On The Confinement Of Quarks Without Applying the Bag Pressure

Transcription

1 On The Confinement Of Quaks Without Applying the Bag Pessue Mohammad Shaifi Depatment of Physics, Univesity of Tehan, Ian Abstact We explain the fatal eo in uantum chomodynamics. By applying this coection to the dynamics of uaks, we can confine uaks in hadons. We will show why uaks do not obey the Pauli exclusion pinciple and why we cannot obseve fee uaks. In addition, we obtain the coect size of hadons. 1 Intoduction Two electons with identical uantum numbes cannot exist in the hydogen atom, because each electon is subluminal and its phase velocity is supeluminal. When thee ae electons with identical uantum numbes in a hydogen atom o with identical enegy levels in a cubic box, the second electon exists at evey location (space-time coodinates) with exactly identical wave function chaacteistics to those of the fist electon. In othe wods, the two electons simultaneously exist at an exact point in eual time. This phenomenon is the conseuence of the pobabilistic chaacteistic of a wave function and uantum mechanics. In othe wods, the wave euation does not povide us with moe infomation about the exact location of each electon. The enegy and absolute value of the momentum of each electon ae exactly detemined, but they do not have a specific location. At a specific time, they ae ubiuitous at evey location whee the wave function does not have a zeo value. Howeve, we can have 3 identical uaks with identical spin states in bayons. To explain this phenomenon, we popose a stange theoem. Theoem 1. Quaks ae Supeluminal paticles. Any specific change of each state of the wave function in its associated Hilbet space will popagate in the space-time coodinate with the phase velocity of the wave function in spacetime. In othe wods, the paticles communicate with one anothe via thei phase velocities [6]. We postulate that uaks ae supeluminal. Because each uak is supeluminal, its phase velocity should be subluminal; thus, uaks with identical spins can occupy the same enegy level in hadons. In othe wods, the fist uak is unawae of the spin and the chaacteistics of the second uak, because thei phase velocities ae subluminal. If we change the wave function of the second uak, this change will popagate with a speed less behshaifi@ut.ac.i 1

2 than c to othe locations of space-time in the bag. The phase velocity is not in spacelike egions. In othe wods, two uaks with identical enegies and momenta ae located at diffeent points in the bag. Quantum mechanics postulate that at a specific time, a subluminal paticle with a specific enegy momentum does not have a specific location. In othe wods, it is ubiuitous in the bag. Howeve, because the phase velocity of a supeluminal paticle is subluminal, supeluminal paticle is no longe ubiuitous. Thus, two supeluminal paticles that ae confined in a cubic box no longe exist at the exact space-time point. Thus, it is not necessay fo them to obey Pauli Exclusion Pinciple. Note that the exclusion pinciple is applicable fo two identical paticles with identical wave function chaacteistics. Theoetically, as we mentioned peviously, the wave function of a single supeluminal paticle cannot collapse, because its phase velocity of collapse is subluminal and obeys causality [6]. Befoe the wave function collapses, the paticle does not have a specific location. We ceate its location by doing an expeiment and measuing its location. Howeve, afte we detemined the location of a paticle, the paticle should not be detected in othe locations even in notably fa space-like locations that have no causal elationship with the location of the collapsed paticle. When ψ space of a subluminal paticle collapses, it communicates via its phase velocity (at infinite velocity in the efeence fame of the collapsed wave function) to othe locations in space-time that the wave function should not collapse at othe locations of the univese. Thus, a paticle cannot be detected in two space-like locations, although two locations do not have causal elations with each othe. Howeve, if the paticle is supeluminal, its phase velocity is subluminal and cannot pefom this communication in space-like egions of space-time. The phase velocity must be supeluminal to allow the collapse of the wave function. Because uaks ae supeluminal, we neve obseve a single fee uak. Wave euation of the hydogen atom with supeluminal electon Thee is a big diffeence between the odinay hydogen atom and a model with supeluminal electon. In the subluminal model, we have negative potential enegy. When we incease the enegy of the electon in the subluminal model, the momentum of the electon deceases; thus, the wavelength of the electon inceases, and the electon inceases its distance fom the poton. In the subluminal model, although the enegy cannot be less than the mass of the paticle, the minimum momentum can be zeo. E = c P + m c 4 (1) Thus, the wave length has no maximum, and it can appoach infinity, which esults in the escape of an electon fom the hydogen atom accoding to the Wilson-Sommefeld ule. The minimum pincipal uantum numbe fo the minimum adius of hydogen atom is n = 1. Howeve, in the supeluminal model, although the minimum amount of elativistic enegy is zeo, the momentum has a specific minimum. It cannot be less than the mass of the electon, which is m s c. c P = E + m sc 4 ()

3 E = m sc β 1 P = m sv β 1 (3) (4) We see that the electon has a maximum wavelength λ = /cm. Thus, by the Wilson- Sommefeld ule, the electon cannot gain infinite wavelength and cannot escape the hydogen atom. This fact sets a limit on the maximum adius of the bag. Thus, the electon in the supeluminal model is confined. Fo the supeluminal model, the pincipal uantum numbe of the maximum adius of the bag is n = 1. (m c 4 + E ) 1/ π = 1 (5) hc When the electon enegy inceases, its momentum inceases, but its wavelength deceases; thus, it becomes inceasingly confined. The electon falls deepe in the hydogen atom o bag, which is contay to ou obsevation in the subluminal model. At this point, we seek to deive and solve the wave function of confined supeluminal electon in the hydogen bag. Fist, we study the adial Diac euation. The Diac euation fo a subluminal paticle with eal mass leads to the euations shown below[3] c dg() d + (1 + κ) c g() [E + m c ]f() = 0 (6) c df() + (1 κ) cf() + [E m c ]g() = 0 (7) d The nomalized solutions ae popotional to 1 f() Γ(γ + 1) (λ)γ 1 e λ {( (n + γ)m c } κ)f ( n, γ + 1; λ) + n F (1 n, γ + 1; λ) E (8) 1 g() Γ(γ + 1) (λ)γ 1 e λ {( (n + γ)m c } κ)f ( n, γ + 1; λ) n F (1 n, γ + 1; λ) E (9) Fo nomalizable wave functions, γ should be positive. κ is the Diac uantum numbe, and λ = (m c 4 E ) 1/ c (10) = λ (11) γ = + κ (Zα) = + (j + 1 ) (Zα) (1) 3

4 Figue 1: eal pat of e ix F (1, 3, ix) To teminate hype geometic seies, we should discad the negative values of n n = n + κ = n + j + 1 n = 1,, 3 (13) The solution fo the hydogen atom povides a hype geometic function, which is an associated Laguee polynomial and is chaacteistic of the wave function fo Coulomb potential. L m (n + m)! n (x) = F ( n, m + 1, x) (14) n!m! whee L m n (x) is the associated Laguee function. look at (8) and (9). We mimic the above pocedue fo the supeluminal model with imaginay mass and obtain c dg() + (1 + κ) c g() [E + im c ]f() = 0 (15) d we define λ as c df() d + (1 κ) cf() + [E im c ]g() = 0 (16) λ = (m c 4 + E ) 1/ (17) c We solve the above euation and exactly mimic the povided method in the efeence fo the solution of Coulomb potential [3]. Finally, we obtain g() (λ) γ 1 e iλ {( (n + γ)m c } κ)f ( n, γ + 1; iλ) n F (1 n, γ + 1; iλ) E (18) f() (λ) γ 1 e iλ {( (n + γ)m c } κ)f ( n, γ + 1; iλ) + n F (1 n, γ + 1; iλ) E (19) 4

5 In the above euations, F ( n, γ + 1; iλ) is nomalized fo only negative values of n if n < γ + 1 (0) Fo example, fo j = 1 ( which gives γ = 1), and n = 1 we have a well behaved wave function (figue 1). Fo n = γ +1, the behavio of the wave function F ( n, γ +1; iλ) is simila to cos(). fo negative n, the above hype geometic euations ae simila to the spheical Bessel function of the fist type. Fom (18) and (19) the elation between the hype geometic seies and the Bessel functions is J ν (x) = e ix ( x ν! )ν F (ν + 1, ν + 1, ix) (1) The spheical Bessel function of the fist type is defined as π j ν (x) = x J ν+1/(x) () We saw that the solution fo subluminal hydogen atom is Laguee polynomial. Howeve, we see that f() and g() fo a supeluminal electon with Coulomb potential is simila to the spheical Bessel function of the fist type. The spheical Bessel functions appea in only two simila cases. The fist case is a paticle tapped in an infinite thee-dimensional adial well potential. The solution to this poblem is the spheical Bessel function of the fist type. Similaly, the solution to the MIT bag model, which postulated the existence of an unknown pessue and the vanishing of the Diac cuent outside the bag, is also spheical Bessel functions of fist type[, 4]. To ceate a supeluminal Diac euation fo uaks, we can use imaginay mass o substitute the following matix β s = iβ to calculate f() and g(). Howeve, when we want to constuct the Diac cuent, we will face a poblem. The coect method is to conside the following non-hemitian matices β s = βγ 5 [1, 5] α = [ ] 0 σ σ 0 β s = [ 0 ] I I 0 This method satisfies all euied popeties of a supeluminal Diac euation. Note that we did not postulate that the stong foce is actually the electomagnetic foce among supeluminal paticles. Howeve, even if the foce among the paticles was epulsive in the above euation o its stength with espect to distance was not 1, the facto that detemines whethe the system is stable and whethe the supeluminal positon can escape the poton is the enegy of the system and not the attactive o epulsive foce among the paticles. Note that the univese fo a supeluminal positon in the hydogen atom is the bag. Its beginning is the bounday of the bag, and its infinity is the cente of the bag. The same law that does not pemit the electon to fall on the poton in the subluminal model pohibits the supeluminal positon o electon to escape fom the hydogen bag. By studying the inte uaks potential, we conside the following conjectue Conjectue. The stong foce is only the supeluminal effect of the electomagnetic foce among supeluminal paticles. Without applying any pessue o infinite potential, we have confined the supeluminal electon with the appopiate bag adius in the hydogen atom. In othe wods, we (3) 5

6 solved a modified Diac euation fo supeluminal paticles and substituted the attactive Coulomb potential in the absence of any infinite potential. The solutions wee spheical Bessel functions of the fist type. The confinement of uaks in hadons has a simila mechanism to the above example. It appeas that we no longe euie SU(3) symmety of the stong foce to confine uaks in hadons. This method indicates that we should conside anothe symmety goup fo QCD. Although it is not clea why the net electic chage of the bag must be an intege value, The autho is completely confident that if we conside the supeluminal coection fo uaks, we can solve QCD at all enegy values. 3 Appendix In the appendix, we solve the Diac euation fo the supeluminal hydogen atom. mimic the method fom efeence [3]. The electic potential is We The adial Diac euations ae V = Ze (4) dg d = k + imc G + [E c df d = k imc F [E c whee we use G = g and F = f, and ]F () (5) ]G() (6) α = e c = (7) fo small nea the oigin, E ± imc is omitted. thus, we have dg d + k G + Zα F () = 0 (8) df d k F G() = 0 (9) We attempt the ansatz G = a γ F = b γ aγ γ 1 + κa γ 1 Zαb γ 1 = 0 (30) bγ γ 1 κb γ 1 a γ 1 = 0 (31) which indicates that a(γ + κ) bzα = 0 (3) azα + b(γ κ) = 0 (33) The deteminant of coefficients must vanish, which yields γ = κ (Zα) (34) 6

7 We choose and which esults in Fo, we have γ = ± κ (Zα) = ± λ = (j + 1 ) Z α (35) = λ (36) E + m c 4 c dg d = kg + [ E + imc λ c df d using (36) and (37), we obtain = [E imc λ c (37) ]F () (38) ]G + k F () (39) dg d = E + imc F (40) λ c df d We have G e ± i, but we choose the negative sign imc = E G (41) λ c d G d = + m c E 4λ c G = 1 4 G (4) d F d = + m c E 4λ c F = 1 4 F (43) G = imc + Ee i (φ1 + φ ) (44) by substituting into euation (38) and (39), we obtain F = imc Ee i (φ1 φ ) (45) imc + E i = k e i (φ1 + φ ) + imc + Ee i (φ 1 + φ ) imc + Ee i (φ1 + φ ) + [ E + imc λ c ][ imc E]e i (φ1 φ )(46) o imc E i e i (φ1 φ ) + imc Ee i (φ 1 φ ) = [ E imc λ c ][ imc + E]e i (φ1 + φ ) + k imc Ee i (φ1 φ )(47) 7

8 imc + E e i [ i = k (φ 1 + φ ) + (φ 1 + φ )] imc + Ee i (φ1 + φ ) + [ E + imc λ c ][ imc E]e i (φ1 φ )(48) imc E e i [ i (φ 1 φ ) + (φ 1 φ )] = [ E imc λ c ][ imc + E]e i (φ1 + φ ) + k imc Ee i (φ1 φ )(49) dividing by e i and futhe dividing the fist euation by (imc + E) 1 and the second euation by (imc E) 1, we obtain [ i (φ 1 + φ ) + (φ 1 + φ )] = k (φ 1 + φ ) + [ E + imc λ c ] imc E imc + E (φ 1 φ ) (50) Howeve, we had [ i (φ 1 φ ) + (φ 1 φ )] = [ E imc λ c ] imc + E imc E (φ 1 + φ ) + k (φ 1 φ ) (51) and Thus imc E imc + E = imc E m c 4 E = imc E imc + E imc E = imc + E m c 4 E = imc + E (5) (53) [ i (φ 1 + φ ) + (φ 1 + φ )] = k (φ 1 + φ ) + [ E + imc E λ c ]imc (φ 1 φ ) (54) [ i (φ 1 φ ) + (φ 1 φ )] = [ E imc λ c + E ]imc (φ 1 + φ ) + k (φ 1 φ ) (55) 8

9 By adding the two above euations o iφ 1 + φ 1 = k φ + ( E + imc λ c E (imc Zα )[φ 1 φ ] [ E imc λ c imc + E (φ 1 + φ ) )( imc E )[φ 1 φ ] ][ imc + E [φ 1 + φ ] = k φ + ( m c 4 E iλ c )(φ 1 φ ) E (imc )(φ 1 φ ) [ E + m c 4 iλ c ](φ 1 + φ ) Zα imc + E (φ 1 + φ ) = kφ 1 i (φ 1 φ ) 1 i (φ 1 + φ ) E (imc i mcλ )(φ 1 φ ) Zα + E (imc )(φ 1 + φ ) (56) iφ 1 + φ 1 = k φ 1 i φ 1 + Zα E (imc )(φ 1 φ ) Zα + E h λc (imc )(φ 1 + φ ) (57) By subtacting two euations, we have o iφ + φ = k φ 1 E + m c 4 i c λ (φ 1 φ ) (imc E) (φ 1 φ ) + (E + m c 4 ) i c λ (φ 1 + φ ) imc + E (φ 1 + φ ) (58) iφ + φ = k φ 1 + φ i Summaizing, we obtain (imc E) (φ 1 φ ) imc + E (φ 1 + φ ) (59) φ 1 = (i ZαE )φ 1 ( k mc )φ (60) 9

10 We use the powe seies. solution fo 0 φ = ( k mc )φ 1 E φ (61) We sepaate a facto γ, which descibes the behavio of the φ 1 = γ α m m (6) Inseting this euation into euations (60) and (61), we obtain φ = γ β m m (63) and (m + γ)αm m+γ 1 = i α m m+γ ZαE αm m+γ 1 (k mc ) β m m+γ 1 (64) βm (m + γ) m+γ 1 = ( k mc ) α m m+γ 1 E βm m+γ 1 (65) By compaing the coefficient, we obtain α m (m + γ) = iα m 1 ZαEα m h Zαmc (k + )β m (66) λc Fom the above euation, we obtain β m (m + γ) = ( k mc )α m E β m (67) β m α m = Zαmc ( k + ) m + γ ZαE = Zαmc (k ) n m (68) Fo m = 0, we obtain n = ZαE γ (69) β = k Zαmc α n = k (n + γ) mc E n (70) 10

11 inseting (68) into (66) and (67), we obtain α m (m + γ) = iα m 1 ZαEα m (k + zαmc Zαmc (k ) ) (m + γ ZαE )α m (71) α m [(m + γ) E zαmc Zαmc (k + )(k (m n ) ) = iα m 1 (7) α m [m + γ E zαmc Zαmc (k + )(k + (n m) ) = iα m 1 (73) α m [(m + γ E )(n m) + k Z α m c 4 c λ ] = iα m 1 (n m) (74) If we expand the backet on the left hand side of the above euation and use euation (69), we obtain Thus, we have (m + γ E )(ZαE γ m) = mγ m γ ( ZαE ) (75) α m [ mγ m γ ( ZαE ) + k Z α m c 4 c λ ] = iα m 1 (n m) (76) with α m [ m(γ + m) + (Zα) ( ZαE ) Z α m c 4 c λ ] = iα m 1 (n m) (77) γ = k (Zα) (78) We conclude that which can be witen as α m [ m(γ + m) + (Zα) (1 E + m c 4 c λ )] = iα m 1 (n m) (79) α m = (n m) m(γ + m) iα m 1 = ( 1)m (n 1)...(n m)α i m m!(γ + 1)...(γ + m) = (1 n )( n )...(m n )(i) m α (80) m!(γ + 1)...(γ + m) 11

12 β m = Zαmc (+k ) n m ( 1) m (n 1)...(n m)α i m m!(γ + 1)...(γ + m) (81) β m = Zαmc (+k )( 1)m (n 1)...(n m + 1)α i m m!(γ + 1)...(γ + m) (8) Using (70), we conclude that β m = Zαmc (+k )( 1)m (n 1)...(n m + 1)i m m!(γ + 1)...(γ + m) n β (+k Zαmc ) (83) β m = n (n 1)...(n m + 1)( 1) m i m β (84) m!(γ + 1)...(γ + m) The above euation is the confluent hype geometic function F (a, c; x) = 1 + a c a(a + 1) x x (85) c(c + 1)! φ 1 = α γ F (1 n, γ + 1; i) (86) φ = β γ F ( n, γ + 1; i) = ( κ Zαmc / n )α γ F ( n, γ + 1; i) (87) Fo negative value of n the above seies is nomalized if we choose the appopiate γ. using (44), (45) and the fact that G = g and F = f, we can constuct the nomalized wave functions f and g. Refeences 1. T. Chang, A new Diac-type euation fo tachyonic neutinos, axiv:hep-th/ v4.. A. Chodos, R. L. Jaffe, K. Johnson and C. B. Thon, Phys. Rev. D 10, 599 (1974). 3. W. Geine, Relativistic uantum mechanics, Spinge (000). 4. K. Johnson, Acta Physica Polonica B6 (1975). 5. U. D. Jentschua, B. J. Wundt, Pseudo-Hemitian uantum dynamics of tachyonic spin- 1/ paticles, J. Phys. A: Math. Theo (01) axiv:hep-th/ v3 6. M. Shaifi, Invaiance of Spooky Action at a Distance in Quantum Entanglement unde Loentz Tansfomation. Quantum Matte Vol. 3, pp (8) (014). axiv: