Jun 4, 2006 - Baleanu[10] has studied the Euler-Lagrange equations for classical .... an euclidean space for N particles in space coordinate represent...

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arXiv:math-ph/0510099v4 4 Jun 2006

(Dated: June 4, 2018) Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schr¨ odinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite potential well and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced by an ordinary Schr¨ odinger equation with an additional potential, which is of the form V ∼ |x| for α < 1, corresponding to the confinement potential, which is introduced phenomenologically to the standard models for a non relativistic interpretation of quarkonium-spectra. The ordinary Schr¨ odinger equation is triple factorized and yields a fractional wave equation with internal SU (3) symmetry. The twofold iterated version of this wave equation shows a direct analogy to the derived fractional Schr¨ odinger equation. The angular momentum eigenvalues are calculated algebraically. The resulting mass formula is applied to the charmonium spectrum and reproduces the experimental masses with an accuracy better than 0.1%. Extending the standard charmonium spectrum, three additional particles are predicted and associated with Σ0c (2455) and Y (4260) observed recently and one X(4965), not yet observed. The root mean square radius for Σ0c (2455) is calculated to be < r >≈ 0.3[fm]. The derived results indicate, that a fractional wave equation may be an appropriate tool for a description of quark-like particles. PACS numbers: 12.39, 12.40, 14.65, 13.66, 11.10, 11.30, 03.65

I.

INTRODUCTION

Since Newton[1] and Leibniz[2] introduced the concept of infinitesimal calculus, differentiating a function f (x1 , ..., xn ) with respect to the variable xi is a standard technique applied in all branches of physics. The derivative operator ∂i , ∂ (1) ∂xi transforms like a vector, its contraction yields the Laplace-operator, using Einstein’s sum convention ∂i =

∆ = ∂ i ∂i

(2)

which is a second order derivative operator, the essential contribution to establish a wave equation, which is the starting point to describe several kinds of wave phenomena. Until now, in high energy physics the derivative operator has only been used in integer steps. We want to extend the idea of differentation to arbitrary, not necessarily integer steps. A natural generalization is to search for an operator Di by setting Dim = ∂in

(3)

where m, n are integers. Formally, this is solved by extracting the m-th root n/m

Di = ∂i

m, n ∈ N

(4)

or, even more general, we will introduce a fractional derivative operator by Di = ∂iα

α ∈ R+

(5)

with the fractional derivative coefficient α being a positive, real number. The concept of fractional calculus has stimulated mathematicians since the days of Leibniz[3]-[5]. In physics, early attempts in the field of applications was studies on non-local dynamics, e.g. anomalous diffusion or fractional Brownian motion [6],[7]. During the last decade, remarkable progress has been made in the theory of fractional wave equations[8]-[16]. Raspini[8],[9] has derived a fractional (α = 2/3) Dirac equation. Baleanu[10] has studied the Euler-Lagrange equations for classical fields and gave the explicit form of a fractional Klein-Gordon-equation and a fractional Dirac-equation, conformal with Raspini’s. Both studies were based on the use of the RiemannLiouville (RL) fractional derivative, which is used by many authors working on the field of fractional derivatives. For practical purposes, the main deficiency of the RL fractional derivative is the fact, that the derivative of a constant function does not vanish. Maybe this is one reason for the fact, that until now, there exists not a single application in the area of high energy physics. Laskin[16] has proposed a hermitean fractional Schr¨odinger equation, based on Feynman’s path integral approach. His applications are based on the semi classical Bohr-Sommerfeld quantization condition only. We will use a different approach. We will apply the concept of fractional derivative to derive a fractional Schr¨odinger type wave equation by a quantization of the classical non relativistic Hamiltonian. We will collect arguments and results which indicate, that this equation is an alternative tool for a appropriate description of the

2 charmonium spectrum, which is normally described by a phenomenological potential. In the following sections, we will explicitely derive exact solutions for the free particle and for particles in an infinite well potential. We will prove, that these solutions show a behaviour, which may be reproduced by an ordinary Schr¨odinger equation with an additional linear potential term for α < 1, indicating that a fractional wave equation and the confinement problem are strongly related. We will derive a fractional multi-component wave equation via threefold factorization of the ordinary non relativistic Sch¨ odinger equation, which contains an internal SU (3) symmetry. We will then study an analytical mass formula in terms of angular momentum multiplets which will reproduce the experimental masses of the charmonium spectrum within an error of better than 0.1%. We will extend the standard charmonium spectrum and predict new, additional particles. Finally, we will give a reasonable estimate for the root mean square radius of Σ0c (2455). II.

FRACTIONAL DERIVATIVE

Let q = [α] be the integer part of α and f a function of n variables xi , i = 1, ..., n with xi > 0. To derive a specific representation of the fractional derivative operator Di , defined by (5) we start with the Cauchy integral + α Ii extended to fractional order

1 Γ(α)

Z

0

+ α i Ii (x )f (x1 , ..., xi , ..., xn ) i x i α−1 1

du(x − u)

f (x , ..., u,

=

(6) ..., xn )

α−(q+1) q+1 ∂i (q+1)−α q+1 ∂i Ii

=

(7)

leads to the definition of the Caputo[17] fractional differential operator + c D, which is the form we will use. + i 1 i n c Di (x )f (x , ..., x , ..., x ) Z xi q+1

1 Γ(q + 1 − α)

0

=

(8)

∂ du f (x1 , ..., u, (xi − u)α−q ∂uq+1

=0

(9)

For a function of the type f (x) = xnα

n∈N

=

Γ(1 + nα) x(n−1)α Γ(1 + (n − 1)α)

(11)

For k > 0, x > 0 we are then able to define Caputo-Taylor series of the form f (kx) =

∞ X

an (kx)nα

(12)

n=0

The corresponding fractional derivatives are given by: + c Df (kx)

= kα

∞ X

an+1

n=0

Γ(1 + (n + 1)α) (kx)nα Γ(1 + nα)

(13)

Since we intend to use x and the fractional derivative on R, the next step is an extension of our definition of x and + c D(x) to negative reals R− . We propose the following mappings for x → χ(x) ¯ and + ¯ c D(x) → D(x): + c D(x)

χ(x) ¯ = sign(x) |x|α ¯ D(x) = sign(x) + c D(|x|)

(14) (15)

Besides a unique mapping from R+ to R+ and R− to R− the behaviour under parity transformations Π is well defined: Πχ(x) ¯ = −χ(x) ¯ ¯ ¯ ΠD(x) = −D(x)

(16) (17)

With these definitions we are able to define series f on R ∞ X

an χ ¯n (kx)

(18)

n=0

¯ with a well defined derivative D ∞ X

Γ(1 + (n + 1)α) n χ ¯ (kx) Γ(1 + nα) n=0 (19) To construct a Hilbert space on functions f (χ(kx)) ¯ we first define the integral operator Z x duα = sign(x)+ I α (|x|α ) (20) ¯ (χ(kx)) Df ¯ = sign(k)|k|α

an+1

−x

..., xn )

For a constant function this fractional derivative vanishes: + c D(x) const

+ nα c D(x) x

f (χ(kx)) ¯ =

a formal split of the partial differential operator into a fractional integral and an integer differential part ∂iα = ∂i

the fractional derivative is:

(10)

with the fractional scalar product Z < f |g >= duα f ∗ (χ(ku)) ¯ g(χ(k ¯ ′ u))

(21)

ˆ > of an operator O ˆ may conAn expectation value < O sequently be defined with Z ˆ ˆ g(χ(k < f |O|g >= duα f ∗ (χ(ku)) ¯ O ¯ ′ u)) (22)

3 to be ˆ ˆ >= < f |O|g >

(23)

The space coordinates and corresponding derivatives, defined by (14) and (15), are the basic input for our derivation of a fractional non relativistic Schr¨odinger type wave equation, the corresponding angular momentum operators and caculation of expectation values. III. QUANTIZATION OF THE CLASSICAL HAMILTONIAN AND FREE PARTICLE SOLUTIONS

By use of the definitions (14),(15) for the space coordinate and for the fractional derivative, we are able to quantize the Hamiltonian of a classical non relativistic particle and solve the corresponding Schr¨odinger equation. We define the following set of conjugated operators on an euclidean space for N particles in space coordinate representation: α ~ ¯ i} Pˆµ = {Pˆ0 , Pˆi } = {i~∂t , −i mcD (24) mc α ~ i = {i~∂t , −i mc sign(xi ) + c D(|x |)} (25) mc (1−α) χ(x ¯ i) ~ ˆ ˆ ˆ } (26) Xµ = {X0 , Xi } = {t, mc Γ(α + 1) (1−α) ~ 1 = {t, sign(xi )|xi |α } (27) mc Γ(α + 1) i = 1, ..., 3N These operators satisfy the following commutator relations on a function set {xnα }: i h ˆi, X ˆj = 0 (28) X i h (29) Pˆi , Pˆj = 0 i h 1 ˆ i , Pˆj = −i~δij X × (30) Γ(1 + α) Γ(1 + (n + 1)α)) Γ(1 + nα) − Γ(1 + (n − 1)α) Γ(1 + nα) = −i~δij c(n, α) (31)

With these operators, the classical, non relativistic Hamilton function Hc , which depends on the classical momenta and coordinates {pi , xi } Hc =

3N X p2i + V (x1 , ..., xi , ..., x3N ) 2m i=1

(32)

is quantized. This yields the Hamiltonian H α 2α ~ 1 ¯ iD ¯ i + V (X ˆ 1 , ..., X ˆ i , ..., X ˆ 3N ) D H α = − mc2 2 mc (33)

Thus, a time dependent Schr¨odinger type equation for fractional derivative operators results 2α 1 2 ~ α ¯ iD ¯i H Ψ = (− mc D (34) 2 mc ˆ 1 , ..., X ˆ i , ..., X ˆ 3N ))Ψ = i~∂t Ψ + V (X For α = 1 this reduces to the classical Schr¨odinger equation. A.

Properties of the momentum operator

We extend the standard series expansion of the exponential function to the fractional case exp(α, χ(kx)) ¯ = =

∞ X

∞ X signn (kx)|kx|αn Γ(1 + αn) n=0

(35)

∞

X |kx|2αn |kx|(2n+1)α + sign(kx) Γ(1 + 2αn) Γ(1 + (2n + 1)α) n=0 n=0 = Eα (χ(kx)) ¯

(36)

where Eα is the Mittag-Leffler function[18]. The functions ψ = exp(α, −iχ(kx)) ¯ are eigenfunctions of the momentum operator with the real eigenvalues α ~ Pˆ ψ = mc sign(k) |k|α ψ (37) mc The Leibniz product rule, which plays an important role for the standard derivative, is not valid any more for the fractional derivative. Instead, with an arbitrary additional function R, we can write: ¯ g) = (Df ¯ )g + f (Dg) ¯ +R D(f

(38)

For the momentum operator it follows Z Z Z α ∗ ˆ α ˆ ∗ du f (P g) = f g − du (P f )g − duα R Z Z = + duα (Pˆ f )∗ g − duα R (39) Consequently, neither Pˆ nor the fractional Schr¨odinger operator with H α in (34) are hermitean operators. A direct consequence is the non orthogonality of the calculated eigenfunctions. In general, hermitean operators are preferred, since their eigenvalues and expectation values are always real. Nevertheless, eigenvalues for momentum, energy and angular momentum as well as expectation values derived with the proposed fractional Schr¨odinger equation (34) turn out to be real, as will be demonstrated in the following sections. Anyhow, we doubt, that a fractional operator should always be hermitean. A typical example was the expectation value of the root mean square radius of a free quark. Indeed, any real value would be a contradiction to the experiment.

4 α=1.10

x

cos(α,x)

α=0.85

α=1.10

sin(α,x)

α=0.85

α

X

FIG. 1: Free particle solutions for the fractional derivative operator Schr¨ odinger type equation, for different values of α. Upper part of figure are cos(α, x), lower part are sin(α, x) with α = 0.85, 0.90, 0.95 (solid thin line), . α = 1 (solid thick line) and α = 1.05, 1.10 (dashed thin lines) each. For α = 1 solutions reduce to the standard cos(x) and sin(x) functions. For α < 1 these functions are increasingly located at x = 0. For α > 1 the amplitudes increase with increasing x.

FIG. 2: Zeroes of eigenfunctions cos(α, π2 x) (solid lines) and sin(α, π2 x) (dashed lines) of the free particle eigenfunctions. For α ≤ 1/2 there are no zeroes. For α = 1 the zeroes are given by kn = n ∈ N. For α ≥ 1 an infinite number of zeroes exists.

following relations are valid: B.

Free particle solutions

We will now present the free particle solutions for the fractional Schr¨odinger type equation (34). We can do ˆ i ) = 0 the commutator [Pˆµ , H α ] vanthis, since for V (X ishes and consequently, energy and momentum are conserved. Let us first consider the one dimensional case. We extend the definition for the standard series expansion for the sine and cosine function cos(α, x) =

∞ X (−1)n |x|2nα Γ(2nα + 1) n=0

= E2α (−χ ¯2 (kx)) ∞ X (−1)n |x|(2n+1)α sin(α, x) = sign(x) Γ((2n + 1)α + 1) n=0 = χ(kx) ¯ E2α,1+α (−χ ¯2 (kx))

(40) (41) (42) (43)

where Eα and Eα,β are Mittag-Leffler[18] and generalized Mittag-Leffler functions[19]. With these definitions, the

¯ sin(α, kx) = sign(k)|k|α cos(α, kx) D ¯ cos(α, kx) = −sign(k)|k|α sin(α, kx) D

(44) (45)

It follows from these relations, that the above functions (40) are the eigenfunctions of the free Schr¨odinger type equation (34) in one dimension. In the stationary case we get the energy relation 1 E = mc2 2

~|k| mc

2α

(46)

This result may easily be extended to the n-dimensional case. In figure 1 the functions sin(α, x) and cos(α, x) are plotted for different values of α. While for α = 1, these functions reduce to the known cos(x) and sin(x), which are spread over the whole x-region, for α < 1 these functions become more and more located at x = 0 and oscillations are damped, a behaviour, which we know e.g. from the Airy-functions. For α > 1 the functions amplitude increases for increasing x. For α < 1 these functions are

5 normalizable on R: There exists an upper bound M Z ∞ dxα cos(α, kx) cos(α, k ′ x) < M −∞ Z ∞ dxα sin(α, kx) sin(α, k ′ x) < M −∞ Z ∞ dxα cos(α, kx) sin(α, k ′ x) = 0

with (47)

α=0.9

α=1.1

n=6

n=6

n=5

n=5

n=4

n=4

n=3

n=3

n=2

n=2

n=1

n=1

(48) (49)

−∞

For α ≥ 1 this integral is not bound any more, instead a box-normalization with a box size L much larger than the dimensions of the system considered is proposed. C.

Particle in an infinite potential well

Now we will give the exact eigenfunctions and eigenvalues for a particle confined in an infinite potential well. We first investigate the one dimensional case. Therefore we define the potential V (x) ( 0 |x| ≤ a V (x) = (50) ∞ |x| > a The corresponding boundary condition for the eigenfunctions ψ(x) is: ψ(−a) = ψ(a) = 0

(51)

In figure 2 the zeroes for the free particle solutions cos(α, π2 x) and sin(α, π2 x) are plotted. For α ≤ 1/2 there are no zeroes. For the interval 1/2 < α < 1 exists only a finite set of zeroes. For α ≥ 1 an infinite number of zeroes exists. Let k 0 be a zero of the free particle solutions. The eigenfunctions of the infinite potential well potential are then given by: (+) 0 x )) (52) ψ2n (x) = cos(α, χ(k ¯ 2n a x (−) 0 ¯ 2n+1 ψ2n+1 (x) = sin(α, χ(k )) (53) a where the sign indicates the parity of the states. The normalization condition is Z a dxα ψn∗ (x)ψn (x) = 1 (54) −a

The energy is then given by 2α ~ k0 1 mc2 | n |2α en = 2 mc a

V/ T

V/ T

(55)

The extension to the N-dimensional case is then Ψn1 n2 ...nN (x1 , x2 , ..., xN ) =

N Y

ψni (xi )

(56)

i=1

and for the energy En1 n2 ...nN

1 = 2

~ mc

2α

mc2

N X k0 | ni |2α ai i=1

(57)

FIG. 3: The six lowest eigenfunctions for the one dimensional infinite square well potential, with α = 0.9 on the left and α = 1.1 on the right side. Below the corresponding potential for equivalent solutions of the ordinary (α = 1) Schr¨ odinger equation, according to (71) is plotted. For α < 1 this potential contains a strong linear admixture, for α > 1 to potential graph behaves like ∼ 1 − x2

6 D.

Radial solutions

In the case of fractional derivative operators there exists no general theory of covariant coordinate transformations until now. We intend to perform a coordinate transformation from carthesian to hyperspherical coordinates in RN f (x1 , x2 , ..., xN ) = f (r, φ1 , φ2 , ..., φN −1 )

(58)

The invariant line element in the case α = 1 ds2 = gij dxi dxj

i, j = 1, ..., N

(59)

for arbitrary fractional derivative coefficient α may be generalized to ds

2α

=

α gij dxiα dxjα

i, j = 1, ..., N

(60)

Consequently a natural definition of the radial coordinate is given by r2α =

N X

x2α i

(61)

i=1

We assume the spherical ground state to be independent of the angular variables, square integrable and of positive parity. Therefore an appropriate ansatz is g(N, α, kr) =

∞ X

(−1)n an (N, α)(|k|r)2αn

(62)

n=0

or in carthesian coordinates g(N, α, kx1 , ..., kxN ) =

∞ X

n=0

N X |kx|2α )n (−1)n an (N, α)( i=1

(63) where the coefficients an depend on the explicit form of the potential. For a free particle, a solution on RN is given with the abbreviation ηj = Γ(1 + 2αj)/Γ(1 + 2α(j − 1))

V (r) =

0 ∞

r ≤ r0 r > r0

j = 1, 2, ... (65)

(66)

The corresponding boundary condition for the ground state wave function g(N, α, r) is: g(N, α, r0 ) = 0

and the ground state energy is then given by !2α 0 1 2 ~ ksph e0 (N, α) = mc 2 mc r0 E.

(67)

(68)

(69)

Remarks on equivalent solutions for the ordinary Schr¨ odinger equation

We have shown, that the free particle solutions and the solutions for the infinite potential square well of the fractional Schr¨odinger equation for α < 1 are localized at the origin and for α > 1 are localized at the boundaries of a given region respectively. We want to deduce a similar behaviour of these functions in terms of the ordinary (α = 1) Schr¨odinger equation. Let us assume, the eigenfunctions of the fractional Schr¨odinger equation may be equivalently interpreted as solutions of the ordinary (α = 1) Schr¨odinger equation with an additional potential V . In order to derive the explicit form of this potential, we use the following relation between the eigenfunctions Ψn , temperature T and a given potential V , which is derived within the framework of thermodynamics and statistical quantum mechanics[25]: Let Ψn and En be the eigenfunctions and energy eigenvalues of the ordinary Schr¨odinger equation with a given potential V : ~2 − ∆ + V Ψn = En Ψn (70) 2m As long as the temperature T is large compared to the average level spacing, the relation (N is the normalization constant)

1 √ exp(−V /T ) = N

An infinite spherical well is described by the potential (

0 g(N, α, ksph r/r0 )

(64)

by the recurrency relation a0 = 1 aj = aj−1 / ((N − 1) j η1 + ηj )

0 Let ksph be the first zero of the free particle ground state wave function, the ground state wave function for the spherical infinite well potential is given by

∞ X

Ψ∗n Ψn exp(−En /T )

n=0 ∞ X

(71) exp(−En /T )

n=0

T ≫ ∆En = En+1 − En is valid. Since eigenfunctions and eigenvalues for the fractional free particle solutions are known, we therefore are able to deduce the explicit form of such a potential. In figure 3 the graph of V /T is plotted for α = 0.9 and α = 1.1 respectively. For α = 0.9 the potential contains a dominant linear term V ∼ |x| , while for α = 1.1 a behaviour like V ∼ 1 − x2 may be deduced. Of course, for α = 1 a potential V = const results.

7 The behaviour of the fractional eigenfunctions may alternatively be interpreted, assuming that α is a measure of charge for a particle moving in an homogenously charged background. For α > 1, we could assume a charged particle, with charge e.g. α moving in a homogeneous background of charged particles with charge 1 − α. Since for a homogeneously charged sphere and approximately for a charged box too, the potential inside the box is V ∼ −Z(1 − r2 ) with Z = 1 − α. This obviously would explain a repulsive force. The energetically favoured positions for a particle with charge α indeed were the boundaries of the box. For α = 1 the background was neutral and therefore no additional interaction with a particle was present. For α < 1 this simple model could explain an attraction, but not the linearity of the potential. Therefore we obtain the remarkable result, that in the case α < 1, the free particle solutions of the fractional Schr¨odinger equation show a behaviour, which is equivalent to the behaviour of solutions of the ordinary Schr¨odinger equation with an additional linear potential term. In other words, the solutions of a free fractional wave equation with α < 1 automatically show confinement, a property, which was first observed for quarks. In order to obtain more properties of the fractional derivative operator Schr¨odinger equation, we will now calculate the eigenvalues of the angular momentum operator. IV.

CLASSIFICATION OF ANGULAR MOMENTUM EIGENSTATES

We define the generators of infinitesimal rotations in the i, j-plane (i, j = 1, ..., 3N ), with N being the number of particles): ˆ j Pˆi ˆ i Pˆj − X Lij = X χ(x ¯ i) ¯ χ(x ¯ j) ¯ = −i~ Dj − Di Γ(α + 1) Γ(α + 1)

(72)

We will derive the angular momentum eigenvalues algebraically. Thus it is necessary, to apply the commutator ˆ i , Pˆj ] (see (30)) repeatedly. In figure 4, this relation [X commutator c(n, α) is plotted. It shows a smooth dependence on n, which we have to eliminate. Ignoring both, the n and α dependence we set as a lowest order approximation: c0 = c(0, 1) = 1

(73)

A more precise statement for c(n, α) can be deduced from the following consideration: Since we will concentrate on the lowest enery levels only , the approximation c1 (α) = c(0, α) = 1 −

1 Γ(1 − α)Γ(1 + α)

(74)

is valid. Of course, this overestimates the commutator for higher values of n. Therefore a more sophisticated

1.0

0.9

0.8 0.7 0.6 0.5 0.4

ˆ i , Pˆi ] from (30) in units of ~ on the FIG. 4: Commutator [X function set {xnα } for different values of α. The dependence on n is a direct consequence of the fact, that the Leibniz product rule is not valid any more for fractional derivatives. For α = 0.6 the positions of the successive approximations on c(n, α) according to (73),(74) and (75) are given as circle, triangle and squares.

treatment fixes c(n, α) for a given j to be: Γ(1 + (j + 1)α) Γ(1 + jα) − Γ(1 + jα)Γ(1 + α) Γ(1 + (j − 1)α)Γ(1 + α) (75) We will consider these three cases, which allows an estimate on the validity of the results. Therefore, as long as the commutator does not depend on n, [Lij , H α ] vanishes and angular momentum is conserved. Commutator relations for Lij are isomorph to an extended fractional SOα (3N ) algebra: c2 (j, α) =

[Lij , Lmn ] = i~ c(α) (δim Ljn +δjn Lim −δin Ljm −δjm Lin ) (76) Consequently, we can proceed in a standard way [20], by defining the Casimir operators k

Λ2k =

1X (Lij )2 2 i,j

,

k = 2, ..., 3N

(77)

which indeed fulfill the relations [Λ23N , Lij ] = 0 and successively [Λ2k , Λ2k′ ] = 0. Consequently the successive group chain SOα (3N ) ⊃ SOα (3N − 1) ⊃ . . . ⊃ SOα (3) ⊃ SOα (2) (78) is established. The explicit form of the Casimir operators is given by ˆ j Pˆj − X ˆ iX ˆ j Pˆi Pˆj ˆ iX ˆ i Pˆ j Pˆj − i~c(α) (δii − 1)X Λ2k = +X (79) We introduce a generalization of the homogeneous Euler operator for fractional derivative operators ¯i Jek (α) = χ(x ¯ i )/Γ(1 + α) D

(80)

8 TABLE I: Eigenvalues in units of ~ for the angular momentum states of a single particle. n is a counter for the eigenvalues of the Euler operator, eigenvalues for Lz (α) and Jc2 (α) are given for α = 1, α = 2/3, α = 0.68 and α = 0.65. Jc2 (α) are listed with c according to (73),(74) and (75). n = Lz (1) 0 1 2 3 4 5 6

Lz (2/3) 0 1 1.460 998 1.860 735 2.222 222 2.556 747 2.870 848

J02 (1) 0 2 6 12 20 30 42

Lz (0.68) 0 1 1.478 157 1.894 649 2.272 597 2.623 332 2.953 417

J02 (2/3) 0 2 3.595 515 5.323 069 7.160 493 9.093 704 11.112 618

With the generalized Euler operator the Casimiroperators are: ˆ iX ˆ i Pˆ j Pˆj + ~2 c(α) (k − 2)Jek + Jek Jek Λ2k = +X

nα

Hα = {f : f (λ~x) = λ

¯ iD ¯ i f = 0} f (x); D

n∈N (82) This is the quantization condition. It guarantees, that solutions are regular at the origin. On this Hilbert space, the generalized Euler operator Jek (α) is diagonal and has the eigenvalues

lk (α, n) =

0

for n = 0

Γ(nα + 1) 1 Γ(α + 1) Γ((n − 1)α + 1)

for n = 1, 2, 3, ...

(83) This is the main result of our derivation. We want to emphasize, that these eigenvalues are different from the degree of homogenity in the general case α 6= 1, or, in other words: only in the case of α = 1 the homogenity degree n of the polynoms considered coincides with the eigenvalues of Jek (α = 1, n). Once the eigenvalues of the generalized Euler operator are known, the eigenvalues of the Casimir-operators Λ2 , Λ2k are known, too: Λ2 f = ~l2 (α, n)f Λ2k f = ~2 lk (α, n)(lk (α, n) + c(α) (k − 2))f

(84) (85)

lk ≥ lk−1 ≥ ... ≥| ±l2 |≥ 0

(86)

with

For the case of only one particle (N = 1), we can introduce the quantum numbers j and m, which now denote the j-th or m-th eigenvalue of the Euler operator. The eigenfunctions are fully determined by these two quantum numbers f =| jm > With the definitions Lz = L12 and J 2 = L212 +L213 +L223

J12 (0.65) 0 1.604 767 3.078 892 4.735 519 6.539 094 8.468 379 10.508 808

J22 (j, 0.65) 0 1.478 157 2.800 590 4.305 776 5.961 779 7.747 796 9.649 033

it follows Lz | jm > = ~l2 (α, m) | jm > (87) m = 0, ±1, ±2, ..., ±j J 2 | jm > = ~2 l3 (α, j) (l3 (α, j) + c(α)) | jm > (88) j = 0, +1, +2, ...

(81)

Now we define a Hilbert space Hα of all homogeneous ¯ iD ¯ if = functions f , which satisfy the Laplace equation D 0 and are normalized in the interval [−1, 1]:

J02 (0.68) 0 2 3.663 108 5.484 346 7.437 298 9.505 205 11.676 094

Please note the fact that Lz remains unchanged for any choice of constant c(α), only J 2 changes. In table I the first seven eigenvalues of Lz and J 2 for a single particle are listed for α = 1, α = 2/3, 0.68 and α = 0.65 and different approximations for c(α). For α 6= 1 the eigenvalues of the generalized Euler operator are not equally spaced any more. For α < 1 the stepsize is strongly reduced. Since the generalized Euler operator eigenvalues contribute quadratically into the definition J 2 , the energy of higher total angular momenta is reduced increasingly. We have derived the full spectrum of the angular momentum operator for the fractional derivative operator Schr¨odinger type wave equation by use of standard algebraic methods. We will get additional information about the properties of this wave equation, if we consider its factorized pendant. We present some results in the next section.

V.

RESULTS FOR THE FACTORIZATION OF A NON RELATIVISTIC SECOND ORDER DIFFERENTIAL EQUATION

Linearization of a relativistic second order wave equation was first considered by Dirac[21]. Starting with the relativistic Klein-Gordon equation his derived Dirac equation gave a correct description of the spin and the magnetic moment of the electron. The concept of linearization is important, since it provides a well defined mechanism to add an additional SU(2) symmetry to a given set of symmetry properties of a second order wave equation. Since linearization may be interpreted as a special case of factorization, namely to 2 factors, a natural generalization is a factorization to n factors.

9 In 2000, Raspini[8] proposed a Dirac-like equation with fractional derivatives of order 2/3 and found the corresponding matrix algebra to be related to generalized Clifford algebras; in 2002 Z´ avada [15] generalized Dirac’s approach, and found, that relativistic covariant equations generated by taking the n-th root of the Klein-Gordon or d’Alembert operator ( 1/n ) are fractional wave equations with an additional SU(n) symmetry. These results indicate, that fractional order wave equations may be appropriate candidates for a description of particles, which own a SU(n) symmetry. The case n = 3, which corresponds to a triple factorization is therefore important for a description of particles with a SU(3) symmetry. Whether or not a factorization of non relativistic wave equations leads to similar results, has not been examined yet. In 1967, Levy-Leblond[22] has linearized the non relativistic Schr¨odinger equation and obtained a linear wave equation with an additional SU(2) symmetry, but until now his approach has not been extended to higher fractional order. In order to obtain additional information on the inherent symmetries of the fractional Schr¨odinger equation, which we proposed in (34), we therefore derive in the following section the explicit form of a fractional operator, which evolves from a triple factorization of the ordinary Schr¨odinger equation:

A.

Triple factorization of the non relativistic Schr¨ odinger equation

We intend to derive a fractional operator R, which, iterated 3 times, conforms with the ordinary, non relativistic Schr¨odinger operator: ~2 RR R = − ∆ − i~∂t 1n 2m ′

′′

(89)

where 1n is the n × n unit matrix. We use the following ansatz: R = aA∂tαt + bB i ∂iαi + cC ′

R = R

′′

=

aA′ ∂tαt + bB ′i ∂iαi + cC ′ aA′′ ∂tαt + bB ′′i ∂iαi + cC ′′

(90) (91) (92)

with matrices A, A′ , A′′ , B, B ′ , B ′′ , C, C ′ , C ′′ , fractional derivative coefficients for time and space derivative αt , αi and scalar factors a, b, c, which will be determined in the following. According to Z´ avada [15], we define a triad of unitary, traceless 3 × 3 Pauli type matrices, which span a subspace of SU (3) with xk = exp(

2πi k) 3

k = 1, 2, 3

(93)

an explicit representation is 0 σ1 = 0 x3 0 σ2 = 0 x3 x1 σ3 = 0 0

x1 0 0 x2 0 0 x2 0 0 x1 0 0 0 0 x2 0 0 x3

(94)

(95)

(96)

These matrices obey an extended Clifford algebra X σ i σ j σ k = 6 δ ijk i, j, k = 1, 2, 3

(97)

all Permutations

Let ⊗ denote the outer product of any two matrices. In order to describe a single particle with the coordinates {t, x, y, z}, we define the following 4 γ matrices, with dimension 9 × 9: γ 0 = 13 ⊗ σ 3 γ i = σi ⊗ σ1

i = 1, 2, 3

(98) (99)

Now we are able to specify the above introduced matrices: 1 A = √ (γ 0 − x1 19 ) 3 1 0 ′ A = √ (γ − x2 19 ) 3 1 A′′ = √ (γ 0 − x3 19 ) 3 i i B = γ B ′i = γ i B ′′i = γ i 1 0 0 C = x1 13 ⊗ 0 0 0 0 0 0 0 0 0 C ′ = x2 13 ⊗ 0 1 0 0 0 0 0 0 0 C ′′ = x3 13 ⊗ 0 0 0 0 0 1

(100) (101) (102) (103) (104) (105) (106)

(107)

(108)

with these specifications R R′ R′′ yields R R′ R′′ =

a2 c ∂t2αt

+ b3

3 X i=1

∂i3αi

!

19

(109)

A term by term comparison with the nonrelativistic Schr¨odinger operator determines the fractional derivative coefficients: αt = 1/2 αi = 2/3

(110) (111)

10 and the scalar factors: a = (−i~)1/2 b = −

~2 2m

1 mc2

1/3

1/6

(112)

prediction 2

µ ∂µ → sign(xµ ) + c D(|x |) ′

2

cR R = a cAA ∂t + b cB

i

m=2

ψ (4040); <31>

m=1

ψ (3770); <30>

m=0

2/3 2/3 B j ∂i ∂j

m=2 m=1 m=0

χ (3556); <22> χ (3511); <21> χ (3415); <20>

(115)

+ additional terms (116)

or, inserting the factors: 1 ~ 4/3 2 1 1/3 i j 2/3 2/3 S (2) = −i~AA′ ∂t − ( ) mc ( ) B B ∂i ∂j 2 mc 2 +additional terms (117) Obviously S (2) and the fractional Schr¨odinger equation (34), we derived in section III, are closely related.

m=1 m=0

J/ψ (3097); <11>

′′

Thus the fractional operators R, R , R are completely determined. As a remarkable fact we note the different fractional derivative coefficients for the fractional time and space derivative. We therefore have proven, that the resulting SU(3) symmetry is neither a consequence of a relativistic treatment nor is it a specific property of linearization (which means, specific to first order derivatives of time and space respectively). It is a consequence of the triple factorization solely. We want to emphasize, that the factorization not only determines the symmetry group of the γ-matrices used, but also determines the dynamics of the system, forcing e.g. for a SU(3) symmetry fractional time (αt = 1/2) and space derivatives(αi = 2/3). Thus, the result of factorization shows two important properties: It yields an additional SU(n) symmetry and simultaneously the corresponding dynamics. Applied to QCD, this seems more consistent than the standard concept based on Yang-Mills field theories, where an arbitrary non abelian symmetry group gauge field, which first has to be deduced from experimental data to be a SU(3) symmetry is coupled to a symmetry independent dynamical (e. g. Dirac) field, neglecting a possible influence of the symmetry onto the dynamics. In that sense, the fractional operator R, defined in (90), would be an alternative starting point for a pure, non relativistic QCD, since it contains a consistent description of both, symmetry and dynamics of a pure SU(3) symmetry, without any additional SU(2) admixture. Finally, besides the fractional wave equation operator R and the triple iterated R R′ R′′ , which corresponds to the ordinary Schr¨odinger operator, an additional type of wave equation, the twofold iterated R R′ emerges, which reads: ′

m=3

(114)

Finally, according to (14) and (8), we extend the derivative operator on R via:

2

Y(4259); <33> ψ (4160); <32>

(113)

c = (mc2 )1/3

′

m=0

ψ (4410); <40>

η (2980); <10> c

Σc (2451); <00>

j=0

j=1

standard spectrum

m=0

prediction 1

j=2

j=3

j=4

FIG. 5: Charmonium spectrum. All observed particles are given with their name, experimental mass from[28] and the proposed SOα (3) conforming quantum numbers j and m, which are the j-th and m-th eigenvalue of the generalized Euler operator. Predicted particles are: in the lower part of the spectrum denoted as prediction 1 < 00 > associated with Σ0c (2455) and in the upper part prediction 2 < 33 > associated with Y(4260).

H α seems nothing else but a scalar version of S (2) for the special case α = 2/3. Therefore, examination of the properties of the scalar fractional Schr¨odinger equation (34) with α = 2/3 should reveal some properties of an inherent SU (3) symmetry. Summarizing all facts collected, we assume, that the fractional derivative operator Schr¨odinger type wave equation with α = 2/3 is an appropriate candidate for a non relativistic description of particles with quark-like properties.

VI. INTERPRETATION OF THE CHARMONIUM SPECTRUM

In the previous sections we have introduced the concept of fractional derivative operators and discussed some properties of the resulting non relativistic fractional Schr¨odinger type equation and its factorized pendant. We have developed a new theoretical concept, which fulfills at least the following three demands: First, available experimental data will be reproduced with a reasonable accuracy. Second, it will give new insights on underlying symmetries and properties of the objects under consideration. Third, we will make predictions, which can be proven by experiment.

11 None of our results, presented so far, does require any information from QCD or a similar theory. All our statements could have been made in the 1930s already, even though they would have been highly speculative. Today, we are in the comfortable position, that there are enough experimental data, our predictions can be compared with. A promising candidate is the charmonium spectrum[26]. In the upper part of figure 5 we have displayed all experimentally observed charmonium-states with experimental masses, which are normally compared with results from a potential model, which tries to simulate confinement and attraction by fitting a model potential[26],[27]. We will assume, that this spectrum is a single particle spectrum for a particle, whose properties are described by the free fractional Schr¨odinger type equation(34). We suppose, the system is rotating in a minimally coupled field, which causes a magnetic field Bj . This leads to the following Hamiltonian H or mass formula H(j, m) | jm >= κJ 2 /~2 + Bj Lz /~ + m0 c2 | jm > (118) where κ, Bj and m0 c2 will be adjusted to the experimental data. The eigenfunctions | jm > are modified spherical harmonics with Di Di | jm >= 0, the eigenvalues for J 2 and Lz are given by (87),(88) and are listed in table 1. A first remarkable observation is the fact, that states with Lz < 0 are missing in the experimental spectrum. Only right-handed particles are realized. This may be due to the fact, that actually L2z is a Casimir-Operator of SOα (2), while Lz is not and therefore the multipletts should more precisely be classified according to the m2 quantum number in the general case α 6= 1. Consequently in the following we will work with Lz ≥ 0. To check the influence of our approximations (73),(74) and (75) of c(α) given in (30) we will proceed in two steps: First, we will consider the case c0 = 1. In a second step, we will test the influence of the successive approximations for c(α) on the accuracy of the proposed mass formula with least square fits on the charmonium spectrum.

A.

Interpretation of the charmonium spectrum in the case c0 = 1

We first consider the case c(α) = 1. The corresponding J 2 and Lz values used are listed in table I as Lz (α) and J02 (α). The first crucial test is the verification of the correct value of the non trivial m = 2 quantum number which corresponds to the n = 2 eigenvalue of the generalized Euler operator. For the set of χ-particles (j = 2) (experimental masses

and errors are taken from [28]), we obtain: χ(3556) < 22 > −χ(3415) < 20 > χ(3511) < 21 > −χ(3415) < 20 > = 1.478 ± 0.007 (119)

Lz (j = 2, m = 2)exp =

Thus, an α from the experimental spectrum is deduced: αj=2 exp = 0.680 ± 0.006

(120)

Which is remarkably close to the theoretically expected αth = 2/3. An alternative approach to determine the experimental value for α or the m = 2 quantum number repectively, follows from the set of Ψ-particles (j = 3): Ψ(4160) < 32 > −Ψ(3770) < 30 > Ψ(4040) < 31 > −Ψ(3770) < 30 > = 1.44 ± 0.09 (121)

Lz (j = 3, m = 2)exp =

A second experimental α is obtained: αj=3 exp = 0.65 ± 0.08

(122)

Within experimental errors, both values are identical. This observation supports the assumption, that the spectrum may be interpreted using one unique α. According to our level scheme, the Ψ < 33 > state is missing in the standard charmonium spectrum. Using αj=2 exp = 0.680 and Lz (α, m) from table I the predicted mass is: Lz (α, m = 3) (Ψ(4160)<32> −Ψ(3770)< 30>) Lz (α, m = 2) = 4268 ± 22[MeV] (123)

Ψ<33> =

In june 2005, the Babar collaboration announced the discovery of a new charmonium state named Y(4260) [30]. The reported mass of 4259[MeV] is in excellent agreement with our mass prediction for Ψ < 33 >. Therefore, we associate the predicted particle with Y (4260). Next we will determine the constants m0 c2 and κ in (118). We choose the two lowest experimental states of the standard charmonium spectrum, ηc (2980) < 10 > and 2 χ(3415) < 20 >. With αj=2 exp = 0.680 and J0 (α, j) from table I we obtain a set of equations m0 c2 + 2κ = ηc (2980) < 10 > m0 c + 3.663108κ = χ(3415) < 20 > 2

(124) (125)

which determine m0 c2 = 2455 ± 3[MeV] κ = 262.4 ± 0.9[MeV]

(126)

Our level scheme predicts a particle with quantum numbers < 00 >, which is beyond the scope of charmonium potential models. According to our mass formula, it has a predicted mass of < 00 >= 2455 ± 3[MeV].

12 Since this is a low lying state, it should already have been observed. Indeed, there exists an appropriate candidate, the Σ0c (2455) < 00 > baryon, with an experimental mass of 2452.2[MeV]. This is a charmed baryon with quark content (ddc). The minimal difference of only 2.8[MeV] between predicted and experimental mass of the Σ0c (2455) < 00 > particle indicates, that the assumed fractional SOα (3) symmetry is fulfilled exactly. Obviously, the fractional SOα (3) multipletts describe mesonic and baryonic states of the charm-quark simultaneously. Due to its experimentally observed properties, the internal structure of the Y (4260)<33> particle is subject of actual discussion[31]. Besides being a conventional c¯ c state, it could alternatively be a tetraquark with constituents (u¯ uc¯ c) or a hybrid charmonium. Finally we have only one experimental candidate for j = 3 and j = 4 respectively. With parameters (126) using the mass formula (118) we obtain the theoretical values Ψ(3770) < 30 >th = 3894 ± 8[MeV] Ψ(4410) < 40 >th = 4406 ± 10[MeV]

(127) (128)

For j = 3 the calculated mass (127) differs by 124[M eV ] from the experimental value. On the other hand, the theoretical Ψ(4410) < 40 > mass (128) matches exactly with the experimental value within the experimental errors. This indicates, that the particles for j = 3, observed in experiment, carry an additional property, which reduces the mass by the amount of e.g. a pion. Of course, if we add an additional (∆τ )δj3 term to the proposed mass formula, we can shift these levels by the necessary amount. Summarizing these results, the charmonium spectrum reveals an underlying SOα (3) symmetry, which agrees with the predictions of our theory in the case of α ≈ 2/3. The eigenvalues of the generalized Euler operator conform within experimental errors with experimental data. Extending the standard charmonium spectrum, two additional particles have been predicted and associated with Σ0c (2455) and Y (4260) observed recently. B.

Least square fits of the charmonium spectrum

In the previous section we gave an interpretation of the charmonium spectrum for the case c(α) = 1. We found, that the spectrum may be described quantitatively, using the proposed mass formula for α = 0.680. Extending the mass formula (118) including a correction term for the j = 3 multiplett, we use H(j, m) | jm >= κJ 2 /~2 + Bj Lz /~ + m0 c2 + δ3 j ∆τ | jm > (129) to find a fit on the experimental charmonium spectrum.

To prove, that for c0 = 1, α = 0.68 indeed is the appropriate choice for an interpretation of the charmonium spectrum, we minimized errors with respect to α and obtained α = 0.681. In table II the optimum parameter sets for α = 2/3 and c0 , c1 , c2 and resulting errors are tabulated. For c1 (α) from (74), α = 0.68 is not the optimum choice any more. We therefore minimized errors with respect to α, finding α = 0.647 for this case. For c2 (j, α) from (75), we observe a minimal shift in α, finding α = 0.649 for this case. This indicates, that a more sophisticated treatment of c(α) will only cause neglible changes for α and corresponding parameter sets. Comparing the optimum parameter sets, the changes in the treatment of c(α) are mainly absorbed by the parameter κ. Parameter ∆τ remains remarkably constant. This supports interpretation for this parameter to be a j = 3 specific quantum number. A comparison of experimental with calculated masses, based on the optimum parameter sets, is given in table III. Mass differences are less than 0.1% and decreasing for c0 , c1 , c2 . With the optimum parameter sets we can predict the mass of the < 50 > state to be X < 50 >th= 4965 ± 10[MeV]

(130)

This state has not been observed in experiments yet. We conclude, that the proposed SOα (3) symmetry is fulfilled exactly. The values of α = 0.68 and α = 0.65 resulting from the least square fits are close to , but differ significantly from the theoretically expected α = 2/3. This indicates, that the inherent SU (3) symmetry is almost fulfilled exactly, with a difference of only 2%.

C.

Size estimate for Σ0c

Up to now we have treated m0 c2 as a simple parameter of the proposed mass formula. As a result of our discussion above, we associate H(0, 0) = m0 c2 with the mass of Σ0c . If H(j, m) was the solution of the Schr¨odinger equation with a given potential V, then H(0, 0) would be interpreted as the zero point energy in this potential plus the rest mass of its constituents. We intend to estimate the size of Σ0c . Therefore we will calculate the expectation value < rˆ > of the radius operator rˆ, which is given by q ˆ2 ˆ2 + X ˆ2 + X (131) X 3 2 1 1−α q ~ 1 2α 2α (132) = x2α 1 + x2 + x3 mc Γ(1 + α)

rˆ =

For a first estimate, we choose the infinite square well potential(50). The energy eigenvalues are given by (57).

13 TABLE II: Optimum parameter sets for a fit of the experimental charmonium spectrum with mass formula (129) in units [MeV]. Errors ∆m are given for the subset m = 0 and the full spectrum ∆mall . The first row corresponds to α = 2/3 fixed, the following three rows correspond to the three approximations for c0 ,c1 and c2 according to (73),(74) and (75) with α optimized. In table III resulting theoretical masses and errors are listed. α 2/3 0.681 0.647 0.649

m0 c2 2439.33 2451.26 2452.67 2451.90

c(α) 1.00 1.00 0.545 c(j,α)

κ 274.66 263.83 336.16 367.41

B1 108.25 117.98 119.79 116.13

B2 87.00 93.39 95.72 98.37

B3 263.69 259.16 270.19 269.46

∆τ -129.04 -124.48 -129.00 -124.39

∆mm=0 5.68 1.86 1.14 0.73

∆mall 8.98 2.02 1.15 0.79

comment α fixed α variation α variation α variation

TABLE III: Comparison of experimental and calculated masses according to mass formula (129) with optimized parameter sets listed in table II in units [MeV]. The last row lists predicted theoretical masses for X <50>. < jm > < 00 > < 10 > < 11 > < 20 > < 21 > < 22 > < 30 > < 31 > < 32 > < 33 > < 40 > < 50 >

symbol Σ0c ηc J/Ψ χ0 χ1 χ2 Ψ Ψ Ψ Y (4260) Ψ X

mexp 2452.2 2979.6 3096.9 3415.2 3510.6 3556.3 3770 4040 4160 4259 4415

α = 2/3, c0 = 1 mth ∆mth 2439.33 -12.87 2988.66 9.05 3096.92 0 3426.89 11.70 3513.89 3.30 3554.00 -2.26 3772.35 2.34 4036.04 -3.95 4157.60 -2.39 4263.01 4.01 4406.07 -8.92 4937.06

α = 0.68, c0 = 1 mth ∆mth 2451.26 -0.93 2978.94 -0.66 3096.92 0 3417.80 2.60 3511.19 0.60 3555.85 -0.40 3773.92 3.92 4033.08 -6.91 4157.02 -2.98 4264.98 5.97 4413.80 -1.20 4959.54

Therefore Σ0c = (2md + mc )c2 + E0 (N = 3, α) (133) 2α 0 3 ~ k = (2md + mc )c2 + mc c2 | 0 |2α 2 mc c a determines the half boxsize a. The wave function was defined in (40). Therefore with the abbreviations α α dV α = dxα (134) 1 dx2 dx3 0 Ψ(x1 , x2 , x3 ) = cos(k0 x1 /a) cos(k00 x2 /a) cos(k00 x3 /a)

the expectation value according to (23) is RRR a dV α Ψ∗ rˆΨ < rˆ(Σ0c ) >= RRR0 a α ∗ 0 dV Ψ Ψ

(135)

= = = =

2452.2[MeV] 300[MeV] 1400[MeV] 1.1648 π/2

(136)

we derive a = 0.81[fm] and therefore we obtain the expectation value for the radius < rˆ(Σ0c ) >cube = 0.32[fm]

α = 0.649, c2 (j, α) mth ∆mth 2451.90 -0.30 2980.78 1.18 3096.92 0 3413.65 -1.54 3512.03 1.43 3555.26 1.00 3770.41 0.41 4039.87 -0.13 4158.30 -1.70 4260.42 1.42 4415.22 0.22 4969.07

Similarly, we can proceed for the infinite spherical well potential (66). For the spherical ground state wave function in carthesian coordinates(63), we obtain Lz g(N = 3, α, x1 , x2 , x3 ) = 0 J 2 g(N = 3, α, x1 , x2 , x3 ) = 0

(137)

(138) (139)

and therefore g indeed is the ground state |00>. The ground state energy e0 is given by (69) Σ0c = (2md + mc )c2 + e0 (N = 3, α) (140) 2α 0 ksph 2α 1 ~ = (2md + mc )c2 + mc c2 | | 2 mc c r0 0 with ksph = 3.1652 π/2 this determines r0 = 1.08[fm]. With (135) we obtain for the expectation value of the radius

< rˆ(Σ0c ) >sphere = 0.33[fm]

Setting α = 2/3 and

Σ0c md c2 mc c2 k00

α = 0.647, c1 (α) mth ∆mth 2452.67 0.47 2977.12 -2.47 3096.92 0 3417.15 1.95 3512.87 2.27 3554.68 -1.58 3770.17 0.17 4040.37 0.37 4158.39 -1.61 4260.08 1.07 4414.36 -0.64 4957.54

(141)

Consequently, both potentials lead to similar expectation values. Since Σ0c is not within the scope of standard charmonium models, there is no direct comparison. Nevertheless, there are radii, derived from charmonium model calculations[26], reported for [32]-[34]: <ˆ r (J/ψ < 11 >)> ≈ 0.2[fm] <ˆ r(χ0 < 20 >)> ≈ 0.3[fm] <ˆ r (Ψ < 30 >)> ≈ 0.4[fm]

(142) (143) (144)

14 Therefore our results are reasonable compared with these calculations. VII.

CONCLUSION

Based on the Caputo fractional derivative, we have defined a fractional derivative operator for arbitrary fractional order α. A Schr¨odinger type wave equation, derived by quantization of the classical non relativistic Hamiltonian, generates free particle solutions, which are confined to a certain region of space. Therefore confinement is a natural consequence of the use of a fractional wave equation. The multiplets of the generalized angular momentum operator have been classified acoording to the SOα (3) scheme, the spectrum of the Casimir-Operators has been calculated analytically. We have also shown, that for α = 2/3, corresponding to a fractional non relativistic Levy-Leblond wave function an inherent SU(3) symmetry is apparent. From a detailed discussion of the charmonium spectrum we conclude, that the spectrum may be understood quantitatively within the framework of our theory. Ap-

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proximately α ≈ 2/3 is valid. The experimental masses are reproduced wih an accuracy better than 0.1%. Extending the standard charmonium spectrum, three new particles have been predicted, two of them associated with Σ0c (2455), a charmed baryon and Y (4260), observed recently. The third particle, labeled X<50> according to the proposed SOα (3) level scheme, with a predicted mass of 4965 ± 10[MeV], has not been experimentally verified yet. Summarizing the results of our considerations, the proposed fractional non relativistic Schr¨odinger type wave equation is a powerful alternative for a discussion of charmonium properties and extends our knowledge beyond the standard achieved with phenomenological models. Therefore fractional wave equations may play an important role in our understanding of particles with quarklike properties, e.g. confinement.

VIII.

ACKNOWLEDGEMENTS

We thank A. Friedrich and G. Plunien from TU Dresden, Germany, for fruitful discussions.

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