The Fine-Structure Constant, Magnetic Monopoles and Dirac Charge. Quantization Condition. T. DATTA. Physics and Astronomy Departme~t, University o/South ...
LETTERE AL NUOVO CIMENTO
VOL. 37, h'. 2
14 Maggio 1983
The Fine-Structure Constant, Magnetic Monopoles and Dirac Charge Quantization Condition. T. DATTA
Physics and Astronomy Departme~t, University o/South Carolina - Columbia, S.C. 29208 (ricevuto il 7 Marzo 1983) PACS. 0 3 . 6 5 . -
Q u a n t u m t h e o r y ; q u a n t u m mechanics.
Summary. - T h e Dirac q u a n t i z a t i o n condition imposes some v e r y peculiar a n d s t r i n g e n t constraints on t h e n a t u r e of monopoles. One of t h e m o s t i n t e r e s t i n g results is t h a t such a p a r t i c l e has a classical radius larger t h a n its q u a n t u m (Compton) or (Bohr) radius. No o t h e r e l e m e n t a r y p a r t i c l e shows this p r o p e r t y . This indicates t h a t t h e monopoles are i n h e r e n t l y relativistic. A t o m like stable b o u n d states of oppositely charged monopoles are u n p h y s i c a l and a p a i r of p r i m o r d i a l monopoles are prone to annihilation, which m a y explain t h e so-called m o n o p o l e p a r a d o x . Conversely, if t h e v a l u e of the S o m m e r f e l d fine-structure c o n s t a n t were g r e a t e r t h a n unity, t h e n unstable. I t appears t h a t in a p h y s i c a l u n i v e r s e for a g i v e n v a l u e of t h e fine-structure constan$ only one k i n d of sources or charges (either electric or magnetic) are permissible. T h i s is a quite general r e q u i r e m e n t i m p o s e d by the laws of eleetrodynamics, q u a n t u m mechanics and r e l a t i v i t y acting s i m u l t a n e o u s l y in conjunction to each other. DIRAC has shown (1) t h a t t h e observed electric-charge q u a n t i z a t i o n can be a c c o u n t e d for if isolated m a g n e t i c charges or rnagnctic monopoles are allowed, p r o v i d e d t h e magnetic-charge u n i t and t h e unit of electric charge satisfy a reciprocal relationship. T h i s is k n o w n as the Dirac q u a n t i z a t i o n condition (2), and is given b y (I)
e
q = n-- , 2~
w h e r e q is t h e m a g n e t i c - p o l e strength, e t h e electronic charge, a = e2/hc is t h e Sommerfeld fine-structure c o n s t a n t and n an integer. F o r simplicity, here we will consider t h e n = 1 ease only. DIRAC also i n d i c a t e d t h a t such a m o n o p o l e will not be b o u n d to an electric charge. Subsequently, on t h e basis of r e l a t i v i s t i c q u a n t u m theory, HARISYI(1) P . A. )/I. DIRAC: Proc. R. Soc. London, S e r . . 4 , 133, 60 (1931); Phys. Rev., 74, 817 (1948). (3) D i r a e o b t a i n e d t h e q u a n t i z a t i o n c o n d i t i o n b y d e m a n d i n g t h e single v a l u e d n e s s of t h e w a v e f u n ction,
51
52
T. DATTA.
CHA~DRA (3) f o r m a l l y e s t a b l i s h e d t h e D i r a c c o n j e c t u r e r e g a r d i n g t h e l a c k of e l e c t r o n m o n o p o l a r b o u n d s t a t e s . SCHWlNGER (4) c o n s i d e r e d p a r t i c l e s w h i c h are s i m u l t a n e o u s l y electrically and magnetically charged and arrived at twice the Dirae quantization v a l u e , i.e., q ~ e/~. SAHA (5) e s t i m a t e d t h e m o n o p o l e m a s s m to b e ~ 2.5 t i m e s t h e p r o t o n m a s s . M o r e r e c e n t l y , some of t h e g r a n d u n i f i e d t h e o r i e s (e) ( G U T ) of element a r y p a r t i c l e s , are b e l i e v e d t o p e r m i t m a g n e t i c m o n o p o l e s . T h e s e m o n o p o l e s a r e assoc i a t e d w i t h t h e s o l u t i o n s of t h e classical t ' H o o f t - P o l y a k o v e q u a t i o n s ( 7 ) , a n d are w i t h m ~ 10 2 ~g (10 ~e GeV) o r ~ 10 -3 t h e P l a n c k m a s s . I n t h i s p u b l i c a t i o n , u s i n g g e n e r a l , m o d e l - i n d e p e n d e n t a r g u m e n t s , we will discuss t h e i m p l i c a t i o n s of t h e D i r a c c o n d i t i o n o n t h e n a t u r e a n d s t a b i l i t y of t h e m a g n e t i c m o n o p o l e s . T h e p e c u l i a r i t i e s b e c o m e i m m e d i a t e l y a p p a r e n t w h e n t h e classical a n d q u a n t u m - m e c h a n i c a l ( C o m p t o n ) sizes of t h e m o n o p o l e s are d e t e r m i n e d . T h e classical size r o is g i v e n b y q~ (2) r o -?D~C2
a n d 2, t h e C o m p t o n , or a c t i o n , r a d i u s b y
(3)
~ --
~C "
i n c.g.s, u n i t s . E m p l o y i n g t h e D i r a c c o n d i t i o n (1) w i t h n = 1 s u b s t i t u t e for q i n (2), we o b t a i n
(4a)
r o --
). 4~. '
or
(4b)
ro > ~.
T h i s s u r p r i s i n g r e s u l t d e p e n d s o n l y on t h e D i r a c c o n d i t i o n a n d is i n d e p e n d e n t of t h e m o n o p o l e m a s s . H e n c e m a g n e t i c m o n o p o l e s will h a v e classical r a d i i l a r g e r t h a n t h e i r q u a n t u m r a d i u s . T h e i n e q u a l i t y (4b) is r a t h e r d i s t u r b i n g , since o n l y a p a r t of t h e m a s s m a y b e d u e t o e l e c t r o m a g n e t i c i n e r t i a t h e classical r a d i u s is e x p e c t e d t o b e t h e s m a l l e s t of a c h a r g e d p a r t i c l e . No o t h e r e l e m e n t a r y p a r t i c l e is k n o w n t o h a v e r o > 4. T h i s a n o m a l y p e r s i s t s w h e n a B o h r r a d i u s , R = h2/mq 2, is c o n s i d e r e d , y i e l d i n g (s) (5)
ro _
R 16~2 '
or
(6)
ro> ~ > R.
(s) ]~[ARISH-CHANDR&" Phys. Rev., 74, 883 (1948). (4) J . SCHWlNGER: Science, 165, 757 (1969) a n d t h e r e f e r e n c e s t h e r e i n . (5) M. 1~r. SAH~_: Ind. J. Phys., 10, 141 (1936). (6) F o r a r e v i e w see 1~ LANGACKER: S L A C p r e p r i n t 2544 (1980). (~) A. 'T HOOFT: Nucl. Phys. B, 79, 276 (1974); A. POLYAKOV: J E T P Left., (0 A. BARUT h a s p r e v i o u s l y o b t a i n e d ( u n k n o w i n g l y to this a u t h o r ) t h a t for A. O. BARUT: Phys. Lett. B, 63, 73 (1976) a n d r e f e r e n c e s t h e r e i n . Also, for d y o n s c h a r g e i n t e r a c t i o n it m a y be possible to r e l a x t h e s t a b i l i t y c o n d i t i o n , p r o v i d e d e l e c t r i c a l l y similar, w i t h l a r g e electric c h a r g e s on t h e m .
20, 194 (1974). d y o n s ~ > R. See d u e to t h e electricthe two dyons are
THE FIN:E-STRUCTURE CONSTANT, MAGNETIC MONOPOLES :ETC.
~
T h e m o s t i m p o r t a n t i m p l i c a t i o n of t h e h i e r a r c h y (6) is t h a t a m o n o p o l e is i n h e r e n t l y a r e l a t i v i s t i c e n t i t y . T h i s w o u l d i m p l y t h a t n o n r c l a t i v i s t i c m o n o p o l e t h e o r i e s are i n a p p r o p r i a t e . T h i s is s o m e w h a t a n t i c i p a t e d in t h e l i g h t of t h e r e l a t i v i s t i c d u a l i t y b e t w e e n e l e c t r o m a g n e t i c i n t e r a c t i o n s . I n a d d i t i o n , r o > R i m p l i e s , t h a t n o t o n l y p o s i t i v e And n e g a t i v e ( n o r t h a n d s o u t h ) m o n o p o l e s in a p a i r are i n t e r p e n e t r a t i n g , b u t m o r e i m p o r t a n t l y , t h i s a l o n g w i t h t h e v i r i a l t h e o r e m shows t h a t s u c h a p a i r c a n f o r m a s t a b l e b o u n d s t a t e o n l y if t h e g r o u n d - s t a t e o r b i t a l speed v = q~ c/~ >> c. Clearly t h i s is u n p h y s i c a l ! I t occurs b e c a u s e ~ 1. T h u s t h e collapse d u e to t h e t r e m e n d o u s i n t e r m o n o p o l e a t t r a c t i o n can bc p r e v e n t e d o n l y if v >> c. As long as e l e c t r o m a g n e t i c c h a r g e is q u a n t i z e d , for p u r e m a g n e t i c charges, t h i s s i t u a t i o n does n o t i m p r o v e e v e n whe.n e l e c t r i c - d i p o l e m o m e n t s a r e i n c o r p o r a t e d a n d r e l a t i v i s t i c c o r r e c t i o n s are a t t e m p t e d . I n p o i n t of f a c t s u c h a (~L a m b - s h i f t effect. )) will t e n d to i n c r e a s e t h e n e t c o l l a p s i n g a t t r a c t i o n . ThAt is, t h e r e l a t i v i s t i c effect w o u l d w o r s e n t h e g r o u n d - s t a t e i n s t a b i l i t y . N e i t h e r , is v >> c a v e r t e d b y s i m p l y r e q u i r i n g t h e m o n o p o l e size to be e x t e n d e d or its m a s s to be s u p e r h e a v y . Since t h e a b o v e c o n d i t i o n s for s t a b i l i t y or b i n d i n g in t h e m o s t s t a b l e ( g r o u n d ) s t a t e are u n a c c e p t a b l e , one is f o r c e d to c o n c l u d e t h a t in t h e p r e s e n t u n i v e r s e
