Fractional Quantum Field Theory on Multifractals Sets

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1 Amercn J. of Engneerng nd Appled Scences 4 (): 33-4, 2 ISSN Scence Publcons Frconl Qunum Feld Theory on Mulfrcls Ses El-Nbuls Ahmd Rm Deprmen of Nucler nd Energy Engneerng, Cheju Nonl Unversy, Ar-dong, Jeju , Souh Kore Absrc: Problem semen: Ths sudy s conrbuon o he generl progrm of descrbng complex dynmcl sysems usng he ool of frconl clculus of vrons. Approch: Followng our prevous work, frconl qunum feld heory bsed on he frconl conlke vronl pproch suppored by Sxen-Kumbh frconl negrls funconls, frconl dervve of order (α, β) nd dynmcl frconl exponen on mul-frcl ses s consdered. Resuls: In order o buld he requred heory, we nroduce he Sxen-Kumbh hypergeomerc frconl funconls deermned on he funcons on mulfrcl ses. We prove, developng he correspondng frconl clculus of vrons, h herrchy of dfferenl equons cn be developed from he exended frconl Lgrngn formlsm. Besdes, generlzon of he resulng Hmlonn nd Lgrngn dynmcs on he complex plne s ddressed. Concluson: The new complexfed dynmcs gudes o new dynmcs whch my dffer olly from he clsscl mechncs crdnlly nd my brng new ppelng consequences. Some ddonl neresng resuls re explored nd dscussed n some dels. Key words: Frconl con-lke vronl pproch, mulfrcl ses, Euler-Lgrnge equons, Sxen-Kumbh frconl negrl, frconl dervve, dynmcl frconl exponen INTRODUCTION Frconl dynmcs s he sudy of complex dynmcl sysems h cn be cs n erms of soluons o frconl dfferenl equons o whch he frconl clculus cn be correcly ppled. Frconl clculus hs become n excng new mhemcl mehod of soluon of dverse problems n mhemcs, scence nd engneerng. Mkng use of frconl dervves nd negrls, one my descrbe more ccurely complex sysem nd ccordngly nvesge more compleely s dynmcl nd physcl properes. Alhough frconl clculus hs been suded for over 3 yers, hs been regrded prncplly s mhemcl curosy unl bou 992, where frconl dynmcl equons were prey much resrced o he relm of mhemcs nd engneerng ncludng hydrology, vscoelsvy, he conducon, polymer physcs, chos nd EL-NABULSI Ahmd Rm frcls, conrol heory, plsm physcs, wve propgon n complex nd porous med, srophyscs, cosmology, qunum feld heory, poenl heory nd so on (Aghe e l., 29; Smko e l., 993; Mller nd Ross, 993; Podlubny, 999; Hlfer, 2; Kluwer, 24; Orguer nd Mchdo, 26; 28; El-Nbuls, 28; 29). Acully, here exs numerous dfferen forms of frconl negrl nd dervves operors nd he 33 defnon of he frconl order dervve nd negrl re no unque where severl defnons exs, e.g., Grunwld-Lenkov, Cpuo, Weyl, Feller, Erdely- Kober, Resz, Sxen, Kumbh, Krykov, Srvsv nd Rn. A heme of presen srong reserch concerns he sudy frconl Lgrngn nd Hmlonn dynmcs sysems bsed on he frconl problems of he Clculus Of Vrons (COV) (Al e l., 29). Dfferen forms of Euler-Lgrnge equons were obned n he lerure dependng on he con nd ype of frconl dervve used. The formulon of he frconl problems of he COV sll needs more elboron s he problem s deeply reled o he frconl qunzon procedure nd o he presence of non-locl frconl dfferenl operors. Mny problems lso occur, e.g.: The occurrence of non-locl frconl dfferenl operors, The djon of frconl dfferenl operor used o descrbe he dynmcs s no he negve of self The derved Euler-Lgrnge equons depend on lef nd rgh frconl dervves even when he dynmcs depend only on one of hem The ppernce of very complced Lebnz rule (he dervve of produc of funcons)

2 The non-presence of ny frconl nlogue of he chn rule. One possble smple nd relsc pproch o beer model non-conservve nd wek dsspve complex dynmcl sysems s he Frconl Acon-Lke Vronl Approch (FALVA) bsed on he concep of lef Remnn-Louvlle frconl negrl funconls wh one prmeer α bu no on frconl-order dervves of he sme order (El-Nbuls, 25; 25b). Mny encourgng resuls were successfully obned nd dscussed (El-Nbuls nd Torres, 28; 27; El-Nbuls, 27; 28b; 28c; 29b; 29c; 29d; 29e; 29f; 29g). In order o smule more neres n he subjec nd o show s uly, hs conrbuon s devoed o new generlzon of he FALVA by nroducng mnly nsde he frconl con-lke negrl he Sxen-Kumbh hypergeomerc frconl operor, ugmened by frconl dervve nd dynmcl frconl exponen defned on mulfrcl me nd spce ses, n prculr when he frconl dmensons of me nd spce re dynmcl, e.g., d = ε(x ). Ths noon s bsed on he Mndelbro des of he frcl geomery of nure nd s expeced o work on smll mulfrcl nervls se S whch s buld from mulfrcl subses S(x ) (Al-Doud, 28; Kobolev, 2). Ech of hem consss n fc of connuous, bu no dfferenble bounded se of smll elemens. Boh me nd spce re consdered s he only merl felds exsng n he mnfold nd hus generng ll oher physcl felds. In hs work, we suppose h he frcl dmensons re slghly dfferen from uny nd hence vld for smll denses of Lgrngns n pons x. The funcon d = d(x ) s expeced o be connuous nd descrbes he frconl dmensons. EL-NABULSI Ahmd Rm: Here, we ce our movons for nroducng he hypergeomerc funcons, he frconl dervves nd dynmcl frconl exponen respecvely: The movon for nroducng he hypergeomerc funcon n he frconl con s h he ler s nmely ssoced wh he Lplce nd wve equons n four-dmensonl spce nd her exensons. Therefore, we expec h he new formlsm wll led o generlzon of he dynmcl dfferenl equons s s presence n he dfferenl equons wll leds o n exremely rch srucure s he hypergeomerc funcon conns lmos ll specl funcons (orhogonl, Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 34 power-ype nd ellpc negrls) s prculr or lmng cse Our movon for nroducng he frconl dervve cme from he fc h he ler porry more precsely he non-rvl behvor of complex physcl sysems whose dynmcs re dsn from equlbrum,.e. delng wh frconl dervves s no more complex hn wh usul dfferenl operors. Dynmcl equons n frconl dervves descrbe normlly he evoluon of physcl sysems wh loss, he frconl exponen of he dervve beng mesure of he frcon of he ses of he dynmcl sysem h re preserved durng he evoluon me. Amzngly, frconl dfferenl operors re n rely globl (non-locl) operors nd lms n he sense of ulr-long me. For h mn reson, dynmcl sysems wh frconl order re non-conservve nd hence he frconl dervve s brodly used for descrbng nermedry physcl processes nd crcl phenomen n non-equlbrum complex non-lner sysems Tme-dependen frconl exponen plys sgnfcn role n dverse brnches of complex dynmcl sysems, e.g. self-ffne me-sequenl d (Sbnl nd Nkgw, 995) Generlzed FALVA ugmened by he sxenkumbh hypergeomerc frconl operor, frconl dervve nd me-dependen frconl dmensons: In order o consruc he requred heory, s desrble o chrcerze he frconl funconls deermned on he funcons, gven on mulfrcl ses. We sr by recllng he followng defnon: Defnon : The Sxen-Kumbh hypergeomerc frconl negrls for funcon f L p (, ) re defned by (Smko e l., 993): I f() τ f( τ) τ ( τ) F,b;c; ζ dτ Γα ( ) () I f() τ f( τ) τ ( τ) F,b;c; ζ dτ Γα ( ) ασ σ α Lef 2 δ δα α Rgh 2 EL-NABULSI Ahmd Rm: Here 2 F s he Gussn hypergeomerc funcon defned hrough s power seres expnson round y = by: () (b) n n n n 2F(,b;c;y) = y = ny (2) n= (c) n n! n= = nd:

3 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 ( n)(b n) n = n,n (c n)(n ) (3) pons of he negrl funconl on mulfrcl me nd spce ses: nd where (x) n = x... (x n ) s sndrd Appell or Pochhmmer symbol, Rα ( ) >, Rσ ( ) > q, Rδ ( ) > p, p q =, p, c,, 2,..., R(c b ) > nd rg( ζ ) <π. Normlly, he seres lwys converges for y < (Grnk, 24). Remrk : As s well-known, mny problems n physcs nd mhemcl physcs requre conforml symmery groups whch n s urn ccouns for mny of he properes of 2 F. I s fundmenl propery of ny dynmcl phenomen or physcl process. I s noeworhy h 2 F plys crucl role n mny brnches of scence, ncludng omc physcs, (negrls n omc collson re very ofen nlyclly expressed wh he 2F funcon), celesl dynmcs (he 3-body problem), srophyscs (n prculr n lnerzed MHD equons), Generl Relvy (grvonl wves emed by bnry srs). Furhermore, one of he mos powerful echnques for clculng Feynmn dgrms s bsed on her presenon n erms of hypergeomerc funcons. Such represenon cn be used for numercl evluon, consrucon of he sympoc expnson (Bev e l., 29). I cn lso provde nlycl wve funcon soluons of he Schrödnger equon nd of he relvsc Klen- Gordon nd Drc equons (Mchel nd Sosov, 28; Gomes nd Olver, 24). Remrk 2: For mhemcl smplcy, we wll defne he new vrble ( τ ),.e., we wll perform our clculon n he rel plne nd we se ζ = for mhemcl smplcy. As he frconl problem of he CoV s bsed on he lef frconl negrl, he Sxen-Kumbh hypergeomerc frconl lef negrl for funcon f L p (, ) wh respec o he new vrble s now defned on he nervl by: I f f(( ))( ) F,b;c; d Γα ασ ( ) (4) Lef 2 ( ) We my propose he followng problem. L(D q(( )),q(( )), Sq() [ ] = α () σ() ( ) ( )) Γα ( (( ))) χ () F,b;c; e d 2 ( ) (5) EL-NABULSI Ahmd Rm:, χ, q = dq/d nd he smooh Lgrngn funcon L(q(( )),q(()),()) s cully weghed α () σ() wh ( ) 2 F(, b;c; ) Γ( α(( ))). Here, he frconl dervve operor of order (α, β),,b,, < s defned by Cresson (27): ( α) ( β) D f() = D( ) f() D(b ) f() γ ( α) ( β) D( ) f() D(b ) f() γ, = nd < αβ<, Here: (6) β ( α) β D( ) f(( )) = f(( )) d Γ( β) (7) β ( β) β D ( ) f (( )) = f (( )) d Γ( β) (8) nd: b α(()) α(()) Γα ( (( ))) = () exp( ())d s he modfed Euler gmm funcon. Remrk 3: The followng properes hold: b b ( βα, ) D f ()g()d = f ()D g()d, b (9) b b ( α) α ( α) ( ) = (b ) D f ()g()d ( ) f ()D g()d () Problem : Consder smooh n-dmensonl mnfold n M nd le L:C ([,] ; ) be he smooh Lgrngn funcon ssfyng fxed boundry condons q() = q nd q() = q. Fnd he sonry 35 ( ) D D f () D f () D f (), k α ( α) ( β) ( αβ) ( σ) () () = () () = = Γ( α) () k q k

4 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 provded h f() = f(b) = nd g() = g(b) =. k n Eq. s whole number. Remrk 4: Snce n our pproch we hve defned frconl dfferenon hrough negron, frconl dervves re no longer locl operons. They re only defned over n nervl. Ths my expln why durng he erly me, Lebnz beleved ws n bsurdy snce hs objecve ws o obn unque nd locl dervve. Theorem : If q((-)) re soluons o he prevous problem, hen ssfes he followng generlzed frconl Euler-Lgrnge equon: L(D q(( )),q(( )), ( )) L(D (, ) q(( )),q(( )),( )) βα D; q dα d σ α () σ() ln( ) d d = d χ Γα ( (( ))) d Γα ( (( ))) ( βα, ) L(D q(( )),q(( )), ( )) EL-NABULSI Ahmd Rm: Sq [ ε h] = S[q] L(D q(( )),q(( )),( )) ε α () σ() ( ) Γα ( (( ))) χ () 2F(,b;c;) e h(( )) α () σ() ( ) χ () 2F(,b;c;) e Γα ( (( ))) L(D q(( )), q(( )), ( )) D ; h(( )) α () σ() ( ) χ () 2F(,b;c;) e Γα ( (( ))) L(D q(( )),q(( )),( )) q d O( ε) D h(( )) Mkng use of he les con prncple nd propery (9) we rrve o (2) fer smple lgebr. Remrk 5: One my subsue no Eq. 2 he hrd erm nsde he brcke by he dgmm funcon defned s he logrhmc dervve of he gmm funcon (Q nd Guo, 29): ( βα, ) d L(D 2F q(( )),q(( )),( )) F d 2 (2) Proof: We execue smll perurbon of he generlzed coordnes: d = Γα ( (( ))) d Γα ( (( ))) d ln Γα ( (( ))) =ψ (( )) d (3) q( ) q( ) εh( ), ε<< for whch: whch s he frs of he polygmm funcons nd hs he negrl represenon: D (q( ) εh( )) D q( ) εd h( ) (, ) (, ) (, ) Consequenly: ψ(( )) =γ ( ) (4) ( ) k(( ) k) L(D q(( )) εd q(( )), γ s he Euler-Mscheron consn defned by: Sq [ ε h] = q(( )) εh(( )),( )) EL-NABULSI Ahmd Rm: α () σ() ( ) χ () 2F(,b;c;) e d Γα ( (( ))) n γ= lm log n (5) n k= k whch mples dong Tylor expnson of L(D q(( )) εd h(( )),q(( )) n ε Corollry : In erms of he polygmm funcon, he εh(( )),( )) generlzed frconl Euler-Lgrnge equon s round zero nd negrng by prs: wren s follows: 36 k=

5 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 L(D q(( )),q(( )),( )) L(D (, ) q(( )),q(( )), ( )) βα D(y); l qx dα dσ ln( ) d d = α () σ () ψ (( )) χ ( βα, ) L(D q(( )),q(( ) ),( )) ( βα, ) d L(D 2F q(( )),q(( )), ( )) F d 2 (6) More generlly, he wo-dmensonl generlzed FALVA s nroduced s follows: Defnon 2: Consder smooh n-dmensonl 2 mnfold Mnd le q: Ω Mbe he dmssble phs ssfyng fxed Drchle boundry condons on Ω. The Lgrngn funcon (q, q, q, (-)), (- ))) L(q, q, q, (-)), (-))) s supposed o be suffcenly smooh wh respec o ll s rgumens. The wo-dmensonl generlzed FALVA con negrl s defned by: ( αδ, ) L(D (,,, ) ( );q, αβδξ Ω γ S [ q] = ( βξ, ) D;q,q, ( )),( ))) α () σ() α () σ() ( ) ( ) Γα ( (( ))) Γα ( (( ))) ( ) ( ) χ () σ () 2F,b;c; 2F,b;c; e e dd (7) (, ) where, nd ; q = ((-), (-)); D δ α ;xnd (, ) D ξβ ;re he frconl dervve operors (2) respecvely of orders (δ, α) nd (ξ, β) wh respec o nd. χσ,. dα dσ ln( ) d d = α () σ () ψ(( )) χ L(D; q,d;q,q,( )),( ))) EL-NABULSI Ahmd Rm: dα d σ α () σ() ln( ) d d ψ (( )) σ L(D; q,d;q,q,( )),( ))) 2 2 d2f L(D ; q,d;q,q,( )),( ))) F d d L(D 2F F d q,d γ q,q,( )),( ))) ( α, δ) ( β, ξ) ; ( ); (8) Proof: We le w be he mnmum soluon ssfyng he Drchle condons w(,) =, (,) Ω, h ny prmeer h nd q be he sonry soluon so h q((-))hw((-)) ssfes he Drchle boundry condons h. The generlzed frconl con s wren ccordngly lke (El-Nbuls nd Torres, 28): ( αβδξ,,, ) [ ] S q hw = L(D q hd w, ( αδ, ) ( αδ, ) Ω ; ; D q hd,q hw,( )),( ))) ( βξ, ) ( βξ, ) ; ; ( ) ( ) Γα ( (( ))) Γα ( (( ))) α () σ() α () σ() ( ) ( ) χ () σ () 2F,b;c; 2F,b;c; e e dd Therefore: Theorem 2: The double-weghed frconl generlzed Euler-Lgrnge equon ssoced o he frconl con (7) s: L(D; q,d;q,q,( )),( ))) L(D (, ) ; q,d;q,q,( )),( ))) δα D; q L(D (, ) ; q,d;q,q,( )),( ))) ξβ D; q L ( αδ, ) w D; dh h= ( βξ, ) w D;w α () σ() α () σ() ( ) ( ) Γα ( (( ))) Γα ( (( ))) d S ( αβδξ,,, ) [ q hw ] = Ω L L ( ) ( ) χ () σ () 2F,b;c; 2F,b;c; e e dd Inegron by prs nd mkng use of he Green's heorems: 37

6 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 2 Ω P P G G2dξdλ = ξ λ G G ξ λ ξ λ ξ λ ( 2 ) 2 P G d G d 2 P d d Ω Where: ( ) ξ= Γα ( (( ))) α () σ() χ () d 2F(,b;c;) e d ( ) λ= Γα ( (( ))) α () σ() σ () d 2F(,b;c;) e d we fnd for ny smooh funcons G nd G 2 : d S ( αβδξ,,, ) [ q hw ] : dh h= L(D (, ) ; q,d;q,q,( )),( ))) αδ Ωw D; EL-NABULSI Ahmd Rm: L(D (, ) ; q,d;q,q,( )),( ))) βξ D; q dα d σ α () σ() ln( ) (( )) d d ψ χ L(D; q,d;q,q,( )),( ))) dα d σ α () σ() ln( ) (( )) d d ψ σ L(D; q,d;q,q,( )),( ))) 2 d2f L(D ; q,d;q,q,( )),( ))) F d d2f L(D ; q,d;q,q,( )),( ))) F d 2 L(D q,d q,q, ( )),( ))) dd ; ; Ω ssume s prevously, h he dmssble phs re N smooh funcons q: Ω M ssfyng gven Drchle boundry condons on Ω (El-Nbuls nd Torres, 28). Defnon 3: The Lgrngn (q,...,q,q, N,..., N) L(q,...,q,q, N,..., N) s o be suffcenly smooh wh respec o ll s rgumens. The N-dmensonl frconl generlzed-falva con negrl s defned by: [ ] S q =... L( q(( )),q(( )),( )) (9) αδ, ( χ) γ Ω( ξ) N α ( ) σ( ) ( ) χ ( ) 2F(,b;c;) e d = Γα ( (( ))) αδ, α, δ αn, δn where, = (D,...,D ) nd dx = dx...dx N. γ γ; γ;n Theorem 3: The N-dmensonl ssoced frconl Euler-Lgrnge equon s gven by: dα dσ ln( ) d d N α, δ L α ( ) σ( ) L Dγ; = ψ (( )) χ EL-NABULSI Ahmd Rm: = 2F d d2f L L (2) where ll prl dervves of he Lgrngn re αδ, evlued ( q(( )),q(( )),( )), Ω. γ Generlzed lgrngn dynmcs: Generlzed Lgrngn dynmcs my be obned f we ssume h he dynmcl frconl exponen s complex,.e.: α () = F() G() nd: σ () = E() D() Becuse of he rbrrness of w(,) nsde he D(),E(),F(),G() re ny rel funcon of x. domn Ω, follows he generlzed frconl Euler- Consequenly, Eq. 2 s enled complex frconl Lgrnge equons. order generlzon of he Euler-Lgrnge equon. All he prevous rgumens cn be repeed, For hs, we my perform he dependen vrble mus munds, o he N-dmensonl problem. We complex nd wre: 38

7 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 q(( )) = q (( )) q 2(( )) We my use nurlly he followng equons: = 2 2 = 2 2 α, δ ( α) ( β) Dγ; = D ( ); D (b ); γ D D D γ D Where: ( α) ( β) α, δ ; α, δ; ( ); (b ); γ; γ; α, δ ; ( α) ( β) ; ( ); (b ); (2) (22) (23) D γ D D (24) whch n s urn my be spled no he equons by seprng he rel from he mgnry prs: N α, δ ; L α, δ ; L Dγ; D γ; = 2 df de ln( ) 2 d d E() F() L ψ(( )) χ q dg dd (D() G() ) ln( ) 2 d d L d2f L L = q 2 2F d q nd: (3) D γ D D (25) α, δ; ( α) ( β) ; ( ); (b ); Choosng γ = b,(,b), we my wre Eq. 23 lke: D = D bd D D D (26) α, δ α, δ ; α, δ; α, δ; α, δ ; α, δ ; γ; γ; γ; γ; γ; γ; Where: D = D bd =R D (27) α, δ ; α, δ ; α, δ; α, δ ; γ; γ; γ; γ; D = D =I D (28) α, δ ; α, δ; α, δ γ; γ; γ; nd ccordngly, Eq. 2 s rewren lke. EL-NABULSI Ahmd Rm: N, ;, ; α δ α δ L L ( Dγ; D ) γ; = 2 2 df de dg dd ln( ) 2 d d d d E() F() (D() G()) L L q q (( )) 2 ψ χ d F L L L L 2 2F d q q 2 2 q 2 2 = (29) 39 N α, δ ; L α, δ ; L Dγ; D γ; = q 2 q df de ln( ) d d E() F() L ψ(( )) χ q2 dg dd (D() G()) ln( ) 2d d L d2f L L = q 2F d q 2 2 (3) These equons requre numercl soluons nd s he uhor s speculon h hey my ply mporn role on complexfed oscllory dynmcl sysems wh me-dependen mss nd me-dependen frequency. Work n hs drecon s lso under progress. Nurlly, one expecs complex generlzon of Hmlonn dynmcs whch cernly wll gudes o new dynmcs whch my dffer compleely from he clsscl mechncs crdnlly nd my brng neresng consequences, e.g. qunum feld heory, conrol heory nd hence we expec h exoc soluons wll be physcl. MATERIALS AND METHODS Recenly, lo of enon hs been pu on he frconl clculus of vrons. In hs work we use he concep of Sxen-Kumbh frconl negrl o

8 explore some new specs of qunum feld heory on mulfrcl ses. The mehod s bsed on he concep of Frconl AconLke Vronl Approch recenly nroduced by he uhor. We hve ld ou he groundwork for generlzng he non-conservve dsspve sysems usng he mehodology of he Sxen-Kumbh hypergeomerc frconl negrl whn he frmework of he frconl con-lke vronl pproch by nroducng he bsc sengs. RESULTS AND DISCUSSION An mporn ddonl feure of he new formlsm s he herrchy of dfferenl equons h cn be obned from he generlzed frconl Euler- Lgrnge equons. Besdes, he resulng frconl Lgrngn nd Hmlonn dynmcs my be complexfed. Ths fc my hve neresng consequences n dfferen specs of qunum dynmcl Hmlonn sysems hvng dscree specr nd exhbng he PT symmery,.e., dynmcl equons nvrn under combned spce nd me. To he bes of our knowledge, he queson of obnng condons on he Lgrngn ssurng he exsence of sonry rjecores s n enrely open queson n he frconl scenery. An mporn ddonl feure of he new formlsm s he herrchy of dfferenl equons h cn be obned from he generlzed frconl Euler-Lgrnge equons. Fuure reserch effors my be dreced owrds formulng predcons h cn be esed numerclly. Furhermore, complex rjecores of qunum mechncl sysems mus be ddressed nd work long hese lnes s presenly underwy. CONCLUSION Conemporneous reserch effors re requred o vlde or flsfy, develop or dsprove he frconl dynmcs dscussed here ncludng our prelmnry fndngs. ACKNOWLEDGMENT I would lke o hnks Professor Mnuel Dure Orguer for nvng me o prcpe on he Symposum on Frconl Sgnls nd Sysems whch wll be held Lsbon-Porugl. REFERENCES Aghe, M.R.S., Z.A. Zukrnn, A. Mm nd H. Znuddn, 29. A hybrd rchecure pproch for qunum lgorhms. J. Compu. Sc., 5: DOI:.3844/jcssp Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 4 Al-Doud, E., 28. A new qunum key dsrbuon proocol. Am. J. Appled Sc., 5: DOI:.3844/jssp Al, S., S. Shrudn nd M.R.B. Whddn, 29. Qunum key dsrbuon usng decoy se proocol. Am. J. Eng. Appled Sc., 2: DOI:.3844/jessp Bev, V.V., M.. Klmkov nd B.A. Knehl, 29. Dfferenl reducon of generlzed hypergeomerc funcons n pplcon o Feynmn dgrms: One-vrble cse. hp:// Cresson, J., 27. Frconl embeddng of dfferenl operors nd Lgrngn sysems. J. Mh. Phys. 48: 34. DOI:.63/ El-Nbuls, R.A. nd D.F.M. Torres, 27. Necessry opmly condon for frconl con-lke vronl pproch wh-louvlle dervves of order (α,β), Mh. Meh. Appl. Sc. 3: El-Nbuls, R.A. nd D.F.M. Torres, 28. Frconl conlke vronl problems, J. Mh. Phys. 49, DOI:.63/ El-Nbuls, R.A., 25. A frconl pproch o nonconservve lgrngn dynmcl sysems, Fzk A, 4: El-Nbuls, R.A., 25b. A frconl con-lke vronl pproch of some clsscl, qunum nd geomercl dynmcs, In. J. Appl. Mh., 7: El-Nbuls, R.A., 27. Geomery of mnfolds on Le endomorphsm spce nd her duls under frconl con-lke vronl pproch, Fzk A, 6: El-Nbuls, R.A., 28. Frconl feld heores from mul-dmensonl frconl vronl problems, In. J. Mod. Geom. Meh. Mod. Phys. 5: 863. DOI:.42/S El-Nbuls, R.A., 28b. Blck hole growh nd ccreon energy from frconl con-lke vronl pproch. Fzk B7, 3: El-Nbuls, R.A., 28c. Subdffuson over frconl qunum phs whou frconl dervve, Fzk A, 7: 7. El-Nbuls, R.A., 29. Frconl qunum Euler- Cuchy equon n he Schrödnger pcure, complexfed hrmonc oscllors nd emergence of complexfed lgrngn nd hmlonn dynmcs. Mod. Phys. Le. B, 23: DOI:.42/S

9 Am. J. Engg. & Appled Sc., 4 (): 33-4, 2 El-Nbuls, R.A., 29b. Frconl con-lke vronl problems n holonomc, non-holonomc nd sem-holonomc consrned nd dsspve dynmcl sysems, Chos, Solons nd Frcls 42 (29) 52 El-Nbuls, R.A., 29c. On he frconl mnml lengh Hesenberg-Weyl uncerny relon from frconl Rcc generlzed momenum operor, Chos, Solons Frcls, 42: DOI:.6/j.chos El-Nbuls, R.A., 29d. Frconl dynmcs, frconl wek bosons msses nd physcs beyond he sndrd model. Chos, Solons Frcls, 4: DOI:.6/j.chos El-Nbuls, R.A., 29e. Frconl Drc operors nd deformed feld heory on Clfford lgebr. Chos, Solons Frcls, 42: DOI:.6/j.chos El-Nbuls, R.A., 29f. Complexfed qunum feld heory nd mss whou mss from muldmensonl frconl conlke vronl pproch wh dynmcl frconl exponen. Chos, Solons Frcls, 42: DOI:.6/j.chos El-Nbuls, R.A., 29g. Frconl lluson heory of spce: frconl grvonl feld wh frconl exr-dmensons, Chos, Solons Frcls, 42: DOI:.6/j.chos Gomes, D. nd E.C.D. Olver, 24. The secondorder Klen-Gordon feld equon. In. J. Mh. Mh. Sc., 69: DOI:.55/S Grnk, A., 24. On loss of nformon n rnson from qunum o qus-clsscl regme. Am. J. Appled Sc., : DOI:.3844/jssp Hlfer, R., 2. Applcons of Frconl Clculus n Physcs. s Edn., Word Scenfc Publshng Co., New Jersey, London, ISBN-: , pp: 463. Kluwer, D., 24. Frconl dervves nd her pplcons. 38: Kobolev, L.., 2. Wh dmensons do he me nd spce hve: neger or frconl. hp://rxv.org/bs/physcs/35 Mchel, N. nd M.V. Sosov, 28. Fs compuon of he Guss hypergeomerc funcon wh ll s enre prmeers complex wh pplcon o he Poschl-Teller-Gnoccho poenl wve funcons. Compu. Phy. Commun., 78: Mller, K.S. nd B. Ross, 993. An Inroducon o he Frconl Clculus nd Frconl Dfferenl Equons. s Edn., John Wley nd Sons Inc., New ork, ISBN-: , pp: 366. Orguer, M.D. nd J.A.T. Mchdo, 26. Frconl clculus pplcons n sgnls nd sysems. Sgnl Processng-Frconl Clculus Applcons Sgnls Sys., 86: DOI:.6/j.sgpro Orguer, M.D. nd J.A.T. Mchdo, 28. Specl Issue on Dsconnuous nd Frconl Dynmcl Sysems. ASME J. Compu. Nonlner Dynmcs, 3:. DOI:.5/ Podlubny, I., 999. Frconl Dfferenl Equons: An Inroducon o Frconl Dervves, Frconl Dfferenl Equons, o Mehods of her Soluon nd some of her Applcons. s Edn., Acdemc Press, New ork-london, ISBN- : , pp: 34. Q, F. nd B.-N. Guo, 29. Two new proofs of he complee monooncy of funcon nvolvng he ps funcon. Clsscl Anlyss ODEs, 47: 3-. DOI:.434/BKMS Sbnl, S. nd M. Nkgw, 995. A sudy of medependen frcl dmensons of vocl sounds, J. Phys. Soc. Jpn., 64: DOI:.43/JPSJ Smko, S., A. Klbs nd O. Mrchev, 993. Frconl Inegrls nd Dervves: Theory nd Applcons. s Edn., Gordon nd Brech Scence Publshers, New ork, ISBN-: , pp:

Jordan Journal of Physics

Jordan Journal of Physics 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.