The Hamilton-Jacobi Treatment of Complex Fields as Constrained Systems. Tamer Eleyan
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1 An - Njh Univ J Res N Sc Vol 1 7 The milon-jcobi Tremen of Complex Fields s Consrined Sysems معالجة نظم المجال المرآبة آا نظمة مقيدة باستخدام طريقة هاميلتيون جاآوبي Tmer Eleyn تامر عليان Deprmen of Mhemics Al-Quds Open Universiy Gz Plesine E-mil: mer_eleyn@homilcom Received: 1/6/6 Acceped: 4/4/7 Absrc The complex sclr field is reed s consrined sysem using he milon-jcobi pproch The reduced phse spce milonin densiy is obined wihou inroducing Lgrnge mulipliers nd wihou ny ddiionl guge fixing condiion The qunizion of his sysem is lso discussed Keywords: milonin nd Lgrngin pproch milon-jcobi pproch ph inegrl qunizion reprmerizion invrin heories qunizion of complex field sysems ملخص ت تم معالج ة نظ م المج ال المرآب ة آ نظم مقي دة باس تخدام طريق ة ه اميلتيون ج اآوبي ي تم الحص ول عل ى الط ور الفراغ ي المص غر ب دون تض مين مض اعفات لاج رانج وب دون تثبي ت أي ة شروط خارجية وأيضا تتم دراسة تكميم هذه الا نظمة باستخدام طريقة التكميم المساري 1 Inroducion There re some fmous heories of prmeerizion which cn be described s invrin Einseins heory of grviion relivisic poin pricle nd relivisic sring heories re good exmples The firs sysemic sudy of mechnicl sysems including field heories wih
2 of The milon-jcobi Tremen 54 consrins ws done by Dirc Dirc 1964 Dirc 195 p19 e showed h he lgebr of poisson brcke deermines division of consrins ino wo clsses: firs- clss consrins h hve vnishing poissons brckes wih ll oher consrins nd second-clss consrins h hve non-vnishing poissons brckes The presence of consrins in such heories mkes one be creful when pplying Dircs mehod especilly when firs-clss consrins rise since he firs clss consrins re generors of guge rnsformions which led o he guge freedom In oher words he equions of moion re sill degenere nd depend on he funcionl rbirriness Recenly he milon-jcobi Guler 199 p1389 Guler 199 p1143 Rbi & Guler 199 p3513 Muslih 1998 p77 mehod hs been developed o invesige consrined sysems In his mehod he disinguish beween he firs nd second clss consrins is no necessry The equions of moion re wrien s ol differenil equion in mny vribles which require he invesigion of he inegrbiliy condiions In oher words he inegrbiliy condiions my led o new consrins Moreover i is shown h he guge fixing which is n essenil procedure o sudy singulr sysems by Dircs mehod is no necessry if he milon-jcobi mehod is used The im of his pper is o nlyze field sysems s singulr sysems by using he milon-jcobi mehod nd by considering h he reprmerizion invrin heories hve vnishing milonins Besides we discuss he qunizion of his sysem using he cnonicl ph inegrl qunizion Muslih p7 Muslih p 3 Muslih p 495 Muslih p1 Muslih p 919 Muslih & el 4 p 119 The pln of his pper is he following: A brief informion of he milon- Jcobi mehod is given in secion In secion 3 he prmeerizion invrin field heory is reed s consrined sysem using he milon-jcobi mehod In secion 4 he milon-jcobi nlysis for he ime "" s n evoluion prmeer is given In secion 5 we obin he ph inegrl qunizion by using he cnonicl ph inegrl formulion In secion 6 conclusions re presened
3 55 Tmer Eleyn The milon-jcobi mehod In his secion we will briefly review he milon-jcobi mehod Guler 199 p 1389 Guler 199 p 1143 Rbi & Guler Muslih 1998 p 77 for sudying he consrined sysems Consider sysem wih n degrees of freedom I my hve r primry consrins in his cse he cnonicl formulion gives he se of milon-jcobi pril differenil equion JPDE Guler 199 p 1389 Guler 199 p 1143 Muslih & Guler 1995 p 37 where S S x X n r 1 n 1 n r 1 nd is he cnonicl milonin The equions of moion is obined s ol differenil equion in mny vrible s follows: Guler 199 p 1389 Guler 199 p 1143 d d d d d d 1 r 3 dz d 4 where z S nd o x x represen he vriions of wih respec The se of equions 34 is inegrble if nd only if Guler 199 p 1143 Muslih & Guler 1995 p 37 d 5
4 of The milon-jcobi Tremen 56 d 1 r 6 If condiions 5 nd 6 re no sisfied ideniclly one considers hem s new consrin nd gin ess he consisency condiions Thus repeing his procedure one my obin se of condiions ence he cnonicl formulion leds us o obin he se of cnonicl phse- spce coordines nd s funcion of besides he cnonicl cion inegrl is obined in erm of he cnonicl coordines The milonin cn be inerpreed s he infiniesiml generors of he cnonicl rnsformion given by prmeers In his cse he ph inegrl represenion my be wrien s Muslih p7 Muslih p 3 Muslih p 495 Muslih p1 Muslih p 919 Muslih & el 4 p D D D exp[ i{ d xd }] 7 One should noice h he inegrl 7 is n inegrion over he cnonicl phse spce coordines nd 3 A remen of complex sclr field s consrined sysem Le us consider complex sclr field described by he Lgrngin densiy L 8 ere nd re funcions of x 1 3nd is funcion of independen prmeer The cion inegrl for his sysem my be wrien s S [ ] dxd[ ] 9 x i x i
5 Tmer Eleyn 57 An - Njh Univ J Res N Sc Vol 1 7 Since L is regulr Lgrngin densiy prmeerize he ime wih hen he cion inegrl 9 my be wrien s ] [ L dxd S 1 where he singulr Lgrngin densiy L is given by L 11 The generlized momen conjuged o he generlized coordine re defined s L 1 L 13 L 14 Using 1 nd 13 he equion 14 cn be rewrien s ] [ 15 ence he primry consrin is 16 where 17
6 of The milon-jcobi Tremen 58 Besides he cnonicl milonin is defined s 3 d x[ L T ] 18 clculions show h is ideniclly zero Now he se of milon-jcobi pril differenil equions JPDE is expressed s S 19 S The equions of moion re obined s ol differenil equion in mny vribles s follow d d d d d d d d d d d [ ] d 1 3 d d d [ ] d 4 d d d 5 To check wheher he se of equions 1-5 is inegrble or no we hve o consider he ol vriion of he consrins Since d d d 6
7 59 Tmer Eleyn vnishes ideniclly he se of equions is inegrble nd he cnonicl phse-spce coordines re obined in erms of 4 milon-jcobi Anlysis for he ime "" s n evoluion prmeer In his secion we shll invesige he model 8 by reing he ime "" s n evoluion prmeer The Lgrngin for his model is given by L x i x 7 i The generlized cnonicl momen re clculed s follows: L L The cnonicl milonin is clculed s The milon-jcobi pril differenil equion is S 31 Besides he equion of moion is he sme s he once obined in equions 1-5 The equivlence beween 31 nd shows h by using he milon-jcobi mehod we obin he cnonicl milonin for he complex sclr field in guge independen mnner
8 of The milon-jcobi Tremen 6 5 Ph inegrl qunizion mehod To obin he ph inegrl qunizion for his sysem we cn use he cnonicl ph inegrl formulion Mking use of 4 he cnonicl cion inegrl is clculed s 3 S[ ] [ ] d xd 3 Then he ph inegrl for he complex sclr field sysem is given s 3 Ou S In dd dd exp[{ i [ ] d xd}] 33 One should noice h he inegrl 33 is n inegrion over he cnonicl phse spce coordines 6 Conclusion Reprmerizion invrin heories hve vnishing cnonicl milonin Gimn & Tyuin 199 Gvrilo & Gimn 1993 p 57 nd enforce here dynmics hrough consrins To obin he correc physicl milonin nd he correc equions of moion by using Dircs mehod Dirc 195 p 19 one hs o impose guge fixing of he form f Such guge fixing is no lwys n esy sk While in he milon-jcobi mehod remen of his sysem here is no need o inroduce Lgrnge mulipliers o he milonin s well s no need o use ny guge fixing condiion On he oher hnd he ph inegrl for he sysem is obined s n inegrion over he cnonicl phse spce coordines wihou using ny guge fixing condiions References - Dirc PAM 1964 Lecures on Qunum Mechnics Belfer Grdue School of Science Yehiv Universiy New York
9 61 Tmer Eleyn - Dirc PAM 195 Cn J Mh 19 - Guler Y 199 Cnonicl Formulion of Consrined Sysems Nuovo Cemeno B Guler Y 199 Inegrion of singulr sysems Nuovo Cimeno B Rbi E & Guler Y 199 milon-jcobi remen of secondclss consrins Phys Rev A Muslih SI & Guler Y 1998 Is guge fixing of consrined sysems necessry? Nuovo Cimeno B Muslih SI & Guler Y 1995 On he inegrion condiions of consrined sysems Nuovo Cimeno B Muslih SI Ph inegrl qunizion of elecromgneic heory Nuovo Cimeno B Muslih SI Ph inegrl formulion of consrined sysems dronic J Muslih SI El-Zln A & El-Sb F Qunizion of Young-Mills heory In J Theor phys Muslih SI Qunizion of prmerizion invrin heories Nuovo Cimeno B Muslih SI Cnonicl qunizion of sysems wih imedependn consrins Czech J phys Muslih SI Eleyn TA & Ayoub FA 4 milon-jcobi remen of Clssicl Field s Consrined Sysems Mu h Lil- Buhuh wd-dirs Gimn DM & Tyuin IV 199 Qunizion of Fields wih consrins spring-verlg Berlin eidelberg - Gvrilo SP & Gimn DM 1993 Qunizion of sysems wih ime dependn consrins Exmple of relivisic pricle in plne wve Clss Qun Grv 1 57
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