Quark-Parton Phase Transitions and the Entropy Problem in Quantum Mechanics

Transcription

1 Quark-Paron Phase Transiions and he Enropy Problem in Quanum Mechanics Y. S. Kim Deparmen of Physics, Universiy of Maryland, College Park, Maryland 20742, U.S.A. Absrac Since Feynman proposed his paron model in 1969, one of he mos pressing problems in high-energy physics has been wheher parons are quarks. I is shown ha he quark model and he paron model are wo differen manifesaions of one covarian eniy. The naure of ransiion from he confined quarks o plasma-like parons is sudied in erms of he enropy and emperaure coming from he ime-separaion variable. According o Einsein, he ime-separaion variable exiss wherever here is a spaial separaion, bu i is no observed in he presen form of quanum mechanics. The failure o observe his variable causes an increase in enropy. Typese using REVTEX elecronic mail: yskim@physics.umd.edu 1

2 I. INTRODUCTION In 1969, Feynman proposed his paron model for hadrons moving wih speed close o ha of ligh [1]. He observed ha he hadron appears as a collecion of infinie number of parons. Since he parons appear o have properies quie differen from hose of he quarks, one of he mos pressing pules in high-energy physics has been wheher he parons are quarks, or wheher he quark model and he paron model are wo differen manifesaions of one covarian formalism. In 1970, a he April meeing of he American Physical Sociey held in Washingon, DC U.S.A.), Feynman gave an invied alk on a model of hadrons. His alk was published in a paper by Feynman, Kislinger and Ravndal in 1971 [2]. There, he auhors aemped o consruc a covarian model for hadrons consising of quarks joined ogeher by an oscillaor force. They indeed formulaed a Loren-invarian oscillaor equaion. They also worked ou he degeneracies of he oscillaor saes which are consisen wih observed mesonic and baryonic mass specra. However, heir wave funcions are no normaliable in he space-ime coordinae sysem. They never considered he quesion of covariance. In his 1972 book on saisical mechanics [3], Feynman says When we solve a quanummechanical problem, wha we really do is divide he universe ino wo pars - he sysem in which we are ineresed and he res of he universe. We hen usually ac as if he sysem in which we are ineresed comprised he enire universe. To moivae he use of densiy marices, le us see wha happens when we include he par of he universe ouside he sysem. Feynman s res of he universe has been sudied in deail in erms of wo coupled oscillaors [4]. In his repor, we combine hese hree componens of Feynman s research effors o show ha he quark and paron models are indeed wo differen manifesaions of he same covarian eniy. In order o achieve his purpose, we fix up firs he mahemaical deficiencies of he paper of Feynman e al. [2]. The idea is o consruc a harmonic oscillaor wave funcion which can be Loren-boosed. We can firs see wheher he wave funcion is applicable o he quark model when he hadron is slow, and hen see wheher he same wave funcion describes he paron model when he hadron is boosed o an infinie-momenum frame. The 1971 paper by Feynman e al. [2] conains very serious mahemaical flaws, bu hey have been all cleaned up wihin he framework of Wigner s lile groups which dicae he inernal space-ime symmeries relaivisic paricles [5,6]. This covarian formulaion solves he covariance problem. However, since we live in he hree-dimensional world, i is possible ha we miss somehing in he four-dimensional world. The ime-separaion variable beween he quarks is a case in poin. In non-relaivisic quanum mechanics, he Bohr radius is spacial separaion beween he quarks or proon and elecron). According o Einsein, here mus be a ime separaion beween he quarks, since oherwise he world will no be covarian. Since we are no dealing wih his ime-separaion variable in he presen form of quanum mechanics, he failure o measure i leads o an increase in enropy [3]. In his repor, we show ha his enropy allows us o define he phase ransiion beween he confined phase of he quark model and he plasma phase of he paron model. In Sec. II, we inroduce he covarian harmonic oscillaor formalism wih normaliable wave funcions which can be Loren boosed. In Sec. III, we use he oscillaor wave func- 2

3 ion o solve he quark-paron pule. In Sec. IV, we deal wih he problems arising from measuring of four-dimensional physics in he hree-dimensional world. The enropy plays a major role. II. COVARIANT HARMONIC OSCILLATORS Le us consider a hadron consising of wo quarks. Then here is a Bohr-like radius measuring he space-like separaion beween he quarks. There is also a ime-like separaion beween he quarks, and his variable becomes mixed wih he longiudinal spaial separaion as he hadron moves wih a relaivisic speed. While here are no quanum exciaions along he ime-like direcion, here is he ime-energy uncerainy relaion which allows quanum ransiions. I is possible o accommodae hese aspecs wihin he framework of he presen form of quanum mechanics. The uncerainy relaion beween he ime and energy variables is he c-number relaion [7], which does no allow exciaions along he ime-like coordinae, as illusraed in Fig. 1 Dirac: Uncerainy wihou Exciaions Heisenberg: Uncerainy wih Exciaions FIG. 1. Presen form of quanum mechanics. There are exciaions along he space-like dimensions, bu here are no exciaions along he ime-like direcion. However, here sill is a ime-energy uncerainy relaion. We call his Dirac s c-number ime-energy uncerainy relaion. I is very imporan o noe ha his space-ime asymmery is quie consisen wih he concep of covariance For a hadron consising of wo quarks, we can consider heir space-ime posiions x a and x b, and use he variables X =x a + x b )/2, x =x a x b )/2 2. 1) The four-vecor X specifies where he hadron is locaed in space and ime, while he variable x measures he space-ime separaion beween he quarks. Since he hree-dimensional oscillaor differenial equaion is separable in boh spherical and Caresian coordinae sysems, he wave funcion consiss of Hermie polynomials of 3

4 x, y, and. If he Loren boos is made along he direcion, he x and y coordinaes are no affeced, and can be emporarily dropped from he wave funcion. Along he space-like longiudinal direcion, here are exciaions. On he oher hand, along he ime-like direcion, here is an uncerainy relaion even hough here are no exciaions. The covarian harmonic oscillaor formalism accommodaes his space-ime asymmery [6]. However, since we are ineresed here only in Loren-boos properies of he wave funcion, we resric ourselves o he ground-sae wave funcion. The wave funcion hen can be wrien as ) 1 1/2 { ψ, ) = exp )}, 2) π 2 which accommodaes he uncerainy relaions along he longiudinal and ime-like direcions. The expression given in Eq.2) is no Loren-invarian. I is covarian. This wave funcion describes he presen form of quanum mechanic if he ime-separaion variable is facored ou, inegraed ou, or ignored. However, he ime-separaion variable is absoluely needed when we consider Loren covariance. The quesion is wheher he above wave funcion can describe he paron model when i is boosed o an infinie-momenum limi. I is convenien o use he ligh-cone variables o describe Loren booss. The ligh-cone coordinae variables are u = + )/ 2, v = )/ 2. 3) In erms of hese variables, he Loren boos along he direcion, akes he simple form ) = ) ) cosh η sinh η, 4) sinh η cosh η u = e η u, v = e η v, 5) where η is he boos parameer and is anh 1 v/c). Indeed, he u variable becomes expanded while he v variable becomes conraced. This is he squeee mechanism illusraed discussed exensively in he lieraure [8,9]. This squeee ransformaion is also illusraed in Fig. 2. Thus, one way o combine quanum mechanics wih relaiviy is o incorporae Fig. 1 ino Fig. 2, and produce he ellipic deformaion illusraed in Fig. 3. If he sysem is boosed, he wave funcion becomes ) 1 1/2 { ψ η, ) = exp 1 e 2η u 2 + e 2η v 2)}. 6) π 2 We noe here ha he ransiion from Eq.2) o Eq.6) is a squeee ransformaion. The wave funcion of Eq.2) is disribued wihin a circular region in he uv plane, and hus in he plane. On he oher hand, he wave funcion of Eq.6) is disribued in an ellipic region. This ellipse is a squeeed circle wih he same area as he circle, as is illusraed in Fig. 3. 4

5 v u A=4u v A=4uv =2 2 2 ) FIG. 2. Furher conens of Loren booss. In he ligh-cone coordinae sysem, he Loren boos akes he form of he lower par of his figure. In erms of he longiudinal and ime-like variables, he ransformaion is illusraed in he upper porion of his figure. III. FEYNMAN S PARTON PICTURE In 1969 [1] Feynman made he following sysemaic observaions on hadrons moving wih speed close o ha of ligh. a). The picure is valid only for hadrons moving wih velociy close o ha of ligh. b). The ineracion ime beween he quarks becomes dilaed, and parons behave as free independen paricles. c). The momenum disribuion of parons becomes widespread as he hadron moves very fas. FIG. 3. Effec of he Loren boos on he space-ime wave funcion. The circular space-ime disribuion a he res frame becomes Loren-squeeed o become an ellipic disribuion. 5

6 d). The number of parons seems o be infinie and much larger han ha of quarks. These observaions consiue Feynman s paron picure. Because he hadron is believed o be a bound sae of wo or hree quarks, each of he above phenomena appears as a paradox, paricularly b) and c) ogeher. If he quarks have he four-momena p a and p b, we can consruc wo independen fourmomenum variables [2] P = p a + p b, q = 2p a p b ). 7) The four-momenum P is he oal four-momenum and is hus he hadronic fourmomenum. q measures he four-momenum separaion beween he quarks. QUARKS PARTONS β=0 BOOST β=0.8 Time dilaion TIME-ENERGY UNCERTAINTY SPACE-TIME DEFORMATION q o β=0 BOOST β=0.8 Weaker spring consan Quarks become almos) free q o q q Energy disribuion MOMENTUM-ENERGY DEFORMATION Paron momenum disribuion becomes wider FIG. 4. Loren-squeeed space-ime and momenum-energy wave funcions. As he hadron s speed approaches ha of ligh, boh wave funcions become concenraed along heir respecive posiive ligh-cone axes. These ligh-cone concenraions lead o Feynman s paron picure. Since we are using here he harmonic oscillaor, he mahemaical form of he above momenum-energy wave funcion is idenical o ha of he space-ime wave funcion, and is ransformaion properies are he same. The Loren squeee properies of hese wave funcions are also he same, as are indicaed in Fig. 4. When he hadron is a res wih η = 0, boh wave funcions behave like hose for he saic bound sae of quarks. As η increases, he wave funcions become coninuously squeeed unil hey become concenraed 6

7 along heir respecive posiive ligh-cone axes. Indeed, his figure provides he answer o he quark-paron pule [6]. The quesion hen is wheher he ellipic deformaions given in Fig. 4 produce any quaniaive resuls which can be compared wih wha we measure in laboraories. Indeed, according o Hussar s calculaion [10], he Loren-boosed oscillaor wave funcion produces a reasonably accurae paron disribuion, as indicaed in Fig. 5 x) Experimenal Harmonic Oscillaor FIG. 5. Paron disribuion. I is possible o calculae he paron disribuion from he Loren-boosed oscillaor wave funcion. This heoreical curve is compared wih he experimenal curve. IV. ENTROPY PROBLEMS The covarian harmonic oscillaor formalism presened in Sec. II produces he Loren squeee propery summaried in Fig. 4. This figure ells us ha he quark model and he paron model are wo differen manifesaions of one covarian formulaion. In his figure, he ime-separaion variable plays he essenial role. However, we are no able o deal wih his variable in he presen form of quanum mechanics. If here is a physical which we canno measure, he variable cerainly belongs o Feynman s res of he universe [3,4]. Then here is a well-defined procedure o deal wih his problem: consruc a densiy marix from he wave funcion and inegrae over he variable which we do no observe. In he presen case, he variable we do no observe is he imeseparaion variable. This process leads o an increase in enropy [11]. I is sraigh-forward o calculae his enropy [4], and he resul is S =2 { cosh 2 η)lncoshη) sinh 2 η) lnsinh η) }. 8) This form is idenical o he enropy caused by hermally excied harmonic oscillaors, if we wrie ) hω anh 2 η) =exp. 9) kt The enropy of Eq.8) akes he form [11,12] 7

8 S = [ hω/kt exp hω/kt ) 1 ln 1 exp )] hω. 10) kt LeusgobackoEq.9). Thevelociy) 2 is ploed agains he emperaure in Fig. 6. Is behavior makes a sudden change as he emperaure rises. If he hadronic velociy is low, he emperaure is relaively insensiive o he velociy, bu for high velociies, i is in he oher way around. We can use his behavior o ell he difference beween he confinemen phase of he quarks and he plasma phase of he parons. FIG. 6. The hadronic velociy versus he hadronic emperaure given in Eq.9). Here we used he uni sysem where hω/k = 1, and anh η = v/c. 8

9 REFERENCES [1] R. P. Feynman, in High Energy Collisions, Proceedings of he Third Inernaional Conference, Sony Brook, New York, edied by C. N. Yang e al. Gordon and Breach, New York, 1969). [2] R. P. Feynman, M. Kislinger, and F. Ravndal, Phys. Rev. D 3, ). [3] R. P. Feynman, Saisical Mechanics Benjamin, Reading, MA, 1972). [4]D.Han,Y.S.Kim,andM.E.No,Am.J.Phys.67, ). [5]E.P.Wigner,Ann.Mah.40, ). [6]Y.S.KimandM.E.No,Theory and Applicaions of he Poincaré GroupReidel, Dordrech, 1986). [7]P.A.M.Dirac,Proc.Roy.Soc.London)A114, 243 and ). [8]Y.S.KimandM.E.No,Phys.Rev.D8, ). [9]Y.S.KimandM.E.No,Phase Space Picure of Quanum Mechanics World Scienific, Singapore, 1991). [10] P. E. Hussar, Phys. Rev. D 23, ). [11] Y. S. Kim and E. P. Wigner, Phys. Le. A 147, ). [12] D. Han, Y. S. Kim, and M. E. No, Phys. Le. A 144, ). 9