between electric charge e and magnetic charge (or (( pole strength >>) g, ... be quantized (or, tipically, the existence of one discrete, magnetic monopole), then.
LETTERE AL NUOVO CIMENTO
VOL. 9, N. 12
23 M~rzo 1974
Do Magnetic Monopoles Exist? Considerations for Theory and Experiments. E. R]~CAM: and R. MIGNANI I s t i t u t o di E i s i e a T e o r i e a dell' U n i v e r s i t ~ - C a t a n i a Centro S i c i l i a n o di .Fisica N u c l e a t e e di S t r u t t u r a della M a t e r i a - C a t a n i a I s t i t u t o N a z i o n a l e di ~ i s i c a N u c l e a t e - S e z i o n e di C a t a n i a
(ricevuto il 4 D i c e m b r c 1973)
1. - The t h e o r e t i c a l m o t i v a t i o n s for h y p o t h e s i z i n g t h e existence of m a g n e t i c monoessentially t h e following ones:
poles (1.2) were
i) T h e lack of s y m m e t r y in Maxwell equations, s u p p o s e d to be due to the absence of m a g n e t i c four-currents. So t h a t t h e wish of g e t t i n g Maxwell equations fully s y m m e t r i c a l led to p o s t u l a t e m a g n c t i c monopoles. ii) The fact t h a t p r e v i o u s h y p o t h e s i s b r o u g h t to t h e r e l a t i o n (~) e g = n h b e t w e e n electric charge e and m a g n e t i c charge (or (( pole s t r e n g t h >>) g, w h e r e n u m b e r is allowed to assume only i n t e g r a l v a l u e s according to SCHWI~GER ('), and also halfi n t e g r a l v a l u e s according to DIRAC (1). T h e r e f o r e , if one assumes m a g n e t i c charge to be q u a n t i z e d (or, tipically, t h e existence of o n e discrete, m a g n e t i c m o n o p o l e ) , t h e n electric charge follows to be q u a n t i z e d too. This second t h e o r e t i c a l m o t i v a t i o n , h o w e v e r , is m u c h w e a k e r t h a n t h e p r e v i o u s o n e - - s i n c e a s s u m p t i o n of magnetic-charge q u a n t i z a t i o a is n o t q u a l i t a t i v e l y different from t h e ~ e r y a s s u m p t i o n of electric charge q u a n t i z a t i o n - - a n d we shall d i s r e g a r d it. 2. - Now, s t a n d a r d special r e l a t i v i t y has been r e c e n t l y recognized (3) to contain as an additional, unjustified p o s t u l a t e t h e a s s u m p t i o n t h a t speeds v are a p r i o r i confined to subluminal (less t h a n e) values. A f t e r eliminating t h a t spurious postulate, special
(1) P . A . M . DIRAC: Proc. Roy. Soc., A133, 60 (1931); Phys. Rev., 74, 817 (1948); N. CABIBBOand E. FERICARI:NUOVO Cimento, 23, 1147 (1962); J. SCHWI~ER: Phys. Rev., 144, 1087 (1966). (2) For an extensive bibliography see, e.g., E. AMALDI: in Old and New Problems in Elementary Particles, edited by G. PuPPI (New York, 1968); E. A~IALDIand N. CABIBBO: in Aspects o! Quantum Theory, edited by A. SALAMand E. P. WINNER (Cambridge, 1972). (a) L. PARKER: Phys. Rev., 188, 9287 (1969). See also O. M. P. BILA~I~_~{,V. K. DESHPANDE o,nd E. C. G. SUDARSHAN:Am. Journ. Phys., 30, 718 (1962); A. F. ANTIPPA:Nuovo Cimcnlo, 10 A, 389 (1972); It. GOLDO~I: Nuovo Cimenio, 14A, 501 (1973). 479
480
~. ~ECAMI a n d R. MIGNANI
r e l a t i v i t y h a s b e e n r e c e n t l y e x t e n d e d (4.~) also t o S u p e r l u m i n a l i n e r t i a l f r a m e s (u2> c 2) a n d t o f a s t e r - t h a n - l i g h t o b j e c t s (or t a c h y o n s , v2> c2). I n o t h e r w o r d s , it h a p p e n s t h a t r e l a t i v i t y (at l e a s t in its e x t e n d e d version) i) p r e d i c t s (4) e x i s t e n c e of S u p e r l u m i n a l electric c h a r g e s (on t h e only basis of exi s t e n c e of s u b l u m i n a l electric charges), r a t h e r t h ~ n of (subluminal) m a g n e t i c m o n o p o l e s ; ii) s h o w s t h a t t h e S u p e r l u m i n a l e l e c t r i c sources themselves ( t o g e t h e r w i t h t h c s u b l u m i n a l ones) allow to w r i t e M a x w e l l e q u a t i o n s in t h e w i s h e d , fully s y m m e t r i c a l f o r m (4.6). To t h i s aim, no r e c o u r s e t o m a g n e t i c m o n o p o l c s r e s u l t s t o be n e c e s s a r y . L e t u s s k e t c h t h c r e s u l t s (a.e) s u m m a r i z e d a t p o i n t ii). Be F#~ t h e u s u a l e l e c t r o m a g n e t i c t e n s o r (7). L e t us i n t r o d u c e t h e autodual e l e c t r o m a g n e t i c t e n s o r (p, v = 0, 1, 2, 3)
T,..-- F,.,+ Pff~..
(1) w h e r e we h a v c d e f i n e d (s)
z
(#,v,~,fl=0,1,2,3), the quuntity s~
b e i n g t h e real, c o m p l e t e l y a n t i s y m m e t r i c Ricci t e n s o r . The p r e s e n t a n d H--->--iE. Since it is
~ duality ,~ effects t h e exch&nges E ~ i H (3)
~t,~ = T~,,,
t h e n our a u t o d u a l t e n s o r is i n v a r i a n t u n d e r one or more than one of t h e following t h r e e exchanges :
Ek->iH k ,
(4)
Hk->--iEk
N o t i c e t h a t w c are u s i n g t h e m e t r i c ( +
(k = 1, 2, 3) .
) ; h o w e v e r , w e use t h e c o n v e n t i o n
x ~ (ct, ix, iy, iz), so t h a t g ~ = 6~v, a n d w e h a v e no d i s t i n c t i o n b e t w e e n c o v a r i a n t a n d c o n t r a v a r i a n t c o m p o n e n t s (*). B y t h e way, t h e t e n s o r T,~ e x p l i c i t l y r c a d s
l (5)
(T~) :
0
iE~--H~
iEv--Hv
iE z-
H~I
H~--iE~
0
iEz--H ~
Hv--iE~
H~--iEy
H~--iE~
0
iE~--Hx]
H~--iE~
iEy--H~
H~--iE~
0
/
(4) R. ~-~IIGNANIand E. RECAMI: preprint IFTUC-PP/387 (Catania, July 1973), to appear iR Riv. Nuovo Cimento; Nuovo Cimento, 14 A, 169 (1973); 16 A, 208 (1973); E. RECAMIand R. MIGNANI: Left. Nuovo Cimento, 4, 144 (1972); 8, 110 (1973). Scc also ref. (5). (5) See also E. RECAMI: ill Encyclopedia EST 21Iondadori, Annuario 1973 (Milano, 1973), p. 85; R. MIGNANI and E. RECAMI: Left. Nuovo Cimento, 7, 388 (1973); 8, 780 (1973); R. MIGNANI, E. RECAMIand U. LOMBARDO: Left. Nuovo Cimento, 4, 624 (1972); V. S. OLKttOVSKYand E. RECAMI:Lett. Nuovo Cimento, 1, 165 (1971); M. BALDO, G. FORTE and E. RECA.~II: Left. Nuovo Cimento, 4, 241 (1970). (~) R. MmNANI and E. RECA.~tI: Left. Nuovo Cimento, 9, 367 (1974). (7) See, e.g., L. D. LA~DAV and E. M. LIFSHITZ: Theorie du champ (Moscow, 1966). (8) See, e.g., J. M. LEII~AAS: Nuovo Cimento, 15A, 740 (1973); D. WEINGARTEX: Ann. o] Phys., 76, 510 (1973). (*) The summation is understood over the repeated indices. Natural units (c = 1) will bc used when convenient. For the electromagnetic quantities, rationalized Gaussian units (i.e. ttcavisidc-Lorcntz's units) will be adopted, for practical purposes.
481
DO MAGNETIC MONOPOLES EXIST ~. CONSIDERATIONS FOI~ THEORY ETC.
Then, s t a n d a r d field e q u a t i o n s - - i n presence of only subluminal electric four-currents
],(s) ~ ( e(s), j(s) ) - - r e a d as u s u a l (6)
~,T/,~ = ]t~ '
~
= TI'~
("'2< e2) "
However, w h e n in presence of both subluminal (s) a n d Superluminal (S) electric four-currents, t h e (extended) Maxwell e q u a t i o n s assume t h e (fully symmetrical) form (4.6) (*) (7)
~,~
= T+,~.
(v~
C2).
Since we have only one f o u r - c u r r e n t - - b o t h s u b l u m i n a l a n d S u p e r l u m i n a l - - w e shall be able to skip t h e adjective (~electric ~) in t h e following. One m a y well call our fourc u r r e n t as the ( u n i q u e ) ( ( e l e c t r o m a g n e t i c ~) c u r r e n t (and, in particular, the electric charge as the (subluminal) (( electromagnetic charge ~)). Let us explicitly notice t h a t first eq. (7) is e q u i v a l e n t - - w h e n we confine ourselves to s u b l u m i n a l inertial f r a m e s - to the set of noticeable e q u a t i o n s divD=
O(s),
d i v B = - - q(S), (8)
~B
rot E = - - - - / + j ( S ) ,
rot H =
(v2 ~ c~; s ~ s u b l u m i n a l ; S ~ s u p e r l u m i n a l ) .
~D
-~- + j(s) .
It is clear t h a t to S u p e r l u m i n a l (electric) c u r r e n t s is given b y relativity the place t h a t was before a t t r i b u t e d to magnetic currents. By the way, a Superluminal, positive (electric) charge--e.g, m o v i n g along t h e positive x-axis w i t h speed V > c - - w i l l bring into the (extended) field e q u a t i o n s t h e c o n t r i b u t i o n t h a t was previously supposed to come from a (subluminal) South magnetic pole (**), m o v i n g along the positive x-axis with t h e speed v = c f / V < c - - a c c o r d i n g to t h e s t a n d a r d conventions, due to historical reasons. We m a y deduce t h a t , at the light o] relativity, t h e theoretical m o t i v a t i o n s (essentially the m o t i v a t i o n i) of Sect. 1 as we saw) for expecting existence of m a g n e t i c monopoles become wery weak. R o u g h l y speaking, we have no reasons for p o s t u l a t i n g (subluminal) magnetic monopoles do exist. On the contrary, supports (4.5) t h e hypothesis t h a t superl u m i n a l do exist, as well as usual (subluminal) electric charges.
3. - Conclusions.
F r o m t h e foregoing, t h e following conclusions m a y be d r a w n :
i) R e l a t i v i t y does n o t predict t h e existence of (subluminal) magnetic monopoles (4.6). On the contrary, it allows to predict S u p e r l u m i n a l (electric) charges do exist (4.5). (*) (**) and is a
Equations (7) hold for - n/2 < ~0< n/2, where ~ has the meaning forwarded in ref. (D. This is the reason of the signs entering Maxwell-Dirac-Schwinger equations. Notice that in cqs. (7) followings, we arc assuming Jt~ ~ 0 u p , where u~=--dx~/dvo is the G-fourvelocity (4), so that ?'/~ G-fourvcctor. The quantity 00 is the proper charge density.
482
n. RECAM~ and R. MIGNANI
ii) Superluminal (electric) charges will contribute to the electromagnetic field as (subluminal) magnetic monopoles were previously supposed to do to give to the Maxwell equations a completely symmetrical form. iii) Tachyonic (electric) charges are predicted by relativity (4.e) to behave, when interacting with an electromagnetic field, as (~magnetic monopoles ~), endowed however with ]aster-than-light speed (*). The details will be forwarded in a forthcoming paper, aimed at suggesting experiments for revealing charged tachyons through their interaction with the electromagnetic field. Here, let us observe that a Superluminal (electric) charge q = me will behave--in the above sense--as a magnetic monopole g ~ - - m e , i n Gaussian u n i t s (**). Therefore, in this case, one would immediately get t h a t (m = positive, integer number) qg
~
__
m 2 e2 ~
--
m2g~
,
the number a being the fine-structure constant. I n general, the product of a charge e by the quantity g ~ ne (in Gaussian units) would yield (9)
eg ~ n ~
(n integer),
which is s i m i l a r to the known relation derived by SCHWINGER(1) on the hypothesis of magnetic-pole existence. iv) Experimellts would be desirable searching for Superluminal charges. Because of what previously clarified, such experiments wouId be interesting with regard to the problems not only of taehyons but also of magnetic monopoles. Actually, B~TLETT and L~ANA (9) already performed a very interesting experimental search for (~tachyon monopoles ~. Those authors, however, looked for electromagnetic ~erenkov radiation supposedly emitted by tachyonie charges, on the basis of theoretical assumptions that we deem to be incorrect (4,~0) (since they violate the requirements of relativity). Therefore, that experiment is not conclusive at all, according to us, and new experiments, on the line forwarded in this note, would be in order. $r
The authors grateftflly acknowledge the useful, kind interest of Profs. A. AGODI, N. CABIBBO, F. CAT~_~A, and A. GIGH and of Drs. L. BALDINI, S. GRECO, D. MA[OLO, C. SPITALE~I and A. ST~AZZERI.
(*) In other words, one may say that the electromagnetic charge behaves as a (subluminal) electric charge when sublumi~al, and as a (Superluminal) magnetic charge when Superluminal. (**) We mean g =--mcv//t~-[o in rationalized ~IKS~ units. (s) D. F. BARTLETTand M. D. L~ANA: Phys. Rev., 6, 1817 (1972). (lo) See, R. MIGNANIand E. RECAMI:Left. Nuovo Cimento, 7, 388 (1973); 9, 362 (1974); Astrophysics and tachyons, preprint IFTUC-PP/389 (Catania, Oct. 1973), submitted Ior publication.
