principle of minimum distance in space of states as new principle...

Romanian Reports in Physics, Vol. 59, No. 4, P. 1045–1080, 2007

Dedicated to Prof. Dumitru Barbu Ion’s 70th Anniversary

PRINCIPLE OF MINIMUM DISTANCE IN SPACEOF STATES AS NEW PRINCIPLE IN QUANTUM PHYSICS

D. B. ION1,2, M. L. D. ION3

1 IFIN-HH, Bucharest, P.O. Box MG-6, Magurele, Romania2 TH-Division, CERN, CH-1211 Geneva 23, Switzerland

3 Faculty of Physics, Bucharest University, Bucharest, Romania

(Received July 16, 2007)

Abstract. The mathematician Leonhard Euler (1707–1783) appears to have been aphilosophical optimist having written: “Since the fabric of universe is the most perfect and is thework of the most wise Creator, nothing whatsoever take place in this universe in which some relationof maximum or minimum does not appear. Wherefore, there is absolutely no doubt that every effectin universe can be explained as satisfactory from final causes themselves the aid of the method ofMaxima and Minima, as can from the effective causes”. Having in mind this kind of optimism in thepapers [1–16] we introduced and investigated the possibility to construct a predictive analytic theoryof the elementary particle interaction based on the principle of minimum distance in the space ofquantum states (PMD-SQS). So, choosing the partial transition amplitudes as the system variationalvariables and the distance in the space of the quantum states as a measure of the system effectiveness,we obtained the results [1–16]. These results proved that the principle of minimum distance in spaceof quantum states (PMD-SQS) can be chosen as variational principle by which we can find theanalytic expressions of the partial transition amplitudes. In this paper we present a description ofhadron-hadron scattering via principle of minimum distance PMD-SQS when the distance in space ofstates is minimized with two directional constraints: dσ/dΩ(±1) = fixed. Then by using the availableexperimental (pion-nucleon and kaon-nucleon) phase shifts we obtained not only consistentexperimental tests of the PMD-SQS optimality, but also strong experimental evidences for newprinciples in hadronic physics such as: Principle of nonextensivity conjugation via the Riesz-Thorinrelation (1/2p + 1/2q = 1) and a new Principle of limited uncertainty in nonextensive quantumphysics. The strong experimental evidence obtained here for the nonextensive statistical behavior ofthe [J, θ]-quantum states in the pion-nucleon, kaon-nucleon and antikaon-nucleon scatterings can beinterpreted as an indirect manifestation the presence of the quarks and gluons as fundamentalconstituents of the scattering system having the strong-coupling long-range regime required by theQuantum Chromodynamics.

Key words: principle of minimum distance, complexity, optimality, angular momentum entropy.

1. INTRODUCTION

From historical point of view the earliest optimum principle was proposed byHeron of Alexandria (125 B.C.) in connection with the behaviour of light. Thus,

1046 D. B. Ion, M. L. D. Ion 2

Heron proved mathematically the following first genuine scientific minimumprinciple of physics: that light travels between two points by shortest path. In factthe Archimedean definition of a straight line as the shortest path between twopoints was an early expression of a variational principle, leading to the modern ideaof a geodesic path. In fact in the same spirit, Hero of Alexandria explained thepaths of reflected rays of light based on the principle of minimum distance (PMD),which Fermat (1657) reinterpreted as a principle of least time, Subsequently,Maupertuis and others developed this approach into a general principle of leastaction, applicable to mechanical as well as to optical phenomena. Of course, amore correct statement of these optimum principles is that systems evolve alongstationary paths, which may be maximal, minimal, or neither (at an inflectionpoint). Laws of mechanics were first formulated in terms of minimum principles.Optics and mechanics were brought together by a single minimum principleconceived by W. R. Hamilton. From Hamilton’s single minimum principle couldbe obtained all the optical and mechanical laws then known. But the effort to findoptimum principles has not been confined entirely to the exact sciences. In moderntime the principles of optimum are extended to all sciences. So, there exists manyminimum principle in action in all sciences, such as: principle of minimum action,principle of minimum free energy, minimum charge, minimum entropy production,minimum Fischer information, minimum potential energy, minimum rate of energydissipation, minimum dissipation, minimum of Chemical distance, minimum crossentropy, minimum complexity in evolution, minimum frustration, minimumsensitivity, etc. So, a variety of generalizations of classical variational principleshave appeared, and we shall not describe them here.

In the last decade, in the papers [1–16] we generalized these ideas to thespace of transition amplitudes by introducing the Principle of Minimum Distance inthe Space of States (PMD-SQS). These results can be grouped in four categories ofresults as follows: New optimum principles [1–5], New entropic uncertaintyrelations [6–10], Limited entropic uncertainty as a new principle of quantumphysics [11, 12, 15], Theory of nonextensive quantum statistics and optimalentropic band [12–16]. To the above results we must add that, using thepion-nucleus (49 sets) and pion-nucleon (88 sets) experimental phase shifts wepresented experimental evidences for nonextensive quantum statistical behaviour[12–16] in hadron-hadron and hadron-nucleus scattering.

In this paper, some new results on the optimal state analysis of hadron-hadron (( , , )N KN KNπ - scatterings, obtained by using the nonextensive quantumentropy [7–9] and principle of minimum distance in the space of quantum states(PMD-SQS)) [1], are presented. Then, using SJ(p), Sθ(q), SJθ(p, q), ( , )JS p qθ -Tsallis-like scattering entropies, the PMD-SQS-optimality as well as thenonextensive statistical behavior of the [J], [θ], [J, θ] quantum systems of statesproduced in hadronic scatterings are investigated in an unified manner.

3 The minimum distance in space of states 1047

A connection between optimal states obtained from the principle of minimumdistance in the space of quantum states (PMD-SQS) [1] and the most stringent(MaxEnt) entropic bounds on Tsallis-like entropies for quantum scattering, isestablished. The nonextensivity indices p and q are determined from theexperimental entropies by a fit with the optimal Tsallis-like entropies. In this waystrong experimental evidences for the p-nonextensivities index in the range p = 0.6with q = p/(2p – 1) = 3, are evidentiated with high accuracy from the experimentaldata of the principal hadron-hadron scatterings.

2. DESCRIPTION OF QUANTUM SCATTERINGVIA PRINCIPLE OF MINIMUM DISTANCE IN SPACE OF STATES

It is well known that any optimizing study [19] ideally involves three steps:(I) The description of the system, by which one should know, accurately and

quantitatively, the essential variable of the system as well as how these systemvariables interact; (II) Finding a unique measure of the system effectivenessexpressible in terms of the system variables; (III) The optimization by which oneshould chose those values of the system variables yielding optimum effectiveness.

In attempting to use the optimization theory for analyzing the interaction ofelementary particles one can reverses the order of these three steps. Then,knowledge about the interacting system can be deduced by assuming that itbehaves as to optimize some given measure of its effectiveness, and thus thebehaviour of the system is completely specified by identifying the criterion ofeffectiveness and applying optimization to it. This approach is in fact known asdescribing the system in terms of an optimum principle.

Therefore, having in mind all successful results, obtained by using optimumprinciples in the entire history of physics from last two centuries, the principle ofminimum distance was extended to the Hilbert space of the quantum states bychoosing the partial transition amplitudes as fundamental physical quantities sincethey are labelled with good quantum numbers (such as charge, angular momentum,isospin, etc.). These fundamental physical quantities are chosen as the systemvariational variables while the distance in the Hilbert space of the quantum states istaken as measure of the system effectiveness expressed in terms of the systemvariables. Thus, the principle of minimum distance in the space of states(PMD-SQS) is chosen as variational principle by which one should obtain thosevalues of the partial amplitudes yielding optimum effectiveness. Of course, thePMD-SQS-optimum principle can be formulated in a more general mathematicalform by using the S-matrix theory of the strong interacting systems andreproducing kernel Hilbert spaces methods [3–5]. Then, a new analytic quantumphysics can be developed in terms of the reproducing functions from thereproducing kernel Hilbert spaces (RKHS) of the transition amplitudes. In this new

1048 D. B. Ion, M. L. D. Ion 4

kind of analytic quantum physics the system variational variables will be the partialtransition amplitudes introduced by the development of S-matrix elements in termsFourier component implied by the fundamental symmetry of the quantuminteracting system. Therefore, all the results will be expressed in terms ofreproducing Kernel functions RKHS [18]. More concretely, applications to thePMD-SQS shown that this optimum principle can give a satisfactory account fordiffraction scattering of any material particle (such as: electrons, neutrons, nuclei,atoms, molecules, etc.) as a universal optimal phenomenon, free of any (dualistic,wave or particle) interpretations.

In the concrete mathematical form, for the two body elastic scattering, thePMD-SQS can be reduced to the determination of the partial scattering amplitudesby solving giving minimization problems subject to some constraints imposed byinteraction.

As example we chose here the case of the meson-nucleon scattering when theconstraints are given just by fixing the differential cross sections in forward(x = +1) and backward (x = –1) directions. So, first we present the basic definitionson the spin (0–1/2+ → 0–1/2+) scatterings

M(0–) + N(1/2+) → M(0–) + N(1/2+) (1)

Therefore, let ( )f x++ and ( ),f x+− x ∈ [–1, 1] be the scattering helicityamplitudes of the meson-nucleon scattering process (see ref. [13]) written in termsof the partial helicities Jf − sand Jf + as follows

( ) ( )( ) ( )

( ) ( )( ) ( )

max

max

1 11 2 22

1 11 2 22

12

12

JJ

J J

J

JJ

J J

J

f x J f f d x

f x J f f d x

++ − +=

+− − + −=

= + +

= + −

∑

∑(2)

where the rotation functions are defined as

( ) ( ) ( )

( ) ( ) ( )

12

11 12 2

12

11 12 2

1 11 2

1 11 2

Jl l

Jl l

xd x P x P xl

xd x P x P xl

• •+

• •+

−

⎡ ⎤+⎡ ⎤= ⋅ −⎢ ⎥⎣ ⎦⎢ ⎥+ ⎣ ⎦

⎡ ⎤−⎡ ⎤= ⋅ +⎢ ⎥⎣ ⎦⎢ ⎥+ ⎣ ⎦

(3)

where Pl(x) are Legendre polinomials, ( ) ( ),o

l ldP x P xdx

= x being the c.m.

scattering angle. The normalisation of the helicity amplitudes ( )f x++ and ( ),f x+−

is chosen such that the c.m. differential cross section is given by


( ) ( ) ( )2 2d x f x f xd ++ +−σ = +Ω

(4)

Then, the elastic integrated cross section is given by

( )max2 2

12

/ 2 (2 1)J

el J J

J

J f f+ −=

σ π = + +∑ (5)

Now, let us consider the following two directional optimization problem:

( ) 2 21min2 j jj f f+ −⎡ ⎤+ +⎢ ⎥⎣ ⎦∑ when ( 1)d

dσ +Ω

and ( 1)ddσ −Ω

are fixed (6)

where ( 1)ddσ ±Ω

are the forward and backward differential cross sections written in

terms of partial amplitudes as follows

( )( )max

22

12

1( 1) ( 1)2

J

J J

J

d f J f fd ++ − +

=

σ + = + = + +Ω ∑ (7)

( )( )max

22

12

1( 1) ( 1)2

J

J J

J

d f J f fd +− − +

=

σ − = − = + −Ω ∑ (8)

Therefore, in this case the variational variables are partial helicity amplitudes

Jf + and Jf − while the measure of the system effectiveness and the constraints arecompletely expressed in terms of these variational variables by eqs. (5)–(8).

We proved that the solution of this optimization problem is given by thefollowing results:

1 12 2

1 12 2

( , 1)( ) ( 1)

( 1, 1)

K xf x f

K++ ++

+= +

+ +,

1 12 2

1 12 2

( , 1)( ) ( 1)

( 1, 1)o

K xf x f

K

−+− +−

−

−= −

+ −(9)

where the functions K(x, y) are the reproducing kernels [1–5 ] expressed in terms ofthe rotation function by

( )1 1 1 1 1 12 2 2 2 2 21/ 2

1( , ) ( ) ( ),2

oJj jK x y j d x d y= +∑ (10)

( )1 1 1 1 1 12 2 2 2 2 21/ 2

1( , ) ( ) ( ),2

oJj jK x y j d x d y− −

= +∑ (11)

1050 D. B. Ion, M. L. D. Ion 6

while the optimal angular momentum is given by

04 1( 1) ( 1) 1

4el

d dJd d

π σ σ⎡ ⎤= + + − + −⎢ ⎥σ Ω Ω⎣ ⎦(12)

Now, let us consider the logarithmic slope b of the forward diffraction peakdefined by

0ln ( , )

t

d db s tdt dt =

σ⎡ ⎤= ⎢ ⎥⎣ ⎦(13)

Then, using the definition of the rotation functions, from (7)–(13) we obtainthe following optimal slope bo

( )2 4 ( 1) ( 1) 14o

el

d dbd d

⎡ ⎤λ π σ σ= + + − −⎢ ⎥σ Ω Ω⎣ ⎦(14)

PMD-SQS-optimal predictions on the differential cross section ( )odx

dσΩ

and

also for the spin-polarization parameters (Po, Ro, Ao), are as follows.

2 21 1 1 11 2 2 2 2

1 1 1 12 2 2 2

( , 1) ( , 1)( ) ( 1) ( 1)

( 1, 1) ( 1, 1)o

K x K xd d dx

d d K d K

−±

−

+ −⎡ ⎤ ⎡ ⎤σ σ σ⎢ ⎥ ⎢ ⎥= + + −Ω Ω + + Ω − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

(15)

1 1 1 11 2 2 2 2

1 1 1 12 2 2 2

( , 1) ( , 1)( ) 2 ( 1) ( 1) sin ( )

( 1, 1) ( 1, 1)o

o

K x K xd d dP x x

d d d K K

−±

−

+ −⎡ ⎤ ⎡ ⎤σ σ σ ⎢ ⎥ ⎢ ⎥= + − φΩ Ω Ω + + − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

(16)

1 1 1 11 2 2 2 2

1 1 1 12 2 2 2

( , 1) ( , 1)( ) 2 ( 1) ( 1) cos ( )

( 1, 1) ( 1, 1)o

o

K x K xd d dR x x

d d d K K

−±

−

+ −⎡ ⎤ ⎡ ⎤σ σ σ ⎢ ⎥ ⎢ ⎥= + − φΩ Ω Ω + + − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

(17)

2 21 1 1 11 2 2 2 2

1 1 1 12 2 2 2

( , 1) ( , 1)( ) ( 1) ( 1)

( 1, 1) ( 1, 1)o

o

K x K xd d dA x

d d K d K

−±

−

+ −⎡ ⎤ ⎡ ⎤σ σ σ⎢ ⎥ ⎢ ⎥= + − −Ω Ω + + Ω − −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

(18)

*Re{[ ( 1)] ( 1)}cos ( ) ,

( 1) ( 1)f f

xf f

++ +−

++ +−+ −φ =

+ −

*Im{[ ( 1)] ( 1)}sin ( )

( 1) ( 1)f f

xf f

++ +−

++ +−+ −φ =

+ −(19)

Finally, it is important to note that all above results can be extended to thescattering of particles with arbitrary spins by using RKHS method presented in thepapers [4, 5], and also to particle production phenomena.


3. QUANTUM DISTANCES BETWEEN TWO ISOSPIN CHANNELSIN PION-NUCLEON SCATTERING

The quantum distances between two isospin channels, in the Hilbert space ofthe hadron-hadron scattering amplitudes, are introduced in our paper [2]. Then, theisospin quantum distances, as well as their isospin bounds, are expressed in termsof the channel cross sections and spin polarisation parameters. The essentialfeatures as well as the energy behavior of the isospin quantum distances, for thepion-nucleon scattering, are presented for all (s, t, u)-channels.

Therefore, let ( , ), ( , )s ss s s sN N N++ +− ++ +−⎡ ⎤Δ Δ Δ⎣ ⎦ and ( , ),u u uN N N++ +−⎡⎣

( , )u u u++ +− ⎤Δ Δ Δ ⎦ be the (s, u) channel helicity πN-scattering amplitudes in the pure

isospin (I = 3/2, 1/2)-scattering states. If the helicity amplitudes are normalized asin Eqs. (4)–(5). Then, the quantum distances are defined by the following relations:

1/ 22 2( , ) 2 ,s ss sv vD N N N⎡ ⎤Δ = Δ + − Δ⎣ ⎦ for v = s, u (20)

12 3/ 2

3/ 211

2 1/ 21/ 2

1

( ) / 2 and

( ) / 2 for ,

vv v

vv v

dx dx

d

dN x dx v s u

d

+

−+

−

σΔ = = σ π

Ω

σ= = σ π =

Ω

∫

∫(21)

1 1* *

1 1

, vv v v v v v vN N dx N N dx+ +

++ ++ +− +−− −

⎡ ⎤Δ = ⋅Δ = Δ + Δ⎣ ⎦∫ ∫ for v = s, u (22)

A similar definition is given for the isospin quantum distance D(F0, F1)between the t-channel isospin states It = 0,1 described by the F0 and F1 vectoramplitudes, respectively. Numerical results on the quantum distances D(Ns, Δs),D(F0, F1) and D(Nu, Δu) as well as on their isospin bounds for the (s, t, u)-channelsin the pion-nucleon scattering are presented in Fig. 1 by white small circles. Allthese results are obtained by using pion-nucleon phase shifs given in Ref. [20].

The estimation of the isospin quantum distances via their isospin bounds ismore suitable since these bounds are expressed only in terms of the integrated crosssections (see Eqs. (41), (45) and (49) in Ref. [1]). The quantum distances D(Ns, Δs)and D(Nu, Δu) attain their absolute maximum values of 1.86 fm at the position ofΔ(1236)-resonance. As is seen from Fig. 1 the maximum value Dmax(F0, F1) == 0.58 fm of the isospin t-channels is also attained in the range of Δ(1236)resonance. In particular, at PLAB = 10 GeV/c, we have 0.01 fm ≤ D(N, Δ) ≤ 0.06 fm,

1052 D. B. Ion, M. L. D. Ion 8

Fig. 1 – The isospin quantum distanceD(ΔS, NS)(o), D(F0, F1)(o) and D(ΔS, NS)(o) andand their isospin bounds calculated by usingEqs. (38)–(41), (42)–(45) and (46)–(49) fromRef. [2] and pion-nucleon phase shifts fromRef. [20]. Due to the isospin invariance in thepion-nuclon scattering, the isospin quantum dis-tances cannot take values outside of their isospin bounds in the hatched regions (see the text).

for (s, u)-isospin channels, 0.247 fm ≤ D(F0, F1) ≤ 0.265 fm for t-channels. At highprimary energies these distances for (s, u)-isospin channels are going to zero atleast as 1/ 2 .LABP− Moreover, the isospin conservation in the pion-nucleon scattering


impose strong constraints on isospin quantum distances, not only in theΔ(1236)-resonance region but also at high energies where the isospin bounds onthese distances are saturated. Consequently, a linear relation between average spinpolarization parameters is expected to be observed (see Ref. in [2]), especially atenergies higher than 2 GeV/c. The estimations of the isospin quantum distances viatheir isospin bounds are more suitable since these bounds are expressed only interms of the integrated cross sections [see Eqs. (41), (45) and (49) in Ref. [1]].

4. TESTS OF THE PMD-SQS OPTIMALITYVIA LOGARITHMIC SLOPE IN THE FORWARD DIRECTION

Let t be usual transfer momentum transfer connected with the c.m scatteringangle (x = cosθ) by the usual relation: t = –2q2(1 – x) where q is the c.m. momen-tum. Then the optimal logarithmic slope (14) is obtained in the following way

( )

2

0 0

2 2

ln ( , ) ( )2

1 4( 1)( 2) ( 1) ( 1) 14 4 4

o oo

t x

o oel

d d d db s t xdt dt dx d

d dJ Jd d

= =

⎡ ⎤ ⎡ ⎤σ σ= = =⎢ ⎥ ⎢ ⎥Ω⎣ ⎦ ⎣ ⎦

⎡ ⎤π σ σ⎡ ⎤= + + − = + + − −⎢ ⎥⎢ ⎥ σ Ω Ω⎣ ⎦ ⎣ ⎦

Now, by the same procedure as in Ref. [1], we proved the followingimportant optimal bound

( )2 4 ( 1) ( 1) 14o

el

d db bd d

⎡ ⎤π σ σ= + + − − ≤⎢ ⎥σ Ω Ω⎣ ⎦(23)

The equality holds in (18) if and only if the helicity amplitudes are theoptimal PMD-SQS solutions (9)–(12).

Therefore, for an experimental test of the PMD-SQS optimal result (9)–(12)the numerical valus of the experimental and optimal slopes are calculated from theexperimental phase shifts (EPS) solutions [20], and [32] and also directly from theexperimental data [25–35]. The results are displayed in Figs. 2–4. In the case ofpion-nucleon scattering for PLAB ≥ 3 GeV/c the experimental data on

exp , (1), eldbdσ⎡ ⎤σ⎢ ⎥Ω⎣ ⎦

are collected mainly from the original fits of Refs. [25]–[29].

To these data we added some values of b from Lasinski et al. [33] and alsocalculate directly from the phase shifts (EPS) [20]. In the low laboratory momentaPLAB ≥ 2 GeV/c, the slope parameters bexp, and bo are determined from the

EPS-solutions [20]. The experimental data for K±P–scatterings are collected from

1054 D. B. Ion, M. L. D. Ion 10

the original fits [25, 27, 29, 31, 32] to which we added those pairs calculated fromthe experimental phase shifts (EPS) solutions of [32]. The experimental data for[ , ]PP PP − scatterings are obtained from Refs. [33]–[35].

Now, it is important to note that the optimal inequality (21) includes in amore general and exact form the unitarity bounds derived by Martin [21] and

Martin-Mac Dowell [22] and Ion [1, 23, 24]. Indeed, since ( 1) 0,ddσ ± ≥Ω

from

Eq. (23) it is easy to obtain

Fig. 2a – The experimental values of the logarithmic slopes bexp (black circles) formeson-nucleon scatterings are compared with optimal state predictions bo(14) (white

circles) (see the text).


Fig. 2b – The experimental values of the logarithmic slopes bexp for[π±P, K±P]-scatterings for pLAB ≥ 2 GeV/c are compared with the optimal

state predictions bo (23) (see Figs. 2a, b).

Fig. 3 – The experimental values of the logarithmic slopes bexp for proton-proton scatterings(black circles) are compared with the optimal state predictions bo (24) (white circles) (see thetext). Dashed curve correspond to an estimation of the Martin-MacDowell unitarity bound (27).

1056 D. B. Ion, M. L. D. Ion 12

Fig. 4 – The experimental values of the logarithmic slopes bexp for PP scatterings (blackecircles) are compared with the optimal state predictions bo (24) (white circles and whitetriangles). Dashed curve correspond to an estimation of the Martin-MacDowell unitarity

bound (27).

21

4 ( 1) 14o

el

db bd+

π σ⎡ ⎤= + − ≤σ Ω⎢ ⎥⎣ ⎦

(proved by D. B. Ion [1]) (24)

21

4 ( 1) 14o

el

db bd−

⎡ ⎤π σ= − − ≤⎢ ⎥σ Ω⎣ ⎦ (proved by in Ref. [5, 24]) (25)

2

4

2

2 14

T

elb

⎡ ⎤σ− ≤⎢ ⎥π σ⎣ ⎦

, (D. B. Ion bound Refs. [5, 23]) (26)

22

22 1

9 4T

Ael

b⎡ ⎤σ

− ≤⎢ ⎥π σ⎣ ⎦, (Martin MacDowell bound [22]) (27)

2

2 14 4

TAb

σ⎡ ⎤− ≤⎢ ⎥π⎣ ⎦ (Martin-bound [21]) (28)

For comparisons with optimal slopes in Figs. 3–4 the estimation of theMartin-MacDowell bound are also presented by dashed curves. Therefore, fromFigs. 3–5, we conclude that the available experimental data on the differential crosssections for the forward peak of all [meson-nucleon, nucleon-nucleon andantinucleon-nucleon scatterings, at all energies are described in an unified way byoptimal logarithmic slope bo (24).


5. SCALING OF HADRON-HADRON SCATTERINGVIA PMD-SQS-OPTIMALITY

The theory of the optimal states from the reproducing kernel Hilbert space(RKHS) of the scattering amplitudes is developed in the papers [1, 3–5]. Then, twoimportant physical laws of hadron-hadron scattering the scaling of angulardistributions and S-channel helicity conservations are derived via optimal statedominance in our papers [1, 5] by using the PMD-SQS minimum principle. Now, itis important to note that all the available experimental data on the differential crosssections for the forward peak of all [meson-nucleon, nucleon-nucleon andantinucleon-nucleon scatterings, at all energies higher than 2 GeV, can bedescribed in an unified way by the following optimal scaling functions

( , )) ( ) ( , ) / ( , 0), 2 ( )od df s t f s t s t b sdt dtσ σ= τ = τ = (29)

21/ 22 ( )( ) ( ) / ( 1) 2

oo

oJd df x t b

d dτ⎡ ⎤σ στ = + = τ = ⎡ ⎤⎣ ⎦⎢ ⎥Ω Ω τ⎣ ⎦

for small τ (30)

The experimental values of the scaling function, obtained by using Eqs. (27)and the experimental data [35, 47–49], are compared with the theoreticalpredictions (30) for [ , , , ]P K PP PP± ±π -scatterings. These results allow us toconclude that a principle of minimum distance in the space of states can be a goodcandidate for an optimum principle in the theory of hadronic interactions.

Next, it is easy to prove that the results (29) include in a more general andexact form the scaling variables 2 /T elt σ σ and Tt σ introduced in Refs. [36–39](see also [40]).

Indeed, as early as 1970 Singh and Roy [36] established that an upper boundfor the absorptive ratio scale in the variable 2 /T eltσ σ for small |t|. Cornille andMarin [37] have examined the rigorous foundation of scaling properties of thedifferential cross section-ratio (see Eq. (29)) in the context of diffraction scattering.They have proved scaling in the variable 2 /T eltσ σ for the case when 2/(ln )T sσ inthe asymptotic. Moreover, Cornille [38] has examined different conditions ofscaling and shown that one of of the scaling variable at asymptotic energies can betb(s), (s, t, u) being the usual Mandelstam variables.

6. THE PMD-SQS OPTIMALITY AND NONEXTENSIVE STATISTICALBEHAVIOUR IN HADRONIC SCATTERING

In the last time there is an increasing interest in the foundation of a newstatistical theory [52, 53] valid for the nonextensive statistical systems which


exhibit some relevant long range interactions, the memory effects or multifractalstructures. It is important to mention here that the Tsallis nonextensive statisticalformalism [53] already has been successfully applied to a large variety ofphenomena such as (see Ref. [54]): Levy-like and correlated anomalous diffusions,turbulence in electron plasma, self-graviting systems, cosmology, galaxy clusters,motion of Hydra viridissima, classical and quantum chaos, quantum entanglement,reassociation in folded proteins, superstatistics, economics, linguistic, etc. Here, isworth to mention the recent applications of nonextensive statistics to nuclear andhigh-energy particle physics, namely: electron-positron annihilations [55, 56],quark-qluon plasma [57], hadronic collisions [6–16, 58], nuclear collisions [59],and solar neutrinos [60, 61]. In general, in hadronic quantum systems thenonextensivity is expected to emerge due to the presence of the quarks and gluonshaving a strong cupling long-range regime of quantum chromodinamics. Hence, wethink that the pion-nucleon, kaon-nucleon, antikaon-nucleon, as well as thepion-nucleus quantum scatterind systems are the most suitable for suchinvestigations since in these cases the experimental nonextensive entropies can beobtained with high accuracy from the available phase shifts analyses [20–32].

6.1. J-NONEXTENSIVE STATISTICSFOR THE QUANTUM SCATTERING STATES

We define two kind of Tsallis-like scattering entropies. One of them, namelySJ(p), p ∈ R is special dedicated to the investigation of the nonextensive statisticalbehavior of the angular momentum quantum states can be defined by

max1

1/ 2

1( ) 1 (2 1) ( ),1

Jp o

J JjJ

S p j p S pp

=

⎡ ⎤= − + ≤⎢ ⎥− ⎢ ⎥⎣ ⎦

∑ p R∈ , (31)

where the probability distribution are given by

( )max

2 2

2 2

1/ 2

,

(2 1)

j jj J

j j

f fp

j f f

+ −

+ −

+=

+ +∑

max

1/ 2

(2 1) 1J

jj p+ =∑ (32)

6.2. θ-NONEXTENSIVE STATISTICSFOR THE QUANTUM SCATTERING STATES

In similar way, for the θ-scattering states considered as statistical canonicalensemble, we can investigate their nonextensive statistical behavior by using anangular Tsallis-like scattering entropy ( )S qθ defined as

1060 D. B. Ion, M. L. D. Ion 16

1

1

1( ) 1 ( ) ,1

qS q dxP x dxq

+θ −

⎡ ⎤= −⎢ ⎥− ⎣ ⎦∫ q R∈ (33)

where the density of probality ( )P x is defined as follows

2( ) ( ),el

dP x xd

π σ=σ Ω

1

1( ) 1dxP x dx

+

−=∫ (34)

with ( )d xdσΩ

and σel are defined by Eqs. (4)–(5) and (6)–(7).

The above Tsallis-like entropies posses two important properties. First, in thelimit p → 1 and q → 1, the Boltzman-Gibbs kind of entropies are recovered

1lim ( ) ( ) ln ( )q S q dxP x P x→ θ = −∫ and max

11/ 2

lim ( ) (2 1) lnJ

p J j jS p j p p→ = − +∑ (35)

and

( ) (1 ) ( ) ( )A B A B A BS k S S k S k S k+ = + + − for ,k p q R= ∈ (36)

for any independent subsystems ( ).A B A Bp p p+ = ⋅ Hence, each of the indices

p ≠ 1, q ≠ 1 from the definitions (31) and (33) can be interpreted as measuring thedegree of nonextensivity.

6.3. [Jθ]-NONEXTENSIVE STATISTICSFOR THE QUANTUM SCATTERING STATES

Also, using the same distribution (32) and (34), we define the combined( , )JS p qθ entropies

max 1

11/ 2

1( , ) 1 (2 1) ( ) ,1

Jp q

J jJ

S p q j p dxP x dxp

+θ

−=

⎡ ⎤= − +⎢ ⎥− ⎢ ⎥⎣ ⎦

∑ ∫ p q R≠ ∈ (37)

( , ) ( )J JS q q S qθ θ= (38)

with the properties

( ) ( ) ( ) (1 ) ( ) ( )J J JS q S q S q q S q S qθ θ θ= + + − (39)

( , ) ( ) ( ) (1 ) ( ) ( ),

1where ( ) ( ),1

J J JS p q S p S p p S p S p

qS p S q p q R

p

θ θ θ

θ θ

= + + −−≡ ≠ ∈−

(40)


6.3. THE EQUILIBRIUM DISTRIBUTIONS FOR THE [J], [θ]AND [Jθ] QUANTUM SYSTEMS OF STATES

We next consider the maximum-entropy (MaxEnt) problem

max ( ), ( ), ( , )JJS p S q S p qθθ⎡ ⎤⎣ ⎦ when σel, ( 1),ddσ +Ω

and ( 1),ddσ −Ω

are fixed (41)

as criterion for the determination of the equilibrium distributions for the quantumstates produced from the meson-nucleon scattering. The equilibrium distributions{ },me

jp ( ),meP x as well as the optimal scattering entropies for the quantum

scattering of the spineless particles were obtained in Ref. [8–9]. For the J-quantumstates, and θ-quantum states in the meson-nucleon scattering case thesedistributions are given by:

12( 1) 1/ 4 for all 1/ 2 and

0 for all 1

me oj j o o

j o

p p J j J

p j J

−⎡ ⎤= = + − ≤ ≤⎣ ⎦

= ≥ +(42)

2( ) ( ) ( )o

me o

el

dP x P x xd

π σ= =σ Ω

(43)

with Jo and ( )od x

dσΩ

given by Eqs. (12) and (15), respectively.

Indeed, solving problem (41) via Lagrange multipliers we get that the singular

solution λ0 = 0 exists and is just given by the ( ), ( ), ( ), ( , )oo o oJJ JS p S q S p S p qθθ θ

⎡ ⎤⎣ ⎦ -

optimal entropies corresponding to the PMD-SQS-optimal state (9)–(11). Therefore,the solution of problem (41) is given by

1

1

1( ) ( ) 1 ( ) ,1

2( ) ( ) [see Eq. (13)]

qo o

oo

el

S q S q P x dxq

dP x xd

+θ θ −

⎡ ⎤⎡ ⎤≤ = − ⎣ ⎦⎢ ⎥− ⎣ ⎦

π σ=σ Ω

∫(44)

1/ 2

2

1( ) ( ) 1 (2 1) ,1

1 for 1/ 2( 1) 1/ 4

oJpo o

J J j

oj o

o

S p S q j pp

p j JJ

⎡ ⎤⎡ ⎤≤ = − +⎢ ⎥⎣ ⎦− ⎢ ⎥⎣ ⎦

= ≤ ≤+ −

∑(45)

1

11/ 2

1( , ) ( , ) 1 (2 1) ( )1

oJp qo o o

J J jS p q S p q j p P x dxp

+θ θ

−

⎡ ⎤⎡ ⎤ ⎡ ⎤≤ = − +⎢ ⎥⎣ ⎦⎣ ⎦− ⎢ ⎥⎣ ⎦

∑ ∫ (46)

1062 D. B. Ion, M. L. D. Ion 18

The equality holds in (42)–(44) if and only if the helicity amplitudes are theoptimal PMD-SQS solutions (9)–(12). Now, for a systematic experimentalinvestigation of the saturation of the optimality limits in hadron-hadron scatteringwe used the available experimental phase-shifts [20, 32] to solve the followingimportant problems: (i) To reconstruct the experimental pion-nucleon,kaon-nucleon and antikaon-nucleon scattering amplitudes; (ii) To obtain numericalvalues of the experimental scattering entropies SJ(p), Sθ(q) from the reconstructedamplitudes; (iii) To obtain the numerical values of the optimal Jo fromexperimental scattering amplitudes, (iv) To calculate the numerical values for thePMD-SQS-optimal entropies ( ),o

JS p ( ).oS qθ So, these results presented in Fig. 6

are compared with the results of the PMD-SQS -optimal model from section 2. Thegrey regions from Figs. 6–7 are obtained by assuming a minimum error of(ΔJo = ±1, ΔLo = ±1) in estimation of the optimal angular momenta (Jo, Lo),respectively.

From Fig. 6a we see that the experimental values of the scattering entropies( )JS p are well described with high accuracy by the optimal values ( )o

JS p only forthe statistical non-extensivitiy p = 2/3 while a significant discrepancy is observedfor p = 1 and p = 2. From Fig. 6b we see that the saturation of the optimality limit

( )oS qθ is evidentiated for the values of ( )S qθ entropy corresponding to the

conjugated nonextensivity q = 2. Significant departures from the optimal entropyoSθ (solid line) is observed for nonextensivities q = 2/3 and q = 1, respectively.

Consequently, the entropies ( , )JS p qθ are reproduced with high accuracyonly for the pairs of nonextensivities (p, q) = (2/3, 2) and (1, 1) (see Fig. 6c).

In particular, for the scattering of spinless particles we proved [9] thefollowing entropic optimal bounds

211

1

[ ( ,1)]1( ) ( ) 11 (1,1)

qo K x

S q S q dxq K

+θ θ −

⎡ ⎤⎡ ⎤⎢ ⎥≤ = − ⎢ ⎥− ⎢ ⎥⎣ ⎦⎣ ⎦∫ (see Ref. [9]) (47)

1 2(1 )1( ) ( ) 1 [ 1]1

o qL L oS q S q L

q−⎡ ⎤≤ = − +⎣ ⎦−

(48)

1( , ) ( , )

oJ JS p q S p qθ θ≤ =

212(1 )

1

[ ( ,1)]1 1 [ 1] 11 (1,1)

qp

oK x

L dxq K

+−−

⎡ ⎤⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥− + − ⎢ ⎥− ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦∫ (49)

01( ,1) (2 1) ( )2

oLlK x l P x= +∑ and 2K(1, 1) = 2( 1)oL + (50)


Fig. 6a – The experimental values of themeson-nucleon scattering entropiesSJ(p) (31), are compared with their

optimal PMD-SQS upper bounds 0( )JS p

(45) (solid curves) for the entropic non-extensivities p = 2/3, 1 and 2, respec-tively. The numerical values for SJ(p)

and 0( )JS p are obtained by using theavailable phase shifts analyses [20] and[32]. The grey region around the optimalentropies are obtained by aasuming anerror of ΔJo = ±1 in the estimation of optimal angular momentum Jo.

1064 D. B. Ion, M. L. D. Ion 20

Fig. 6b – The experimental values of themeson-nucleon scattering entropies Sθ(q)(33), are compared with their optimalPMD-SQS upper bounds Sθ(q) (44) (solidcurves) for the entropic nonextensivitiesp = 2/3, 1 and 2, respectively. The nume-rical values for Sθ(p) and 0( )S pθ are ob-tained by using the available phase shiftsanalyses [20] and [32]. The grey regionaround the optimal entropies are obtainedby asuming an error of ΔJo = ±1 in theestimation of optimal angular momentum Jo.


Fig. 6c – The experimental values of themeson-nucleon scattering entropies

( , )JS p qθ (37), are compared with theiroptimal PMD-SQS upper bounds Sθ(q)(46) (solid curves) for the entropic non-extensivities p = 2/3, 1 and 2, respec-

tively. The numerical values for ( , )JS p qθ

and 0( )S pθ are obtained by using theavailable phase shifts analyses [20] and[32]. The grey region around the optimalentropies are obtained by aasuming anerror of ΔJo = ±1 in the estimation of optimal angular momentum Jo.

So, the problem to find an upper bound for all Tsallis-like scattering entropies(47–49) when the elastic integrated cross section σel and the forward differential cross

section dσ/dΩ (1) are fixed is completely solved in terms of the optimal entropiesobtained from PMD-SQS. The results on experimental tests of these optimal upperbounds are presented for both extensive and nonextensive statistics cases in Fig. 2by using 49 sets of the available pion-nucleus (He, C, O, and Ca) phase shifts.

1066 D. B. Ion, M. L. D. Ion 22

Fig. 7 – The experimental values of the scattering entropies Sθ(q) and SL(q), are compared with the theoreticaloptimal state predictions (full curve) for pion-nucleus (He, C, O, and Ca) scatterings (see the paper [9] for

details).

7. EVIDENCE FOR 1 12 2p q

⎛ ⎞+ −⎜ ⎟⎝ ⎠

NONEXTENSIVITY CONJUGATION

IN QUANTUM NONEXTENSIVE SCATTERING

A natural but fundamental question was addressed in Refs. [9–11], namely,what kind correlation (if it exists) is expected to be observed between thenonextensivity indices p and q corresponding to the (p, J)-nonextensive statisticsdescribed by SJ(p)-scattering entropy and (q, θ)-nonextensive statistics describedby Sθ(q)-scattering entropy? In order to get an consistent mathematical answer tothis question we first present the following important remarks.

Remark 1. Any Tsallis-like entropy SJ(p) (31)–(33) can be written in thefollowing equivalent form

( )1

2

21( ) 1 ,

1p

J j pS p p R

p

⎡ ⎤⎢ ⎥= − ϕ ∈

− ⎢ ⎥⎣ ⎦(51)


max 2 2

1/ 2

2 22

,

(2 1)

, { [ , ]}

jj J

j j

j j j j j j p

f

j f f

p L

±±

+ −

+ − + −

ϕ =

⎡ ⎤+ +⎢ ⎥⎣ ⎦

= ϕ + ϕ ϕ = ϕ ϕ ∈

∑ (52)

[ ]max

12 1

22

1/ 2

(2 1) 1 (1 ) ( )J p

pj j Jpj p p S p

⎡ ⎤ϕ = + = + −⎢ ⎥

⎢ ⎥⎣ ⎦∑ (53)

and consequently the nonextensivity index p can be interprted as being directlyconnected with the dimension 2p of the Hilbert space L2p of the vector valuedfunctions ϕj = [ϕj+, ϕj–].

Remark 2. Any scattering entropy Sθ(q) (31)–(32) can be written in thefollowing equivalent form

( )221( ) 1 ,

1

q

qS q q Rqθ

⎡ ⎤= − φ ∈⎢ ⎥− ⎣ ⎦(54)

12 2

1

2 22

( )( ) ,

( ) ( )

( ) ( ) ( ) , { ( ) [ ( ), ( )]} q

f xx

dx f x f x

P x x x x x x L

+±+± +

++ +−−

++ +− ++ +−

φ =

⎡ ⎤+⎣ ⎦

= φ + φ φ = φ φ ∈

∫ (55)

[ ]1

1 2 12

21

( ) ( ) 1 (1 ) ( )q

q qpx dxP x q S q

+

θ−

⎡ ⎤φ = = + −⎢ ⎥

⎢ ⎥⎣ ⎦∫ (56)

and consequently the nonextensivity index q can be interpreted as being directlyconnected with the dimension 2q of the Hilbert space L2q of the vector valuedfunctions [ ( ), ( )].x x++ +−φ = φ φ

Riesz-Thorin (1/2p + 1/2q)-correlation. If the Fourier transform defined byEqs. (2)–(3) is a bounded map T : L2p → L2q, then the nonextensivity indices (p, q)of the J-statistics described by the Tsallis-like entropy SJ(p) and θ-statisticsdescribed by the scattering entropy Sθ(q), must be correlated via the Riesz-Thorinrelation

1 1 12 2p q

+ = or 2 1

pq

q=

−(57)

1068 D. B. Ion, M. L. D. Ion 24

while the norm estimate of this map is given by

2 2 1

2

2p

q q

p

TfM

f−= ≤ (58)

Then, the results (57)–(58) can be obtained as a direct consequence of theRiesz-Thorin interpolation theorem extended to the vector-valued functions (seeRef. [15] for a detailed proof). Next, the nonextensivity indices p and q aredetermined [12] from the experimental entropies by a fit with the optimal entropies

o1 o1( ), ( )JS p S qθ⎡ ⎤⎣ ⎦ obtained from the principle of minimum distance in the space of

states. In this way strong experimental evidences for the p-nonextensivities in therange 0.5 ≤ p ≤ 0.6 with q = p/(2p – 1) > 3 are obtained [with high accuracy(CL > 99%)] from the experimental data of pion– nucleon and pion–nucleusscatterings.

An experimental verification of the correlation (49) is illustrated in Figs. 8–9.In Fig. 9 the experimental values of the Tsallis-like entropies Sθ(q), SJ(p) and

( , ),JS p qθ calculated by using Eqs. (31), (33) and (37) p = 0.6 and q = 3 andexperimental pion-nucleon [20] and kaon-nucleon [32] phase shifts, are comparedwith the theoretical optimal state predictions (full curves) for pion-nucleon andkaon-nucleon scatterings. The grey regions from Figs. 8–10 are obtained byassuming a minimum error of ΔJo = ±1 in the estimation of the optimal angularmomentum from the experimental data.

8. LIMITED ENTROPIC UNCERTAINTY PRINCIPLE [15]

In the paper [14] by introduction of the nonextensivity conjugated entropy( , )JS p qθ we proved the following new nonextensive conjugated entropic

uncertainty relations (NC-EUR) as well as new nonextensive conjugated entropic

uncertainty bands (NC-EUB). If σel and (1)ddσΩ

are known from experiments then

the nonextensivity conjugated entropy ( , ),JS p qθ defined by Eqs. (37)–(40), mustobey the following entropic bands:

min max( , ) ( , ) ( , )J J JS p q S p q S p qθ θ θ≤ ≤ q = p/(2p – 1) (NC-EUB) (59)

21 1 1

min 2 21

1 11 2 2

( ,1)1( , ) 1 [2]

1 (1,1)

q

pJ

K x

S p q dxp K

−θ

−

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= −⎢ ⎥⎜ ⎟− ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∫ (NC-EUR) (60)

1070 D. B. Ion, M. L. D. Ion 26

Fig. 9 – The experimental values ofthe Tsallis-like entropies Sθ(q), SJ(p)

and ( , ),JS p qθ calculated by usingEqs. (31)–(37) and experimental pion-nucleon [20] and kaon-nucleon [32]phase shifts, are compared with thetheoretical optimal state predictions(full curves) for pion-nucleon and kaon-nucleon scatterings.


21 1 1

max 1 2 211 1

1 12 2 1 2 2

( ,1)1( , ) ( , ) 1 [2 (1,1)]

1 (1,1)

PMD-SQS-prediction

q

o pJ J

K x

S p q S p q K dxp K

−θ θ

−

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= = −⎢ ⎥⎜ ⎟− ⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

∫ (61)

for any (1/ 2, )p∈ ∞ and q = p/(2p – 1).For numerical investigation of the optimal NC-EUB (59) we calculated

nonextensive scattering entropies by using the available phase shifts [20] for allcharge πP → πP-channels.

So, in Figs. 10a, b, c we presented the first experimental test of theuncertainty entropic inequalities (53)–(55) in pion-nucleon scattering for threekinds of nonextensivity conjugated statistics: (a) [p = 2/3, q = 2]; (b) [p = 1, q = 1]and (c) [p = 2, q = 2/3], So, the experimental results for [ , /(2 1)]JS p p pθ − arecompared with the PMD = SQS predictions (full curve). Hence, by the numericalresults displayed in Fig. 10a,b,c, we obtained not only consistent experimental testsof the optimal NC-EUR and NC-EUB, but also strong experimental evidences forthe principle of minimum distance in space of quantum states (PMD-SQS). The greyregions from Figs. 10 are obtained by assuming a minimum error of ΔJo = ±1 in the

estimation of the optimal angular momentum 1/ 2

4int (1) 1/ 4 1oel

dJ half egerd

⎡ ⎤π σ= + −⎢ ⎥σ Ω⎣ ⎦from the experimental data [20].

From Figs. 10a, b, c, we see that the experimental data on the nonextensivityconjugated entropies [ , /(2 1)]JS p p pθ − are in excellent agreement (CL > 99%)with the PMD-SQS optimal predictions if the nonextensivities p and q of thescattering entropies SJ(p) and S(q) are correlated via Riesz-Thorin relation

1 1 12 2p q⎡ ⎤+ =⎢ ⎥⎣ ⎦

[or q = p/(2p – 1)]. So, the best fit is obtained for the conjugate

pairs [p, q = p/(2p – 1)] with the values of nonextensivity p in the range1/2 ≤ p ≤ 2/3. In fact a significant departure from PMD-SQS-optimality is observedonly for conjugate pairs with p ≥ 2 and 1/2 ≤ q ≤ 2/3 where 2 / 500.Dnχ >

The results obtained in Fig. 10 for [ , /(2 1)]JS p p pθ − can be compared withthose for the entropies ( , ),JS p qθ for p = q presented in Fig. 4 in Ref. [15]. Then,

we see that the values of the usual ( , )JS p qθ are far from optimal values 1( , ),oJS p qθ

p = q, and in some cases violate the entropic uncertainty band.These results alow us to conclude thar the new entropy [ , /(2 1)]JS p p pθ − is

the best candidate for description of entropic uncertainty relations when thenonextensivity conjugation is taken into account.

1072 D. B. Ion, M. L. D. Ion 28

Fig. 10a – Experimental tests of NC-entropic uncertainty band (59)–(61)for the pion-nucleon quantum scatte-ring corresponding to nonextensivityp = 0.6 and q = 3. The full curve theoptimal state PMD-SQS predictionswhile their errors are given by by the grey regions (see the text).


Fig. 10b – Experimental tests of NC-entropic uncertainty band (59)–(61)for the pion-nucleon quantum scatte-ring corresponding to nonextensivityp = 2/3 and q = 2. The full curve theoptimal state PMD-SQS predictionswhile their errors are given by by the grey regions (see the text).

1074 D. B. Ion, M. L. D. Ion 30

Fig. 10c – Experimental tests of NC-entropic uncertainty band (59)–(61)for the pion-nucleon quantum scatte-ring corresponding to nonextensivityp = 2 and q = 2/3. The full curve theoptimal state PMD-SQS predictionswhile their errors are given by by the grey regions (see the text).


9. CONCLUSIONS AND OUTLOOK

The significant results obtained in Ref. [1–16], can be summarized asfollows: (i) A description of quantum scattering via principle of minimum distance in

space of states.[1] (ii) Optimal state description of hadron-hadron via RKHS methods. (iii) A proof of two important physical laws: Scaling and S-channel helicity

conservation in hadron-hadron scattering. (iv) A proof of Limited Entropic Uncertainty Principle (LEU-Principle) as new

principle in quantum physics. (v) Experimental evidences for optimal conjugated nonextensive statistics in

hadronic scatterings. (vi) Introduction of a new nonextensive quantum entropy and discovery of

equilibrium of the quantum hadronic states.(vii) The strong experimental evidence obtained here for the nonextensive statistical

behavior} of the $(J,\theta )-$ quantum scatterings states in the pion-nucleon,kaon-nucleon and antikaon-nucleon scatterings can be interpreted as anindirect manifestation the presence of the quarks and gluons as fundamentalconstituents of the scattering system having the strong-coupling long-rangeregime required by the Quantum Chromodynamics.The theory of diffraction obtained via PMD-SQS in our paper [1] is free of

any paradox of the old quantum physics in the sense that diffraction andinterference are universal phenomena independent of particle-wave duality inter-pretation, possessing not only universal character but also “scaling” properties.Such universal diffraction or universal interference phenomenon can occur for anymassive particle. Therefore a continuation of the papers [1–17] is needed since thetheory based on the PMD-SQS-optimality includes the heart of quantumphenomena.

Universal PMD-SQS-optimal bound states. Special investigations will beconducted for the development of a theory of the PMD-SQS-optimal bound states.We prove that the PMD-SQS-optimal “universal atomic levels” are all of form:

2/ ,nE R n= where R = ma + mN – MA is a Rydberg-like energy while optimalprincipal quantum number is ( 1).on L= + We will determine the analyticexpression of the PMD-SQS-optimal “atomic” amplitudes in uni-directional, aswell as, in bidirectional optimal models. For atomic transitions, we expect to prove

that the optimal spectral series 2 21 1o o

Rn m

⎡ ⎤ω = −⎢ ⎥

⎣ ⎦ are direct consequences of the

energy conservation in that transition. Such spectral series are just those observedfrom the atomic experimental data. The construction of a possible optimal

1076 D. B. Ion, M. L. D. Ion 32

Mendeleyev-like Table will also be investigated. We try to get an answer to thefollowing important questions: Can atoms be completely described by the PMD-SQS-optimum principle without Coulomb potential and without using Schrödingeror Dirac equation? Also, similar problems of optimality can be formulated just inthe same way for the nuclear structures, chemical structures or even for thegravitational bound states structures.

Universal optimal resonances. Special PMD-SQS-optimality investigationsalso will be dedicated to the development of a theory of the PMD-SQS-optimalresonances. Here, many important results are expected to be proved. (i) The veryhigh widths for the optimal resonance, where: 1 ,o

n nΓ = Γ n = Lo + 1. (ii) A definitecorrelation between “positions” and “widths” of “elementary” resonances whichare constituents of the optimal resonances, (iii) A “diffractive” character of theangular distributions, (iv) A total intensity proportional with (Lo + 1)2. Moreover,our investigation will be conducted to get from the available data the experimentalevidences for the optimal resonances in hadron-hadron scattering as well as inhadron-nucleus scattering. We must underline that not only so called “diffractive(Morrison) resonances” (A1, A2, et al.) are candidates for to be optimal resonances,but also, the “dual diffractive resonances” discovered by us and published in thepaper [17].

Finally, we remark that the physics based on the PMD-SQS minimumprinciple is a new branch of the quantum physics interconnecting, in a creativeway, quantum physics, functional analysis, group theory, as well as quantuminformation theory. All these new ideas and theoretical developments will allow usto look at natural phenomena in a radically new and original way, eventuallyleading to unifying concepts independently of the detailed structures of systems notonly in physics, but also in other sciences such as: Astrophysics, Biology,Psychology, Genetics, Economy, etc.

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