Volume, Number, 00. pp. 47-54 RTICLE Jordn Journl of Physcs Frconl Cnoncl Qunzon of he Free Elecromgnec Lgrngn ensy E. K. Jrd, R. S. w b nd J. M. Khlfeh eprmen of Physcs, Unversy of Jordn, 94 mmn, Jordn. b eprmen of Physcs, Muh Unversy, l-kr, Jordn. Receved on: 8/0/00; cceped on: 7/6/00 bsrc: We reformuled he frconl free elecromgnec Lgrngn densy usng he rdon (Coulomb guge nd Lorenz guge. We lso obned frconl Euler- Lgrnge (E-L equons resulng from hese Lgrngn denses. Then we found frconl mlonn densy n generl form nd used rc lgebrc mehod o deermne he creon nd nnhlon operors o consruc he Cnoncl Commuon Relons (CCRs. Keywords: Cnoncl qunzon; Coulomb guge; Lorenz guge; Frconl dervve; Free elecromgnec lgrngn densy. Inroducon The heory of dervves of non neger order goes bc o Lebnz, Louvlle, Remnn nd Lenov [-9]. Frconl clculus generlzes he clsscl clculus nd hs mny pplcons n vrous felds of physcs. These pplcons nclude clsscl nd qunum mechncs, feld heory nd elecromgnec heory formuled mosly n erms of lef Remnn- Louvlle frconl dervve [0-5]. The frconl vronl prncple represens n mporn pr of frconl clculus nd s deeply reled o he frconl qunzon procedure by obnng he frconl Euler-Lgrnge equon nd he correspondng frconl mlonn. The qunzon of sysems wh frconl dervves s n mporn re n he pplcons of frconl dfferenl nd negrl clculus. Cnoncl qunzon s he procedure by whch clsscl heory, formuled by usng he Lgrngn-mlon formlsm, cn be mde no qunum heory. The process of qunzng he mlonn srs wh chngng he coordnes nd he conuge momenum no operors, hose ssfyng commuon relons whch correspond o he Posson brce relon of clsscl heory [6]. In he usul pproch o he qunzon of he free elecromgnec feld, he guge of he elecromgnec poenls s frs fed n eher he rdon (Coulomb guge or he Lorenz guge. If he rdon guge s used, hen Fourer epnson of he rnsverse vecor poenl s mde. When he mlonn s epressed n erms of he vecor poenl, reduces o sum of uncoupled hrmonc oscllor mlonns. The hrmonc oscllors re hen cnonclly qunzed. If he Lorenz guge s used for qunzon, subsdry condons mus be mposed nd n ndefne merc used o vod conrdcons. I mus hen be shown h he wo procedures yeld he sme resuls, so h guge nvrnce s ensured [7-9]. The mn m of hs pper s o qunze he elecromgnec Lgrngn densy wh Correspondng uhor: J. M. Khlfeh. Eml: lf@u.edu.o
rcle Jrd e l. Rdon nd Lorenz guge usng lef Remnn-Louvlle frconl dervve nd o obn he frconl cnoncl commuon relons nd compre hem wh he sndrd CCRs n clsscl clculus. The pln of hs pper s s follows: n he followng secon Remnn-Louvlle frconl dervves re brefly revewed. Then, he frconl elecromgnec Lgrngn densy nd he cnoncl qunzon n rdon guge re del wh. Then, he elecromgnec Lgrngn densy nd s cnoncl qunzon n Lorenz guge re presened. n ppend s nsered o show h he elecromgnec Lgrngn densy s nvrn under guge rnsformon. Fnlly some concludng remrs re gven. Bsc efnons Severl defnons of frconl dervve hve been proposed. These defnons nclude Remnn-Louvlle, Cpuo, Mrchud nd Resz frconl dervves [4-5]. In he followng pr of he pper, we brefly presen some fundmenl defnons used n hs wor. The lef nd rgh Remnn-Louvlle frconl dervves re defned s follows: The lef Remnn-Louvlle frconl dervve f ( n d d n τ f τ dτ n ( Γ ( The rgh Remnn-Louvlle frconl dervve f b ( n n b d n ( τ f ( τ dτ d Γ ( where represens he order of he dervve such h n- < n nd Γ represens he Euler's gmm funcon. If s n neger, hese dervves re defned n he usul sense;.e., d f d f b d d,, Remnn-Louvlle frconl dervves hve mny properes. One of hese properes s h he R-L dervve of consn s no zero, nmely: ( Γ( ( noher propery s h he R-L dervve of power hs he followng form: Γ ( + ( (4 Γ + > -, 0 Fnlly, he frconl produc rule s gven below: d g ( f g f 0 d (5 Frconl Cnoncl Qunzon n Rdon (Coulomb Guge Cnoncl qunzon procedure mouns o he mposon of cnoncl commuon relons for he feld vrbles nd her cnonclly conuge momen. To qunze he free EM Lgrngn densy n rdon (Coulomb guge, we wll sr o reformule hs Lgrngn densy n frconl form usng LRLF procedure. ( ( φ φ L + ( ( (6 B + ( φ ( where B 48
Frconl Cnoncl Qunzon of he Free Elecromgnec Lgrngn ensy, φ re he frconl grden of sclr poenl nd vecor poenl, respecvely. Usng rdon (Coulomb guge 0, ϕ 0, we ge L ( ( ( ( (7 From hs defnon of Lgrngn densy, we obn he frconl (E-L equon by pplyng he generl formul gven by grwl [8] s: L β bϕ L ρ + 0 ϕρ β L + b ϕ ρ (8 For feld vrbles, we ge he equon: { φ } { } (9 + 0 Snce φ0, we ge { } { } + (0 0 Equon (9 represens he second nonhomogeneous Mwell's equon n frconl form, where φ nd ( ( re he frconl elecrc nd mgnec felds, respecvely. Equon (0 cn be represened s wve equon: ( 0 ( where s he vecor poenl whch es he plne wve soluon., Then (, d + + π ω + ε e.. ( where ε s he polrzon vecor whch hs he followng properes: ε 0 ( ε ε δ (4 + ere, s he polrzon se nd, + re he creon nd nnhlon operors. To sr he qunzon process of he free EM Lgrngn densy, we hve o nroduce he mlonn densy n frconl form usng LRLF s: π + π L (5 Bu, π 0, π Then π + (6 Usng he defnonπ, we ge: π + (7 We cn generlze hs formulon n frconl form n erms of, s:, ( + π (8 where, nonneger numbers. Usng lgebrc mehod n qunum mechncs:, ( + ( π (9 ( ( π 49
rcle Jrd e l. where ε ( + ( π (0 + + ε ( ( π ( We consruc he cnoncl commuon relons CCRs n frconl form: + + ε, ε + + + + εε ε ε ( Usng he defnons n equons (0 nd (, we obn: + + εε, + π, + + ε ε, π, ( (4 Subsung hese resuls n equon (, we ge: ε, ε π, ε, ε, π (5 (6 Snce π s he cnoncl momenum conuge o he wre s [, 4] ( π coordne, we cn h Then, he CCRs become le: ε, ε h (, Rerrngng hs equon, we ge:. (7 ε, ε F h ( (, F (8 Ths mens h + + ε, ε F ( F h ( ( F Usng Lebnz rule o rewre he second erm n he squre brces, we obn s n [ref. 0]:, r h r ε + ε + F F ( r F r 0 r r β r s specl cse, ng, he CCRs reduce o he orgnl relons le ε, ε h F F F (9 50
Frconl Cnoncl Qunzon of he Free Elecromgnec Lgrngn ensy + + ε, ε h F Fnlly, le us wre he mlonn densy n erms of creon nd nnhlon operors. Snce π + Usng hese defnons we ge (,,, d + + π ω + ε e π (,.. d { } ω + + π ω ε e... β β d β (, { } + +, ( β π ω. + ε e Then + + + +, εε + ε ε (0 lso, we cn obn oher CCRs le: + + +, ε ε ε ε, ε + + + + + + ε, ε ε ε, ε + + + +, ε ε ε, ε 0 Frconl Cnoncl Qunzon n Lorenz Guge s n he prevous secon, le us rewre he elecromgnec Lgrngn densy n Lorenz guge 0. ere, we need erm connng he me dervve of φ n order o nsure he esence of he cnonclly conuge feld π o. Ferm dded hs erm o he elecromgnec Lgrngn densy, so he elecromgnec Lgrngn densy cn be wren n frconl form usng LRLF s: φ + φ φ φ L + l l ξ φ φ+ l l ( where ξ s freely chosen prmeer. The Euler- Lgrnge equon for hs Lgrngn formulon cn be obned usng equon (8. For feld vrble φ, (E-L equon es he form: ( ξ φ+ φ 0 ( Ths s smlr o he frs nonhomogeneous Mwell's equon n free feld ecep h here s n ddonl erm comng from he dded erm o he EM Lgrngn densy. Now, for he felds,, we obn he oher equon by he sme mehod. Then we ge: ( φ { } + l + ξ l l 0 ( Ths equon s smlr o he second Mwell's equon n free feld, nd he only dfference s he dded erm comng from he dded erm o he Lgrngn densy. 5
rcle Jrd e l. fer he preprons gven bove, he sge now s se for frconl qunzon of hs Lgrngn densy. Usng Feynmnn gugeξ, he EM Lgrngn densy es he form: L (4 Now, usng he defnon of he mlonn π L we obn { } - π π + (5 Ths formulon cn be generlzed n frconl form s: {( π ( } +, (6 {( } + π, (7 Usng he lgebrc mehod n qunum mechncs, we ge:, ( + ( π ( ( π (8 Le ε, ε + + be he creon nd nnhlon operors: ε + π (9 + + ε π (40 Now consruc he cnoncl commuon relons s: + + + + + + ε, ε εε ε ε Usng he sme procedure s n he prevous secon, we ge: + +, ε ε, π (4 + +, ε ε, π (4 Snce( π s he cnoncl conuge o he ( π, we cn wre: h ε, ε + + h, ( Rerrngng hs equon, we ge: (4 ε, ε + + F h, F ( (44 Usng Lebnz rule, we obn hs equon s n [0] r r + + h F, ε F F 0 r r r ( ( ε (45 5
Frconl Cnoncl Qunzon of he Free Elecromgnec Lgrngn ensy s specl cse, hen ε, ε + + F h F Fnlly, o obn he mlonn densy n erms of creon nd nnhlon operors, we sr wh he defnon of, π, where he vecor poenl cn be epnded no plne wves s: (, d + 0 π ω + + (46 The cnonclly conuge vrble es he form: π (, d w + 0 π ω + (47 where ω o, 0,,, (polrzon se, ε s se of 4 lnerly ndependen vecors whch my ssume rel. Then we ge: g (48 0 + where g, -,-,- We lso found oher commuon relons le:, ε ε ε, ε + + + + +, + + + ε, ε ε ε + + + +, ε ε ε, ε 0 ppend Guge Invrnce of Lgrngn ensy The Lgrngn densy for he frconl elecromgnec feld L s gven by equon (6. Vron of L wh respec o he frconl poenl yelds he frconl nhomogeneous Mwell's equons. The poenl s no unquely deermned. chnge of he poenl of he ype + Λ leves he elecromgnec feld unchnged nd herefore s clled guge nvrn L L []. We cn rewre equon (6 usng he defnon of he vecor poenl n 4 dmensons s:,,,, Λ sclr funcon φ L (- L (- where + Λ, + Λ,, spce-me dmenson,, spce dmenson ( + Λ ( + Λ L ( + Λ ( + Λ Then 5
rcle Jrd e l. ( + ( Λ + ( Λ + ( Λ Λ ( ( Λ ( Λ ( Λ Λ L (- whch cn be smplfed no L (-4 So LL References []Blenu,., Golmnhneh,.K. nd Blenu, M., In. J. Theor. Phys. 48 (009 4. []Muslh, S.I., grwl, O.P. nd Blenu,., In. J. Theor. Phys. (00 OI 0.007/s077-00-054-. []Muslh, S.I., In. J. Theor. Phys. (00 OI 0.007/s077-00-096-0. [4]Podlubny, I., Frconl fferenl Equons, (cdemc Press, NewYor, 999. [5]Oldhm, K.B. nd Spner, J., The Frconl Clculus, (cdemc Press, New Yor, 974. [6]Rewe, F., Phys. Rev. E, 5 (996 890. [7]Rewe, F., Phys. Rev. E, 55 (997 58. [8]grwl, O.P., J. Mh. nl. ppl. 7 (00 68. [9]Blenu,. nd Muslh, S.I., Czech. J. Phys. 55(6 (005 6. [0]lfer, R., pplcons of Frcon Clculus n Physcs, (World Scenfc Publshng Compny, Sngpore, NewJersy, London nd ong Kong, 000. Th s he Lgrngn densy. I s nvrn under guge rnsformon, whch s he curren (chrge s conserved. Concluson The frconl qunzon of feld heory s no n esy s, especlly when he frconl mlonn s complced. ere, we hve qunzed he free EM Lgrngn densy n boh rdon (Coulomb guge nd Lorenz guge. For he wo cses, we obned he mlonn n erms of vecor poenl nd lso n erms of creon nd nnhlon operors, hen we consruced he frconl cnoncl commuon relons. We hve shown h he wo guges yeld he sme resuls, snce he mlonn reduces no sum of uncoupled hrmonc oscllor mlonns for wo cses. []Blenu,. nd Muslh, S.I., Czech. J. Phys. 55(9 (005 06. []grwl, O.P., Nonlner ynmcs, 8 (004 9. []grwl, O.P., Nonlner ynmcs, 8 (004. [4]Muslh, S.I. nd Blenu,., Nuovo Cmeno, 0 (005 507. [5]rc, P..M., (Yeshv Unversy, New Yor, 964. [6]Gry, R.. nd Kobe,.., J. Phys. : Mh. Gen. 5 (98 45. [7]Lsn, N., Phys. Le., 68 (000 98. [8]Lsn, N., Phys. Rev. E, 66 (00 05608. [9]Jrd, E.K., Ph.. Thess, Unversy of Jordn (009. [0]Erson, E. nd Lenss, J.M., Physc Scrp, (980 99. []Jcson J.., Clsscl Elecrodynmcs, JohnWley, second edon (980. 54