NON-SINGULAR MAGNETIC MONOPOLES

Transcription

1 THE PUBISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Sris A OF THE ROMANIAN ACADEMY Volu 8 Nubr 3/2007 pp NON-SINGUAR MAGNETIC MONOPOES S KHADEMI * M SHAHSAVARI ** and A M SAEED ** * Dpartnt of Physis Zanjan Univrsity Zanjan Iran ** Institut of Gophysis Thran Univrsity Air-Abad Strt Thran Iran Corrsponding author: skhadi@ailznuair ; siaakhadi@yahooo Magnti Monopol is a onsqun of th xistn of th duality sytry in ltrodynais Although no onlusiv xprintal vidns hav so far bn found but th subjt is still of uh intrst to physiists Th thory of agnti onopols was first proposd by Dira in 93 and soon aftr it was studid by physiist of any disiplins spially partil physis quantu fild thory and th non-linar Soliton quations On iportant onsqun of th agnti onopol thory is th quantization of th ltri harg whih was first drivd by Dira In th dfinition of th lassial agnti onopol th onpt of Dira string is usd Dira string is th lous of th points whr th vtor potntial is not wll-bhavd On th othr hand by introduing th ida of agnti onopols Maxwll's quations ba sytrial with rspt to th agnti and ltri filds but thy still rain unsytrial as far as th salar and th vtor potntials ar onrnd In this work th ltri and agnti filds ar rdfind in trs of so nw salar and vtor potntials rsptivly as a rsult of whih th Maxwll s quations bo sytrial with rspt to th potntials too On advantag of using this forulation is that on an disard th Dira string all togthr Finally dfinition of th nw potntials guarants th orntz invarian of quations Ky words: Magnti onopol ltrodynais quantu fild thory INTRODUCTION Th onpt of sytry plays a ntral rol in physis and othr filds of sin [] Morovr duationally it is ssntial for studnts to know th various aspts of th phnona Sytry apart fro its asthti aspts is a powrful tool in prditing and forasting physial laws and ffts sothing whih aks it all or iportant for studnts to larn Th appliation of sytry in prditing nw partils is a wll known thniqu g orntz invarian of th quantu hanial quations of partils ld to th prdition of anti-partils in 930 In 993 anti-ltrons (positrons) wr disovrd by C D Andrson [2] Magnti onopol is also an iportant fundantal partil whos xistn is postulatd basd on th duality sytry Although so far no xprints hav rvald suh partils but so physiists ar hoping that ths partils an xprintally b dttd in futur Th quantization of th ltri hargs is an iportant onsqun of th invstigation of th dual sytry in quantu ltrodynais by Dira [3] In his work th vtor potntial was singular and h onsidrd th onpt of Dira string in th forulation [4] In this work th duality sytry is r-xaind and it is shown that th singularitis originat fro th lak of th dual sytry in th dfinition of th ltroagnti filds in trs of potntials Sytrization of th Maxwll s quations with rgards to th potntials would rsult in th dfinition of nw salar and vtor potntials 2 DUA SYMMETRY OF MAXWE'S EQUATIONS WITH RESPECT TO EECTROMAGNETIC FIEDS Th wll-known Maxwll s quations onsist of two isotropi and two non-isotropi quations

2 S KHADEMI M SHAHSAVARI A M SAEED 2 E = 4πρ B E = 4π j B = 0 E + B = 0 If th agnti onopols xist in xatly th sa way as that of th ltri hargs thn th agnti fild du to suh onopols is givn by r B = q 3 (2) r In Eq (2) q is th agnti harg Th sign of N and S agnti hargs would diffr as sign ( q N S ) = sign( q ) (3) With th assuption of th xistn of th agnti onopols th Maxwll s quations would thn hav a sytrial for [5] E = 4πρ B E = 4π j B = 4πρ E + B = and thy would also b invariant undr th following transforations j j ρ ρ ρ ρ E B B E j j 4π j whih ar alld dual transforation [5] If on xprsss th ltroagnti filds in trs of th salar and vtor potntials by B = A E = A ϕ thn th Gauss law for th ltri hargs would no longr hold baus B = A) = 0 4πρ (7) ( To rsolv this Dira usd a singular vtor potntial H first onsidrd th intgral for of th Gauss law for a agnti fild ovr an opn surfa A Gaussian opn surfa is dfind as a sphr whih has a hol of radius R ut out of it (s Fig ) S R Opn 0 B ds = 4πq Now using th Eq (6) and th Stok s law Eq (8) is transford into a lin intgral around th ut out ara of radius R S Opn R 0 B ds = ( A) ds = A dl = 4πρ S R 0 Now th radius R is lt to tnd to zro till a singular point is obtaind on th surfa In th obtaind singular point th vtor potntial is not wll-dfind As an xapl in a onvnint gaug or with a suitabl hoi of a vtor potntial g C R 0 () (4) (5) (6) (8) (9)

3 3 Non-singular agnti onopols + Cosθ A = q φˆ (0) rsinθ th singularity would nd up on th Z-axis ( θ = 0 ) Howvr on an find any Gaussian sphrial surfas whih would hav th agnti harg in thir ntr and a singular point on thir surfa Th lous of suh singular points would for a lin or string whih is known as Dira string In anothr word in this thod a singular string is attahd to ah agnti onopol whih strths fro th onopol to infinity Unlik ltroagnti filds potntials ar not asurabl quantitis and in spit of having singularitis in th potntials th ltroagnti filds ar not singular thslvs Howvr th Hailtonian of an intrating hargd partil with xtrnal ltroagnti filds xpliitly dpnds on th ltroagnti potntials q 2 ϕ 2 H = ( P A) + q () thrfor in quantu ltrodynais th Shrödingr's quation for a hargd partil intrating with th ltroagnti filds of a agnti onopol would b a singular diffrntial quation On th othr hand using th dfinition of ltroagnti filds in trs of ordinary potntials Eq (6) rsults in non-sytrial Maxwll s quations In anothr word th Maxwll's quations do not hav th duality sytry with rspt to th ordinary potntials 3 DUA SYMMETRY OF MAXWE'S EQUATIONS WITH RESPECT TO THE EECTROMAGNETIC POTENTIAS In ordr to sytriz th Maxwll s quations two nw potntials a vtor potntial potntial ψ in addition to th ordinary potntials ar introdud suh that G and a salar G B = + ψ + A A E = ϕ + G Equations (2) ar invariant undr th dual transforation ϕ ψ ψ ϕ A G G A E B B E Siilarly Eqs (2) ar invariant undr th Gaug transforations G G + g ψ ψ g t A A + f ϕ ϕ too In ths transforations in addition to th hoi of th wll-bhavd funtion f th hoi of funtion g is also arbitrary Th gaug invarian of th forulation in trs of th two arbitrary funtions f and g rsults in th onsrvation of ltri and agnti hargs rsptivly Th orntz ovariant for of Eq (2) is obtaind as f t Fμν = μ Aν ν Aμ + ε μναβ ( G G ) (5) α β β α 2 (2) (3) (4) In Eq (5) th fild strngth tnsor F μν is dfind in trs of A μ ( ϕ A) and G ν ( ψ G) Also

4 S KHADEMI M SHAHSAVARI A M SAEED 4 ε μναβ is fourth ordr vi-civita tnsor Now onsidring th nw potntials in dfinition of ltroagnti filds Eq(2) on ay obtain th divrgn of th agnti fild whih is no longr zro and is givn by ( ) ( G 2 ) G B = + ψ + A = + ψ = ρ (6) With an spial gaug G = 0 th Poisson quation for th potntial ψ is obtaind as 2 ψ = ρ (7) Th solutions of th Poisson quation (7) for a agnti onopol with harg is thn givn by q Th agnti fild is givn by th gradint of th salar potntial ψ q ψ = (8) r B = ψ (9) Evidntly Eqs (8) and (9) lad to Eq (2) Fro th sytri Maxwll s quations on ay onlud that a oving ltri harg partil givs ris to an ltri fild and sytrially a oving agnti hargd partil would also giv ris to an ltri fild (s Figurs 3 and 4) Th ltri and agnti filds in Eq (2) would now split in to longitudinal and transvrsal filds as givn blow B E G = A = + ψ ϕ B E T T GT = + AT AT = + GT In Eq (20) th indis and T dnot th longitudinal and transvrs oponnts rsptivly Th longitudinal ltri and th transvrsal agnti filds ar produd by a oving ltri harg whil th longitudinal agnti and th transvrsal ltri filds ar produd by a oving agnti hargd partil Th duality sytry (5) an larly b sn in figurs 3 and 4 (20) Fig 3 Transvrsal ltri fild B E T and longitudinal agnti fild du to a oving agnti harg q Fig 4 ongitudinal ltri fild E and transvrsal agnti fild B T du to a oving of th ltrially hargd partil q In studis onrning th agntially as wll as th ltrially hargd partils bsid th gaug

5 5 Non-singular agnti onopols rlatd to th ordinary potntials A and ϕ on has an additional gaug rlatd to th nw agnti potntials G and ψ Thrfor in addition to th hoi of a oon gaug th gaug rlatd to th nw potntials ust also b spifid For xapl th gaug A = 0 G = 0 whih on ay all it ϕ th Coulob Coulob gaug and G = 0 A + = 0 th orntz-coulob gaug As it is slf-xplanatory th first half of th gaug na is rlatd to th ltri potntials whil th sond half to th agnti potntials 4 ORENTZ FORCE On ay divid th ltri(agnti) filds in to two parts: th fild whih is du to th ltri hargs E ( B ) and on whih is du to agnti hargs E ( B ) A E = ϕ + G = E + E G B = + ψ + A = B + B Th subsripts and dnot th filds du to th ltri and agnti hargs rsptivly In Eq G (2) th agnti and ltrial filds du to a agnti harg is givn by B = + ψ and E = G rsptivly Th agnti and ltri filds rsulting fro an ltri harg is also obtaind fro Eq (6) Th for ating on an ltri harg q du to th ltri and agnti filds whih ar in turn du to an ltri harg is givn by (2) F = q E + V B ] (22) [ Th ltroagnti for ausd by a agnti harg and ating on anothr agnti harg obtaind by dual transforation of Eq (22) F = q B V E ] (23) [ In xaination of th intration btwn th agnti hargs and th ltroagnti filds ratd by ltri hargs and vi vrsa it is iportant to not that thr is no diffrn btwn th ltroagnti fild ratd by ithr typs of hargs and hn thy ar indistinguishabl As a rsult th orntz for ating on a agnti or ltri harg an rsptivly b writtn as [( G ) ( A F A V G = q + ψ + ϕ + )] (24) [( A ) ( G F G V A = q ϕ ψ + )] (25) Equations (24) and (25) ar dual transforation of ah othr q is 5 CONCUSIONS Equations (4) (24) and (25) for a oplt st of lassial quations whih dsrib th dynais of both th ltri and agnti hargs as wll as th ltroagnti filds Thrfor thy ould b usd in takling all fors of th lassial ltrodynais probls in th prsn of a agnti harg Also sin

6 S KHADEMI M SHAHSAVARI A M SAEED 6 thr is no singularity involvd in this forulation thrfor thr is no nd for th Dira string This is a onsqun of th introdution of th duality sytry into th potntials Equations (2) whih giv th nw potntials whih ar invariant undr th duality transforation rain invariant undr orntz transforation too REFERENCES W GREINER W MUER B Quantu Mhanis Sytris Springr-Vrlag (995) 2 ANDERSONC D Phys Rv 43 (933) 3 DIRAC P A M Pro Roy So ondon Sr A (93) 4 OHANIAN H C Classial Eltrodynais Prnti Hall Collg Div (988) 5 JACKSON J D Classial Eltrodynais 3 rd Edition John Wily & Sons In (998) Rivd May