UBC Theses and Dissertations

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UBC Theses and Dissertations

Considerations regarding the duality rotation. Levman, Garry 1970

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C O N S I D E R A T I O N S R E G A R D I N G T H E D U A L I T Y R O T A T I O N b y G A R R Y L E V M A N ' B . S c . , U n i v e r s i t y o f T o r o n t o , 1 9 6 9 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF. T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF S C I E N C E i n t h e D e p a r t m e n t o f P h y s i c s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A A u g u s t , 1 9 7 0 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study . I f u r t h e r agree t h a t permiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date A u g u s t , 1970 i i A B S T R A C T ^ M a x w e l 1 ' s e q u a t i o n s f o r t h e v a c u u m a r e i n v a r i a n t u n d e r t h e d u a l i t y r o t a t i o n ; h o w e v e r , t h e s i g n i f i c a n c e o f t h i s i n v a r i a n c e i s n o t u u e l l u n d e r s t o o d . T h e p u r p o s e o f t h i s t h e s i s i s t o c o n s i d e r t h e d u a l i t y r o t a t i o n i n g r e a t e r d e t a i l t h a n h a s b e e n d o n e p r e v i o u s l y . T h e d u a l i t y i n v a r i a n c e o f M a x w e l l ' s e q u a t i o n s i s d i s c u s s e d , a n d i t i s s h o w n . t h a t t h e o n l y d u a l i t y i n v a r i a n t s b i l i n e a r i n t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e a r b i t r a r y l i n e a r c o m b i n a t i o n s o f t h e c o m p o n e n t s o f t h e s t r e s s - e n e r g y - m o m e n t u m t e n s o r . I t i s a l s o s h o w n t h a t t h e m o s t g e n e r a l l i n e a r f i e l d t r a n s f o r m a t i o n w h i c h l e a v e s M a x w e l l ' s v a c u u m e q u a t i o n s i n v a r i a n t i s t h e d u a l i t y r o t a t i o n . T h e u s u a l L a g r a n g i a n d e n s i t y f o r t h e e l e c t r o m a g n e t i c f i e l d d o e s n o t e x h i b i t d u a l i t y i n v a r i a n c e . I t i s s h o w n , h o w e v e r , t h a t i f o n e t a k e s t h e c o m p o n e n t s o f t h e e l e c t r o -m a g n e t i c f i e l d t e n s o r a s f i e l d v a r i a b l e s , t h e n t h e m o s t g e n e r a l L o r e n t z i n v a r i a n t L a g r a n g i a n d e n s i t y b i l i n e a r i n t h e e l e c t o -m a g n e t i c f i e l d s a n d t h e i r f i r s t d e r i v a t i v e s i s d e t e r m i n e d u n i q u e l y b y t h e r e q u i r e m e n t o f d u a l i t y i n v a r i a n c e . T h e e n s u i n g f i e l d e q u a t i o n s a r e i d e n t i c a l w i t h t h e i t e r a t e d M a x w e l l e q u a t i o n s . I t i s f u r t h e r s h o w n t h a t i n n e u t r i n o t h e o r y t h e P a u l i t r a n s f o r m -a t i o n o f t h e s e c o n d k i n d c o r r e s p o n d s t o t h e d u a l i t y r o t a t i o n . i i i C O N T E N T S A B S T R A C T . . i i C O N T E N T S i i i A C K N O W L E D G M E N T S " i v 0. I N T R O D U C T I O N 1 1. D U A L I T Y P A I R S AND D U A L I T Y I N V A R I A N T S 3 2. A C O V A R I A N T F O R M U L A T I O N OF T H E D U A L I T Y R O T A T I O N 8 3. C O O R D I N A T E AND F I E L D T R A N S F O R M A T I O N S 11 4. T H E C O N S T R U C T I O N OF F I E L D T H E O R I E S 15 5. A D U A L I T Y I N V A R I A N T A C T I O N P R I N C I P L E 21 6 . T H E I T E R A T E D M A X W E L L E Q U A T I O N S AND C O N S E R V E D C U R R E N T 27 7. T H E D U A L I T Y R O T A T I O N I N Q U A N T U M M E C H A N I C S . . 32 8. C O N C L U S I O N - . 37 N O T E S 38 L I T E R A T U R E C I T E D • 39 A P P E N D I X A: T H E C O N F O R M A L G R O U P 41 A P P E N D I X B: ON E Q U A T I O N S (3.8) AND (3.9) 47 A P P E N D I X Cs ON E Q U A T I O N S (5.9) ' 49 iv A C K N O W L E D G M E N T S I t h a n k D r . F . A . K a e m p f f e r f o r s u g g e s t i n g t h i s r e s e a r c h t o p i c , a n d " f o r g u i d i n g me t h r o u g h i t s c o m p l e t i o n . I a l s o t h a n k B . G o g a l a , B . K e e f e r , a n d F . P e e t f o r t h e i r e n c o u r a g e m e n t a n d s u p p o r t . T h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a p r o v i d e d me w i t h f i n a n c i a l a s s i s t a n c e . 0 . INTRODUCTION \ After the introduction of the complex f i e l d vector 1 F = E + iB Maxwell's equations for the vacuum , V x E + 2B/dt = 0 S7-E = 0 ( 0 .1) V X B - ^E/Jt = 0 <?'B = 0 , take the form V x- F - i\?/lt = 0 (0/2) V" • F = 0 . These equations are invariant under the transformation F '-*'£• = F e " 1 8 (0.3) p * —* p * • = p * e + i t i where F* = E - iB.. Misner and Wheeler (1957) have named t h i s transformation the duality r o t a t i o n . Milner (1927) used the duality rotation to simplify the solution of Maxwell's equations with sources; however, his paper seems to have been forgotten. Recently, interest in t h i s invariance property of Maxwell's vacuum equations has been revived by Witten (1962), Calkin (1965), Schwinger (1969), and Kaempffer (1970). Schwinger has developed a theory of strongly int e r a c t i n g p a r t i c l e s by requiring that the duality invariance of Maxwell's equations without sources be maintained in the equations with sources. Kaempffer has 2 c o n s i d e r e d t h e q u a n t u m f i e l d t h e o r y a s s o c i a t e d w i t h d u a l i t y i n v a r i a n t a c t i o n p r i n c i p l e s . T h i s t h e s i s c o n t a i n s s o m e e l e m e n t a r y c o n s i d e r a t i o n s r e g a r d i n g ' ' t h e d u a l i t y r o t a t i o n . T h e m a j o r r e s u l t o f t h e r e s e a r c h o u t l i n e d h e r e c a n b e s u m m a r i z e d i n t h e f o l l o w i n g t h r e e p r o p o s i t i o n s : 1 . T h e o n l y d u a l i t y i n v a r i a n t s b i l i n e a r i n t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e a r b i t r a r y l i n e a r c o m b i n a t i o n s o f t h e c o m p o n e n t s o f t h e s t r e s s - e n e r g y - m o m e n t u m t e n s o r . 2. T h e m o s t g e n e r a l l i n e a r f i e l d t r a n s f o r m a t i o n l e a v i n g M a x w e l l ' s v a c u u m e q u a t i o n s i n v a r i a n t i s , u p t o a n o r m a l -i z a t i o n f a c t o r , t h e d u a l i t y r o t a t i o n . 3. T h e m o s t g e n e r a l L o r e n t z i n v a r i a n t L a g r a n g i a n d e n s i t y b i l i n e a r i n t h e e l e c t r o m a g n e t i c f i e l d t e n s o r a n d i t s f i r s t d e r i v a t i v e s i s u n i q u e l y d e t e r m i n e d b y t h e r e q u i r e m e n t o f d u a l i t y i n v a r i a n c e . 3 1 . D U A L I T Y P A I R S AND D U A L I T Y I N V A R I A N T S I n t e r m s o f t h e e l e c t r i c f i e l d E a n d t h e m a g n e t i c f i e l d B t h e d u a l i t y r o t a t i o n ( 0 . 3 ) b e c o m e s E — E ' = E c o s B • B s i n 0 ( 1 . 1 ) " B > B ' = - E s i n 8 + B c o s 9 . ( E , § ) f o r m s a " d u a l i t y p a i r " w i t h " a n g l e o f r o t a t i o n " 8 . A n o t h e r i m p o r t a n t d u a l i t y p a i r e x i s t s . F r o m E a n d B o n e c a n 2 2 3 f o r m t h e L o r e n t z s c a l a r ^ ( B - E ) a n d t h e L o r e n t z p s e u d o s c a l a r E - B . U n d e r t h e d u a l i t y r o t a t i o n t h e y t r a n s f o r m a s E ' - B ' = E - B c o s 2 8 «• J ? ( B 2 - E ? ) s i n 2 B • ' ( 1 . 2 ) i ( B ' 2 - E ' ) = - E - B s i n 2 8 + i ( § 2 - E 2 ) c o s 2 8 ; t h e r e f o r e , t h e y f o r m a d u a l i t y p a i r w i t h a n g l e o f r o t a t i o n 2 8 . F r o m e q u a t i o n ( 1 . 2 ) i t f o l l o w s t h a t t h e u s u a l L a g r a n g i a n d e n s i t y f o r t h e e l e c t r o m a g n e t i c f i e l d ( 1 . 3 ) L = - ^ ( B 2 - E 2 ) i s n o t d u a l i t y i n v a r i a n t , M i s n e r a n d W h e e l e r ( 1 9 5 7 ) h a v e i n t r o d u c e d ( 1 . 4 ) = ( p § ) 2 <- | ( B 2 - E 2 ) w h i c h i s d u a l i t y i n v a r i a n t } h o w e v e r , L|YI_W i s f o u r t h o r d e r i n t h e e l e c t r o m a g n e t i c f i e l d s . • B e s i d e s L^_ ( A j t h e r e a r e s e v e r a l o t h e r d u a l i t y i n v a r i a n t s . 4 T h e P o y n t i n g v e c t o r S = E x B , t h e e l e c t r o m a g n e t i c e n e r g y d e n s i t y 2 2 <f = ^ ( E + B ) , a n d M a x w e l l ' s s t r e s s t e n s o r 4 a r e a l l d u a l i t y i n v a r i a n t . I t s h o u l d b e n o t e d h e r e t h a t t h e e l e c t r o m a g n e t i c f i e l d p o t e n t i a l s A a n d <P , t r a n s f o r m i n a v e r y c o m p l i c a t e d m a n n e r . T h e y a r e n o t f u n d a m e n t a l q u a n t i t i e s , h o w e v e r , s i n c e t h e y c a n n o t b e d e f i n e d u n i q u e l y . A l t h o u g h c l a s s i c a l e l e c t r o d y n a m i c s i s s i m p l i f i e d b y t h e i n t r o d u c t i o n o f p o t e n t i a l s , e l e c t r o m a g n e t i c p h e n o m e n a c a n b e s t u d i e d , a t l e a s t i n p r i n c i p l e , w i t h o u t t h e i n t r o d u c t i o n o f p o t e n t i a l s . T h e q u e s t i o n a r i s e s w h e t h e r a n y o t h e r d u a l i t y i n v a r i a n t s e x i s t . A s a p a r t i a l a n s w e r t o t h i s q u e s t i o n c o n s i d e r a n a r b i t r a r y f u n c t i o n o f t h e f i e l d s ' " f = fiE^ ) . U n d e r a n i n f i n i t e s i m a l d u a l i t y t r a n s f o r m a t i o n E a n d B t r a n s f o r m a s T = I - ( E E + B B ) ( 1 . 5 ) E = - v - | - 3 A / H B = V X A = E + 8 B ( 1.6) B = B - 9 E w h e r e 6 « 1 . T h e n if = f C E ' r f , B ' ^ ) - f ( E ^ , t y ) ( 1.7) I f f i s a d u a l i t y i n v a r i a n t , £ f = 0 a n d C o n s i d e r t h e c a s e w h e r e f i s b i l i n e a r i n t h e e l e c t r i c a n d m a g n e t i c f i e l d s ; t h e r e f o r e , ( 1 . 9 ) f = + b ^ B ^ + c ^ E ^ w h e r e , b « ^ , c ^ ^ a r e r e a l c o n s t a n t s . W i t h o u t l o s s o f g e n e r a l i t y w e m a y t a k e a ^ ^ = a ^ , ^ a n d b ^ = . E q u a t i o n s ( 1.8) n o w g i v e s ( 1 . 1 0 ) 2 a ^ B ^ E ^ f c ^ B ^ B ^ ^ b ^ E ^ B ^ • c ^ E^ E ^ . I n e q u a t i o n ( 1 . 1 0 ) E a n d B m a y b e c o n s i d e r e d t o b e a s e t o f s i x a r b i t r a r y r e a l n u m b e r s . S e t t i n g E = 0 g i v e s c ^ ^ + c ^ = 0 s i n c e B B i s s y m m e t r i c . E q u a t i o n ( 1 . 1 0 ) n o w b e c o m e s w h i c h y i e l d s a ^ = b ^ ; t h e r e f o r e , f c a n b e w r i t t e n ( 1 . 1 2 ) f = ( E ^ E ; )*ce,/aZ«B/S w h e r e a ^ i s s y m m e t r i c a n d c^^is s k e w s y m m e t r i c . I n s p e c t i o n o f e q u a t i o n ( 1 . 1 2 ) r e v e a l s t h a t t h e o n l y d u a l i t y i n v a r i a n t s b i l i n e a r i n t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e a r b i t r a r y l i n e a r c o m b i n a t i o n s o f £ a n d t h e c o m p o n e n t s o f S a n d T . I f t h e e l e c t r o m a g n e t i c s t r e s s - e n e r g y - m o m e n t u m t e n s o r T , 6 T = i s i n t r o d u c e d , t h e a b o v e r e s u l t c a n b e s t a t e d : t h e . o n l y d u a l i t y i n v a r i a n t s b i l i n e a r i n t h e e l e c t r i c a n d m a g n e t i c f i e l d s a r e a r b i t r a r y l i n e a r c o m b i n a t i o n s o f t h e c o m p o n e n t s o f T . I n S c h w i n g e r ' s t h e o r y , o f s t r o n g l y i n t e r a c t i n g p a r t i c l e s M a x w e l l ' s v a c u u m e q u a t i o n s a r e e x t e n d e d t o i n c l u d e b o t h e l e c t r i c a n d m a g n e t i c c h a r g e s b y w r i t i n g V X z + 2 § / 3 t = - J (1.13) m = P* V * B - P E/H = J e iP'e a n d Pm a r e t h e e l e c t r i c a n d m a g n e t i c c h a r g e d e n s i t i e s ; J e a n d J m a r e t h e e l e c t r i c a n d m a g n e t i c c u r r e n t d e n s i t y v e c t o r s . I t i s p o s t u l a t e d t h a t u n d e r t h e d u a l i t y r o t a t i o n b o t h (j>e 1(ftm ) a n d ( j e , J m ) t r a n s f o r m a s d u a l i t y p a i r s w i t h a n g l e o f r o t a t i o n 8 . P' 2 2 "7 2 2 I n t h i s case ( p e f i m , h\P m - ft B\) a n d ( J e ' J m , * [ j m - J e J ) a r e d u a l i t y p a i r s w i t h a n g l e o f r o t a t i o n 2 8 . T h e f a l l o w i n g a r e 2 2 2 2 d u a l i t y i n v a r i a n t s : J ° e + (j°m> Je + J m » J e * J m » a n d Je-jm + J m J e • I n c o n c l u s i o n w e n o t e f o r c o m p l e t e n e s s t h a t S = i i F XV* € = h I'I* T = I - i ( F F * + F * F ) 7 i F - F = i ( E 2 - B 2 ) + i E - B The transformation properties of the above quantities follow now from equation ( 0 . 3 ) . 8 2 . A C O V A R I ANT F O R M U L A T I O N OF T H E D U A L I T Y R O T A T I O N M a x w e l l ' s e q u a t i o n s f o r m t h e b a s i s o f a L o r e n t z c o v a r i a n t t h e o r y . A s a r e s u l t i t i s c o n v e n i e n t t o w r i t e t h e d u a l i t y r o t a t i o n i n a m a n i f e s t l y L o r e n t z c o v a r i a n t f o r m . I n s u c h a f o r m t h e s i g n i f i c a n c e o f t h e t e r m " d u a l i t y " a l s o b e c o m e s a p p a r e n t . I n t h e f o l l o w i n g . t h e M i n k o w s k i m e t r i c t e n s o r g m n h a s t h e d i a g o n a l e l e m e n t s - 1 , - 1 , - -1 , +1 . T h e d u a l o f a s e c o n d r a n k c o n t r a v a r i a n t t e n s o r A m n i s d e f i n e d a s ^ ( 2 . 1 ) A m n V i ( - g ) - V m n r S A r s w h e r e A m n = g m r 9 n s A r S i s t h e a s s o c i a t e d c o v a r i a n t t e n s o r o f A mn a n d g = d e t g m n . £ m n r s | s a c o m p l e t e l y s k e w s y m m e t r i c r e l a t i v e t e n s o r o f w e i g h t w = +1 w i t h £ = + 1 . T h e f a c t o r ( - g ) e n s u r e s t h a t A m n i s a p s e u d o t e n s o r " ^ r a t h e r t h a n a r e l a t i v e t e n s o r S i m i l a r l y , t h e d u a l o f a s e c o n d r a n k c o v a r i a n t t e n s o r B m n i s d e f i n e d a s ( 2 . 2 ) B m n = - ^ - ( - g ) 7 £ m n r s B r w h e r e B m n = g r n r g ' ! S B r s i s t h e a s s o c i a t e d c o n t r a v a r i a n t t e n s o r o f B m n a n d g m n = g m n . £ m n r s i s a c o m p l e t e l y s k e w s y m m e t r i c r e l a t i v e t e n s o r o f w e i g h t w = - 1 w i t h £ " 1 2 3 4 = - 1 • W i t h t h e a b o v e c o n v e n t i o n s ~* r s ~* ( 2 . 3 ) g m r g n s C = C r n n . I t c a n n o w b e s h o w n t h a t f o r a n a r b i t r a r y a n t i s y m m e t r i c t e n s o r D m n = _ n n m Q n e h a s g m n = _ D m n # i 1-2 g W i t h t h i s p r e p a r a t i o n c o n s i d e r t h e e l e c t r o m a g n e t i c f i e l d t e n s o r ( 2 . 4 ) • mn mn T h e d u a l o f F i s g i v e n b y ( 2 . 5 ) . m n 0 B 3 - B 2 - E l - B 3 0 B 1 - E 2 B 2 - B T 0 " E 3 E l E 2 E 3 0 0 ^ 3 - E 2 01 0 E l 0 2 - E l 0 B 3 - B i - B 2 - 0 3 0 \ F m n a n d F m n a r e s e c o n d r a n k , c o n t r a v a r i a n t , s k e w s y m m e t r i c t e n s o r s . M a x w e l l ' s v a c u u m e q u a t i o n s ( 0 . 1 ) c a n n o w b e w r i t t e n ( 2 . 6 ) F m n - 0 r , n - u F , n = 0 i _ ~% i \ m c J. t • n m n r m n , r m n w h e r e , _ = <>/ox . S e t t i n g G = F + I F g i v e s G M N , N = 0 ( 2 . 7 ) E q u a t i o n ( 2 . 7 ) i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n ( 2 . 8 ) G M N — > G ' M N = G m n - e - - i B w h i c h c a n b e r e w r i t t e n a s ( 2 . 9 ) r , mn mn 'Tmn . F ' = F c o s 9 + F s i n © / r f n n v , - r m n . ^ m n ( F ) = - F s i n B + F c o s 8 10 _. ..run _mn / r m n \ , / r - i m n \ , , . _ S i n c e F = -F , (F )' = ( F ' ) . When expanded i n t e r m s o f F and B e q u a t i o n ( 2 . 9 ) i s e x a c t l y t h e d u a l i t y r o t a t i o n ( 1 . 1 ) . A mn ~mn p r o b l e m a r i s e s b e c a u s e F i s a. t e n s o r w h i l e F i s a p s e u d o -t e n s o r . I f F , m n i s t o be a t e n s o r and F ' m n i s t o be a p seudo-t e n s o r , t h e n 8 must be a p s e u d o s c a l a r . With t h i s a s s u m p t i o n t h e d u a l i t y r o t a t i o n ( 2 . 9 ) i s a L o r e n t z c o v a r i a n t t r a n s f o r m a t i o n . A l l t h e r e s u l t s o f s e c t i o n 1 c a n now be p r o v e d u s i n g t h e c o v a r i a n t n o t a t i o n . One need o n l y o b s e r v e t h a t i r m r v m n = - n (2.10) i F m n F m n = E.B mn m _ mr ^ l C m r r s r T n F n r " ^ n F F r s * A l t h o u g h M a x w e l l ' s a q u a t i o n s a r e f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s t h e y a r e e q u i v a l e n t t o t h e s e c o n d o r d e r s y s t e m (2.11) nr = ° , nr nr = ° , nr o b t a i n e d by d i f f e r e n t i a t i n g ( 2 . 6 ) by x r . The i n t e g r a t i o n o f e q u a t i o n s ( 2 .11) g i v e s mn m p ,n = c (2.12) ~mn m F _ = D ' n where C m and D m a r e v e c t o r c o n s t a n t s o f i n t e g r a t i o n w h i c h c a n be s e t e q u a l t o z e r o by t h e p r i n c i p l e o f i s o t r o p y o f s p a c e t i m e . They must be s e t t o z e r o i f t h e vacuum i s t o be L o r e n t z i n v a r i a n t . 11 3. C O O R D I N A T E A N D F I E L D T R A N S F O R M A T I O N S A n y c o o r d i n a t e t r a n s f o r m a t i o n i n d u c e s a t r a n s f o r m a t i o n o f t h e e l e c t r i c a n d m a g n e t i c f i e l d s . U n d e r a L o r e n t z t r a n s f o r m -a t i o n , f o r e x a m p l e , t h e e l e c t r o m a g n e t i c f i e l d s F m n t r a n s f o r m a s a s e c o n d r a n k c o n t r a v a r i a n t t e n s o r . T h e d u a l i t y r o t a t i o n d i f f e r s f u n d a m e n t a l l y . f r o m t h e L o r e n t z t r a n s f o r m a t i o n s i n c e i t d o e s n o t p o s s e s s a n a s s o c i a t e d c o o r d i n a t e t r a n s f o r m a t i o n . T h e d u a l i t y r o t a t i o n i s a f i e l d t r a n s f o r m a t i o n r a t h e r t h a n a c o -o r d i n a t e t r a n s f o r m a t i o n . T h r e e q u e s t i o n s i m m e d i a t e l y o c c u r : 1 . W h a t i s t h e m o s t g e n e r a l g r o u p C o f c o o r d i n a t e t r a n s f o r m -a t i o n s u n d e r w h i c h M a x w e l l ' s v a s c u u m e q u a t i o n s a r e c o v a r i a n t ? ' 2. W h a t i s t h e m o s t g e n e r a l g r o u p D o f f i e l d t r a n s f o r m a t i o n s l e a v i n g M a x w e l l ' s v a c u u m e q u a t i o n s i n v a r i a n t ? t 3. F o r a g i v e n f i e l d t r a n s f o r m a t i o n d e D d o e s t h e r e e x i s t a c o o r d i n a t e t r a n s f o r m a t i o n c e C w h i c h i n d u c e s d ? I t i s w e l l k n o w n ( A p p e n d i x A ) t h a t t h e f u l l c o v a r i a n c e g r o u p o f M a x w e l l ' s e q u a t i o n s i s t h e f i f t e e n p a r a m e t e r g r o u p o f 7 c o n f o r m a l t r a n s f o r m a t i o n s . I n t h i s s e c t i o n a p a r t i a l a n s w e r i s o b t a i n e d t o q u e s t i o n 2. I t i s s h o w n t h a t t h e m o s t g e n e r a l l i n e a r f i e l d t r a n s f o r m a t i o n l e a v i n g M a x w e l l ' s v a c u u m e q u a t i o n s i n v a r i a n t i s , u p t o a n o r m a l i z a t i o n f a c t o r , t h e d u a l i t y r o t a t i o n . T h e a n s w e r t o q u e s t i o n 3 i s n o t y e t k n o w n . T h e m o s t g e n e r a l l i n e a r e l e c t r o m a g n e t i c f i e l d t r a n s f o r m a t i o n 12 c a n b e w r i t t e n ( 3.1) F ' m n = A m n r s F r s w h e r e A r _ a r e r e a l c o n s t a n t s . S i n c e F = - F , F = - F r s a n d F m n = - F n m t h e c o n s t a n t s A m n r s s a t i s f y t h e s y m m e t r y c o n d i t i o n ( 3.2) A r s • A r s = A s r + A r s n r s - " s r " s r T h e i n v a r i a n c e o f f Y l a x w e l l ' s v a c u u m e q u a t i o n s u n d e r t r a n s f o r m a t i o n ( 3 . 1 ) i m p l i e s ( 3 . 3 ) w h e r e mn r s _ H r s r » n - u mn r s . / M r s ' » n u ( 3 - 4 ) ^ " r s = * ( - 9 ) ' " ^ n U , k l A k l r s / \ r s I n e q u a t i o n ( 3 . 3 ) t h e c o e f f i c i e n t s o f f , n c a n n o t b e s e t e q u a l . r s t o z e r o s i n c e t h e F , n a r e d e p e n d e n t . T h e f o l l o w i n g c o n d i t i o n s r s e x i s t o n t h e F , n s S r ^ s ^ ' n 1 ^ ^ / o r- \ i / N - i _ k n a b r s ( 3 . 5 ) f ( - g ) € • g a r g b s F , n = 0 / r - a <- b r b c a \ , - r s ( S j S S + S r S g ) F , R = 0 E q u a t i o n s ( 3 . 3 ) c a n n o w b e s i m p l i f i e d b y L a g r a n g e ' s m ' e t h o d o f u n d e t e r m i n e d m u l t i p l i e r s . E a c h e q u a t i o n o f c o n s t r a i n t i n ( 3 . . 5 ) i s m u l t i p l i e d b y a n u n d e t e r m i n e d r e a l c o n s t a n t . . T h e r e s u l t i n g e q u a t i o n s a r e a d d e d t o ( 3 . 3 a ) g i v i n g . . 13 1 /nmn m r k r n X / m -knab ( A rs " a k ^ r * s ~ ? ( - g ) b k cf g a r g b s where a m k , b m k , and f ~ , m n a b are the undetermined m u l t i p l i e r s . By choosing the constants so that the c o e f f i c i e n t s of the r s dependent F , n disappear one obtains «mn m r k c n j . / \ - i ,m knab A rs = a k » r * s + 2(-g) b k £ g a r g b s. (3.7) ^ r i mn / r a C b c b r a \ + 1 ab ( >^ r S s + ^ r * s> • Transformation (3.1) now becomes ti o\ ' t - i m n ~ m rkn ,m ^kn (3.8) F ' = a k F / + b k F , the P m n r g not appearing due to the skew symmetry of F m n . S i m i l a r l y , equations (3.3b) and (3.5) y i e l d (3.9) ? ' m n = c m k F k n • d m k ? k n where' c m k and d m k are real constants. Equations (3.8) and (3.9) can be s i m p l i f i e d further by use of equation (3.4); however, i t i s more convenient to work d i r e c t l y with the equations themselves. From the skew symmetry of F * m n and F ' m n one can'easily show (Appendix B) that the offr diagonal elements of a m k , b m k , c m k , and d m k are zero -so that equations (3.8) and (3.9) read (3.10) / A 1 F 14 i _ where a^ , t>m , c m and d m are real constants. Expanding thes equations in terms of £ and B gives four equations for each component of E' and B ' . In Appendix B i t is shown that thes equations are consistent only i f (3.11) a = d = a , b = - c = b m m m m where a and b are constants. Setting a=rcosB , b=rsinB , and 2 . 2 2 . a + b = r gives i m n _ _ _• _ n p"in . _ _ . _ Q r-mn F' = r cos 8 F + r s i n 8 F (3.12) "p", mn . n rmn n ~mn F = -r s i n 8 F + r cos 8 F Equations (3.12) are the duality rotation (2.9) with a normal iz a t i o n factor r . 1 5 4 . T H E C O N S T R U C T I O N OF F I E L D T H E O R I E S A p h y s i c a l s y s t e m w i t h i n f i n i t e d e g r e e s o f f r e e d o m i s d e s c r i b e d b y a s e t o f f i e l d f u n c t i o n s = y A ( x m ) a n d b y a s e t . o f d i f f e r e n t i a l e q u a t i o n s f o r t h e f i e l d s y ^ . I n t h e L a g r a n g i a n f o r m u l a t i o n o f f i e l d t h e o r y a l l t h e p r o p e r t i e s o f t h e s y s t e m a r e a s s u m e d t o b e e m b o d i e d i n a L a g r a n g i a n d e n s i t y L =L ( /A< r\m>> r \ m n ' ••• ) F r o m t h e a c t i o n ( 4 . 2 ) . IAI = ^ - d 4 x o n e c a n o b t a i n t h e f i e l d e q u a t i o n s a b y a p p l i c a t i o n o f H a m i l t o n ' s p r i n c i p l e £ Ul = 0 . I f t h e a c t i o n Ul i s i n v a r i a n t u n d e r a g r o u p o f t r a n s f o r m a t i o n s a n y c o n s e r v e d c u r r e n t s w h i c h e x i s t m a y b e o b t a i n e d f r o m N o e t h e r ' s t h e o r e m s ( s e e S c h r o e d e r , . 1 9 . 6 8 ) . H i s t o r i c a l l y f i e l d e q u a t i o n s a r e u s u a l l y d i s c o v e r e d b e f o r e t h e i r c o r r e s p o n d i n g L a g r a n g i a n d e n s i t y . N e v e r -t h e l e s s , t h e L a g r a n g i a n d e n s i t y i s c o n s i d e r e d t o b e f u n d a m e n t a l . I n t h i s s e c t i o n a m e t h o d i s o u t l i n e d f o r o b t a i n i n g a s y s t e m ' s L a g r a n g i a n d e n s i t y w i t h o u t c o m p l e t e k n o w l e d g e o f t h e s y s t e m ' s f i e l d e q u a t i o n s . , T h e f o r m o f a L a g r a n g i a n d e n s i t y s u i t a b l e f o r t h e d e s c r i p -t i o n o f a p h y s i c a l s y s t e m i s s e v e r e l y r e s t r i c t e d i f t h e o r d e r 1 6 a n d t h e d e g r e e o f t h e s y s t e m a r e k n o w n . A s y s t e m i s o f o r d e r k i f t h e f i e l d e q u a t i o n s c o n t a i n n o d e r i v a t i v e s o f t h e f i e l d s h i g h e r t h a n k t h o r d e r . A s y s t e m i s o f d e g r e e m i f t h e f i e l d s a n d t h e f i e l d g r a d i e n t s h a v e m a x i m u m t o t a l p o w e r m i n t h e f i e l d e q u a t i o n s . T h e s i m p l e s t . p o s s i b l e c a s e i s a s y s t e m o f o r d e r a n d d e g r e e 1 ; h o w e v e r , t h i s c o n d i t i o n p r o v e s t o b e t o o r e s t r i c t i v e . I t h a s b e e n f o u n d t h a t m o s t p h y s i c a l s y s t e m s h a v e o r d e r 2 a n d d e g r e e 1 . S u c h a s y s t e m h a s l i n e a r , s e c o n d o r d e r f i e l d e q u a t i o n s s o t h a t t h e L a g r a n g i a n d e n s i t y m u s t b e b i l i n e a r i n t h e f i e l d s a n d t h e i r f i r s t d e r i v a t i v e s . O n l y s y s t e m s o f t h i s t y p e w i l l b e c o n s i d e r e d i n t h e f o l l o w i n g . I f a p h y s i c a l s y s t e m p o s s e s s e s a n i n t r i n s i c s y m m e t r y ' , w e e x p e c t , b y N o e t h e r ' s t h e o r e m s , t h a t t h e L a g r a n g i a n d e n s i t y f o r t h e s y s t e m e x h i b i t s t h i s s y m m e t r y b y b e i n g i n v a r i a n t u n d e r a n a s s o c i a t e d g r o u p o f t r a n s f o r m a t i o n s . T h e u n i v e r s a l v a l i d i t y o f t h e p r i n c i p l e o f s p e c i a l r e l a t i v i t y i m p l i e s , f o r e x a m p l e , t h a t a p h y s i c a l s y s t e m s L a g r a n g i a n d e n s i t y m u s t b e L o r e n t z i n v a r i a n t . B y N o e t h e r ' s f i r s t t h e o r e m t h e c o n s e r v a t i o n o f l i n e a r m o m e n t u m , a n g u l a r m o m e n t u m , a n d e n e r g y f o l l o w s . T h e i m p o s i t i o n o f i n v a r -i a n c e p r o p e r t i e s o n a L a g r a n g i a n d e n s i t y o f t e n d e t e r m i n e s t h e L a g r a n g i a n d e n s i t y u p t o a m u l t i p l i c a t i v e f a c t o r . T w o e x a m p l e s w i l l b e c o n s i d e r e d : a p h a s e i n v a r i a n t c o m p l e x L o r e n t z s c a l a r f i e l d a n d a g a u g e i n v a r i a n t L o r e n t z v e c t o r f i e l d . C o n s i d e r a c o m p l e x L o r e n t z s c a l a r f i e l d = Y*\ * i ^ a n d V * = / . 1 - i ^ 2 w i l l b e t a k e n a s t h e f i e l d v a r i a b l e s . I f t h e L a g r a n g i a n d e n s i t y i s L o r e n t z i n v a r i a n t a n d b i l i n e a r i n . 17 t h e f i e l d s a n d t h e i r f i r s t d e r i v a t i v e s , t h e n i t m u s t b e a l i n e a r c o m b i n a t i o n o f t h e f o l l o w i n g L o r e n t z s c a l a r s : m (4.4) j 2 = y * 2 J 5 = v * , m ^ * , m I f t h e L a g r a n g i a n d e n s i t y i s a l s o i n v a r i a n t u n d e r t h e p h a s e t r a n s f o r m a t i o n V - » y • = y e " i B ( 4 . 5 ) . „ t h e n o n l y a n d J g c a n a p p e a r ; t h e r e f o r e , f o r a p h a s e i n v a r i a n t c o m p l e x s c a l a r f i e l d t h e L a g r a n g i a n d e n s i t y h a s t h e f o r m (4.6) j _ - s = a Y * i> * b . y * f [ T ) y • * ' • w h e r e a a n d b a r e c o n s t a n t s . T h e f i e l d e q u a t i o n s b y m m - a ^ =0 (4.7) f o l l o w f r o m H a m i l t o n ' s p r i n c i p l e . T h e p h a s e t r a n s f o r m a t i o n ( 4 . 5 ) i s a o n e p a r a m e t e r c o n t i n u o u s t r a n s f o r m a t i o n . B y N o e t h e r ' s f i r s t t h e o r e m a c o n s e r v a t i o n l a w e x i s t s . U n d e r t h e i n f i n i t e s i m a l p h a s e t r a n s f o r m a t i o n v = y - i 9 i>. (4.8) y *• = y** + ie y * 18 t h e L a g r a n g i a n d e n s i t y L g t r a n s f o r m s a s S L S = L- - L 5 (4,9) - I B ( 3 L s / ^ ) r - I B ( 3 L5/3f,J<-ffm + iB O L g / ^ * ) / * + 18 ( ? L s / c ? V * m ) ^ m w h e r e e q u a t i o n (4.3) h a s b e e n u s e d f o r s i m p l i f i c a t i o n . S i n c e b y c o n s t r u c t i o n £> L g = 0 , o n e h a s t h a t (4.10) 2 j m/ax m =0 w h e r e (4.11) j m = ( ^ / ^ . m ^ " ^'Ls/3^*,m ) V * I t i s e a s i l y s h o w n f r o m e q u a t i o n ( 4 . 6 ) t h a t (4.12) j m = b ( y * ' m < / - Lf> m(f*) . I n s p e c t i o n o f t h e s e r e s u l t s s h o w s t h a t t h e p h a s e i n v a r i a n t c o m p l e x s c a l a r f i e l d i s c o m p l e t e l y e q u i v a l e n t t o t h e c o m p l e x K l e i n -G o r d o n f i e l d . A s a f i n a l e x a m p l e c o n s i d e r a L o r e n t z v e c t o r f i e l d A w h o s e s m L a g r a n g i a n d e n s i t y i s i n v a r i a n t u n d e r t h e g a u g e t r a n s f o r m a t i o n (*-13) A m — * A ' m = A m • h f m w h e r e h = h ( x m ) i s a s c a l a r f u n c t i o n . F o r l i n e a r , s e c o n d o r d e r , L o r e n t z c o v a r i a n t f i e l d e q u a t i o n s t h e L a g r a n g i a n d e n s i t y L y m u s t b e o f t h e f o r m 1 9 a ( 4 . 1 4 ) I = ^ c m K where t h e c a r e r e a l c o n s t a n t s and m K 1 = A m A m K, = A m m A n 1 m 3 , m , n ( 4 . 1 5 ) M _ H = A ' A">.n 'K4 = A m . n A n > m a r e L o r e n t z s c a l a r s . Under t r a n s f o r m a t i o n ( 4 . 1 3 ) Ly •—* Ly where ( 4 . 1 6 ) L J = ^ c m K ; and K' = K, + 2 A m h _ + h » m h m • 1 • t TI , m K' = K 7 + 2 A * h m _ *- h' h 2 * , mn , mn ( 4 . 1 7 ) «• . K 3 • 2 A " > - " N • . , • " > • " „ K - « 4 ^ m , n H , m n • S i n c e t h e L o r e n t z s c a l a r s K m a r e l i n e a r l y i n d e p e n d e n t , Ly = Ly i m p l i e s ( 4 . 1 8 ) c^ = C 3 = 0 , C 2 = - C 4 I t i s c o n v e n t i o n a l t o c h o o s e c ^ = . T h i s c h o i c e g i v e s The r e s u l t i n g f i e l d e q u a t i o n s a r e ( * - 2 0 ) . ( A M P N - A N P J ' N = 0 . • I f we s e t F m ° = A M , N - A n , f " t h e n e q u a t i o n ( 4 . 2 0 ) becomes F m n = 0 . 2 0 S i n c e F m n = A m , n - A n , m i m p l i e s t h a t F m n _ = 0 uie s e e t h a t t h e gauge i n v a r i a n t v e c t o r " f i e l d i s e q u i v a l e n t t o t h e e l e c t r o m a g n e t i c f i e l d . Note t h a t ( 4 . 2 1 ) L = 1 F m P F L U 4 h mn = * (E 2 - B 2 ) The gauge t r a n s f o r m a t i o n ( 4 . 1 3 ) depends a n a l y t i c a l l y on t h e f i r s t d e r i v a t i v e s o f a s c a l a r f u n c t i o n h. A c o n s e r v e d c u r r e n t e x i s t s , but t h e c o n s e r v a t i o n law i s s a t i s f i e d i d e n t i c a l l y . 2 1 5 . A D U A L I T Y I N V A R I A N T A C T I O N P R I N C I P L E 8 S c h w i n g e r 1 s t h e o r y ( 1 9 6 9 ) o f s t r o n g l y i n t e r a c t i n g p a r t i c l e s r e q u i r e s t h e i n t r o d u c t i o n o f m a g n e t i c a s w e l l a s e l e c t r i c c h a r g e s a n d a l t h o u g h t h i s p r o c e d u r e s y m m e t r i z e s M a x w e l l ' s e q u a t i o n s i n a p l e a s i n g m a n n e r , i t i n t r o d u c e s a n e w p r o b l e m . I f m a g n e t i c a n d e l e c t r i c c h a r g e s a n d c u r r e n t s a r e a l l o w e d , e l e c t r o m a g n e t i c f i e l d p o t e n t i a l s m a y n o t e x i s t ? t h e r e f o r e , a r e f o r m u l a t i o n o f t h e c l a s s i c a l t h e o r y i n d e p e n d e n t o f t h e e x i s t e n c e o f p o t e n t i a l s i s d e s i r a b l e . I n t h i s s e c t i o n a d e v e l o p m e n t i n t h i s d i r e c t i o n i s a t t e m p t e d . T h e u s u a l L a g r a n g i a n d e n s i t y f o r t h e e l e c t r o m a g n e t i c f i e l d ( 1 . 3 ) L = * ( E 2 - B 2 ) , a l t h o u g h L o r e n t z i n v a r i a n t , i s n o t d u a l i t y i n v a r i a n t . T h e q u e s t i o n a r i s e s w h e t h e r a L a g r a n g i a n d e n s i t y L Q e x i s t s w h i c h i s b o t h L o r e n t z a n d d u a l i t y i n v a r i a n t . T h i s s e c t i o n c o n t a i n s a -d e r i v a t i o n , u s i n g t h e m e t h o d s o f s e c t i o n 4 , o f t h e m o s t g e n e r a l L o r e n t z a n d d u a l i t y i n v a r i a n t L a g r a n g i a n d e n s i t y b i l i n e a r i n t h e e l e c t r o m a g n e t i c f i e l d t e n s o r F m n a n d i t s f i r s t d e r i v a t i v e s F m n ^ T h e r e q u i r e m e n t o f d u a l i t y i n v a r i a n c e i s s u f f i c i e n t t o d e t e r m i n e t h e L a g r a n g i a n d e n s i t y L Q u p t o a m u l t i p l i c a t i v e c o n s t a n t . T h e c o m p o n e n t s o f t h e f i e l d t e n s o r c a n t h e n b e t r e a t e d a s f i e l d v a r i a b l e s . I f t h e t h e o r y i s t o b e L o r e n t z c o v a r i a n t , L Q m u s t b e f o r m e d f r o m t h e s c a l a r s a n d s c a l a r d e n s i t i e s o f F m n a n d i t s d e r i v a t i v e s . F o r l i n e a r f i e l d e q u a t i o n s c o n t a i n i n g n o d e r i v a t i v e s h i g h e r t h a n 22 t h e s e c o n d , L Q m u s t b e b i l i n e a r i n F m n a n d F"" 1 "^ > t h e r e f o r e , L Q m u s t b e a l i n e a r c o m b i n a t i o n o f s c a l a r s a n d s c a l a r d e n s i t i e s b i l i n e a r i n F a n d F • k ' . mn F o r a n a r b i t r a r y t e n s o r A t h e r e a r e f o u r t e e n l i n e a r l y i n d e p e n d e n t L o r e n t z s c a l a r s o f t h e r e q u i r e d t y p e ( K a e m p f f e r , 1968): , m n . J1 = A A mn I - AMNA J2 ~ nm J Q = A m n ' k A k m , n A m n , ~k A A A k n , m U - AM AN  J 3 " M m M n J 1 0 = A m , k f l , n m M n k ( 5 . 1 ) J 4 = A m n » k A m n , k I* «. = Am n A i • k  J11 H , n H m k J c = A m n ' k A n m , k J12 = A r a n f n A k m ' k J c = A m » k A n m H n , k J n = Am n m A , m " k n i _ flmn,kfl  J 7 " H H m k , n J 14 flm , k f l , n H m H k n F o r a n a r b i t r a r y s k e w s y m m e t r i c t e n s o r t h e r e a r e o n l y f o u r l i n e a r l y i n d e p e n d e n t s c a l a r s : = F m n F mn ( 5.2) J , = F m n ' k F m k , n Y _ F m n , k r -J2 " h h m n , k p- mn r- , k w h e r e F m n = - F n m . F o r a n a r b i t r a r y t e n s o r A m n t h e r e a r e t e n l i n e a r l y i n d e p e n d e n t s c a l a r d e n s i t i e s o f t h e r e q u i r e d t y p e ( P e l l e g r i n i a n d P l e b a n s k i , 1963): K1 * M k l H m n i / _ c - k l m n r s Q fl k l , m n r , s ( 5 . 3 ) 23 K 3 = £ 9 A k l , m A r n , s K 4 = E 9 H k l , m H r s , n K 5 = £ 9 A k l , r A m n , s K 6 = € 9 A k r , l A m n , s u _ k l m n n r s A . A K 7 - <f 9 H r k , 1 m n f s 1/ _ J< l m n , _ r s n a K g = € 9 M k r , l H m s , n k l m n r s . fl K g = <f 9 A r k , l m s , n k l m n r s f l fl  K 1 0 = <f 9 r k , l A s m , n F o r a s k e w s y m m e t r i c t e n s o r F m n t h e a b o v e s c a l a r d e n s i t i e s r e d u c e t o t h e f o l l o w i n g f i v e s c a l a r d e n s i t i e s : v - _ k l m n r r * k l m n r s F F ,^ k l r n n r s r F ( 5 . 4 ) K 3 = cf ? h k l , r f m n . s • t ^ k l m n n r s F , F K 4 - £ 9 ^ k r , l r m n , s £ _ k l m n r s F r K5 = £ - 9 ^ k r , l m s . n T h e L a g r a n g i a n d e n s i t y lQ i s t h e r e f o r e o f t h e f o r m (5.5) L D ~ m s , d m J m = i ™ m w h e r e a a n d b a r e r e a l c o n s t a n t s . m m O n e c a n n o w r e q u i r e t h a t L Q b e i n v a r i a n t u n d e r t h e d u a l i t y 24 rotation (2,9). It is mathematically convenient, however, to require only that L Q be invariant under the transformation (5.6) F M N — * 7 m n and F — * 7 v ' mn mn obtained by setting 8 = ^~rT in equations (2.9). As w i l l be seen, the invariance of L Q under th i s special case of the duality rotation guarantees i t s invariance under the f u l l duality r o t a t i o n . Under transformation (5.6) L ^ — * L Q where (5.7) L ' = a m J ' + b K ' v ' D t v , = - i m m ^ » i mm and >/ —' .— J 1 r r mn Yi _ rmn,kr-J2 ~ ' fmn,k r. _ f m n , k r •3 ~ hmk,n \ = F M N , nUk' k (5.8) K ' = £ k l m n F K L F M N K - = * k l m V s r ~ k l , / n r , s v. = £ k l m n g r s ~ ~ '37 y kl,rmn,s • _ -klmn rsT T K4 ~ f 9 k r . l mn.s K' =£klmnqTSf:r ,?.n 5 y kr,1 ms,n After a short c a l c u l a t i o n (see Appendix C) one finds (5.9) 2 5 K ^ — K1 K ^ ~ "~ ^ 2 K 3 = - K 3 = -K4 K5 = ~ ^ ( K 4 + K 2 ) ' . • ' T h e r e q u i r e m e n t L Q = L Q g i v e s 2 8 ^ ^ + ^ ( 4 a 2 + a 3 + a 4 ) J 2 + ( a 3 - a 4 ) J 3 ( 5 . 1 0 ) + ( a 4 - a 3 ) J 4 + 2 b 1 K 1 + i ( 4 b 2 + b 5)i< 2 + 2 b 3 * < 3 + i ( 4 b 4 + b 5 ) K 4 + b ' 5 K 5 = 0 . S i n c e J m a n d K a r e l i n e a r l y i n d e p e n d e n t m m . 2 a ^ = 0 2 a 2 + ? ( a 3 + a 4 ) = 0 ( 5 . 1 1 ) a 3 - a 4 = 0 2 b 1 = 0 2 b 3 = 0 2 b 2 + ^ b 5 = 0 b 5 = 0 2 b 4 + ^ b 5 = 0 . T h e s o l u t i o n o f e q u a t i o n s ( 5 . 1 1 ) i s a-j = b-| = b 2 = b 3 = b 4 = b ^ = 0 ( 5 . 1 2 ) a 2 = - ' ? a 3 ~ - ^ a 4 B y c h o o s i n g a 2 = 1 o n e c a n w r i t e L Q a s ( 5 . 1 3 ) L Q = F m n » k F m n . - 2 F m n ' k F . n - 2 F m n F ' k m n , k m k , n , n m k 26 O n l y s c a l a r s a p p e a r i n t h e L a g r a n g i a n d e n s i t y L Q , a n d i t s L o r e n t z i n v a r i a n c e i s t h u s m a n i f e s t . A l t e r n a t i v e l y , i f L Q i s w r i t t e n i n t e r m s o f t h e c o m p l e x f i e l d v e c t o r s F a n d F * , . _ 2 £ * . l f - ' . f 2 f *. (v X . F ) _ l f . ( v x F * ) 7 L D ~ 5 t dt + x L ^ t ~ at ~ J (5.14) + • ( v X F * ) ' ( * 7 X F ) - ( V . F * ) ( V « F ) , i t s d u a l i t y i n v a r i a n c e b e c o m e s m a n i f e s t o n a c c o u n t o f t r a n s f o r m a t i o n ( 0 . 3 ) . 2 7 6 . T H E I T E R A T E D M A X W E L L E Q U A T I O N S AND C O N S E R V E D C U R R E N T . I f t h e c o m p o n e n t s o f t h e e l e c t r o m a g n e t i c f i e l d t e n s o r F m n a r e t r e a t e d a s f i e l d v a r i a b l e s , o n e o b t a i n s f r o m H a m i l t o n ' s p r i n c i p l e ( 6 . 1 ) S ^ L D ( p m n , k ) d * x = 0 t h e f i e l d e q u a t i o n s ( 6 . 2 ) . F m"' k k - 4 F m k ' n k = 0 . S i n c e L Q i s d u a l i t y i n v a r i a n t , e q u a t i o n s ( 6 . 2 ) i m p l y t h a t ( 6 . 3 ) 7 m n ' k k - 4 F m k ' n k = 0 . F m n , k k a n d F m n ' k k a r e s k e w s y m m e t r i c i n ( m , n ) ; t h e r e f o r e , e q u a t i o n s ( 6 . 2 ) a n d ( 6 . 3 ) c a n b e r e w r i t t e n a s r m n , k , m k , n n k , m F k - 2 ( F k - F k ) = 0 ( 6 . 4 ) 7 mn'\ - 2 ( ? m k ' n k - ? n k ' m k ) = 0 . : • -I n t e r m s o f t h e c o m p l e x f i e l d v e c t o r F t h e f i e l d e q u a t i o n s ( 6 . 4 ) b e c o m e ( 6 . 5 ) ~d F / 3 t 2 - V 2 F - 2 ^ x ( K 7 * r - i ^ f / 3 t ) = 0 . T h e s e e q u a t i o n s p e r m i t l o n g i t u d i n a l m o d e s a s s o l u t i o n s . I f t h e s e m o d e s a r e e l i m i n a t e d b y t h e i m p o s i t i o n o f t h e s u b s i d i a r y c o n d i t i o n ( 0 . 2 b ) V . _ F = 0 , . -2 8 t h e n e q u a t i o n ( 6 . 5 ) c a n b e w r i t t e n a s ( 6 . 6 ) (vy - i ^ t ) ( v x - i 3 / * t ) F = 0 w h i c h i s i d e n t i c a l w i t h t h e i t e r a t e d M a x w e l l ' s v a c u u m e q u a t i o n s . I t i s c l e a r t h a t M a x w e l l ' s v a c u u m e q u a t i o n s i m p l y t h e d u a l i t y i n v a r i a n t f i e l d e q u a t i o n s . S i m i l a r l y , i f e q u a t i o n s ( 6 . 2 ) a n d ( 6 . 3 ) a r e s o l v e d w i t h t h e s u b s i d i a r y c o n d i t i o n ( 6 . 7 ) F m n ' k k = 0 M a x w e l l ' s e q u a t i o n s a r e o b t a i n e d i n t h e f o r m ( 2 . 1 2 ) . I n t h e u s u a l f i e l d - t h e o r e t i c a l f o r m u l a t i o n o f v a c u u m e l e c t r o d y n a m i c s t h e L a g r a n g i a n d e n s i t y i s ( 6 . 8 ) L = - 1 / 4 F m n F m n v ' ' mn * w i t h t h e s u b s i d i a r y c o n d i t i o n ( 6 . 9 ) F m P > = A m n - A „ m  x ' mn m , n n , m w h e r e A m i s t h e f i e l d 4 - p o t e n t i a l . T h i s c o n d i t i o n h o l d s i f a n d o n l y i f v ~ m n ( 2 . 6 b ) F n = 0 . T h e s u b s i d i a r y c o n d i t i o n ( 6 . 9 ) i s t h u s e q u i v a l e n t t o a s s u m i n g e q u a t i o n ( 2 . 6 b ) . T h e c o m p o n e n t s o f t h e f i e l d p o t e n t i a l A m a r e t r e a t e d a s t h e f i e l d v a r i a b l e s t o o b t a i n M a x w e l l ' s e q u a t i o n s ( 2 . 6 a T h i s a p p r o a c h i s f a r f r o m s a t i s f a c t o r y s i n c e t h e A m a r e n o t m e a s u r a b l e q u a n t i t i e s . T h e f i e l d p o t e n t i a l i s d e f i n e d o n l y u p t o a g a u g e t r a n s f o r m a t i o n , a n d a p p e a r s o n l y a s a n i n t e r m e d i a r y 2 9 q u a n t i t y i n c a l c u l a t i o n s . T h e a p p r o a c h u s i n g t h e d u a l i t y i n v a r i a n t L a g r a n g i a n d e n s i t y L Q a v o i d s t h i s p r o b l e m b y t r e a t i n g t h e c o m p o n e n t s o f t h e f i e l d ' t e n s o r F m n a s t h e f i e l d v a r i a b l e s . I t i s i n t e r e s t i n g , n e v e r t h e l e s s , t o c o n s i d e r t h e e f f e c t o f t h e s u b s i d i a r y c o n d i t i o n ( 6 . 9 ) o n t h e d u a l i t y i n v a r i a n t f o r m u l a t i o n . S i n c e e q u a t i o n ( 6 . 9 ) i m p l i e s ( 2 . 6 b ) o n e o b t a i n s f r o m e q u a t i o n ( 6 . 3 ) ( 6 . 1 0 ) - - p m n , ! ^ = Q a n d a s a r e s u l t ( 6 . 7 ) F k = 0 } t h e r e f o r e , c o n d i t i o n ( 6 . 9 ) a l s o g i v e s M a x w e l l ' s v a c u u m e q u a t i o n s i n t h e f o r m ( 2 . 1 2 ) . T h e f i e l d p o t e n t i a l A m c a n a l s o b e t r e a t e d a s t h e f i e l d v a r i a b l e i n t h e d u a l i t y i n v a r i a n t a c t i o n p r i n c i p l e ( 6 . 1 ) . F r o m e q u a t i o n ( 5 . 1 3 ) o n e h a s ( 6 1 1 ) l - A A k ' m A R 2 A m ' k A n 7 A k ' m A k ^ • ' • J L D 4 H k M m , n l R k H m , n - l R k M n ^ m • H a m i l t o n ' s p r i n c i p l e n o w y i e l d s ( 6 . 1 2 ) 2ig . 2 . l i D ' • , -2— 1±0 = o o r ( 6 . 1 3 ) A m ' s n n m - A S ' m n m n = 0 v ' nm mn o r 30 Equation (6.14) i s implied by the duality invariant equation (6.2) since i f equation (6.2) is d i f f e r e n t i a t e d with respect to x m one obtains /, .- \ _mn,k , mk,n „ ( 6- 1 5) F km " * F km = 0 • The skew symmetry of F implies that F km = 0 so that (6.14) resu l t s from equation (6.15). The duality rotation is a one parameter continuous trans-formation; therefore, by Noether's f i r s t theorem a conserved cur-rent e x i s t s . From the i n f i n i t e s i m a l duality rotation / r «r \ r,!nn rmn "\ (6.16) F = F + 8 F one obtains (6.17) & L Q = B ILLD T M N + elL.D 7 M N "c>Fmn ^ p m n , k the f i e l d equations allow equation (6.17) to be rewritten as (6.18) g L n = 8 ^ I ijrD F * ° dx k I ^ F m n mn Since the Lagrangian density L n i s duality invariant % L Q = 0 and ( 6 . 1 9 ) = o or (6.20) ~%jk/dxk = 0 31 w h e r e j = ^ . j j ) n F m n . A s i m p l e c a l c u l a t i o n s h o w s t h a t ( 6 . 2 1 ) l ! : D ' _ 2 F , k , 4 F k - , l r k " r > F m n > k " m n s o t h a t ( 6 . 2 2 ) ) k = 2 F m n F ' k + 4 F m n F k m n - 4 F m k F ' n v 7 J m n T . m , n m n 3 2 7 . T H E D U A L I T Y R O T A T I O N I N Q U A N T U M M E C H A N I C S M a x w e l l ' s e q u a t i o n s ( 0 . 2 ) c a n b e w r i t t e n a s a S c h r o e d i n g e r e q u a t i o n f o r m o m e n t u m - h a n d e d n e s s . I f t h i s i s d o n e , t h e q u a n t u m m e c h a n i c a l s i g n i f i c a n c e . o f t h e d u a l i t y r o t a t i o n b e c o m e s a p p a r e n t a n d a d u a l i t y o p e r a t o r c a n b e i n t r o d u c e d . T h e r e i s t h e n a n e x a c t c o r r e s p o n d e n c e t h e d u a l i t y r o t a t i o n a n d t h e P a u l i t r a n s f o r a t i o n o f t h e s e c o n d k i n d i n n e u t r i n o t h e o r y . C o n s i d e r t h e q u a n t u m m e c h a n i c a l m o m e n t u m o p e r a t o r P = - i V , E q u a t i o n ( 0 . 2 ) c a n b e w r i t t e n ( 7 . 1 ) o r ( 7 . 2 ) w h e r e 0 - i p 3 i P - 0 i P 2 i P i H F H = S « P = l at A i \ F 3 ( 7 . 3 ) S , = S o = S 3 = T h e 3X" 3 m a t r i c e s S ^ a r e a n g u l a r m o m e n t u m o p e r a t o r s o f s p i n J = 1 . T h e y s a t i s f y t h e c o m m u t a t i o n r e l a t i o n s 33 S^S* = 21 where c^^y * s the three dimensional L e v i - C i v i t a tensor.. For F: the equation corresponding to ( 7 . 2 ) i s (7.5) H F* = - i ^F*/3t C l a s s i c a l l y , S'P i s the magnitude of the momentum P times the component of the spin angular momentum S in the di r e c t i o n of • the momentum. To convert to quantum mechanics one uses the well-known recipe of replacing c l a s s i c a l quantities by th e i r cor-responding quantum mechanical operators; therefore, (7.6) § « P — » S « P H i s seen to be the momentum-handedness operator, and equations ( 7 . 2 ) and (7.5) are Schroedinger equations in momentum-handedness with F and F* as state vectors. Consider a beam of photons with energy and momentum k One has (7.7) F = l ^ > ei ( k ' x - w t } F* = |<*> e - i ( k * x " w fc) whereto = k . T h e substitution of (7.7) into equations ( 7 . 2 ) and (7.5) yields S »k F = coF 3 4 T h u s F r e p r e s e n t s a r i g h t - h a n d e d p h o t o n a n d F * r e p r e s e n t s a l e f t - h a n d e d p h o t o n . T h e d u a l i t y r o t a t i o n ( 0 . 3 ) i s n o w s e e n t o • b e a p h a s e t r a n s f o r m a t i o n o f t h e q u a n t u m m e c h a n i c a l s t a t e v e c t o r s ^ F a n d F * . N o t e , h o w e v e r , t h a t t h e p h a s e f a c t o r o f F * i s t h e c o m p l e x c o n j u g a t e o f t h e p h a s e f a c t o r o f _F . T h i s f a c t d i s t i n g u i s h e s t h e d u a l i t y r o t a t i o n f r o m a t r u e p h a s e t r a n s f o r m a t i o n w h e r e a l l t h e s t a t e v e c t o r s a r e m u l t i p l i e d b y t h e s a m e p h a s e f a c t o r . I t i s c o n v e n i e n t t o f i n d a d u a l i t y o p e r a t o r f o r t h e p u r p o s e o f c o m p a r i n g t h e d u a l i t y r o t a t i o n a n d t h e P a u l i t r a n s f o r m a t i o n o f t h e s e c o n d k i n d . L e t ' F F * . .o< / 0 - S , 4 \ 3 « x U * / 0 I I 0 f i s a 6 X 1 m a t r i x a n d t h e £ m a r e 6 X 6 m a t r i c e s . E q u a t i o n s ( 7.2) a n d ( 7 . 5 ) n o w r e a d ( 7 . 1 0 ) £ m 2 / 3 x m * = 0 . T h e £ m s a t i s f y t h e c o m m u t a t i o n r e l a t i o n s < 7 ' 1 1 ) C i", S*l = - ' S V o< w h e r e ^ = i 0 S rf 35 The matrix I 0 0 - I has the same properties as the imaginary unit i since (7.12) ( S 5 ) 2 = - I , ( S 5 ) 3 = - f, and ( J 5 ) 4 = I . Therefore, (7.13) . e~ H° = I cos 8 - £ sin 8 . By writing f r y F 0 0 ' F* one obtains -9$ 5 fr\ _ „-i9 (F) = e " i 8 fF) (7.14) 5 e- B (F*) = e + i 9 (V*) e" i s the desired duality operator. It can now be shown that the duality transformation corresponds to the Pauli transformation of the second kind in neutrino theory. The f i e l d equations for the neutrino (see Lurie^, 1968; Kaempffer, 1965) are given by (7.15) y k ) - V 7 ^ x k = 0 3 6 w h e r e t h e a r e 4 * 4 m a t r i c e s s a t i s f y i n g t h e a n t i c o m m u t a t i o n r e l a t i o n s ( 7 . 1 6 ) Wk , = 2 g k l = ( L ! i s a 4 x 1 m a t r i x , R a n d L b e i n g t h e s t a t e f u n c t i o n s o f t h e r i g h t - a n d l e f t - h a n d e d n e u t r i n o r e s p e c t i v e l y . I f a s a b o v e o n e s e t s - * t ( 11) ( 7 . 1 7 ) ( K l = { * ) t h e n e - Q ^ ( f O = e - i f l ( R ; ( 7 . 1 8 ) 5 e " 9 t ( O = e + i 9 ( O E q u a t i o n s ( 7 . 1 8 ) a r e c a l l e d t h e P a u l i t r a n s f o r m a t i o n o f t h e s e c o n d k i n d . T h e a b o v e t r a n s f o r m a t i o n c l e a r l y c o r r e s p o n d s t o t h e d u a l i t y r o t a t i o n . i 37 8 . C O N C L U S I O N T h i s t h e s i s d e m o n s t r a t e s t h a t t h e d u a l i t y i n v a r i a n c e o f M a x w e l l ' s v a c u u m e q u a t i o n s c o n t a i n s g r e a t e r s i g n i f i c a n c e t h a n i s g e n e r a l l y b e l i e v e d . T h e c o n s i d e r a t i o n s o u t l i n e d h e r e h a v e b e e n m a i n l y c l a s s i c a l i n n a t u r e . M o s t i n t e r e s t i n g i s t h e p o s s i b i l i t y o f e x t e n d i n g t h e s e r e s u l t s b y t h e m e t h o d s o f q u a n t u m f i e l d t h e o r y B o t h S c h w i n g e r ( 1 9 6 9 ) a n d K a e m p f f e r ( 1 9 7 0 ) h a v e u s e d t h e d u a l i t y i n v a r i a n c e o f M a x w e l l ' s v a c u u m e q u a t i o n s a s a s t e p p i n g s t o n e t o a t h e o r y o f e l e m e n t a r y p a r t i c l e s . W i t h t h e e v i d e n c e n o w a t h a n d s u c h e f f o r t s m u s t b e c o n s i d e r e d t o b e i n t h e r e a l m o f s p e c u l a t i o n I t i s t h e a u t h o r ' s h o p e , h o w e v e r , t h a t t h i s t h e s i s p r o v i d e s s o m e j u s t i f i c a t i o n f o r f u r t h e r TXiork a l o n g t h e s e l i n e s . 3 8 N O T E S 1 . N a t u r a l u n i t s (rf = c = 1 ) a r e u s e d t h r o u g h o u t t h e t h e s i s . E l e c t r o m a g n e t i c q u a n t i t i e s a r e r a t i o n a l i z e d . 2 . T h e a u t h o r i s i n d e b t e d t o F . P e e t f o r t h i s r e f e r e n c e . 3 . F o r t h e s i g n i f i c a n c e o f t h e t e r m " p s e u d o " s e e A p p e n d i x A . 4 . I i s a l w a y s a u n i t m a t r i x w h o s e d i m e n s i o n a l i t y i s d e t e r m i n e d c A c A b y c o n t e x t . T h e c o m p o n e n t s o f I a r e o g w h e r e & g i s t h e K r o n e c k e r s y m b o l . 5 . G r e e k i n d i c e s r u n f r o m o n e t o t h r e e ; l a t i n i n d i c e s , f r o m o n e t o f o u r . R e p e a t e d i n d i c e s a r e s u m m e d . T h e i n d i c e s 1 , 2 , 3 , 4 r e f e r , r e s p e c t i v e l y , t o t h e s p a t i a l c o o r d i n a t e s x , y , z , a n d t o t h e t i m e c o o r d i n a t e t . 6 . T h e d u a l i s v a r i o u s l y ' d e f i n e d i n P h y s i c s l i t e r a t u r e . T h e d e f i n i t i o n h e r e f o l l o w s W i t t e n ( 1 9 6 2 ) . R o b e r t s o n a n d N o o n a n ( 1 9 6 8 ) o m i t t h e f a c t o r ( - g ) . C o r s o n ( 1 9 5 3 ) r e p l a c e s ( - g ) w i t h ( g ) 2 . F o r t h e s i g n i f i c a n c e o f t h e f a c t o r s e e A p p e n d i x A . 7 . T h e c o n f o r m a l g r o u p c o n t a i n s t h e L o r e n t z t r a n s f o r m a t i o n s , t h e t r a n s l a t i o n s , t h e d i l a t i o n s , a n d t h e s p e c i a l c o n f o r m a l t r a n s -f o r m a t i o n s . 8 . T h i s s e c t i o n i s b a s e d o n a p a p e r a c c e p t e d f o r p u b l i c a t i o n i n t h e C a n a d i a n J o u r n a l o f P h y s i c s ( L e v m a n , 1 9 7 0 ) . 3 9 L I T E R A T U R E C I T E D C a l k i n , HI. G . 1 9 6 5 . A m . J . P h y s . 3 3 , 9 5 8 C o r s o n , E . M . 1 9 5 3 . I n t r o d u c t i o n t o t e n s o r s , s p i n o r s a n d r e l a t i v i s t i c w a v e e q u a t i o n s ( B l a c k i e a n d S o n L i m i t e d , L o n d o n ) I s h a m , C . J . , S a l a m , A . a n d S t r a t h d e e , J . 1 9 7 0 . P h y s . L e t t . • 31_B, 3 0 0 K a e m p f f e r , F . A . 1 9 6 5 . C o n c e p t s i n q u a n t u m m e c h a n i c s ( A c a d e m i c P r e s s , New Y o r k ) K a e m p f f e r , F . A . 1 9 6 8 . P h y s . R e v . 1 6 5 , 1 4 2 0 K a e m p f f e r , F . A . 1 9 7 0 . U n p u b l i s h e d n o t e ( U n i v e r s i t y o f B r i t i s h C o l u m b i a ) L e v m a n , G . IY|. 1 9 7 0 . C a n , J . P h y s . ( t o b e p u b l i s h e d ) L u r i e , D . 1 9 6 8 . P a r t i c l e s a n d f i e l d s ( I n t e r s c i e n c e P u b l i s h e r s I n c . , New Y o r k ) miner, S . R . 1 9 2 7 . P r o c . R o y . S o c . A 1 1 4 , 2 3 M i s n e r , C . a n d W h e e l e r , J . 1 9 5 7 . A n n a l s o f P h y s i c s 2 , 5 2 5 P e l l e g r i n i , C . a n d P l e b a n s k i , J . 1 9 6 3 . K g l . D a n s k e V i d e n s k a b . S e l s k a b M a t . - P h y s . 2, n o . 4 R o b e r t s o n , H . P . a n d N o o n a n , T . W. 1 9 6 8 . R e l a t i v i t y a n d c o s m o l o g y ( W . B . S a u n d e r s C o m p a n y , T o r o n t o ) 40 S c h o u t e n , J . A. 1 9 5 1 . T e n s o r c a l c u l u s f o r p h y s i c i s t s ( C l a r e n d o n P r e s s , O x f o r d ) S c h r o e d e r , U . E. 1 9 6 8 . F o r t s c h r i t t e d e r P h y s i k 1_6, 3 5 7 S c h w i n g e r , J . 1 9 6 9 . S c i e n c e 1 6 5 , 7 5 7 W i t t e n , L . 1 9 6 2 . G r a v i t a t i o n ; a n i n t r o d u c t i o n t o c u r r e n t r e s e a r c h , L . W i t t e n e d . , p . 3 8 2 ( J o h n W i l e y a n d S o n s , I n c . N e w Y o r k ) 41 A P P E N D I X A : T H E C O N F O R M A L G R O U P B e f o r e p r o c e e d i n g t o a d i s c u s s i o n o f t h e c o n f o r m a l g r o u p o f t r a n s f o r m a t i o n s C s o m e t e r m i n o l o g y w h i c h i s o f t e n c o n f u s e d i n P h y s i c s l i t e r a t u r e m u s t b e c l a r i f i e d ( s e e S c h o u t e n , 1 9 5 1 ) . C o n s i d e r a g r o u p A o f c o o r d i n a t e t r a n s f o r m a t i o n s x , m _. x . m ( x n ) ( A . 1 ) J = d e t ( 3 x / 3 x ' ) \ 0 . A g e o m e t r i c a l o b j e c t w i t h r e s p e c t t o A i s a s e t o f f u n c t i o n s G = G a t 5 C ( x m ) t r a n s f o r m i n g u n d e r A a c c o r d i n g t o ~\ i ~\ f / n 0 \ r , i j k . . . _ I . i T ,W . . . SJL . . . a b c . . . ( A - 2 ) G l m n . . . = t J l J } x a ^ x ' n G d e f . . . ( i , j , k , . . . ) a r e c o n t r a v a r i a n t i n d i c e s ; ( l , m , n , ) a r e c o v a r i a n t i n d i c e s . I f T = Ul = 0, t h e n G i s s a i d t o b e a t e n s o r w i t h r e s p e c t t o A . I f T = 0, G i s a r e l a t i v e t e n s o r o f w e i g h t U l . F o r T = 0 a n d W = 1 G i s a t e n s o r d e n s i t y s i n c e §Gdx. i s a t e n s o r . I f UJ = 0, G i s a n a b s o l u t e r e l a t i v e t e n s o r o f w e i g h t T . L a s t l y , i f UJ = - T = ± 1 , G i s c a l l e d a p s e u d o t e n s o r . A p s e u d o t e n s o r t r a n s f o r m s l i k e a t e n s o r u p t o a s i g n . C o n s i d e r , a s a n e x a m p l e , t h e L e v i - C i v i t a s y m b o l £" . W i t h r e s p e c t t o t h e g e n e r a l g r o u p o f n o n - s i n g u l a r c o o r d i n a t e t r a n s f o r m a t i o n s £ i s a r e l a t i v e t e n s o r o f w e i g h t + 1 ; h o w e v e r , £ m a y b e c o n s i d e r e d a L o r e n t z p s e u o d t e n s o r a n d a t r a n s l a t i o n t e n s o r . I t i s t h e p u r p o s e o f t h i s a p p e n d i x t o f i n d t h e m o s t g e n e r a l g r o u p o f t r a n s f o r m a t i o n s u n d e r w h i c h M a x w e l l ' s v a c u u m e q u a t i o n s a r e c o v a r i a n t . I f M a x w e l l ' s e q u a t i o n s (2.6) 42 F m n = 0 F m n . n = 0 a r e t o b e c o v a r i a n t u n d e r a g r o u p o f t r a n s f o r m a t i o n s R, t h e n F n a n d F , n m u s t b e g e o m e t r i c a l o b j e c t s w i t h r e s p e c t t o R . T h i s r e q u i r e m e n t a c t u a l l y d e f i n e s w h a t i s m e a n t b y t h e c o v a r i a n c e o f (2.6) w i t h r e s p e c t t o R . N o t e t h a t M a x w e l l ' s e q u a t i o n s a r e L o r e n t z c o v a r i a n t a n d t h a t F m n i s a L o r e n t z c o n t r a v a r i a n t t e n s o r ; t h e r e f o r e , w e r e q u i r e t h a t R c o n t a i n t h e L o r e n t z g r o u p a n d t h a t t h e t r a n s f o r m a t i o n p r o p e r t y o f Fmn u n d e r R r e d u c e t o t h e t e n s o r t r a n s f o r m a t i o n u n d e r t h e L o r e n t z g r o u p . S i n c e ( A . 3 ) j ( L o r e n t z ) = i 1 t h e a b o v e c o n d i t i o n r e q u i r e s UI = 0 f o r R s o t h a t F m n i s a t m o s t a n R a b s o l u t e r e l a t i v e t e n s o r . C o n s i d e r t h e s c a l e t r a n s f o r m a t i o n ( d i l a t i o n ) / n / \ tffi m ( A .4) x ' = s x . ; : U n d e r t h i s t r a n s f o r m a t i o n c h a r g e a n d c u r r e n t a r e i n v a r i a n t . S i n c e 2 d i m e n s i o n (F_) = c h a r g e / ( d i s t a n c e ) ( A . 5 ) 2 d i m e n s i o n ( B ) = c u r r e n t / ( d i s t a n c e ) o n e h a s 43 M a x w e l l ' s v a c u u m e q u a t i o n s ( 0 . 1 ) a r e c l e a r l y i n v a r i a n t u n d e r ( A . 4 ) a n d ( A . 5 ) . T h e d i l a t i o n g r o u p i s t h e r e f o r e a s u b g r o u p o f R . F r o m e q u a t i o n ( A . 6 ) i t f o l l o w s t h a t ( A . 7 ) F m n - » F ' m n = ( s ) " 2 F m P ; h o w e v e r , f r o m e q u a t i o n ( A . 2 ) w i t h W = 0 ( A . 8 ) ' F m n F ' m 0 = ( s - 4 ) T s 2 F m n s i n c e ( A . 9 ) J ( d i l a t i o n ) = s - 4 . . v c m • ^ x n ( d i l a t i o n ) = s n F o r e q u a t i o n s ( A . 7 ) a n d ( A . 8 ) t o b e c o n s i s t e n t T = 1 . S i m i l a r l y , mn ( A . 1 0 ) . F m r , = q g F . mn y m r y n s r s g m n m u s t h a v e w e i g h t T = - 7 . - A s a r e s u l t , u n d e r t h e g r o u p R g t r a n s f o r m s a s ^ m n ( A . 1 1 ) g ' m n =\J| * ^ . m ^ . n g r s T h i s e q u a t i o n d o e s n o t r e d u c e c o r r e c t l y t o t h e L o r e n t z f o r m u n l e s s q ' = q . T h i s c o n d i t i o n i s a l s o n e c e s s a r y i f i n d i c e s y mn J m n ' a r e t o b e r a i s e d a n d l o w e r e d i n a c o n s i s t e n t m a n n e r . H e n c e e q u a t i o n ( A . . 1 1 ) b e c o m e s 44 T h e g r o u p o f t r a n s f o r m a t i o n s C s a t i s f y i n g e q u a t i o n ( A . 1 2 ) i s c a l l e d t h e c o n f o r m a l g r o u p ( s e e I s h a m ; 1 9 7 0 ) . I t w i l l n o w b e s h o w n t h a t M a x w e l l ' s v a c u u m e q u a t i o n s ( 2.6) a r e c o v a r i a n t u n d e r a n y g r o u p o f t r a n s f o r m a t i o n s f o r w h i c h F m n i s a n a b s o l u t e r e l a t i v e t e n s o r o f w e i g h t T = 1 . F r o m t h e c o n s i d e r a t i o n s a b o v e , t h e o n l y p h y s i c a l l y a d m i s s a b l e t r a n s f o r m a t i o n s a r e t h o s e s a t i s f y i n g e q u a t i o n ( A . 1 2 ) ; t h e r e f o r e , t h e c o n f o r m a l g r o u p C f o r m s t h e f u l l c o v a r i a n c e g r o u p o f M a x w e l l ' s e q u a t i o n s . I f F m n i s an a b s o l u t e r e l a t i v e c o n t r a v a r i a n t t e n s o r o f w e i g h t T = 1 , t h e n F ' m n , n = (!J^ x ' n ) ( 3 x'7^ x r ) ( ^ x ' n / ^ S ) F r S + i J l O a x « m / ^ x r 3 x s ) F r s ( A . 1 3 ) • + U l ( 3 x'7 ^ x ' n ) ( 3 x * m / 3 x r ) ( ^ x ' n / a x s ^ x r ) F + U H 3 x ' 7 3 x r ) ( ^ x ' n / a x s ) ) F r S / ^ x ' n . T h e s e c o n d t e r m on t h e r i g h t o f e q u a t i o n ( A . 1 3 ) v a n i s h e s d u e t o t h e s k e w s y m m e t r y o f F m n . S i n c e ( A . 1 4 ) -J"1 ^j/dxm = ( ^ x k / > x ' n ) ( / x ' n / ^ x m ^ x k ) t h e f i r s t a n d t h i r d t e r m s u b t r a c t t o z e r o . E q u a t i o n ( A . 1 3 ) n o w b e c o m e s ( A . 1 5 ) F ' m n , n =U\ p x ' m / 2 x r ) F r n , n so t h a t F m n , n i s a n a b s o l u t e v e c t o r o f w e i g h t T = 1 . O n e c a n n o w c o n c l u d e t h a t F m n , n = 0 i m p l i e s F o m n , n = 0 . S i m i l a r l y , 45 F = 0 i s a covariant equation. Note, however, that F m n transforms as a r e l a t i v e tensor of weight UI = 1 . The factor (-g) ^ appearing in the d e f i n i t i o n of F m n is unnecessary for Lorentz covariance, but is. important in t h i s context. Note also that the duality rotation (2.9) is conformal covariant i f 8 i s a conformal pseudoscalar. In summary, the f u l l covariance group of Maxwell's vacuum equations i s the conformal group defined by equation (A.12) . Under the conformal group F m P is an absolute r e l a t i v e tensor of weight T = 1, F i s a tensor density, and F m n i s a tensor. The duality rotation parameter 8 transforms as a pseudoscalar. If equation (A.12) is solved one finds that the conformal group consists of Lorentz transformations, t r a n s l a t i o n s , d i l a t i o n s , and the special conformal transformations (A.16) x' m = ( x m • x n x N / 3 M )( 1 2x h /* n- + / « > n x r x r ) ' 1 where/3 m are constants. It is int e r e s t i n g to note that the conformal group can be defined as the group of transformations for which (A.17) dx m dx m = 0 s ' m is a covariant equation (see Robertson and.Noonan, 1968). This d e f i n i t i o n of C implies that the conformal group i s the group of transformations which leaves the light-cone invariant. F i n a l l y , the term "conformal" is used since the group C i s also the group of transformations leaving angles invariant. In conclusion note that Maxwell's equations with charge ( A . 1 8 ) 46 p m n _ .m rmn „ = o » n a r e a l s o c o n f o r m a l c o v a r i a n t . J m , t h e c h a r g e c u r r e n t d e n s i t y , m u s t b e , b y e q u a t i o n ( A . 1 8 ) , a n a b s o l u t e r e l a t i v e t e n s o r o f w e i g h t T = 1 . T h i s t r a n s f o r m a t i o n p r o p e r t y f o r J m i s c o n s i s t e n t w i t h e q u a t i o n ( A . 5 ) s i n c e a v o l u m e \l t r a n s f o r m s u n d e r d i l a t i o n s 3 a c c o r d i n g t o V ' = s V . 4 7 A P P E N D I X B : ON E Q U A T I O N S ( 3 . 8 ) AND ( 3 . 9 ) S i n c e p , m n = _ F ' m o n e o b t a i n s f r o m e q u a t i o n ( 3 . 8 ) . 1 1 1 11 1 21 1 31 1 41 0 = F ' = a 1 1 F M + a 2 F + a 3 F + a ^ F 1 . ~ 1 1 1 - 2 1 1 - 3 1 . 1 ~ 4 1 + c ( B . 1 ) + b 1 F + b 2 F + b 3 ( r + b 4 F = - a 1 B 3 + a 3 8 2 + a 4 E<| - b 2 E 3 1 1 + b 3 E 2 - b 4 B-| . A n y t w o c o n s t a n t f i e l d s E a n d B s a t i s f y M a x w e l l ' s v a c u u m e q u a t i o n s . A s a r e s u l t , f o r e q u a t i o n ( 8 . 1 ) t o b e v a l i d f o r a r b i t r a r y c o n s t a n t v e c t o r s E a n d B_ w e m u s t h a v e ( B . 2 ) a 1 2 = a 1 3 = a 1 4 = b 1 2 = b 1 3 = b 1 4 = 0 . S i m i l a r l y , a l l o t h e r o f f - d i a g o n a l e l e m e n t s o f a m ( < , b 1 7 1 ^ , c™^ a n d d r T ' | < a r e z e r o ; e q u a t i o n ( 3 . 1 0 ) r e s u l t s . T h e e x p a n s i o n o f e q u a t i o n ( 3 . 1 0 ) i n t e r m s o f E, B , E' a n d B ' g i v e s F ' 4 1 = E i = 'a 4 ET - b 4 B 1 ( B . 3 ) • F ' 1 4 = E.J = a-, E<| - b 1 B<i 23 F * - = d 2 E-| + c 2 .B1 . F ' 3 2 » E^ = d 3 ET • c 3 E q u a t i o n s ( B . 3 ) a r e c o n s i s t e n t o n l y i f ( B . 4 ) a-j = a 4 s d 2 = d 3 , b i = b 4 = - c 2 = - c 3 48 Continuing in the same manner one easily v e r i f i e s equation (3.11) . With t h i s step the reduction of equation (3.1) to equation (3.12) i s completed. 49 A P P E N D I X C : . ON E Q U A T I O N S (5.9) I n t h e d e r i v a t i o n o f e q u a t i o n s (5.9) t h e f o l l o w i n g i d e n t i t i e s a r e u s e f u l : _ k l m n _ , . r l m n £ ^ k r s t - - 1 ! ^ r s t ( C . 1 ) ^ k l m n £ - , , = - 2 ! v ' c c k l r s ° r s k l m n c  g r k g s l g t m Q u n £ = " g ^ r s t u • £ r s t e c l u a l s + 1 ( l > m , n ) i s a n e v e n p e r m u t a t i o n o f ( r , s , t ) , e q u a l s - 1 i f ( l , m , n ) i s a n o d d p e r m u t a t i o n o f ( r , s , t ) , a n d e q u a l s 0 i n a l l o t h e r c a s e s . S ™g i s d e f i n e d s i m i l a r l y , a n c l ^ r s t c a n h G r e p r e s e n t e d i n t e r m s o f t h e K r o n e c k e r s y m b o l a s r mn a> = d e t c m r m o r * s r s \ n c n o r * 5 ( C . 2 ) r l m n * r s t o <1 » r » s * t r m r m r m d r a s 4 t >^ r S s A s a n e x a m p l e , c o n s i d e r JI = F m n F . F r o m e q u a t i o n s ( C.1) ^ 1 mn a n d ( C . 2 ) i / \ - 2 ^ . k l m n , . _ . J 1 = <^  F m n * ( - Q r <^klrsF r s ( C 3 ) = -* ^ n F F r s v ' w r s mn 

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