Strong evidences for correlated nonextensive quantum statistics in hadronic scatterings

1 June 2000

Physics Letters B 482 Ž2000. 57–64

Strong evidences for correlated nonextensive quantum statistics in hadronic scatterings M.L.D. Ion, D.B. Ion National Institute for Physics and Nuclear Engineering, IFIN-HH, P.O. Box MG-6, Bucharest, Romania Received 29 February 2000; accepted 6 April 2000 Editor: R. Gatto

Abstract

™

In this paper, by introducing w S J Ž p ., Su Ž q .x Tsallis-like entropies, the nonextensiÕe statistics of quantum states in the case of Ž0y 12 0y 12 . scatterings are investigated. The nonextensivity indices p and q are determined from the experimental entropies by a fit with the optimal entropies w S Jo1 Ž p ., Suo1 Ž q .x obtained from the principle of minimum distance in the space of states. In this way strong experimental evidences for the p-nonextensivities in the range 0.5 F p F 0.6 with q s prŽ2 p y 1. ) 3, are obtained with high accuracy ŽCL ) 99%. from the pion-nucleon experimental phase shifts. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj

1. Introduction In the last time there has been an increasing interest Žsee Ref. w1x. in the investigation of the quantum entropy not only by proving new entropic uncertainty relations Žsee e.g. Refs. w1–4x. for the standard additive systems but also by generalization of such results to the nonextensive statistical systems w5–10x. In Ref. w3x Žthe angle and angular momentum. information entropies as well as the entropic angle-angular momentum uncertainty relations are introduced. Using Tsallis-like entropies and the Riesz theorem w11x, in Ref. w4x are proved the state independent angle-angular momentum entropic lower bounds for the quantum scattering of the spinless particles. Then, it is shown that the experimental pion-nucleus scattering entropies are well described

by the optimal entropies corresponding to the optimal states which were introduced in Ref. w12x, via reproducing kernel Hilbert space methods ŽRKHS.. Also, in Refs. w8,9x, it was proved that extremal properties of entropy, such as maximum entropy, is very important property of the optimal states derived from the principle of minimum distance in the space of states Žsee Refs. w8,12x.. All these results on the quantum entropy was specifically designed to be applicable to the scattering of spinless particles. Hence, the generalization of these results to the scattering of particle with spins, as well as, the experimental determination of nonextensivity degrees Ž p,q . for the quantum systems in the hadronhadron scattering are of great interest. In this paper the degrees of nonextensivity of the quantum systems composed from Ž J or u .-quantum

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 4 6 0 - 3

58

™

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

q q states in the spin Ž0y 12 0y 12 . hadron-hadron scatterings are investigated. So, first we present a complete statistical description of the nonextensive q q quantum systems from the Ž0y 12 0y 21 .scatterings in terms of the scattering entropies S J Ž p . and Su Ž q ., p,q g R. Then, the nonextensivity indices of quantum states from the pion-nucleon and pionnucleus scatterings, determined by the fit of the scattering entropies S J Ž p . and Su Ž q . with the optimal scattering entropies S Jo1 Ž p . and Suo1 Ž q . are obtained from the available pion-nucleon w13x experimental phase shifts. A two-parameter Ž p,q .-fit of the scattering entropies S J Ž p . and Su Ž q . with the optimal scattering entropies S Jo1 Ž p . and Suo1 Ž q . advanced the conclusion that the best values for p-nonextensiÕity are found in the range Ž0.5 F p F 0.6. with q s prŽ2 p y 1. ) 3 Žfixed by Riesz–Thorin relation : 1rp q 1rq s 2..

™

Then, the elastic integrated cross section sel is given by ds

q1

sel s 2p

Hy1

dV

q1

s 2p

Hy1

Ž x . dx

N fqq Ž x . N 2 q N fqy Ž x . N 2 dx

Ž 3.

Since we will work at fixed energy, the dependence of sel and ddVs Ž x . and of f Ž x ., on this variable was suppressed. Hence, the helicities of incoming and outgoing nucleons are denoted by m , mX , and was written as Žq.,Žy., corresponding to Ž 12 . and Žy 12 ., respectively. In terms of the partial waves amplitudes f Jq and f Jy Žsee Ref. w13x. we have Jmax

Ý Ž J q 1r2. Ž f Jyq f Jq . dJ Ž x . Ž 4.

fqq Ž x . s

1 1 2 2

Js 12 Jmax

Ý Ž J q 1r2. Ž f Jyy f Jq . dyJ Ž x .

fqy Ž x . s

™

H 2. Optimal nonextensive entropies for (0 I 12 I 1 H) 0 2 -scatterings

A nonextensive-statistical description of quantum state in the case of scattering of particles with spins is mathematically more complicated due to the fact that we must work with the vector-valued complex functions or even with the spin-density matrices. Here, for sake of simplicity we restrict our consideraq q tions only to Ž0y 12 0y 12 .-scattering case. Therefore, let fqq Ž x . and fqy Ž x ., x g wy1,1x, be the scattering helicity amplitudes of the meson– nucleon scattering process:

™

M Ž 0y . q N Ž 12

q

. ™ M Ž 0y . q N Ž . 1q 2

Ž 1.

1 1 2 2

Js 12

Ž 5. J Ž . where the dmn x -rotation functions are given by

1

J

d Ž x. s 1 1 2 2

P

lq1

ds dV

2

Ž x . sN fqq Ž x . N q N fqy Ž x . N

2

Ž 2.

1 (

2

(

Plq1 Ž x . y Pl Ž x .

2

Ž 6. dJy 12 12

Ž x. s

1

P

lq1

1yx 2

1 2

(

(

Plq1 Ž x . q Pl Ž x .

Ž 7. and prime indicates differentiation with respect to x ' cos u . Now, the elastic integrated cross section Ž3. for the meson–nucleon scattering can be expressed in terms of partial amplitudes f Jq and f Jy q1

x s cosŽ u ., u being the c.m. scattering angle. The normalization of the helicity amplitudes fqq Ž x . and fqy Ž x . is chosen such that the c.m. differential cross section ddVs Ž x . is given by

1qx

sel r2p s

ds

Hy1 dx d V Ž x . Jmax

s

Ý Ž 2 J q 1. Ž N f JqN 2 q N f JyN 2 .

Ž 8.

Js 12

J-nonextensiÕe statistics for the quantum scattering:y We define two kind of Tsallis-like scattering entropies. One of them, namely S J Ž p ., p g R, is

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

special dedicated to the investigation of the nonextensive statistical behavior of the angular momentum J-quantum states, and can be defined by S J Ž p . s 1 y Ý Ž 2 l q 1 . p lp r Ž p y 1 . ,

pgR,

Ž 9. where the probability distributions p J , is given by N f JqN 2 q N f JyN 2

pJ s

,

Jmax

Jmax

Ý Ž 2 J q 1. pJs1

Ž 10 .

Js 12

u-nonextensiÕe statistics for the quantum scattering:y In similar way, for the u-scattering states considered as statistical canonical ensemble, we can investigate their Žnonextensive. statistical behavior by using an angular Tsallis-like scattering entropy Su Ž q . defined as q1

Hy1 dx

PŽ x.

q

r Ž q y 1. ,

the definitions Ž9. andror Ž11. can be interpreted as measuring the degree nonextensiÕity. We next consider the maximum-entropy ŽMaxEnt. problem max  S J Ž p . ,Su Ž q . 4 ds

when sel s fixed and

qgR

Ž 11 .

PŽ x. s

sel

P

ds dV

1

Ž x. ,

Hy1P Ž x . dx s 1

ds dV

Ž 12 .

with Ž x . and sel defined by Eqs. Ž2., Ž3. and Ž8.. The above Tsallis-like scattering entropies posses two important properties. First, in the limit k 1,k ' p,q, the Boltzmann–Gibbs kind of entropies is recovered:

™

™1

lim S J Ž p . s S J Ž 1 . s y Ý Ž 2 J q 1 . p J ln p J Ž 13 .

p

q1

™

p Jm e s p Jo1 s

1

1 s

2 K Ž 1,1 . 1 1 2 2

™1

Hy1 dxP Ž x . ln P Ž x .

Secondly, these entropies are nonextensiÕe in the sense that SAq B Ž k . s SA Ž k . q SB Ž k . q Ž 1 y k . SA Ž k . SB Ž k . ,

Ž 15 .

for any independent sub-systems A, B Ž pAq B s pA P pB .. Hence, each of the indices p / 1 or q / 1 from

,

q for 12 F J F Jo , and p Jm e s 0,

for J G Jo q 1

Ž 17 .

while, for the u-quantum states, these distributions are as follows K2 Ž x ,1 . 1 1 2 2

P m e Ž x . s P o1 Ž x . s

Ž 18 .

K Ž 1,1 . 1 1 2 2

where d J Ž x . are the d-spin( rotation functions Ž6. for the spin 1r2 particles, Pl Ž x . ' dPl Ž x .rdx, the reproducing kernel K Ž x, y . is given by 1 1 2 2

1 1 2 2

Jo

K Ž x , y . s 12 1 1 2 2

Ý Ž 2 J q 1. dJ Ž x . dJ Ž y . 1 1 2 2

1 1 2 2

Ž 19 .

1r2

while the optimal angular momentum Jo is 1 1 2 2

Ž 14 .

k s p,q

2 Ž Jo q 1 . y 1r4

2 Ž Jo q 1 . y 1r4 s 2 K Ž 1,1 . s

lim Su Ž q . s Su Ž 1 . s y

q

Ž 16 .

as criterion for the determination of the equilibrium distributions p lm e and P m e Ž x . for the quantum states from the Ž0y 12 0y 12 .-scattering. The equilibrium (optimal) distributions, as well as the optimal scattering entropies for the quantum scattering of the spinless particles was obtained in Ref. w9x Žsee Eqs. Ž21. – Ž25... For the J-quantum states, in the spin Ž0y 12 0y 12 . scattering case, these distributions are given by:

where 2p

Ž 1 . s fixed

dV

™

Ý Ž 2 J q 1. Ž N f JqN 2 q N f JyN 2 . Js 12

Su Ž q . s 1 y

59

4p d s

sel d V

Ž 1 . Ž 20 .

Proof: In this case solving the problem Ž16. via Lagrange multipliers we obtain that the singular solution l 0 s 0 exists and is just given by the w S Jo1 Ž p .,Suo1 Ž q .x-optimal entropies corresponding to the PMD-SQS-optimal state: o1 fqq Ž x . s fqq Ž 1 . K Ž x ,1 . rK Ž 1,1 . , 1 1 2 2

o1 fqy Ž x . s 0.

1 1 2 2

Ž 21 .

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

60

Indeed, the problem Ž16. is equivalent to the following unconstrained maximization problem: £

™ max

Ž 22 .

where the Lagrangian function is defined as £ ' l0  S J Ž p . ,Su Ž q . 4 q l1  selr4p y Ý Ž 2 J q 1 . N f JyN 2 q N f JqN 2 q l2 y

½

ds dV

Ž 1. y

4

Ý Ž 2 J q 1. Re f Jq

Ý Ž 2 J q 1. Im f Jq

2

5

2

Ž 23 .

Hence, the solution of the problem Ž16. in the singular case l0 s 0 is reduced just to the solution of the minimum constrained distance in space of quantum states:

Ý Ž 2 J q 1. when

ds

N f JyN 2 q N f JqN 2

Ž 1 . s is fixed

Ž 24 .

definitions Ž9. and Ž11. the Ž p,q .-nonextensivities can take any value in the interval Žy`,q `.. So, in general, an answer at this question is difficult to give for all values of the nonextensivities p,q g R. However, if the Fourier transform defined by Eqs. Ž4. and Ž5. is considered a bounded map from the space L2 p to the space L2 p of the vector valued functions Ž f Jq, f Jy ., J s 12 , 32 , . . . 4 and Ž fqq, fqy ., respectively, then, the answer can be given by the following important result. The nonextensiÕity indices p and q corresponding to the J-statistics and u-statistics, respectiÕely, are expected to be correlated Õia the Riesz–Thorin relation 1 1 q s 1, or q s pr Ž 2 p y 1 . Ž 27 . 2p 2q while the norm M of the Fourier transform wEqs. Ž4. and Ž5.x is bounded by < < Tf < < L 2 q

py 1 2p

dV Therefore, by a straightforward calculus we obtain that the solution of the problem Ž16. is given by

M'

S Jmax

Proof: In our case we show that the result given by Eq. Ž27. is a direct consequence of the Riesz– Thorin interpolation theorem w11x extended to the Õector-Õalued functions. Indeed, let T be the Fourier transform defined by the helicity scattering amplitude Ž4., Ž5. where the partial amplitudes are expressed as follow

Ž p.

s S Jo1 Ž p . s 1 y 2 K Ž 1,1 .

1y p

1 1 2 2

r Ž p y 1. ,

Ž 25 . Sumax Ž q . s Suo1 Ž q . q1

s 1y

Hy1 dx

ž

K Ž x ,1 . 1 1 2 2

K Ž 1,1 . 1 1 2 2

q

2

/

r Ž q y 1.

1 2

1 1 2 2

Ž x,1. are given

F2

f J "s 12

q1

Hy1

Ž 28 .

fqq Ž x . dJ Ž x . 1 1 2 2

"fqq Ž x . dJy

1 1 2 2

Ž x . dx

Ž 29 .

Then, it is easy to prove that: sup  < f Jq < 2 q < f Jy < 2 4

3. State independent inequalities and possible correlations of nonextensive statistics

,

- p - 1 and q s pr Ž 2 p y 1 .

Ž 26 . where the reproducing kernels K by Eq. Ž19..

< < f < < L2 p

1 F

'2

q1

Hy1

1r2

< fqq Ž x . < 2 q < fqy Ž x . < 2

1r2

dx

Ž 30 . Now, a natural but fundamental question can be addressed, namely, what kind correlation Žif it exists. is expected to be observed between the nonextensivity indices p and q corresponding to the Ž p, J .-nonextensive statistics described by S J Ž p . and Ž q, u .nonextensive statistics described by Su Ž q .? By the

and the Parseval’s formula

Ý Ž 2 J q 1. q1

s

Hy1

< f Jq < 2 q < f Jy < 2

< fqq Ž x . < 2 q < fqy Ž x . < 2 dx

Ž 31 .

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64 1r2

™ ™

J since < dJ Ž x . < 2 " < dy F 2 1r2 . This Ž x. <2 means that T : L1 L` with the norm M1 s 2y1 r2 and T : L2 L 2 with the norm M2 s 1. Then, using the Riesz–Thorin interpolation theorem for the vector-valued functions Žsee Ref. w11x. T : L pX L qX with the norm M with Ž1rpX . s Ž1 y t .r1 q tr2, Ž1rqX . s Ž1 y t .r` q tr2, and 0 - t - 1. Hence, eliminating the parameter t w t s Ž1r2 q . y Ž1r2 p . q 1x and using the relations pX s 2 p and q X s 2 q, we get not only the condition (27) but also the norm-estimate (28) since according to Riesz–Thorin theorem M F M11y t M2t . Now it is important to remark that, by the result Ž28. and a similar result for the inverse Fourier q transform, we proved for the case of spin Ž0y 12 y 1 q. 0 2 -scatterings the following important inequalities: 1 1 2 2

1 1 2 2

™

™

1 2q

1 q Ž 1 y q . Su Ž q .

F2

py1

1

2p

2p

1 q Ž 1 y p . SJ Ž p .

61

4. Determination of nonextensivity from experimental data The essential objective in determination of nonextensivity Ž p,q . is to fit experimental data on S J Ž p . and Su Ž q . with optimal state (or MaxEnt) predictions (25), (26) and to get the values of p and q for which we obtain the best fit. So, we determine the value of the nonextensiÕity indices p and q for which x J2 Ž p . orrand xu2 Ž q .-test functions are minimum, where n exp

x X2 Ž k . s

Ý

S X Ž k . i y S Xo1 Ž k .

D S Xo1

is1

X ' J ,u ;

2 i

,

i

k ' p,q

Ž 35 .

For the investigation of this important problem we use the experimental pion-nucleon w13x phase-shifts for to obtain the experimental entropies w S J Ž p .x i and w Su Ž q .x i for different values of the pair Ž p,q .-nonextensivities indices, as well as, for the optimal angular-momentum Jo wEq. Ž20.x. So, by using the values

Ž 32 .

1 2q

1 q Ž 1 y q . SJ Ž q .

F2

py1

1

2p

2p

1 q Ž 1 y p . Su Ž p .

Ž 33 .

Table 1 The values of the x J2 Ž p . and xu2 Ž q . obtained from a separate p-fit and q-fit of the scattering entropies S J Ž p . and Su Ž q . with the optimal state predictions S Jo1 Ž p . wEq. Ž25.x and Suo1 Ž q . wEq. Ž26.x, for p N-scatterings. p

Proof: Indeed by dividing Eq. Ž30. by Ž31. we obtain sup w p J x

1r2

F 2y1r2

q1

Hy1

PŽ x.

1r2

dx ,

q1

Ý Ž 2 J q 1. pJ s H

P Ž x . dx s 1

Ž 34 .

y1

This means that by the result Ž28. we proved the inequality Ž32. since by definitions of the L pX-norms from Eqs. Ž9. – Ž12. we have < < Tf < < L 2 q s w1 q Ž1 y q . Su Ž q .x and < < f < < L 2 p s w1 q Ž1 y p . S J Ž p .x . Of course, the results Ž32., Ž33. can be presented in the form of the state independent (p,q)-entropic bound q q for the spin Ž0y 12 0y 12 .-scatterings just as in the scattering of the spinless particles wsee Eqs. Ž29. and Ž30. in Ref. w10xx. 1 2q

1 2p

™

0.538 0.545 0.550 0.556 0.563 0.571 0.583 0.600 0.625 0.667 0.700 0.750 0.833 1.00 1.33 1.50 2.00 3.00

pq p

py p p 0 p

x J2 Ž p .

x J2 Ž p .

87

0.17 m 0.18 0.18 0.19 0.20 0.21 0.23 0.26 0.32 0.43 0.53 0.70 1.05 2.02 5.72 9.16 36.8 766.

p

q s 2 p y 1 pq p

py p

p 0p

x J2 Ž p .

xu2 Ž q .

xu2 Ž q .

xu2 Ž q .

87

87

87

87

0.14 0.13 0.13 0.13 m 0.13 0.13 0.13 0.14 0.16 0.20 0.25 0.34 0.53 1.09 3.22 5.14 19.6 319.

0.16 m 0.16 0.16 0.17 0.17 0.18 0.19 0.22 0.26 0.34 0.41 0.55 0.84 1.63 4.69 7.53 30.0 580.

7.0 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.75 1.50 1.25 1.00 0.80 0.75 0.67 0.60

0.007 m 0.012 0.017 0.020 0.03 0.04 0.06 0.09 014 0.24 0.32 0.44 0.64 1.06 1.93 1.94 2.45 3.01

0.001m 0.002 0.002 0.002 0.004 0.006 0.009 0.014 0.024 0.046 0.068 0.11 0.22 0.51 1.24 1.16 1.61 2.12

87

0.004 m 0.006 0.008 0.009 0.012 0.016 0.023 0.033 0.049 0.080 0.11 0.15 0.25 0.54 1.28 1.23 1.70 2.22

62

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

of Jo we can calculate the numerical values of the optimal entropies w S Jo1 Ž p .x i and w Suo1 Ž prŽ2 p y 1..x i wEqs. Ž25. and Ž26.x, and finally, the values of x J2 Ž p .,

and x J2 Ž q . for each p and q fixed. The values of Ž D S Xo1 . i are calculated by assuming an error of D Jo s "1 in the estimation of the optimal angular

™

™

™

Fig. 1. The experimental results on w S J Ž p ., Su Ž q .x-scattering entropies, calculated from the experimental phase shifts analysis w13x, are compared with the optimal predictions wEqs. Ž25. and Ž26.x Žthe full curves., for: Ža. pq p pq p, Žb. p 0 p p 0 p, Žc. py p py p. The grey regions around the full curves are obtained from the optimal entropies S Jo1 Ž p . or Suo1 Ž q . by assuming errors of D Jo s "1 in the estimation of Jo .

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

63

entropies S J Ž p . and Su Ž q . are in excellent agreement Ž CL ) 99%. with the w S Jo1 Ž p ., Suo1 Ž q .x-optimal state predictions wEqs. Ž25. and Ž26.x for the pairs w p,q s prŽ2 p y 1.x with values of p below 0.6, while significant departures from ’’optimality’’ are observed for those pairs with p ) 2.

5. Conclusions The main results and conclusions obtained in this paper can be summarized as follows: Ži. In this paper we obtained a complete nonextensive-statistical description of the quantum state in the Ž0y 12 q 0y 21 q .-scatterings. The equilibrium (or MaxEnt) distributions and optimal Ž J or u .-scattering entropies are obtained in terms of the spin J Ž . dmn x -functions and of corresponding reproducing kernel functions. Moreover, by extension of the Riesz–Thorin interpolation theorem to the vector valued complex function, we proved the results Ž27. q q and Ž28. for the spin Ž0y 12 0y 12 .-scattering cases. All these results, together the results Ž25., Ž26., allow us to obtain a complete extension of the previous results, obtained on the entropic bands w8,9x as well as on the entropic uncertainty relations w10x, q to the nonextensive statistics of the Ž0y 12 q 0y 12 .-scattering states; Žii. The experimental determination of statistical behavior of the quantum scattering states in the pion-nucleon scattering was performed by the experimental fit of the (p,q)-nonextensiÕity indices using the available pion-nucleon experimental phase shifts w13x. Hence, a separate p-fit of the experimental data on the scattering entropies S J Ž p ., p g Ž1r2,`. with the ‘‘equilibrium’’ entropy S Jo1 Ž p . wEq. Ž25.x allowed us to conclude that the Ž J, p .-statistics of the system of J-quantum states are subextensiÕe with an index p slightly above 1r2, while a separate q-fit of the experimental entropies Su Ž q .,q g Ž1r2,`. with the optimal entropy Suo1 Ž q . wEq. Ž26.x proved that Ž u ,q .-statistics of the u-quantum states are superextensiÕe with q G 2 Žsee Table 1 and Fig. 1. ; Žiii. A two-parameter Ž p,q .-fit of the scattering entropies S J Ž p . and Su Ž q . with the optimal scattering entropies S Jo1 Ž p . and Suo1 Ž q . advanced the conclusion that the best values for p are also in the range Ž0.5 F p F 0.6. but with the q-nonextensiÕities

™

™

™

Fig. 1 Ž continued ..

momentum Jo from the experimental data. The numerical results obtained in this way for different nonextensivities k in the interval 0.5 F k F 7.00, are presented in the Table 1 and Fig. 1 for the p ",0 pscattering, respectively. Hence, from Fig. 1 and Table 1 we see that the experimental data on the scattering

64

M.L.D. Ion, D.B. Ion r Physics Letters B 482 (2000) 57–64

correlated by the Riesz–Thorin relation : 1rp q 1rq s 2 Žsee Tables 1 and Fig. 1.. This important experimental discovery require the introduction of a new concept in quantum physics, namely, that of conjugate pair of nonextensiÕe quantum statistics; Živ. The strong experimental evidence obtained here for the nonextensiÕe statistical behaÕior of the Ž J, u .-quantum scatterings states in the pion-nucleon scattering can be interpreted as an indirect manifestation the presence of the quarks and gluons as fundamental constituents of the scattering system having the strong-coupling long-distance regime required by the Quantum Chromodynamics. However, a possible presence of long-time memory effects in the quantum scattering system, or, the presence of the fractal space-time structures during the scattering process, are not excluded as being possible sources of nonextensivity of the pionic scattering Ž J and u .-quantum systems. Finally, we note that further investigations are needed since this nonextensiÕe statistical behaÕior of the quantum scattering, discovered here with high accuracy Ž CL ) 99%., can be a signature of a new uniÕersal law of the quantum scattering. This suggestion is also well supported by the recent results of Ref. w14x, about the observed radial density profiles

in pure-electron plasmas in Penning traps, which are also consistent with a value of the nonextensivity index around p s 1r2.

References w1x M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, Berlin, 1993, p. 279, and references therein. w2x I. Bialynicki-Birula, J. Mycielski, Commun. Math. Phys. 44 Ž1975. 129; I. Bialynicki-Birula, J.L. Madajczuk, Phys. Lett. A 108 Ž1985. 384; K. Kraus, Phys. Rev. 35 Ž1987. 3070. w3x D.B. Ion, M.L. Ion, Phys. Lett. B 352 Ž1995. 155. w4x D.B. Ion, M.L.D. Ion, Phys. Rev. Lett. 81 Ž1998. 5714. w5x C. Tsallis, J. Stat. Phys. 52 Ž1988. 479; C. Tsallis, Phys. Lett. A 195 Ž1994. 329; See also the review paper C. Tsallis, Brazilian J. Phys. 29 Ž1999. 1. w6x A.K. Rajagopal, Phys. Lett. A 205 Ž1995. 32. w7x E.P. Borges, I. Roditi in Phys. Lett. A 246 Ž1998. 399. w8x D.B. Ion, M.L.D. Ion, Phys. Rev. E 60 Ž1999. 5261. w9x M.L.D. Ion, D.B. Ion, Phys. Rev. Lett. 83 Ž1999. 463. w10x D.B. Ion, M.L.D. Ion, Phys. Lett. B 466 Ž1999. 27. w11x J. Bergh, J. Lofstrom, Interpolation Spaces, Springer, Berlin, 1976, p. 2-5. w12x D.B. Ion, Phys. Lett. B 376 Ž1996. 282. w13x G. Hohler, F. Kaiser, R. Koch, E. Pietarinen, Physics Data, ¨ Handbook of Pion-Nucleon Sattering, 1979, Nr. 12-1. w14x B.M. Boghosian, Phys. Rev. E 53 Ž1996. 4754; C. Anteneodo, C. Tsallis, J. Mol. Liq. 71 Ž1997. 255; B.M. Boghosian, Brazilian J. Phys. 29 Ž1999. 91.