Green Dyadic for the Proca Fields. Paul Dragulin and P. T. Leung ( 梁培德 )*

Transcription

1 Gn Dyadic fo th Poca Filds Paul Dagulin and P. T. Lung ( 梁培德 )* Dpatmnt of Physics, Potland Stat Univsity, P. O. Box 751, Potland, OR Abstact Th dyadic Gn functions fo th Poca filds in f spac a divd to includ th singula tms. Both th lctic typ and th magntic typ will b obtaind with th sults ducd back to thos fo th Maxwll filds in th limit of zo photon mass. Moov, th singula tms a idntical in both masslss and massiv lctodynamics. As an illustation, th sults a applid to obtain th xact dynamical filds fo an oscillating dipol which duc back to th wll-known xpssions fo static filds divd pviously in th litatu fo massiv lctodynamics. PACS numb(s): p, z, 0.50.D *Cosponding autho

2 Intoduction Although th two fundamntal focs lctomagntic and gavitational which govn most of th physical phnomna at a macoscopic scal a blivd to b tuly long-ang in natu, a non-vanishing valu fo ith th photon mass o th gaviton mass will hav many significant implications fom th dispsion of light in vacuum to ctain fundamntal issus in cosmology [1]. In paticula, th possibility of having a finit photon mass has bn studid intnsivly fo almost a cntuy, with both sious thotical and xpimntal invstigations dvotd to it []. Thotically, th simplst Lontz invaiant gnalization of Maxwll s thoy to includ a finit photon mass was fist wokd out by Poca []; and th quantization of th Poca fild is also possibl dspit th loss of gaug invaianc in massiv lctodynamics []. At th classical lvl, th Poca quations lad to modifications to only th two Maxwll quations with souc, and th consquncs of ths hav bn studid by many sachs in th litatu to includ poblms such as ffcts on spcial lativity, multipol adiations fom localizd soucs, tc. Expimntally, th two main appoachs to st an upp limit fo th photon mass hav bn basd on xamination of th accuacy of Coulomb s law (static appoach) and th dispsion of light in vacuum (dynamic appoach). Whil th static appoach can usually st mo stingnt limits on th upp bound of th photon mass, th dynamic appoach can nabl on to go byond tstial xpimnts so that possibl dispsion can b studid with light popagating ov a lag scal of distancs. To dat, on of th bst upp limits stablishd is blivd to b in th od of g fo th mass of th photon. [].

3 Ou intst h in th psnt wok is to div th most gnal xpssions fo th Gn dyadic functions of th classical Poca fild quations to includ th singula tms in th dyadic. As is wll-known in convntional Maxwllian lctodynamics, th knowldg of such dyadic functions will allow on to calculat th gnal timdpndnt filds fo any abitay localizd cunt soucs [4]. Nw sults fo th dyadic Gn functions a always significant whnv a nw vcto fild thoy is intoducd [5]. As fo th Poca fild, pvious wok in th litatu had divd th scala Gn function fo th Poca wav quation. Th sults obtaind had thn bn applid to th analysis of th adiations fom binay pulsa, lading to nw limits on th lctic chag of vaious astophysical bodis as wll as on th stngth of possibl nw focs of wak intaction [6]. Howv, nith th dyadic no th singula tm fo th Poca fild has bn stablishd in th pvious studis. As fo th latt, it has bn cognizd fom tim to tim th impotanc of tating th singula bhavios of vaious filds and soucs to th xtnt that a complt txt has bn wittn fo th tatmnt of this poblm [7]. Fo xampl, fo th Maxwll fild, both lctostatic [8] and lctodynamic dipol filds hav bn obtaind with th coct singula tms includd [9, 10]. Mo gnally, th Gn dyadic (lctic typ) with th coct singulaity has also bn pviously divd in th litatu [9, 11]. H w shall div in th following both th lctic and magntic dyadic functions fo th Poca filds to includ all th appopiat singula tms. W shall thn apply ths gnal sults to div th dynamical lctic and magntic filds fom dipol soucs accoding to th Poca thoy, and cov som wll-known sults stablishd in th litatu und ctain limiting conditions.

4 Gnal Dyadic Fomulation Lt us bgin by capitulating th following wll-known Maxwll-Poca quations in vacuum (in Gaussian units): iε = 4πρ μ φ, (1) iβ = 0, () 1 Β Ε + = 0, () c t 1 Ε 4π c t c μ Β = J A, (4) mc wh μ = is th invs Compton wavlngth of th photon with m th photon mass; and th filds a givn via th usual xpssions in tms of th potntials as vald fom () and () as follows: B= A, (5) 1 A E = φ. (6) c t As is wll-known [], Eqs. (1) and (4) imply that th Lontz gaug condition must b implmntd so that th consvation of chag mains valid. In addition, it has bn thooughly studid in th litatu [6] fo th lctomagntic wavs and adiations via th divation of th following wav quations fo th potntials: 1 μ φ 4πρ =, (7) c t 1 4π J μ A =. (8) c t c 4

5 Fo hamonic soucs and filds (i.. i t ω ), ths duc to th Hlmholtz quations with th wll-known tadd Gn s function givn by: ' G0( ', ) = ', (9) wh w hav intoducd an ffctiv wav numb ω β = μ k μ, (10) c and G 0 satisfis ( β ) + G 0 (, ' ) = δ( '. ) (11) Basd on th intoduction of this ffctiv wav numb togth with th appopiat modifications of th vaious quantitis lik th ngy-momntum and Poynting vcto fo th Poca fild, many sults calculatd in Maxwll (masslss) lctodynamics fo th adiation poblm can b tanslatd ov to th cas of massiv lctodynamics [6]. Howv, in spit of having many of ths pvious sults stablishd in th litatu, it sms that th poblm of th Gn dyadic and its singula bhavio fo th Poca fild hav not bn studid in most of th pvious woks. Whil th sam poblm has bn studid xtnsivly in th cas of convntional (masslss) lctodynamics [4, 9], it has bn known fom ths studis that th sult fo th singulaity of th filds is significant which not only povids consistncy with th fild quations, but also lads to intsting applications [8, 10]. Futhmo, th knowldg of th Gn dyadics nabls on to calculat th lctomagntic filds dictly fom a givn souc without having to sot to th potntials. Hnc w fl justifid to psnt an xplicit divation fo thm and 5

6 illustat thi applications to obtain nw sults fo dipol filds which cov som wll-known sults fo th Poca filds availabl in th litatu. To div th Gn dyadics, w hav fist to obtain th vcto wav quations fo th lctic and magntic filds just lik th cas with th odinay Maxwllian lctodynamics [4]. Ths can b obtaind in a staightfowad way fom (1) to (4) in th following fom: and 4π ik E k E+ ikμ A= J, (1) c 4π B β B= J. (1) c Hnc w s that whil th quation fo th magntic fild (with th us of th ffctiv wav numb ) has th sam fom as that in Maxwll s lctodynamics, th on fo th lctic fild is diffnt and has th vcto potntial appad xplicitly. Howv, w shall s blow that this will not affct at all th usual way fo th dtmination of th dyadics in tms of th scala Gn s function in (9). To show xplicitly this is th cas, lt us intoduc th lctic ( G ) and magntic ( m ) fild by making th following ansatz: and 0 G dyadic Gn function fo th Poca ( ' ) G k G + μ IG = Iδ, (14) ( ' ) Gm β Gm = Iδ, (15) wh I is th unit (idntity) dyadic and G 0 is as givn in Eq. (9). Not that whil (15) is compltly analogous to that fo th Maxwll fild, th sult in (14) is uniqu fo th 6

7 Poca fild which mixs th lctic dyadic Gn function with th scala Gn function (intoducd fo solving th vcto potntial) though th mass tm ( μ ). Although th fom of Eq. (14) is stongly suggstd fom that of Eq. (1) togth with th sults in Eqs. (8) (11) --- by noting that achg and G0 a simply latd to th cosponding lctic fild and vcto potntial fo a point souc --- it is indd possibl to show xplicitly th consistncy of this nw schm of intoducing th lctic Gn dyadic with th convntional appoach in Maxwll lctodynamics [4], in th dtmination of ths dyadics in tms of th scala Gn function as don in th following. To bgin, w fist not that sinc Eqs. (), (), (5) and (6) a idntical to thos in Maxwll lctodynamics, w hav thfo th wll-known lations btwn th dyadics and th scala Gn function G 0 as follows [4]: 1 G = I + G 0, (16) k G ( I ) =. (17) m G 0 Nxt w show that th sult in (16) is indd a solution to (14) with th hlp of th wav quation in (11). Thus w hav fom (16): G k G = IG k G 0 ( i ) = IG IG k IG G, (18) ( k ) = I sinc th scond tm in (16) is culss, and th fist and last tms cancl in th scond ow in (18). On th oth hand, w hav fom (11): G ( + ) ' = ( + β + μ ) ( ) G0 (, ) 0 k G0(, ) G0(, ' ). (19) = δ ' + μ ' 7

8 Substitution of (19) into (18) lads immdiatly to th sult in (14). Hnc th sult in (14), which mixs th scala and dyadic Gn functions, indd povids a consistnt schm fo th intoduction of G in th cas of massiv lctodynamics, and is dtmind th usual way via Eq.(16). Onc this is stablishd, th dyadics intoducd in (14) and (15) will b th coct dyadics fo th Poca filds though th uniqunss thom fo Gn functions. With th dyadics givn in (16) and (17), th filds can thn b obtaind dictly in tms of th souc cunt in th following foms: and E 4π ik = c G 'J', (0) () (, )( ) dx' B Calculation of Gn Dyadic 4π = c G 'J'. (1) () m(, )( ) dx' In this sction, w show how th dyadics in (16) and (17) can b calculatd with th singula tms xplicitly includd. Fo th cas of th Maxwll filds, th singulaity of th lctic dyadic G has bn obtaind pviously in th litatu [4, 9, 11]. H w shall follow a diffnt appoach by diving both G and G m via a dict calculation, with th application of th vaious diffntiation idntitis fo th (1/) potntial function involving th δ -function which a wll-stablishd in th litatu [1]. Fo simplicity, lt us st ' = 0 and consid a componnt of th dyadic tnso. Thus fom (9) and (16), w hav: 8

9 1 = + k 4π ( G ) δ pq pq p q = δ pq + p q + p q + q p + p q 4π 4πk i xx xx δ = δ + β + δ β δ δ + 4π 4πk 1 β p q 4π p q pq pq ( ) xx p q pq i 4 pq 5 1 = k 4π k ( ) ( ) δ( ) δ 1 pq k δ 5 pq β xpxq (), wh w hav usd th idntity (1/ ) = (4 π / ) δ δ( ) + ( x x δ ) / 5 p q pq p q pq stablishd in Rf. [1]. W also not that th singula tm will b th sam as that fo th cas of static fild ( k 0 ) and/o masslss photon ( μ 0), fo th tm i β δ () is actually quivalnt to δ ( ) sinc = 1 at = 0. Thus th gnal fom of th lctic dyadic fo th Poca fild including th singulaity tm can b finally obtaind in th following fom: ' ( k ) 1 G (, ' ) = δ( ' ) I + ' 1+ ' + ' I 5 k 4π k '. + ( ' β ' )( ' )( ' ) Not that Eq. () can b applid to calculat th dynamical lctic fild fom an abitay localizd souc in massiv lctodynamics. In th limit of masslss Maxwll lctodynamics, β k and th sult in () poducs th pvious sult obtaind in th litatu [9]. Nxt w div th cosponding magntic dyadic fo th Poca fild. Th possibl singulaity in this cas has not bn claifid pviously vn fo th masslss () 9

10 Maxwll cas [9], but it is ath staightfowad and w shall s that th is no xplicit singula tms associatd with it. Howv, singula tms can still mg duing th calculation of th magntic filds using this dyadic (s blow). Again, w st ' = 0 and apply (9) to a componnt of (17) as follows: ( Gm) = ε pst s( δtqg0 ) pq, (4) = ε psq s 4π so that th dyadic can finally b xpssd in th following fom: ( ) i m(, ) ( 1 i ) β ' ' G ' = β ' I. (5) ' Application to th Calculation of Dipol Filds As an illustation of th usfulnss of th sults in () - (5), w shall apply thm to (0) and (1) to div th fully dynamical dipol filds fo massiv lctodynamics accoding to th Poca quations. (i) Elctic dipol filds Fo a hamonic oscillating lctic dipolp at th oigin, w hav th cunt dnsity givn by J' ( ) = iωpδ( ' ) [8]. Thus fom (0), w obtain th following xpssion fo th lctic fild: E G p, (6) () = 4 π k (,0) which, on using th sult in () yilds th following sult fo th lctic dipol Poca fild: 4π δ β β β E () = p () + 1+ i + k p + i p 5 ( ) ( )( ) i. (7) 10

11 Lt us xamin th vaious limiting cass divd fom (7). Fo th cas of massiv lctostatics, w hav k β μ 0 and, (7) yilds th following lctostatic Yukawa fild: 4 π δ μ μ μ μ E () = p () + 5 ( 1 ) p + ( + + )( p ) i, (8) which was obtaind pviously in th litatu [1] (xcpt fo th singula tm and a small o). It is cla that in th cas of zo photon mass, μ = 0 and (8) ducs back to th wll-known lctic dipol fild with th singula tm as can b found in standad lctodynamics txt [8]. On th oth hand, if w st μ = 0 in (7), w will cov th following wll-known dynamical lctic dipol fild [8] togth with th sam singula tm as fo th static fild [9, 10]: ik E 4 () = π δ () 5 ( 1 ik k ) ( ik k )( ) p p + p i. (9) (ii) Magntic dipol fild Fo a magntic dipol m at th oigin, w hav th cunt dnsity givn by [ ] J' ( ) = c ' M = c ' mδ ( ' ). To calculat th magntic fild du to this cunt, it is saf if w go back to calculat on of its componnt using th sult in (4) so that w will not miss any possibl singula tms fom diffntiating twic th potntial function (1/) [1]. Thus w hav By convting s to ' 4π Bp() = ε psq s c ε qab ' a[ mb δ ( )] d x' c i '. (0) 4π ' ' and intgating by pats, (0) can b valuatd to obtain: s 11

12 ' Bp() = ( δpaδsb δpbδsa) mb ' a s' δ( ' ) d x'. (1) ' Th valuation of th doubl divativ in th intgand has alady bn calculatd bfo in diving Eq. (), using th sult obtaind th and intgating ov th dlta function in (1), w finally obtain β 1 xx B () = ( δ δ δ δ ) m β x x + δ i a s p pa sb pb sa b a s as 4 4π xx a s δ as δ ( ) δ as + 5, 8π 1 β β = δ ( ) mp m p m i ( ) xp () which lads to th following xpssion fo th dynamical Poca fild du to a hamonically oscillating magntic dipol at th oigin: 8π B () = m δ() β m + β m 5 ( ) ( )( ) i. () Not that () is a gnalization of a sult fist obtaind by Schoding [14] fo th static cas to th dynamical cas with th xplicit inclusion of th singula tm. Again, on can chck all th quid limits as follows. Fo xampl, by stting k β μ 0 and, on covs th magntostatic sult fist obtaind by Schoding [14]: 8π μ B () = m δ() + 1+ μ+ μ m + + μ+ μ m 5 ( ) ( )( ) i, (4) with th singula tm includd. In th limit of zo photon mass, (4) ducs back to th wll-known xpssion obtaind by Jackson [8]. On th oth hand, by stting 1

13 th mass to zo in (), β k and on covs th dynamic magntic dipol fild in Maxwll s thoy with th singula tm xplicitly includd [10]. Discussion and Conclusion In this wok, w hav divd th dyadic Gn functions fo massiv lctodynamics by using Poca s quations to includ th singula tm in th lctic dyadic, whil non is ndd fo th magntic dyadic [Eqs. () (5)]. As in convntional lctodynamics, th knowldg of ths dyadics is vy usful fo thy nabl on to calculat th lctomagntic filds dictly fom any givn localizd souc [4], spcially whn ths soucs a xpssd in multipol foms [15]. As an illustation, w hav wokd out in dtails both th lctic and magntic dipol filds, which duc to th appopiat sults wll-stablishd in th litatu in th cas of statics and/o Maxwll (masslss) lctodynamics. W also not that th singulaity of th dyadic is unaffctd by ith th photon mass o th dynamic bhavio of th filds. Th physics bhind this is that whil th singula bhavio is dominatd by th na filds; th Yukawa facto ( μ ) bcoms unity at th location of th souc in th cas of th Poca filds. Howv, w notic that, unlik th Maxwll cas wh th dipol lctic and magntic filds hav xactly th sam fom away fom th souc, this symmty is dstoyd in th psnc of th photon mass as can b sn fom Eqs. (7) and (). Such asymmty aiss sinc in th Poca thoy, th mass tm affcts only th souc pai of Maxwll s quation (1) and (4), whil th oth pai mains homognous and is not affctd. This occus du to th absnc of magntic monopols and th associatd cunts vn in th Poca thoy [16]. It is th sam oigin fo th diffnc in th two singula tms in Eqs. (7) and () [8, 17]. Ths singula tms can b of high 1

14 significanc in quantum physics sinc a chagd paticl in th fom of a wav can hav nonzo psnc at th oigin of th souc fo th filds. A famous xampl of ths is th application of th dlta function tm in th magntic dipol fild to th calculation of th hypfin tansition fo gound stat H-atom, lading to th famous 1 cm mission in astonomy [8]. It will b of intst to find futh applications of th gnal sults obtaind fo th Gn dyadics in Eq.s ()- (5) in ou psnt wok such as in th calculation of high od multipol filds. Acknowldgmnts On of us (PTL) would lik to thank Pofsso G.- J. Ni fo usful discussions, and th lat Madam Luk Han-Yin fo h liflong suppot in all his woks. 14

15 Rfncs [1] S,.g., th cnt poposal of th possibility to hav massiv gaviton as a candidat fo cold-dak-matt in th Univs by S. L. Dubovsky, P. G. Tinyakov, and I. I. Takchv, Phys. Rv. Ltt. 94, (005). [] S th cnt vy comphnsiv viw by L.-C. Tu, J. Luo, and G. T. Gillis, Rp. Pog. Phys. 68, 77 (005). [] A. Poca, Compt. Rnd. 190, 177 (190); many oth oiginal fncs by Poca can b found fom Rf. [] abov. [4] C.-T. Tai, Dyadic Gn Function in Elctomagntic Thoy (IEEE pss, Nw Yok, 1994). [5] As an xampl in th litatu, such dyadic was divd fo th Bltami- Maxwll filds in a gnal matial continuum by W. S. Wiglhof and A. Lakhtakia in Phys. Rv. E 49, 57 (1994). [6] D. E. Kaus, H. T. Kloo, E. Fischbach, Phys. Rv. D 49, 689 (1994). [7] J. V. Bladl, Singula Elctomagntic Filds and Soucs (Clandon Pss, Oxfod, 1991). [8] J. D. Jackson, Classical Elctodynamics (Wily, Nw Yok, 1998). [9] W. Wiglhof, Am. J. Phys. 57, 455 (1989). [10] P. T. Lung, Eu. J. Phys. 9, 17 (008). [11] S also, V. M. Finkl bg, Sovit Phys. Tch. Phys. 9, 96 (1964); Y. A. Ryzhov, Sovit Phys. JETP 1, 4 (1965); and C. T. Tai, Poc.IEEE 61, 480 (197). [1] C. P. Fahm, Am. J. Phys. 51, 86 (198). 15

16 [1] G. Omanovic, Found. Phys. Ltts. 10, 47 (1997). [14] E. Schoding, Poc. R. Iish Acad. A 49, 4 (194); ibid. 49, 15 (194). [15] E. G. P. Row, J. Phys. A 1, 45 (1979). [16] Th is som discussion in th litatu on th incompatibility of a finit photon mass with th xistnc of magntic monopol. S,.g., A. Y. Ignativ and G. C. Joshi, Mod. Phys. Ltts. A 11, 75 (1996). [17] P. T. Lung and G. J. Ni, Eu. J. Phys. 7, N1 (006). 16