A New Formulation of Quantum Mechanics

Transcription

1 Journal of Modern Physis hp://dxdoiorg/436/jmp3 Published Online February (hp://wwwsirporg/journal/jmp) A New Formulaion of Quanum Mehanis A I Arbab Faisal A Yassein Deparmen of Physis Fauly of Siene Universiy of Kharoum Kharoum Sudan Deparmen of Physis Fauly of Siene Alneelain Universiy Kharoum Sudan {arbabibrahim fayassein}@gmailom aiarbab@uofkedu Reeived July 9 ; revised Sepember ; aeped Oober 6 ABSTRACT A new formulaion of quanum mehanis based on differenial ommuaor brakes is developed We have found a wave equaion represening he fermioni parile In his formalism he oninuiy equaion mixes he Klein-Gordon Shrodinger probabiliy densiy while keeping he Klein-Gordon Shrodinger urren unalered We have found ime spae ransformaions under whih Dira s equaion is invarian The invariane of Maxwell s equaions under hese ransformaions shows ha he eleri magnei fields of a moving harged parile are perpendiular o he veloiy of he propagaing parile This formulaion agrees wih he quaernioni formulaion reenly developed by Arbab Keywords: Mahemaial Formulaion; Quanum Mehanis; Differenial Commuaor Brakes Inroduion Shrodinger s equaion was used o explain desribe all phenomena in aomi physis However afer he developmen of he heory of speial relaiviy by Einsein here was a need o unify quanum mehanis speial relaiviy ino a single Relaivisi Quanum Theory Despie he suess of Shrodinger s equaion in desribing quie auraely he Hydrogen sperum giving orre prediions for a large amoun of speral daa his equaion is no invarian under Lorenz ransformaions In oher words Shrodinger s equaion is no relaivisi is only an approximaion valid a he non-relaivisi limi when he veloiies of he pariles involved are muh smaller han he speed of ligh Quanum mehanis has been formulaed by assigning an operaor for any dynamial observable In Heisenberg formalism he operaor is governed by a ommuaor brake The fundamenal ommuaor brake relaes o he posiion momenum is given by x px i The ommuaor brake generalizes he Poisson brake of lassial mehanis If an operaor ommues wih Hamilonian of he sysem hen he dynamial variable orresponding o ha operaor is said o be onserved An equaion ompaible wih Lorenz ransformaion guaranees is appliabiliy o any inerial frame Suh an equaion is symmeri in spae-ime Thus a symmeri spae-ime formulaion of any heory will generally guaranee he universaliy of he heory However Shrodinger equaions doesn exhibi his feaure beause i is no symmeri in spae ime To remedy his problem Klein Gordon looked for an equaion whih is seond order in spae ime onsequenly obained he Klein-Gordon equaion (KG) The probabiliy densiy in his heory is found o be non-posiive definie Consequenly Dira hough for a linear equaion in spae ime ha has no suh a problem He obained he familiar Dira equaion wih a posiive definie probabiliy However he probabiliy in KG formalism is laer on (from a heorei field poin of view) inerpreed as a harge densiy raher han a probabiliy densiy whih ould be posiive or negaive [-3] Wih hese moivaion we adop a differenial ommuaor brake involving firs order spae ime derivaive operaors o formulae he Maxwell equaions quanum mehanis This is in addiion o our reen quarernioni formulaion of physial laws where we have shown ha many physial equaions are found o emerge from a unified form of physial variables [4] Moreover using quaernions we have reenly shown ha quanum mehanis an be formulaed in a se of hree equaions [5] In suh a formulaion he Dira Klein-Gordon equaions emerge from a se of hree equaions obained from he appliaion of an eigen-value problem of he linear momenum We aim in his paper o derive he equaion of moion of he quanum sysem by applying he vanishing differenial ommuaor brakes I is ineresing o noe ha hese ommuaor brakes are Lorenz invarian Moreover x x where is ime ˆ i ˆ j kˆ x x x x x y z are he Copyrigh SiRes

2 64 A I ARBAB F A YASSEIN moving ime spae oordinaes We know ha he seond order parial derivaives ommue for spae-spae variables We don assume here ha his propery is a priori for spae ime To guaranee his we eliminae he ime derivaive of a quaniy ha is aed by a spae ( ) derivaive followed by a ime derivaive vie versa In exping he differenial ommuaor brakes we don ommue ime spae derivaive bu raher eliminae he ime derivaive by he spae derivaive vie versa These linear differenial ommuaor brakes may enlighen us o quanize hese physial quaniies By employing he differenial ommuaor brakes of he veor A salar poenial we have derived Maxwell equaions wihou invoking any a priori physial law [6] We would like here o apply he differenial ommuaor brakes o explore quanum mehanis Differenial Commuaors Algebra Define he hree linear differenial ommuaor brakes as follows [6]; () where are he spae ime derivaives For a salar a veor A one finds ha: () A A A (3) A A A Moreover one an show ha: A A A A A A A B B A A B (4) The differenial ommuaor brake saisfies he disribuion rule: where Aˆ BC ˆ ˆ AB ˆ ˆ Cˆ ABC ˆ ˆ ˆ ˆ ˆ ˆ AC B s for eiher or (5) I is eviden ha he differenial ommuaor brakes ideniies follow he same ordinary veor ideniies We all he hree differenial ommuaor brakes in Equaion () he grad-ommuaor brake he do- ommuaor brake he ross-ommuaor brake respeively The prime idea here is o replae he ime derivaive of a quaniy by he spae derivaive of anoher quaniy vie versa so ha he ime derivaive of a quaniy is followed by a ime derivaive wih whih i ommues We assume here ha spae ime derivaives don ommue Wih his minimal assumpion we have shown here ha all physial laws are deermined by vanishing differenial ommuaor brake 3 The Coninuiy Equaion Using quaernioni algebra [7-9] we have reenly found ha generalized oninuiy equaions an be wrien as [5] J (6) J (7) J (8) where J are he speed of ligh urren densiy probabiliy densiy respeively Now onsider he do-ommuaor brake of J J J J (9) Using Equaions (6-8) he veor ideniies G G G G G G one obains J J J J () () For arbirary J Equaion () yields he wo wave equaions () J J (3) Equaions () (3) are also obained uilizing he Copyrigh SiRes

3 A I ARBAB F A YASSEIN 65 quaernioni formulaion following referene [5] Hene he wave equaions of J in our presen brakes formulaion are equivalen o J (4) Equaions () (3) show ha he harge urren densiies saisfy a wave raveling a he speed of ligh in vauum I is remarkable o know ha hese wo equaions are already obained in referene [5] 4 Quanum Mehanis Consider a parile desribed by he four veor i This is equivalen o spinor represena- ion of ordinary quanum mehanis We have reenly developed a quaernioni quanum mehanis dealing wih suh a four veor [7-9] The evoluion of his four veor is given by he hree equaions [7-9] m (5) m (6) (7) where m are he quasi-parile mass Plank onsan respeively Equaions (5-7) yield he wo wave equaions [5] m m (8) m m Using he ransformaion (9) m () so ha Equaions (5) (6) beome () Employing Equaion () Equaions (8) (9) are ransformed ino he wave equaions () Equaions (8) (9) an be obained from he Einsein s energy equaion by seing E E im where E p using he familiar quanum mehanial operaor replaemens viz pˆ i ˆ E i E p is an equaion for a massless parile This is also eviden from Equaion () Thus i is ineresing ha a massive parile an be ransformed ino a massless parile using Equaion () Sine energy is a real quaniy his equaion is physially aepable if i desribes a parile wih imaginary mass In his ase he energy equa- ions spli ino wo pars; one wih E E m he oher wih energy E E m Suh energies an desribe he sae of a parile aniparile A hypoheial parile wih an imaginary mass moving a a speed higher han he speed of ligh in vauum is known as ahyon [] Hene our above equaion an be used o rea he moion of ahyons This implies ha our equaions Equaions (4) (7) an be applied o ahyons Some sieniss propose ha neurino an be a ahyoni fermion [] We know ha he Cherenkov radiaion is emied from a parile moving in a medium wih a speed larger han he speed of ligh in vauum When he speed exeeds he speed of ligh in a vauum he exra energy aquired by he parile is ransformed in radiaion This an happen momenarily for a parile keeping is oal energy onserved Thus he exess energy (speed) is suh ha i ompensaes he dissipaions Now onsider he ross-ommuaor brake of (3) Using Equaions (5) (6) (7) he veor ideniies G G G yield he wave equaion (4) m m (5) Similarly he do-ommuaor brake of (6) Upon using Equaions () (5) (6) one obains he wave equaion of m m (7) I is ineresing o see ha Equaions (5) (7) are he same as Equaions (8) (9) obained from quaernioni manipulaion We hus wrie Equaions (5) (7) as Copyrigh SiRes

4 66 A I ARBAB F A YASSEIN (8) 5 Dira s Equaion Dira s equaion an be wrien in he form [-3] im (9) Consider he differenial ommuaor brake (3) Using Equaion (9) Equaion (3) yields m mi (3) where we have used he fa ha are he Pauli maries Equaion (3) an be obained from Equaion (9) by squaring i This equaion an be ompared wih he Klein- Gordon equaion of spin- pariles m Equaion (3) is anoher form of Dira s equaion exhibiing he wave naure of spin-/ pariles expliily Using he ransformaion m i Equaion (3) an be wrien as (3) (33) This is a wave equaion for a massless parile Thus a parile annihilaes (loses is mass) afer a ime inerval of hen reaed (aquired a mass) I is m ineresing o noie ha during suh a period of ime energy an be violaed as endorsed by he Heisenberg s unerainy relaion ( E )where E m This also applies o he pariles as defined by Equaion () Equaion (3) desribes he behavior of a parile of a definie mass m Afer a ime of he par- m ile beomes a wave wih energy E p governed by Equaion (33) The parile ineras wih he vauum in suh a way ha when he parile beomes a wave (anni- hilaes) gives is mass energy o he vauum resores i afer a ime of as defined before beoming a parile one again This is he essene of he osillaory moion as known as zierbewegung moion [-3] This resul suppors he fa ha here is a vauum fluuaion assoiaed wih he parile This means when a parile beomes a wave i gives is mass o he vauum resores i when beomes a orpusule Thus he orpusular wave naure (dualiy) of a parile is onomian wih he parile moion Sine is a four omponens spinor we an wrie i in erms of wo omponens dou- bles viz Subsiuing hese deomposed spinors in Equaion (3) one obains he wo equaions mi m mi m (34) (35) Equaions (34) (35) imply wo energy soluions one wih E E m he oher wih energy E E m This is also eviden from using he Ein- 4 sein energy-momenum equaion ( E p m ) The wo energy saes may define a parile an aniparile Sine he ime faor in he wavefunion is of he form expie he new wavefunion wih he new ime ( ) will beome exp ie where m i (36) is a omplex ime as eviden from Equaion (33) I an be seen as a roaion of he real ime by a phase ino a omplex plane Suh an effe arises from he very naure of he parile when propagaing in spae-ime The hird erm in Equaion (3) represens a dissipaion ha may resul from he moion of he parile in spae (eher) Hene any massive parile should exhibi his sor of propagaion when ravels in spae-ime This erm is vanishingly small ompared wih he mass erm in Equaion (3) bu very fundamenal Moreover Our Equaions (5) (7) are equivalen o Dira equaions Equaions (34) (35) if we replae m by im Consider now he ase when is spae independen so ha Equaion (3) beomes wih d mi d m d d his yields he wo equaions (37) Copyrigh SiRes

5 A I ARBAB F A YASSEIN 67 d mi d m d d d mi d m d d (38) (39) These wo equaions have an osillaory behavior ie where A exp i m A A A exp i (4) are onsans This means ha he parile wih he wavefunion has wo energy eigen saes one for a parile he oher one for an aniparile Hene Equaion (4) reveals ha he parile is desribed by a sing wave having posiive negaive energy This is he essene of Dira s heory The wo saes are separaed by an amoun of energy E m Using Equaion (9) Equaion (3) an be wrien in he form m mi (4) his an be wrien as where (4) m ' i (43) Equaion (4) is a wave equaion in he new oordinae defined by Equaion (43) Equaion (43) an be wrien as p' p m (44) This an be ompared wih he ovarian derivaive ha resuls from he ineraion of a parile wih a phoon field A viz p' p ea Equaion (3) an be wrien as E E m (45) Equaions (3) (43) an be ombined ino a single equaion as m D i (46) We all here he derivaive D he spinor derivaive Wih his derivaive he Dira equaion akes he simple forms where P i D (47) P I is ineresing o noie ha Equaions (47) looks like massless Dira equaion The seond erm in Equaion (44) represens a self ineraion of he parile due o is spin Sine he veor poenial is a gauge field he spin of he parile will aordingly beome a gauge quaniy In presen ase he eleron ineras wih is spin ha is relaed o α This effe represens a self ineraion of he parile The algebra of he P s ommuaor brake is i P P mi (48) Thus unlike parial derivaive spinor derivaives do no ommue The momena ommue for a massless parile Equaion (4) desribes a parile wih definie mass whih afer a haraerisi disane of m beomes a wave as desribed by Equaion (4) Hene he orpusular naure of he parile is exhibied afer a disane of X he wave naure afer a ime m of T The parile s veloiy mus be in suh a m way o reah he nex poin in he same ime required o be in he oher sae This requires is veloiy o be v α Thus he parile remains in a oninuous dual sae (parile + wave) This dualiy is manifesed during a ime of T a a disane of X This may usher ino a quanizaion of spae ime in unis of T X as fundamenal unis Wih he definiion v α Dira s equaion an be wrien as im d v (49) d whih implies ha d i m (5) d Hene Equaion (5) is a varian form of Dira s equaion Bu sine an be wrien as wo-omponens olumn viz he above equaion implies ha i d d d m i d m (5) d This shows ha he operaor Aˆ i is he res mass d energy operaor of he individual spinor omponens The Coninuiy Equaion Taking he omplex onjugae of Equaion (3) muliplying i by one from righ subra i from * Equaion (3) afer muliplying i by from lef we obain he oninuiy equaion Copyrigh SiRes

6 68 A I ARBAB F A YASSEIN where T J T S KG * S * i * KG m * * J mi (5) (53) I is undersood here ha is a spinor is he T harge densiy J is he urren densiy I is ineresing ha Equaion (3) obained from Dira s equaion using he differenial operaor brake in Equaion (3) yields a oninuiy equaion sharing boh he Dira Klein-Gordon feaures of he harge (probabiliy) densiy This inerplay exiss despie he fa ha Dira s equaion represens a fermioni parile while Klein-Gordon equaion represens a bosoni parile 6 The Spae Time Invariane of Dira s Equaion If we apply he ransformaions in Equaions (3) (43) o Equaion (9) Dira s equaion will be invarian Thus he spae ime ransformaion represened by Equaions (3) (43) ushers ino a new ransformaion of Dira s equaion ha were never known before Wih some sruiny we know from he heory of relaiviy ha he kinei energy ( E K ) is relaed o he oal energy (E) by EK E m In quanum mehanis E i so ha m Eˆ K i m i i i (54) This is he relaivisi kinei energy operaor Alernaively using Equaion (9) his an be wrien as Eˆ α pˆ (55) K This equaion implies ha Dira s equaion an be obained from he relaivisi energy equaion E E m K (56) This equaion suggess ha here are wo possible en- ergy equaions These are E EK m Hene a Dira s parile has in priniple wo energies E E E E K m K m Using Equaion (3) Dira s equaion Equaion (9) beomes α (57) Thus in he ime oordinae Dira s equaion represens a oninuiy-like equaion However in he real ime he oninuiy equaion in Dira s formalism is defined as J (58) where J α Using Equaion (43) Dira s equaion is ransformed ino a oninuiy-like equaion in he new spae oordinae viz ' α (59) Noie here ha J has he same form in boh oordinaes 7 Spae Time Invariane of Maxwell s Equaions We would like here o apply he spae ime ransformaions in Equaions (3) (43) o explore heir impliaions in Maxwell s equaions These ransformaions leave Dira s equaion invarian We know ha quanum elerodynamis inorporaes he ineraion of an eleron wih a phoon Quanum elerodynamis beomes invarian under gauge ransformaion if we replae he parial derivaive wih a ovarian derivaive inorporaeing he phoon field Analogously we assume here ha Maxwell s equaions are invarian under he new spae ime ransformaions in Equaions (3) (43) Applying Equaions (3) (43) o he Ampere s Faraday s equaions [] Β Ε (6) Ε Β J (6) yield v Ε Β v α (6) Ε v Β v α (63) The remaining wo Maxwell s equaions Ε Β (64) yield v Ε v Β v α (65) I is ineresing o noie ha Equaion (65) is ompaible wih Equaions (6) (63) Moreover Equaions (6) (63) define he relaions beween he eleri magnei fields produed by he moving harge If he eleri (magnei) field is known one an obain Copyrigh SiRes

7 A I ARBAB F A YASSEIN 69 he orresponding magnei (eleri) field Equaion (63) shows ha he harge moving wih onsan veloiy experienes no ne fore The eleri field lines of a moving harge rowded in he direion perpendiular o v are given by [3] q r 4 v (66) 3 3 r sin q v r v Β (67) 3 3 4π r sin Equaion (66) shows ha he eleri magnei fields of a moving harge in he forward direion ( ) are less han he eleri field of saionary harge However he eleri magnei fields in he perpendiular direion ( ) are bigger han he eleri field of saionary harge Equaions (66) (67) give he relaions beween he eleri magnei fields produed by a moving harged parile wih onsan veloiy v This veloiy is given by v α This oinides wih he quanum mehanis definiion of he parile veloiy [-3] Equaion (65) shows ha he eleri magnei fields produed by he harged parile are always perpendiular o he parile s direion of moion Equaion (67) gives a low veloiy he Bio-Savar law The power delivered by he fermioni harged parile by is eleri magnei fields is given by P Fv qvε qvvβ The appliaion of he ransformaions (3) (43) in he generalized oninuiy Equaions (6) (8) yields vj J v v α (68) Sine in Dira formalism > he urren ushers in a direion opposie o he veloiy direion Moreover for a onsan veloiy one has v Equaion (68) is very ineresing sine i defines he harge densiy (salar) in erms of he urren densiy (veor) Aordingly one an define he four veors in erms of veorial quaniies only 8 Conluding Remarks By inroduing hree vanishing differenial ommuaor brakes for spinor fields we have derived a varian form of Dira s Klein-Gordon wave equaions Dira s equaion yields a modified Klein-Gordon wave equaion This equaion yields direly wo energy saes for he parile in quesion Moreover Dira s equaion is found o be similar o he oninuiy equaion We have found ime spae ransformaions under whih Dira s Maxwell s equaions are invarian In erms of hese oordinaes Dira Klein-Gordon equaions desribe a mass-less parile The invariane of hese ransformaions under Maxwell s equaions shows ha he eleri magnei fields produed by a moving harge are perpendiular o veloiy of he parile Hene here is no power assoiaed wih hese fields The spae ime ransformaions show ha spae ime are quanized in erms of haraerisi unis of T X The fermioni harged parile exhibis is wave orpusular naure on periodi spae ime basis 9 Aknowledgemens This work is suppored by he universiy of Kharoum researh fund We graefully aknowledge his suppor The riial useful ommens by he anonymous referees are highly aknowledged REFERENCES [] J D Bjorken S D Drell Relaivisi Quanum Mehanis MGraw-Hill New York 964 [] L D Lau E M Lifshiz Quanum Mehanis 3rd Ediion Pergamon Press Oxford 977 [3] V B Bereseskii L P Piaevskii E M Lifshiz Quanum Elerodynamis nd Ediion Vol 4 Elsevier Amserdam 98 [4] A I Arbab Z Sai On he Generalized Maxwell Equaions Their Prediion of Elerosalar Wave Progress in Physis Vol No 8 9 pp 8-3 [5] A I Arbab H M Widaallah The Generalized Coninuiy Equaions Chinese Physis Leers Vol 7 No 8 Arile ID 8473 doi:88/56-37x/7/8/8473 [6] A I Arbab F A Yassein A New Formulaion of Eleromagneism Journal of Eleromagnei Analysis Appliaions Vol No 8 p 457 doi:436/jemaa86 [7] A I Arbab A Quaernioni Quanum Mehanis Applied Physis Researh Vol 3 No p 6 [8] P G Tai An Elemenary Treaise on Quaernions nd Ediion Cambridge Universiy Press Cambridge 873 [9] H F Harmuh T W Barre B Meffer Modified Maxwell Equaions in Quanum Elerodynamis World Sienifi River Edge doi:4/ [] G Feinberg Possibiliy of Faser-Than-Ligh Pariles Physial Reviews Vol 59 No pp 89-5 doi:3/physrev5989 [] J Ciborowski Hypohesis of Tahyoni Neurinos Aa Physisa Polonia B Vol 9 No pp 3- [] J D Jakson Classial Elerodynamis nd Ediion Wiley New York 975 [3] F Zbigniew Leure Noes in Eleromagnei Theory Universiy of Queensl Brisbane 5 Copyrigh SiRes