A.: Comm. Math. Phys. 40 (1975) 153. [9] Chakraborty, B., Gazeau, J.-P., Youssef, A.: arXiv:0805.1847v1. [10] Ali, S. T., Balková, L., Curado, E. M. F., Gazeau, J.-P., Rego-Monteiro, M. A., Ro- drigues, Ligia M. C. S., Sekimoto, K.: J. Math. Phys., 50, 043517-1-28 (2009). [11] Baldiotti, M. C., Gazeau, J.-P., Gitman, D. M.: Phys. Lett. A, 373, 1916–1920 (2009); Erratum: Phys. Lett. A, 373, 2600 (2009). [12] Baldiotti, M. C., Gazeau, J.-P., Gitman, D. M.: Phys. Lett. A 373, 3937–3943 (2009). [13] Stieltjes, T.: Ann. Fac. Sci. Univ. Toulouse 8 (1894–1895), J1-J122; 9, A5-A47. [14] Simon, B.: Adv. in Math. 137 (1998), 82–203. 35 Acta Polytechnica Vol. 50 No. 3/2010 [15] Malkin, I. A., Man’ko, V. I.: Zh. Eksp. Teor. Fiz. 55 (1968) 1014 [Sov. Phys. – JETP 28, no. 3 (1969) 527]. [16] Kowalski, K., Rembielinski, J.: J. Phys. A 38 (2005) 8247. [17] Kowalski, K., Rembielinsk, J., Papaloucas, L. C.: J. Phys. A 29 (1996) 4149. J. P. Gazeau E-mail: gazeau@apc.univ-paris7.fr Laboratoire APC Université Paris Diderot Paris 7 10 rue A. Domon et L. Duquet 75205 Paris Cedex 13, France M. C. Baldiotti, D. M. Gitman E-mail: baldiott@fma.if.usp.br, gitman@dfn.if.usp.br Instituto de F́ısica, Universidade de São Paulo Caixa Postal 66318-CEP 05315-970 São Paulo, S.P., Brazil 36