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Nuclear PhysicsB (Proc. Suppl.) 6 (1989) 9-64North-Holland, AmsterdamO N U N I T A R Y R E P R E S EN T A T I O N S O F T H E I N H O M O G E N EO U S

L O R E N T Z G R O U P *B y E . W I G N E R

( R e c e i v e d D e c e m b e r 2 2 , 1 9 3 7 )

I . O m < ) m a ND ~ C T ~ - P . L ~ T I O N o F TH E PR OW .F.~I t i s pe r h ap s t h e m o s t f u n d a m e n t a l p r in c ip le o f Q u a n t u m M e c h a n i c s t h a t . t h e

s y s t e m o f s t a t e s f o r m s a l i near man i fo ld , t i n w h i c h a u n i t a r y s c a / a r p r o d u c t i sd e f i n e d , z T h e s t a t e s a r e g e n e r a l l y re p r e s e n t e d b y w a v e f u n c t i o n s ~ i n s u c h a w a yt h a t ~o a n d c o n s t a n t m u l t i p l e s o f ~o r e p r e s e n t t h e s a m e p h y s i c a l s t a t e . I t i sp o s si b le , t h e r e f o r e , t o n o r m a l i z e t h e w a v e f u n c t i o n , i .e ., t o m u l t i p l y i t b y ac o n s t a n t f a c t o r s u c h t h a t i t s s c a l a r p r o d u c t w i t h it se l f b e c o m e s 1 . T h e n , o n l y ac o n s t a n t f a c t o r o f m o d u l u s 1, t h e s o - c a ll e d p h a s e , w i ll b e l e f t u n d e t e r m i n e di n t h e w a v e fu n c t i o n . T h e l i n e a r c h a r a c t e r o f t h e w a v e f u n c t i o n is c a l l e d t h es u p e r p c e i t i o n p r in c ip l e. T h e s q u a r e o f th e m o d u l u s o f t h e u n i t a r y s c ~ h rp r o d u c t ( ~, ~ ) o f t w o n o r m a l i z e d w a v e f u n c t i o n s ~ a n d ~o i s c a l l ed t h e t r a n s i t i o np r o b a b i l i t y f r o m t h e s t a t e i n t o ~,, o r c o n v e r s e l y . T h i s i s s u p p o s e d t o g i v e t h ep r o b a b i l i t y t h a t a n e x p e r i m e n t p e r f o r m e d o n a s y s t e m i n th e s t a t e , t o se ew h e t h e r o r n o t t h e s t a t e i s ~ , g iv e s t h e r e s u l t t h a t i t is ~ . I f t h e r e a r e t w o o rm o r e d i f f e r e n t e x p e r i m e n t s t o d e c i d e t h i s ( e .g . , e s s e n t i a l ly t h e s a m e e x p e r i m e n t ,

" P a r t s o f t h e p r e s e n t p a p e r w e r e p r e s en t e d a t t h e P i t t e b u r g h S y m p o s i um o n G r o u pT h e o r y a n d Q - A - t u m M e c h a n ic s . C f . B u l l. A m e r . M a t h . S o t . , 41, p. 306, 1935.t T h e p c m i b i l i ty o f a f u t u r e n o n l i n e ar c h a r a c t e r o f th e q u a n t u m m e c h a n ic s m u s t b ea d m i t t e d , o f c o u r s e. A n i n d i c a t i o n in t h i s d i re c t i o n i s g i v e n b y t h e t h e o r y o f t h e p o s i t r o n ,u deve lop ed by P . A. M. Di rac (P roc . Cam b. Ph i l . Soc . S0, 150 , 1934, e l . a l so W. He isenherg ,Ze i t s . f . Phy s . 90, 209, 1934; 9t , 623, 1934; W . H e i s e n b e r g a n d H . E u l e r , i b i d . 98, 714, 1936and R . 8erber , Phys . R ev . /~ , 49 , 1935; ~9 , 545 , 1936) wh ich does not us e wave funct ionsa n d i s a n o n l i n e a r t h e o r y .s Cf . P . A . M. Di rae , T#~ Pr im =ip~ o f Q ~z ~u m M ed ~n i~ , O xford 1935, Chapters I andI I ; J . v . N e ,, m ~n n, M ~ c h e . G ~ u z ~ g o f f e n d~rr ~t~z~enmec]uznik, Bedin IW2, pages19-24.' T h e w a v e f u n c t io n s r e p r e s e n t t h r o u g h o u t t h i s p a p e r s t a t e s i n t h e s en s e o f th e " H ~ n -b e r g p i c t u r e , " i . e . a s in g l e w a v e f u n c t i o n r e p r e s e n t s t h e s t a t e f o r al l p a s t a n d f u t u r e . O nt h e o t h e r h a n d , t h e o p e r a t o r w h i c h r e f e rs to s m e a s u r e m e n t a t a c e r t a i n t im e I m m t a i n st h i s f u a imram eter. (Cf . e .g . D irac , l . c . ref . 2 , pages I15-123) . O n e o b t a i n s t h e w a v ef u n c t i o n q ' ,( 0 o f t h e 8 c h r 6 d i n p r p i c t u r e f ro m t h e w a v e f u n c t io n cw o f th e H e i s e n b er zp i c t u r e b y ~ , ( 0 - e x p ( - - iH t /A )wl T h e o p e r a t o r o f t h e H e i s e n b e r s p i c tu r e is O ( 0 -ezp(~H t/~) Qexp --~H g/A),w h e r e O i s t h e o p e r a t o r i n t h e S c h r O d i n s e r p i c t u r e w b i e h d o e s n o td e p e n d o n t i m e . C f . a l s o E . S e i z e r , S i t s. 4 i . K 6 n . P r e u a s . A k s d . p . 4 18 , 1 ~ 0 .T h o w a v e f u n c ti o n s a r e c o m p l e x q u a n t i t i M a n d t h e u n d e t e r m i n e d f ~ t o r s m t ~ m ~r~e om p im x a l so . Re c m a t l y a t t e m p t s h a v e b e e n m a d e t o w a r d a t h e o r y w i t h r e a l w a v e r u n e -t/ or aL Cf . E . M s j o r s n s , N u o v o C i m . 1 4 ,1 71 ,1 9~ 7 a n d P . A . M . D i r s c , i n p r i n t .

Reprinted from Annals of Mathematics,Volume40, No. 1, p.149 (1939).

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10 E.P. Wigner / Unitary representations o f Lo ren tz grou p

p e r f o r m e d a t d i f f e n m t t i m e s ) t h e y a r e a l l s u p p o s e d t o g i v e t h e s a m e r e s u l t ,i .e ., t h e t a m ~ t i n a p r o lm b i l it y h a s a n i n v a r i a n t p h y s i c a l s e in e .T h e w a v e f u n et im m fo r m a d e s c ri p ti o n o f t h e p h y s i c a l s ta t e , n o t a n i n v a r i a n t

h o w e v e r , si n c e t h s m i n e s t a t e w i l l b e d e s c r i b e d in d i f f e r e n t co6rdlnm.te systemsb y d i f f e r e n t w a v e f u n c t io n s . I n o r d e r t o p u t t h i s i n t o e v i d e n c e , w e s h a ll a f f ~a n i n d e x t o o u r w a v e fu n c t io n s , d e n o t in g t h e L o r e n t z f r a m e o i r d e r e n c e f o r w h i e ht h e w a v e f u n e t i ~ i s g iv e n. T h u s ~ a n d ~ . r e p r e s en t t h e s a m e s t at e , b u t t h e ya r e d i f fe r e n t f u n c t io n s . T h e f ir s t i s t h e w a v e f u n c t i o n o f t h e s t a t e i n t h e c o -o r d l n A t e s y s t e m / , t h e s e c o n d i n t h e c o 6 r d i n A t e s y s t e m l '. I f ~ = ~ ,v t h es t a t e 9 b e h a v e s i n t h e e o 6 r d i n a t e s y s t e m l e x a c t l y a s ~ b e h a v e s i n t h e c o 6 r d l n ~ t es T s t em P . I f ~ i s g i v e u , A l l ~ , a r e d e t er m i n e d u p t o a e o u s t a n t f ac to r . B e c au s eo f t h e i n v a r i a n c e o f t h e t r a n d t i o n p r o b a b i l it y w e h a v e( 1 ) l ( ~ , ~ ) I f = I (~ . , , . ) I "a n d i t c a n b e s ho w n * t h a t t h e a f o r e m e n t i o n e d c o n s t a n t s i n t h e ~ , c a n b e c h o s e ni n s u c h a w a y t h a t t h e ~ , a r e o b t a in e d f ro m t h e ~ . b y a l i n e a r u n i t a r y o p e r at io n ,d e p e n d i n g , o f c o u r se , o n I a n d I( 2 ) ~ , . = z ) ( t ' , 0 ~ .T h e u n i t a r y o p e r a t o rs D a r e d e t e r m i n e d b y t h e p h y s i c a l c o n t e n t o f t h e t h e o c yu p t o a c o n s t a n t f a c to r a g a in , w h i c h c a n d e p e n d o n I a n d I '. A p a r t f r o m t h i sc o n s t a n t h o w e v e r , t h e o p e r a t i o n s D ( l ' , l ) a n d D ( ~ , l ~ ) m u s t b e i d e n t i e a l i f l 'a r is e s f r o m ! b y t h e s a m e L o r m t s t m n d o r m a t i o n , b y w h i c h ~ a r is e s f r o m S t.I f t h i s w e r e n o t t r u e , t h e r e w o u l d b e a r e al d i ff e re n c e b e t w e e n t h e f r a m e s o fr e f er e n c e I a n d / i . T h u s t h e u n i t a r y o p e r a t o r D ( l ' , l ) = D ( L ) i s i n e v e r yL o r e n tz i n v a r i a n t q n ~ n t u m m e c h a n ic a l t h e o r y ( a p a r t f r o m t h e c o n s t a n t f a c t o rw h i c h h a s n o p h y s i c al si gn if ic a nc e ) c o m p l e t e l y d e t e r m i n e d b y t h e L o r e n t st r a n s f o r m a t i o n L w h i c h carr ies l in to I ' = L / . O n e c a n w r i t e , i n s t e a d o f ( 2 )(2a) u = D ( L ) ~ .B y g o i n g o v e r f r o m a f ir st s y s t e m o f r e f e r e n c e I t o a s e c o n d l ' - - L J a n d t h e n t o at h i r d l " ffi L ~ l o r d i r e c t l y t o t h e t h i r d l " = - ( / 4 ~ ) l , o n e m u s t o b t a in - - a p a r tf r o m t h e a b o v e m e n t i o n e d e m m t s n t - - ~ h e ~ a m e s e t o f w a v e f u n c ti o n s. H e n c ef r o m

follows

~ . . f f i D ( r ' , r ) D ( ~ ' ,~ , .. = 1 ) ( I " , r ) ~

D(e ' , 190( l ' , 0 = .D(~" ,' g . WiEner, Gr,~plm, lm ~ z mwl ihre Aswey,d u ~ . n a u fd ie Ouanten=~./~m/E d ~ A ~ -pd~hrem. Brmmwhwei l[ Im l , l a l l e s 2M -2 5 4 .

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E.P. Wigner / Unitary representations o f Lor entz group 11

or(3 a ) D ( /4 )D (L , ) f fi o ~ D(1 4L~ ),where co is a number of modulus 1 and can depend o n / 4 and Lt . Thu~ theD ( L ) form, up to a factor, a re ~ n t a t i o n of the inhomogeneou~ Lorentzgroup by linear, -nitary operators.We see thuss that there corresponds to every invariant quantum meclmnicalsystem of equations such a representation of the inhomogeneotLq Lorentz group.This representation, on the other hand, though not sutKcient to replace thequantum mechanical equations entirely, can replace them to a large extent.If we knew, e.g , the operator K corresponding to the measurement of a physicalquantity at the t ime t ffi 0, we could follow up the change of thi-~ quanti tythroughout time. In order to obtain its value for the time t =. t~, we couldtr~ndorm the or;~n~! wave function vn by D(I ' , l ) to a co6rdlnnte system l"the time scale of which begins a time tl later. The measurement of the quan tit yin question in this co6rdinate system for the t ime 0 is given- -as in the originalo n e - b y the operator K. This measurement is indentical, however, with themeamuement of the quantity at time tt in the or~'n,d system. One can say tha tthe representation can replace the equation of motion, it cannot replace, ho~-ever, connections holding between operators at one instant of time.It may be mentioned, finally, that these developments apply not only inquant~tm mechanics, but also to all linear theories, e.g., the MJc[well equationsin empty space. The only difference is that there is no arbit rary factor in thedescription and the o, can be omit ted in (3a) and one is led to real representationsinstead of representations up to a factor. On the other hi nd , the ~__nltary char-acter of the representation is not a consequence of the basic assumptions.The increase in generality, obtained by the present calculus, as comparedwith the usual tensor theory, consists in that no assumpticms regarding thefield nature of the undedying equations are n _ece~m___ry. Thus more generalequations, as far as they exist (e.g., in which the eo6rdinate is quantized, etc.)are also included in the present treatment. It must he realized, however.tha t some assumptions concerning the continuity of space have been made byassuming Lorents frames of reference in the classical sense. We should like tomention, on the other hand, that the previous remarks concerning the time-parameter in the observables, have only an explanatory character, a nd we do notmake assumptions of the kind that measurements can be performed instan-tanemmly.We shall endeavor, in the ensuing sections, to determine all the continuo~~unitary representations up to a factor of the inhomogeneous Lormts group.i.e., all continuous systems of linear, unitary operators satisfying (3a).

' E. Wiper, I.e. G~m4~erXX.' T h e e ~ d e i m i ~ o n o f the eontinuoume ~ t e r of a rq.pr~ent~xtion up t o a fartor w i l lbe given in 8eetiou SA. The definition of the inhomo~neous I~>r~ntz group is eontninedin Seetiom4A.

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12 L:P. Wigner / Unitary representations o f Lore ntz group

2 . C O M PA R IS O N W I T H P R E V I O U S T RE .a.T M EN T 8 A ND S O M E I ~ E D I A T ESIMPLIFICATIONS

A . P r e v i o u s t r e a t m e n t sT h e r e p r e ~ n t a t i o n s o f th e L o r e n t z g r o up h a v e b e e n i n v e s t i g a te d r e p e a t e d ly .T i l e fi rs t i n v e s t ig a t i o n is d u e t o M a j o r a n a / w h o i n f a c t f o u n d a ll r e p r e s e n t a t i o n s

o f t h e c la s s t o b e d e a l t w i t h i ll t h e p r e s e n t w o r k e x c e p t i n g t w o s e t s o f r e p r e s e n t a -t i on s . D i r a c ~ a n d P r o c a 8 g a v e m o r e e l e g a n t d e r i v a t i o n s o f M a j o r a n a ' s r e s u lt sa n d b r o u g h t th e m i n t o a f o r m w h i c h c a n b e h a n d l e d m o r e e a si ly . K l e i n ' sw o r k 9 d oes. n o t e n d e a v o r t o d e r i v e i r r e d u c i b l e r e p r e s e n t a t i o n s a n d s e e m s t o b ei n a l e s s c l o s e c o n n e c t i o n ~ d t h t h e p r e s e n t w o r k .

T h e d if fe re n ce b e t w e e n t h e p r e s e n t p a p e r a n d t h a t o f M a j o r a n a a n d D i r a cl i e s - - a p a r t f r o m t h e f i n d i n g o f n e w r e p r e s e n t a t i o n s - - m a i n l y i n i t s g r e a t e rm a t h e n m t i c a l r ig o r . M a j o r a n a a n d D i r a c fr e e l y u s e t h e n o t i o n o f i n f in i t es i m a lo p e r a t o r s a n d a s e t o f f u n c t i o n s t o a l l m e m b e r s o f w h i c h e v e r y in f i n it e s im a lo p e r a t o r c a n b e a p p l i e d . T h i s p r o c e d u r e c a n n o t b e m a t h e m a t i c a l l y j u s ti f ie da t p r e s e n t , a n d n o s u c h a s s u m p t i o n w i ll b e u s e d i n t h e p r e s e n t p a p e r . A l s o t h ec o n d i t i o n s o f r e d u c i b i l i ty a n d i r r e d u c i b i li t y c o u ld b e , i n g e n e r a l, s o m e w h a t m o r ec o m p l i c a te d t h a n a s s u m e d b y M a j o r a n a a n d D i r a c . F i n a l ly , t h e p r ev i o u st r e a t m e n t s a s s u m e f r o m t h e o u t s e t t l m t t h e s p a c e a n d t i m e c o 6 r d i n a te s w i ll b ec o n t i n u o u s v a r ia b l e s o f t h e w a v e f u n c t i o n in t h e u s u a l w a y . T h i s w i l l n o t b ed o n e , o f c o u r se , in t h e p r e s e n t w o r k .

B . S o m e i m m e d i a t e s i m p l i f i c a t i o n sT w o r e p r e s e n t a t i o n s a r e physical ly equivalent i f t h e r e i s a o n e t o o n e c o r -

r e s p o n d e n c e b e t w e e n t h e s t a t e s o f b o t h w h i c h i s 1 . i n v a r i a n t u n d e r L o r e n t zt r a n s f o r m a t i o n s a n d 2 . o f s u c h a c h a r a c t e r t h a t t h e t r a n s i t i o n p r o b a b i l i t i e sb e t w e e n c o r r es p o n d i n g s ta t e s a r e t h e s a m e .

I t f o ll o w s f r o m t h e s e c o n d c o n d i t i o n s t h a t t h e r e e i t h e r e x i st s a u n i t a r y o p e r a t o rS b y w h i c h t h e w a v e f u n c t i o n s ~ c,~ o f t h e s e c o n d r e p r e s e n t a t i o n c a n b e o b t a i n e df r o m t h e c o r r e s p o n d i n g w a v e f u n c t io n s ~ m o f t h e f ir st r e p r e s e n t a t i o n( 4 ) ~("~ -- 8 ~ ("o r t h a t t h i s i s t r u e f o r t h e c o n j u g a t e i m a g i n a r y o f ~ ( 2~ A l t h o u g h , i n t h el a t t e r c a se , t h e t w o r e p r e s e n t a t i o n s a r e s ti ll e q u i v a l e n t p h y s i c a l l y , w e s h a ll , i nk e e p i n g w i t h t h e m a t h e m a t i c a l c o n v e n t i o n , n o t c al l t h e m e q u i v a l e n t.T h e f ir s t c o n d i t i o n n o w m e a n s t h a t i f t h e s t a t e s ~ ( i) , ~ ) = ~ g4 ,m c o r r e s p o n dt o e a c h o t h e r i n o n e c o S r d i n a t e s y s t e m , t h e s t a t e s D m ( L ) ~ (~) a n d D m ( L ) ~ , cs)c o r r e s p o n d t o e a c h o t h e r a ls o . W e h a v e t h e n(4 a) D(2)(L)4,(~) --- SD ( I ~(L) ,~ ( I) = 8D m ( L) 8 - 1 , ~ (z).

: E . M a j o ra n a , N uov o C i m . 9 , 335, 1932 .P . A . M . D i r a c , P r o c . R o y . S o c . A . 155, 447, 1936 ; AI . P ro c a , J . de P h ys . R a d . 7 , 347 ,1936.' K l e i n , A r k i v f . l ~ t e m . A s t r . o c h F y s i k , 2 5 A , .N o . 1 5, 1 93 6. I a m i n d e b t e d t o M r .D a r l i n g f o r a n i n t e r e s t i n g c o n v e r s a t i o n o n t h i s p a p e r .

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E.P. Wigner Unitary representations of Lorentz group 13

A s t h i s s h a l l h o l d f o r e v e r y 4 , 2 ) , t h e e x i s t e n c e o f a u n i t a r y ,5 w h i c h t r a n s f o r m sD ~" i n t o D ~" is t h e c o n d i t i o n f o r t h e e q u i v a l e n c e o f t h e s e t w o r e p re . -~ ' n ta t io n s .E q u i v a l e n t r e p r e s e n t a t i o n s a r e n o t c o n s i d e r e d t o b e r e a l l y d if f er e n t a n d i t w i llb e s u f f i c i e n t t o f i n d o n e s a m p l e f r o m e v e r y i n f i n i t e c l a s s o f e q u i v a l e n t r e p r e -s e n t a t i o n s .I f t h e r e i s a c l o s e d li n e a r m a n i f o l d o f s t a t e s w h i c h i s in ~ , a ri a n t u n d e r a l lL o r e n t z t r a n s f o r m a t i o n s , i .e . w h i c h c o n t a i n s D (L)~ / i f i t c o n t a i , ~ ~ . t h e l i n e a rm a n i f o l d p e r p e n d i c u l a r t o t h i s o n e w i ll b e i n v a r i a n t a l ~ . I n f a c t. i f ~ b e l o n g st o t h e s e c o n d m a n i f o l d , D(L)~o ~ 1 1 b e , o n a c c o u n t o f t h e u n i t a r y c h a r a c t e r o fD ( L ) , p e r p e n d i c u l a r t o D ( L ) ~ ' if ~ ' b e l o n g s t o t h e f ir st m a n i f o l d . H o w e v e r ,D(L-S)~ b e l o n g s t o t h e f ir st m a n i f o l d if d o e s a n d t h u s D(L)~o w i ll b e o r t h o g o n a lt o D(L)D(L-~)~ - - ~ ,~ i .e . t o a ll m e m b e r s o f t h e f ir s t m a n i f o l d a n d b e l o n g i t s e lf t ot h e s e c o nd m a n i f o l d als o. T h e o r ig i na l r e p r e s e n ta t i o n t h e n " d e c o m p o s e s "i n t o t w o r e p r e s e n t a t i o t m , c o r r e s p o n d i n g t o t h e t w o l i n e a r m au ifo lc L~ . I t i sc le a r t h a t , c o n v e r s e ly , o n e c a n f o r m a r e p r e s e n t a t i o n , b y s i m p l y " a d d i n g "s e v e r a l o t h e r r e p r e s e n t a t i o n s t o g e t h e r , i .e . b y c o n s i d e r i n g a.~ s t a t e s l i n e a rc o m b i n a t i o n s o f t h e s t a t e s o f s e v e r a l r e p r e s e n t a t i o n s a n d a s s u m e t h a t t h e s t a t e sw h i c h o r i g in a t e f r o m d i f f e re n t r e p r e s e n t a t i o n s a r e p e r p e n d i c u l a r t o e a c h o t h e r .R e p r e s e n t a t io n s w h i c h a re e q u i v a l e n t t o s u m s o f a l r e a d y k n o ~ m r e p r e se n t a -t io n s a r e n o t r e a l l y n e w a n d , i n o r d e r t o m a s t e r a l l r e p r e s e n t a t io n s , i t w i ll b es u ff ic ie n t t o d e t e r m i n e t h o se , o u t o f w h i c h a ll o t h e r s c a n b e o b t a i n e d b y " a d d i n g "a f i n it e o r i n f i n it e n u m b e r o f t h e m t o g e t h e r .T w o s i m p l e t h e o r e m s s h a ll b e m e n t i o n e d h e r e w h i c h w i ll b e p r o v e d l a t e r( S ec t io n s 7 A a n d 8 C r e s p e c t iv e l y ). T h e f i rs t o n e r e fe r s t o u n i t a r y r e p r e s e n t a -t i o n s o f a n y c l o s e d g r o u p , t h e s e c o n d t o i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s o fa n y ( c lo s e d o r o p e n ) g r o u p .T h e r e p r e s e n t a t i o n s o f a c l o s e d g r o u p b y u n i t a r y operators c a n b e t r a n s f o r m e di n t o t h e s u m o f u n i t a r y r e p r e s e n t a t i o n s w i ~ h m a t r i c e s o f f i n it e dim e rL ~io ns .

G i v e n tw o n o n e q u i v a l e n t i rr e du c ib l e u n i t a r y r e p r e ~ n t a t i o n s o f a n a r b i t r a r yg r ou p . I f t h e s c a l a r p r o d u c t b e t w e e n t h e w a v e f u n c ti o n s is i n v a r i a n t u n d e r t h eo p e r a t io n s of t h e g r o u p , t h e w a v e f u n c t i o n s b e l o n g i n g " t o t h e f ir st r e p r e s o n t a -t io n a r e o r t h o g o n a l t o a ll w a v e f u n c ti o rm b e l o n g in g t o t h e s e c o n d r e p r e ~ n t a t i o n .

C . C l a s s i f i c a t i o n o f ,, - ; t a r y r e p r e s e n t a t i o n s a c c o r d in s t o y o n N o - , r e , n -a n d M u r r a y ~

G i v e n t h e o p e r a t o r s D (L) o f a u n i t a r y r e p re s e n ta t io m ~ , o r a r e p r e s e n t a t i o nu p t o a f a c t o r , o n e c a n c o n s i d e r t h e a l g e b r a o f t h e s e o p e r a t o r s , i .e . a l l l i n e a rc o m b i n a t i o n s a,D (LO -4- a2D (I4) -4- asD(14) + . . .o f t h e D ( L ) a n d a l l l i m i t s o f s u c h l i n e a r c o m b i n a t i o n s w h i c h a r e b o u n d e do p e r at o rs . A c c o r d i n g t o t h e p r o p e r t i e s o f t h i s r e p r e s e n t a t i o n a l g e b r a , t h r e ee l& ~ e s o f u n i t a r y r e p r e s e n t a t i o n s e a n b e d i s t i n g u i sh e d .

t , F . J . M u r r a y a n d J . v . N e u m a n n , A n n . o f M a t h . $ 7 , 116, 1 93 6; J . ~ '. N e u m a n n , t o b ep u b l i s h e d s o o n .

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14 E.P. Wigner / Unitary representations o f Lo ren tz grou p

The f i r s t c l a s s o f i n ' e duc i b l e r e p r e s e n t a t i o n s h a s a r e p r e s e n t a t i o n - a l g e b r aw h i c h c o n t a i n s a l l b o u n d e d o p e r a t o r s , i . e. i f q , a n d ~ a r e t w o a r b i t r a r y sta tes ,t h e r e i s a n o p e r a t o r A o f t h e r e p r e s e n t a t i o n a l g e b r a f o r w h i c h A ~ ~ ,p a n dA ~ ' = 0 i f ~ ' is o r t h o g o n a l t o q,. I t i s c l e a r t h a t t h e c e n t e r o f t h e a l g e b r a c o n -r a i n s o n l y t h e u n i t o p e r a t o r a n d m u l t i p l y t h e r e o f . I n f ac t., if C is i n t h e c e n t e ro n e c a n d e c o m p o s e C ~ = o t , + ,~ ' s o t h a t ~ ' s h a l l b e o r t h o g o n a l t o ,. H o w e v e r ,~b' m u s t v a n i s h s in c e o t h e r w i s e C w o u l d n o t c o m m u t e w i t h t h e o p e r a t o r w h i c hl e av e s / i n v a r i a n t a n d t r a n s f o rm s , e v e r y fu n c t i o n o r t h o g o n a l t o i t i n t o 0. F o r. si m il ar r e a s o n s , a m u s t b e t h e s a m e f o r a l l ~ . F o r i r r e d u c i b l e r e p r e s e n t a t i o n st h e r e i s n o c lo s e d li n e a r m a n i f o l d o f s t a t e s , ( e x c e p t i n g t b e m a n i f o l d o f a l l s t a t e s )w h i c h is i n v a r i a n t u n d e r a ll L o r e n t z t r a n s fo r m a t i o n s . I n f a c t, a c c o r d i n g t o t h ea b o v e d e f i n i t i o n , a @ ' a r b i t r a r i l y c l o se to a n y ~ c a n b e r e p r e s e n t e d b y a f i n i t el i n e a r c o i n b i n a t i o n

a , D ( L , ) ~ + a . ,D ( I a ) , 4 - . . . 4 - a ,,D ( L ,, )~ b .H e n c e , a c l o s e d l i n e a r i n v a r i a n t m a n i f o l d c o n t a i n s e v e r y s t a t e i f i t c o n t a i n s o n e .T h i s is, i n f a c t , t h e m o r e c u s t o m a r y d e f i n i t i o n f o r i r r e d u c i b l e r e p r e s e n t a t i o n sa n d t h e o n e w h i c h w i l l b e u s e d s u b s e q u e n t l y . I t i s w e l l k n o w n t h a t al l f in it ed i m e n s i o n a l r e p r e s e n t a t i o n s a r e s u m s o f i r r e d u c i b l e r e p r e s e n t a t i o n s . T h i s i sn o t t r u e , i n g e n e r a l , i n a n i nf in it e u m b e r o f d i m e n s i o n s .

T h e s e c o n d c l a s s o f r e p r e s e n t a t i o n s w i l l b e c a l l e d f a c t o r / a / . F o r t h e s e , t il ec e n t e r o f t h e r e p r e s e n t a t i o n a l g e b r a s ti ll c o n t a i n s o n l y m u l t i p l e s o f t h e u n i to p e r a t o r . C l e a r l y , t h e i r r e d u c i b l e r e p r e s e n t a t i o n s a r e a ll f a c t o r i a l , b u t n o tc o n v e r s e l y . F o r f i ni t e d i m e n s i o n s , t h e f a c t o r i a l r e p r e s e n t a t i o n s m a y c o n t a i no n e i r r e d u c i b l e r e p r e s e n t a t i o n s e v e r a l t i m e s . T h i s i s a l s o p o s s i b l e i n a n i n fi n it en u m b e r o f d i m e n s i o n s , b u t i n a d d i t i o n t o t h i s , t h e r e a r e t h e " c o n t i n u o u s "r e p r e s e n t a t i o n s o f M u r r a y a n d y o n N e u m a n n ) T h e s e a r e n o t i r r e d u c i b l e a st h e r e a r e i n v a r i a n t li n e a r m a n i f o l d s o f s t a t e s . O n t h e o t h e r h a n d , i r i s m p o s s i b l et o c a r r y t h e d e c o m p o s i t i o n s o f a r a s t o o b t a i n a s p a r t s o n l y i r r e d u c i b l e r e p r e -s e n ta t i on s . I n a ll t h e e x a m p l e s k n o w n s o f a r , t h e r e p r e s e n t a t i o n s i n t o w h i c ht h e s e c o n t i n u o u s r e p r e s e n t a t i o n s c a n b e d e c o m p o s e d , a r e e q u i v a l e n t t o t h eo r i g i n a l r e p r e s e n t a t i o n .

T h e t h i r d c l a s s c o n t a i n s a l l p o s s i b l e u n i t a r y r e p r e s e n t a t i o n s . I n a f i n i t en u m b e r o f d i m e n s i o n s , t h e s e c a n b e d e c o m p o s e d f i r s t i n t o f a c t o r i a l r e p r e -s e n t a t i o n s , a n d t h e s e , i n t u r n , i n i r r e d u c i b l e o n e s . V o n N e u m ~ n n T M h a s s h o w nt h a t t h e f i rs t s t e p s ti ll i s p o s s i b l e i n i nf i n i te d i m e n s i o n s . W e c a n a s s u m e ,t h e r e fo r e , f r o m t h e o u t s e t t h a t w e a r e d e a l i n g w i t h f a c t o r ia l r e l ~r e s en t a ti o n s.

I n t h e t h e o r y o f r e p r e s e n t a t i o n s o f f i ni t e i m e n s i o n s , i t i s s u f f ic i e n t t o d e t e r -m i n e o n l y t h e i r r e d u c i b l e o n e s , a ll o t h e r s a r e e q u i v a l e n t t o s u m s o f t h e s e . H e r e ,i t w i l l b e n e c e s s a r y t o d e t e r m i n e a ll f a c to r i a l r e p r e s e n t a t i o n s . H a v i n g d o n et h a t , w e s h a l l k n o w f r o m t h e a b o v e t h e o r e m o f y o n N e m n a n n , t h a t a l l r e p r e -~ n t a t i o n s a r e e q u i v a l e n t t o fi n it e r i nf i ni t e u m s o f f a c t o ri a l r e p r e s e n t a t i o n s .

I t w i l l b e o n e o f t h e r e s u l t s o f t h e d e t a i l e d i n v e s t i g a t i o n t h a t t h e i n h o m o -g e n e o u s L o r e n t z g r o u p h a s n o " c o n t i n u o u s " r e p r e s e n t a t i o n s , a l l e p r e s e n t a t i o n s

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c al l b e d e c o m p o s e d i n to i r re d u c i b l e o n e s. T h u s t h e w o r k o f M a j o r a n a a n dD i r a c a p p e a r s t o b e j u s t i f i e d f r o m t h i s p o i n t o f v i e w a p o s t e r i o r i.

D . C l a s si f ic a ti o n o f u n i t a r y r e p r e s e n t a t i o n s f r o m t h e p o i n t o f v i e w o fi n f i n i t e s i m a l o p e r a t o r sT h e e x i s t e n c e o f a n i n f i n i te s i m a l o p e r a t g r o f a c o n t i n u o u s o n e p a r a m e t r i c

( cy c li c, a b e l i an ) u n i t a r y g r o u p h a s b e e n s h o w n b y S t o n e . " H e p r o v e d t h a t t h eo p e r a t o r s o f s u c h a g r o u p c a n b e w r i t t e n a .~ e x p ( i H t ) w h e r e H i s a ( b o u n d e d o ru n b o u n d e d ) h e r m i t e a n o p e r a t o r a n d t is t h e g r o u p p a r a m e t e r . H o w e v e r , t h eL o r e n t z g r o u p h a s m a n y o n e p a r a m e t r i c s u b g r o u p s , a n d t h e c o r r e s p o n d i n gi n fi n it e si m a l o p e r a t o r s H ~ , H 2 , - - - a r e al l u n b o u n d e d . F o r e v e r y H~ a ne v e r y w h e r e d e n s e s e t of f u n c t i o n s ~ c a n b e f o u n d s u c h t h a t H , ~ c a n b e d e fi n e d .I t i s n o t c l e a r, h o w e v e r , t h a t a n e v e r y w h e r e d e n s e s e t c a n b e f o u n d , t o a llm e m b e r s o f w h i c h e v e r y H c a n b e a p p l i e d . I n f a c t, it is n o t c l e a r t h a t o n es u c h ~ c a n b e f o u n d .I n d e e d , i t m a y b e i n t e r e s ti n g t o r e m a r k t h a t f o r a n i r re d u c i b le re p r e s e n t a t i o nt h e e x i s t e n c e o f o n e f u n c t i o n ~ , t o w h i c h a l l i n fi n i te s i m a l o p e r a t o r s c a n b e a p p l i e d ,e n t a i ls t h e e x i s t e n c e o f a n e v e r y w h e r e d e n s e se t o f s u c h f u n c t i o n s . T h i s a g a i nh a s t h e c o n s e q u e n c e t h a t o n e c a n o p e r a t e u S t h in f i n it e s im a l o p e r a t o r s t o a l a r g ee x t e n t i n t h e u s u a l w a y .PROOF: Le t Q ( t ) b e a on e p a r a m e t r i c s u b g r o u p s u c h t h a t Q ( t ) Q ( t ' ) ffi Q ( / + t ') .I f t h e i n f i n i t e s im a l o p e r a t o r o f a l l s u b g r o u p s c a n b e a p p l i e d t o ~ , t h e(5) l im C ~ ( Q ( t ) - 1)~

t - - 0

e x is ts . I t f o ll o w s , t h e n , t h a t t h e i n f i n i te s i m a l o p e r a t o r s c a n b e a p p l i e d t o R a ls o w h e r e R is a n a r b i t r a r y o p e r a t o r o f t h e r e p r e s e n t a t i o n : S i n c e R - ~ Q ( t ) R i sa l s e a o n e p a r a m e t r i c s u b g r o u p

l ir a C t ( R - ~ Q ( t ) R - - 1 ) ~ -- ll m R - ~ . C t ( Q ( t ) - 1 ) R ~#--0 t--0

a ls o e x is t s a n d h e n c e a ls o ( R i s u n i t a r y )-

t - - 0

E v e r y i n f in i te s i m a l o p e r a t o r c a n b e a p p l i e d t o P ~ i f t h e y a ll c a n b e a p p l i e d t o ~ ,a n d t h e s a m e h o l d s f or s u m s o f t h e k i n d(6 ) atR ia , + ~ W . . . - t- a , ,R , s , .T h e s e f o r m , h o w e v e r , a n e v e r y w h e r e d e n s e se t o f f u n c t io n s if t h e r e p r e s e n t a t i o ni s i r r e d u c i b l e .I f t h e r e p r e s e n t a t i o n i s n o t i r r e d u c i b l e , o n e c a n c o n s i d e r t h e s e t N o o f s u c hw a v e f u n c t io n s t o w h i c h e v e r y i n t ~ u i t e ~ n a l o p e ra t o r c a n b e a p p l ie d . T h i s s e t is

t~ M . H . Stone , P roc. N at . Aead . I6, 173, 1930, Ann . of Math . 35, 643, 1932, also J . v.Neumann, ibid, 35, 567, 1932.

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clearly linear and, according to the previous paragraph, invariant under theoperations of the group (i.e. contains every Pun if it contains ~). The sameholds for the closed set N generated by No and also of the set P of functionswhich are perpendicular to all functions of N. In fact, if ~,, is perpendicular toall ~, of N, it is perpendicular also to all R-t~, and, for the unitary character ofR, the R~p is perpendicular to all ~, , i.e. is also contained ill the set P.We can decompose thus, by a unitary transformation, every uni tary repre-sentation into a "normal" and a ':pathological" part. For the former, there isan everywhere dense set of functions, to which all infinitesimal operators can beapplied. There is no single wave functions to which all infinitesimal operatorsof a "pathological" representat ion could be applied.According to Murray and yon Neumazm, if the original representation wasfactorial, all representations into which it can be decomposed will be factorialalso. Thus every representation is equivalent to a sum of factorial repre-sentations, part of which is "normal," the other part "pathological."It will turn out again that the inhomogeneous Lorentz group has no path-ological representations. Thus this assumption of Majorana and Dirae alsowill be justified a posteriori. Every unitary representation of the inhomogenousLorentz group can be decomposed into normal irreducible representations. Itshould be stated, however, that the representations in which the unit operatorcorresponds to every translation have not been determined to date (el. alsosection 3, end). Hence, the above statements are not proved for these repre-sentations, which are, however, more truly representations of the homogeneousLorentz group, than of the inhomogeneous group.

While all these points may be of interest to the mathematic ian only, the newrepresentation of the Lorentz group which will be described in section 7 mayinterest the physicist also. It describes a particle with a continuous spin.Acknowledgement. The subject of this paper was suggested to me as early as1928 by P. A. M. Dirae who realised even at tha t date the connection of repre-sentations with quantum mechanical equations. I am greatly indebted to himalso for many fruitful conversations about this subject, especially during theyears 1934/35, the outgrowth of which the present paper is.I am indebted also to J. v. Neumann for his help and friendly advice.3, SUMMARY OF ENSUINO SECTIONS

Section 4 will be devoted to the definition of the inhomogeneous Lorentzgroup and the theory of characteristic values and characteristic vectors of ahomogeneous (ordinary) Lorentz transformation. The discu.~sion will followvery closely the corresponding, well-known theory of the group of motions inordinary space and the theory of characteristic values of orthogonal trans-formations.12 I t will contain only a straightforward generalization of themethods usually applied in those discussions.x2 Cf. e.g .E. Wigner, l.c. Chapter II I. O. Veblen an dJ . W. Young, P r o j e c t i o e G e o m e t r y ,Boston 1917. Vol . 2, especially Chapter VII.

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E.P. Wigner / Unitary representations of Lorentz group 17

I n S e c t i o n 5 , i t w i l l b e p r o v e d t h a t o n e c a n d e t e r m i n e t h e p h y s i c a l l y m e a n i n g -l e s s c o n s t a n t s i n t h e D ( L ) i n s u c h a w a y t h a t i n s t e a d o f ( 3 a ) t h e m o r e s p e c i a le q u a t i o n(7 ) D (L~)D (L2) = -t-D (L~L~)w i l l b e v a l id . T h i s m e a n s t h a t i n s t e a d o f a r e p r e s e n t a t i o n u p t o a fa c t o r, w ec a n c o n s i d e r r e p r e s e n t a t i o n s u p t o t h e s i g n . F o r t h e c a s e t h a t e i t h e r L~ o r L 2i s a p u r e t r a n s l a t i o n , D i r a c t~ h a s g i v e n a p r o o f o f (7 ) u s i n g i n f i n i t e s i m a l o p e r a t o r s .A c o n s i d e r a t i o n v e r y s i m i l a r t o h i s c a n h e c a r r i e d o u t , h o w e v e r , al so u s i n g o n l yf i n i t e t r a n s f o r m a t i o n s .

F o r r e p r e s e n t a t i o n s w i t h a f i n i t e n u m b e r o f d i m e n s i o n s ( c o r r e s p o n d i n g t o a no n l y f i n i t e n u m b e r o f l i n e a r l y i n d e p e n d e n t s t a t e s ) , ( 7 ) c o u l d b e p r o v e d a l s o i fb o t h L ~ a n d L ~ a r e h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n s , b y a s t r a i g h t f o r w a r da p p l i c a t i o n o f th e m e t h o d o f W e y l a n d S c h r e ie r . ~' H o w e v e r , t h e L o r e n t z g r o u ph a s n o f i n it e d i m e n s i o n a l r e p r e s e n t a t i o n ( a p a r t f r o m t h e t r i v i a l o n e i n w h i c h t h eu n i t o p e r a t i o n c o rr e s p o n d s t o e v e r y L ) . T h u s t h e m e t h o d o f W e y l a n d S c h r e ie rc a n n o t b e a p p l ie d . I t s f ir s t s t e p i s t o n o r m a l i z e t h e i n d e t e r m i n a t e c o n s t a n t s ine v e r y m a t r i x D ( L ) i n s u c h a w a y t h a t t h e d e t e r m i n a n t o f D ( L ) b e c o m e s 1 .N o d e t e r m i n a n t c a n b e d e f in e d fo r g e n e ra l u n i t a r y o p e r a t or s .

T h e m e t h o d t o b e e m p l o y e d h e r e w il l b e t o d e c o m p o s e e v e r y L i n t o a p r o d u c to f t w o i n v o l u t i o n s L = M N w i t h M 2 = N ~ = 1 . T h e n D ( M ) a n d D ( N ) wi l l ben o r m a l i z e d so t h a t t h e i r s q u a r e s b e c o m e u n i t y a n d D( L ) ffi D ( M ) D( N ) s e tI t w i l l b e p o s s i b le , t h e n , t o p r o v e ( 7) w i t h o u t g o i n g b a c k t o t h e t o p o l o g y o f t h eg r o u p .

S e c t i o n s 6 , 7 , a n d 8 W IU c o n t a i n t h e d e t e r m i n a t i o n o f t h e r e p r e s e n t a t i o n s .T h e p u r e t r a n s l a ti o n s f o r m a n i n v a r i a n t s u b g r o u p o f t h e w h o l e i n h o m o g e n e o u sL o r e n t , . g r o u p a n d F r o b e n i u s ' m e t h o d t~ w i ll b e a p p l i e d i n S e c t i o n 6 t o b u i l du p t h e r e p r e s e n t a t i o n s o f t h e w h o l e g r o u p o u t o f r e p r e s e n t a t i o n s o f t h e s u b g r o u p ,b y m e a n s o f a " l i t t le g r o u p . " I n S e c t i o n 6 , i t w i l l b e s h o w n o n t h e b a s is o f a n a sy e t u n p u b l i s h e d w o r k 2' o f J . v . N e u m a n n t h a t t h e r e i s a c h a r a c t e r i s t ic ( i n -v a r i a n t ) s e t o f " m o m e n t u m v e c t o r s " fo r e v e r y i rr e d u ci b le r e p r e se n t a ti o n . T h ei r r e d u c i b l e r e p r e s e n t a t i o n s o f t h e L o r e n t z g r o u p w i ll b e d i v i d e d i n t o f o u r c l as se s .T h e m o m e n t u m v e c t o r s o f t h e1 st class a r e t i m e - l i k e ,2 n d c / as ~ a r e n u l l - v e c t o r s , b u t n o t a l l t h e i r c o m p o n e n t s w i l l b e z e r o ,3 r d c / a s s v a n i s h ( i .e . , a l l t h e i r c o m p o n e n t s w i l l b e z e r o ) ,4 t h c / a s s a r e s p a c e - l i k e .O n l y t h e f i r s t t w o c a s e s w i l l b e c o n s i d e r e d i n S e c t i o n 7 , a l t h o u g h t h e l a s t c a s e

iz p . A . M . D i r a e , m i m e o g r a p h e d n o t e s o f le c t u r e s d e li v e r e d a t P r i n c e t o n U n i v e n d t y ,1 9 3 4 / 3 5 , p a g e 5 1 .t t H . W e y l , M a t h e m . Z e i t s . t S , 2 7 1 ; 2 4 , 3 2 8 , 3 7 7 , 7 8 9 , 19 2 5 ; O . S e h r e i e r , A b h a n d l . M a t h e m .S e m i n a r H a m b u r g , 4 , 1 5, 19 2 6; 5, 233, 1927.

t s G . F r o b e n i t m , S i t z . d . K 6 n . P r e t u m . A k a d . p . 5 0 1 , 1 89 8 , I . S e h u r , i b i d , p . 1 6 4 , 1 9 0 6 ;F . S e i t z , A n n . o f M a t h . S 7 , 1 7, 19 3 6.

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18 E.P. Wigner / Unitary representations o f Lor entz group

m a y b e t h e m o s t i n t e re s t in g f r o m th e m a t h e m a t i c a l p o i n t o f v i ew . I h o p e t or e t u r n t o i t i n a n o t h e r p a p e r . I d i d n o t s u c c e e d so f a r in g i v i n g a c o m p l e t ed i s c u s s io n o f t h e 3 r d c l as s . ( A ll t h e s e r e s t r i c t i o n s a p p e a r i n th p p r e v i o u st r e a t m e n t s a l s o . )

I n S e c t i o n . 7 , w e s h a ll f in d a g a i n a l l k n o w n r e p r e s e n t a t i o n s o f t h e i n h o m o -g e n e o u s L o r e n t z g r o u p ( i . e . , a l l k n o w n I , o r e n t z i n v a r i a n t e q u a t i o n s ) a n d t w on e w s e t s .

S e c t i o n s 5, 6, 7 w i l l d e a l w i t h t i le " r e s t r i c t e d . L o r e n t z g r o u p " o n l y , i . e . L o r e n t zt r a n s fo r m a t i o n s w i t h d e t e r m i n a n t 1 w h i c h d o n o t r e v e r s e t h e d i r e c t io n o f t h et i m e a x is . I n s e c t i o n 8 , t h e r e p r e s e n t a t i o n s of t h e e x t e n d e d L o r e n t z g r o u p w i l lb e c o n s id e r e d , t h e t r a n s f o r m a t i o n s o f w h i c h a r e n o t s u b j e c t t o t h e s e c o n d i t i o n s .

4 . DESC RIFrION OF THE [NHOMOGENEOUS LORENTZ G R o u PA .

A n in h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n L = ( a, A ) is t h e p r o d u c t o f at r a n s l a t i o n b y a r e a l v e c t o r a!(8 ) x~ = x , + a , (i: = 1 . 2 , 3 , 4 )

a n d a h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n A m fith r e a l c o e ff i ci e n ts4

( 9 ) & = ) -' :k s l

T h e t r a n s l a t i o n s h a l l b e p e r f o r m e d a f t e r t h e h o m o g e n e o u s t r a n s f o r m a t i o n .T h e c o e f fi ci e nt s o f t h e h o m o g e n e o u s t r a n s f o r m a t i o n s a t i s f y t h r e e c o n d i t i o n s :

2( I ) T h e y a r e r e a l a n d h l e a v e s t h e i n d e f i n i te q u a d r a t i c f o r m - x ~ - x ] - x3 + x~i n v a r i a n t :(10) A F A ' = Fw h e r e t h e p r i m e d e n o t e s t h e i n t e r c h a n g e o f r o w s a n d c o l u m n s a n d F i s t h ed i a g o n a l m a t r i x w i t h t h e d i a g o n a l e l e m e n t s - 1 , - 1 , - 1 , + 1 . - - ( 2 ) T h e d e t e r -m i n a n t l A ~ I = 1 a n d - - ( 3 ) A ~ > O .

W e s h a l l d e n o t e t h e L o r e n t z - h e r m i t e a n p r o d u c t o f t w o v e c t < ~ r s x a n d y b y(11) {x, y} = - - x ~ y ~ - - x ~ y ~ - - x ~ y 3 + x ~ y 4 .( T h e s t a r d e n o t e s t h e c o n j u g a t e i m a g i n a r y . ) I f { x, x } < 0 t h e v e c t o r x i sca l l ed space - l i ke , if { x , x} = 0 , i t i s a nu l l ve c to r , i f { x, x} > 0 , i t i s c a l l ed t im e -l ik e . A r e a l t im e - l i k e v e c t o r l ie s i n t h e p o s i t i v e l i g h t c o n e if x~ > 0 ; i t l i e s i n t h en e g a t i v e l i g h t c o n e i f x4 < 0 . T w o v e c t o r s x a n d y a r e c a l l e d o r t h o g o n a l ifI z , Y l = 0 .

O n a c c o u n t o f i ts l in e a r c h a r a c t e r a h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n isc o m p l e t e l y d e f in e d if A v is g i v e n f o r fo u r l i n e a r l y i n d e p e n d e n t v e c t o r s @ ~,V 2), !/(z), V 4).F r o m ( 1 t ) : r o d ( lO ) i t f o l lo w s t h a t I v, w l = { A v, A w } f o r e v e r y p a i r o f v e c t o r sv , w . Th i s w i l l be s a t i s f i ed fo r e ve ry p a i r i f i t i s s a t i s f i ed fo r a l l pa i r s v ~), vok)

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E.P lCigner / Unitary representations o f Lo ren tz group 19

o f fo u r l i n e a r l y i n d e p e n d e n t v e c t o r s . T h e r e a l i ty c o n d i t i o n i s s a t is f ie d if(Av~) * ffi A(v *) h old s for f our s uch vec tors .T h e s c a l a r p r o d u c t o f t w o v e c t o r s z a n d F i s p o s i t i v e i f b o t h l ie in t h e p o s i t i v e

l ig h t c o n e o r b o t h i n t h e n e g a t i v e li g h t c o n e . I t i s n e g a t i v e i f o n e l ie s i n t h ep o s i t i v e , t h e o t h e r i n t h e n e g a t i v e . l i g h t c o n e . S i n c e b o t h z a n d y a r e t im e - l i k e~ x4 t 2 > l z l l * + l x z l ~ + l x s l ~ ; ly 41~ ~ l y t l 2 + l y : l s + l Y a l 2. H e n c e , b yS c h w a r z ' s i n e q u a l i t y I z ~y 41 > I z ~ Y ~ - F z i y 2 - F z ~ y s ] a n d t h e s i g n o f t h e s c a l a rp r o d u c t o f t w o r e a l ti m e - l i k e v e c t o r s is d e t e r m i n e d b y t h e p r o d u c t o f t h e i rt i m e c o m p o n e n t s .

A t i m e - l i k e v e c t o r i s t r a n s f o r m e d b y a L ore nt~ . tr a n s f o r m a t i o n i n t o a t i m e - l i k ev e c t o r . F u r t h e r m o r e , o n a c c o u n t o f t h e c o n d i t io n A ~ > 0 , t h e v e c t o r v )w i t h t h e c o m p o n e n t s 0 , 0 , 0 , 1 r e m a i n s i n t h e p o s i ti v e l i g h t co n e , s in c e t h e f o u r t hc o m p o n e n t o f A v ) i s a , . I f v s~ i s a n o t h e r v e c t o r s i n t h e p o s i t i v e l i g h t c o n e{vcu, v~~} > 0 an d he nc e a l so {Av ~ , Av ) } > 0 an d av ~t~ i s in th e p os i t iv e l igh tc o n e a l so . T h e t h i r d c o n d i t i o n f o r a Lo re nt~ . t r a n s f o r m a t i o n c a n b e f o r m u l a t e da l so a s t h e r e q u i r e m e n t t h a t e v e r y v e c t o r i n ( o r o n ) t h e p o s i ti v e li g h t c o n e sh a l lr e m a i n i n ( o r, r e s p e c t i v e l y , o n ) t h e p o s i t i v e l i g h t c o n e .

T h i s f o r m u l a t i o n o f t h e t h i r d c o n d i t i o n s h ow s t h a t t h e t h i r d c o n d i t i o n h o l d sf o r t h e p r o d u c t o f tw o h o m o g e n e o u s L o r e n t s t r a n ~ o r m a t i o n s i f i t h o l d s f o r b o t hf a c t o r s . T h e s a m e i s e v i d e n t f o r t h e f i rs t t w o c o n d i ti o n s .F r o m A F A ' = F o n e o b t a i n s b y m u l t ip l y i n g w i t h A ~ f r o m t h e l e ft a n dA~ 1 = ( A - t ) f r o m t h e r i g h t F = A - t F ( A - ~ ) s o t h a t t h e r e c i p r o c a l o f a h o m o -g e n e o u s L o r e n t z t r a n s f o r m a t i o n i s a g a i n s u ch a t r a n s fo r m a t i o n . T h e h o m o -g e n e o u s L o r e n t z t r a n s f o r m a t i o n s f o r m a g r o u p, t h e r ef o re .

O n e e a s i ly c a lc u l a te s t h a t t h e p r o d u c t o f t w o i n h o m o g e n e o u s L o r e n t z t r A n s-f o r m a t i o n s (b , ~ a n d (c, N ) i s a g a i n a n i n h o m o g e n e o u s L o r e n t z tr a n s f o r m a t i o nA )

(12) (b, M )(c , N ) = (a , A)w h e r e( 1 2 a ) = = b , + Y ' .i io r , s o m e w h a t s h o r t e r( 12b) A = M N ; a = b + M e .B . T h e o r y o f c h a r a c t e r i s t ic v a l u e s a n d c h a r a c t er i s ti c v e c t o rs o f a h o m o g e a e o u s

L o r e a t z t r a n M o r m a t i o nL i n e a r h o m o g e n e o u s t r a n s f o r m a t i o n s a r e m o s t s i m p l y d e s c ri b e d b y t h e i rc h a r a c t er i s ti c v a l u e s a n d v e c t o rs . B e f o r e d o i n g t h is f o r t h e h o m o g e n e o u sL o r e n t s g r o u p , h o w e v e r , w e s h a ll n e e d t w o r u le s a b o u t o r t h o g o n a l v ec t o rs .~" Wherever s confusion between vec tors and veetor eomponents appears to be po~ib] e,upper indices will be used for distinguishi ng different vectors and lower indices for denoting

the components of a vector.

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20 E.P. W igner Un itary represen tationsof Lorentz group[11 I/" Iv, w} = 0 an d Iv, v} > O, th en { w , to} < O; i f I v , to} = O, It ,, t ,} = O,t h e n to i s d . t h e r s p a ~ - l i k e , o r 1 ~ a r a l ld t o v ( ~ , r { to , to } < O , o r t o = c v ) .P s o o F :

(13) V4to4 = * ,l t o t "~ t , ~ " ] - t , * ~ .B y 8 e h w a r z ' s i n e q u a l i t y , t h e n( 1 4 ) I v , I ' I t o , l" - ( I v , I' + I ~ I ' + I v , I ' ) ( I t o , 1 ' + l ~ l" + ! ~ I ' ) .F o r I v, 1' > I vt I~ q- I t~ 12 -t- I t~ 1' it fo llo w s th a t I w, [' < [ wt ]~ -t- 1 o, [ ' -t- [ to, 1'.I f l t,, 1' --- Iv , [2 -I- [t~ I~ + I v , [~ t he s econd i ne qu a l i t y s t i l l fo l l ows i f t he i n -e q u a l i t y s i g n h o l d s i n ( 14 ). T h e e q u a l i t y s ig n c a n h o l d o n l y , h o w e v e r , i f t h ef ir s t t h r e e c o m p o n e n t s o f t h e v e e t o r s t, a n d to a r e p r o p o r t i o n a l. T h e n , o na e e o u n t o f ( 1 3 ) a n d b o t h b e i n g n u l l v e c t o r s , t h e f o u r t h c o m p o n e n t s a r e i n t h esam e r a t i o a l so .

[21 I f f o u r v e c t o rs t ,< u , v c '.) , v< a), v c o a r e m u t u a l l y o r t h o g o n a l a n d l i n e a r l y i n d e -p e n d e n t , o n e o f th e m i s t i m e - l i k e , t h re e a r e sp a c e - l ik e .P R o o F : I t f o ll o w s f r o m t h e p r e v i o u s p a r a g r a p h t h a t o n l y o n e of f o u r m u t u a l l y

o r t h o g o n a l , l i n e a r l y i n d e p e n d e n t v e c t o r s c a n be ti m e - l i k e o r a n u ll v e c t or . I tr e m a i n s t o b e s h o w n t h e r e f o r e o n l y t h a t o n e o f t h e m i s t i m e - li k e . S i n c e t h e ya r e l i n e a r l y i n d e p e n d e n t , i t is p o s s ib l e t o e x p re s s b y t h e m a n y t im e - l ik e v e c t o r

4V(t) ~ E ~/~ v(k)-t - - I

Th e s ca l a r p r od uc t o f t he l e f t s i de o f t h i s equ a t i on wi t h i t s e l f i s poe i t i ve andt he r e f o r e

o r( 1 5 )

t ~ ~zkV(k)~ ~ ak v k~} > 0

[ ~ . r { v ' ~ ' , v c " ] > 0ka n d o n e { v, v

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E.P. l~igner Unitary representations o f Lore ntz group 21PROOF:

{ v , , ~ } = { a n , , A ~ } = { x , v , , x ~ } f f i x ~ x f l n , , ~ } .T hu s if {vt, vs} # 0, X~X, ffi I .

[5] I f t h ~ m o d u l u s o f a c h a r a d e r i a i c m l u e X i s l x l ~ 1, the cor re spond i ngeh ar ad er ~ie vector v i s a n ul l aee tor and X i t se l f real and pos i t ive .F ro m Iv, v} ffi {A n, An} ffi [ X l f l n , v } the {v, v} f fi 0 fo l lows im m ed ia te ly forI X ! ~ 1 . I f h we r e com plex , X * wou l d be a cha r a c t e r i s t i c va l u e a l so . T hecha r ac t e r i s t i c v ec t o r s o f X an d X * wou l d be t wo d i f f e r en t nu l l vec t o r s and ,

because o f [4 ], o r t hogona l t o e ach o t he r . Th i s i s i m poss i b l e on a cco un t o f [1 ].T h u s X i s r e a l a n d v a r e al n u l l v e c t or . T h e n , o n a c c o u n t o f t h e t h i r d c o n d i t i o nf o r a h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n , X m u s t b e p o s i t i v e .[6] T h e c h a r a d e r i s t i c v a l ue X o f a c h a r a d e r i s l i c v e ct o r v o f ~ n g t h n u l l i s r e a l a n dpositive.

I f X wqr e no t r e a l , X * wou l d be a cha r ac t e r i s t i c va l u e a lso . Th e co r r e spo nd i ngcha r ac t e r i s t i c vec t o r v* wou l d be d i f f e r en t f r om v , a n u l l vec t o r a l so , and pe r -pend i cu l a r t o v on accoun t o f [4 ]. Th i s i s i m poss i b l e because o f [1 ].[7 ] The e h a r a d z ~ / e vec to r n o$ a eomp/c .z eh a r aa e r ~ / e m / ue X ( the m odu l us o ft o h ic h i s 1 o n a c c o u r~ o f [5]) is ~rpace-|/ke- Iv, v} < 0.PR oo f : X* i s a cha r ac t e r i s t i c va l ue a l so , t he co r r e spond i ng cha r ac t e r i s t i cvec tor is v*. Sinc e 0.*)*X = )I ~ 1, {n*, v} ffi 0. Sin ce th e y ar e diffe ren t , a tl eas t one is space- l ike . On acco un t of {v, n} ffi [v*, v*} bo th a re space - l ike . I fa l l f o u r c h a r a c te r i s ti c v a l u e s w e r e c o m p l e x a n d t h e c o r r e s p o n d i n g c h a r a c t e r i s t i cv e c t o rs l i n e a r l y i n d e p e n d e n t ( w h i c h is t r u e e x c e p t i f A h a s e l e m e n t a r y d i v i s o rs )w e s h o u l d h a v e f o u r s p a ce - li k e, m u t u a l l y o r t h o g o n a l v e c t o r s. T h i s i s i m p o s s i b le ,o n a c c o u n t o f [ 2 ] . H e n c e[8] There i s not more than on~ pair o f conjugate comple . z eAarader i s t i c mlus~,i f A has no e l em en t ary d i dsor s . 8 i m i la r I F , under the m n ~ cond i ti on , the re i srug more than one pair X, X t o f charader i s t i e va lues whose mo dulu s i s d i f f eremf r o m 1 . Ot he r w i se t he i r cha r ac t e r i s t i c vec t o r s wou l d be o r t hog ona l , w h i cht h e y o~mnot be, being null VectOle8.

F o r h o m o g e n e o u s L o r e n t z t r s n d o r r a a t i o u s w h i c h d o n o t h a v e e l e m e n t a r yd i v iso r s , t h e f o l lowi ng pos s i b i li t ie s r em a i n :( a ) T he r e i s a pa i r o f com pl ex cha r ac t e r i s t i c va l ue s , t h e i r m odu l u s i s .1 , ona c c o u n t o f [5 1( l e ) x , ffi x : = x ~ ' ; I x , I = I x 1 1 f fi 1 ,and a l so a pa i r o f. cha r ac t e r i s t i c va l ue s k s , X ~ , t he m o du l u s o f wh i c h i s no t 1"T h e s e m u s t b e r e a l a n d p o s i t i v e :(16n) M f fi X T x; M - -- - X ~ > 0 .The ch~r ac t e r i a t i c vec t o r s o f t he eon j ugLt e com pl ex cha r ac t e r i s t i c va l ue8 a r ec o n j u g a t e c o m p le x , p e r p e n d i c u l a r t o e a c h o t h e r a n d s p e x~ - li k e s o t h a t t h e y c a nb e n o r m ~ s e d t o - -1

, , = ~ ; I ~ , v , I = I , ' , , n ~ ' t f f i o(17)I n , , ~ } = l ~ , ~ l f f i - I

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2 2 E.P. Wigner / Unitary representations of Lorentz group

t h o s e o f t h e re a l c h a r a c t e r i s t ic v a l u e s a r e r ea l n m l v e c t o r s , t h e i r s c a l a r p r o d u c tc a n b e n o r m a l i z e d t o 1

v ~ = v a v , - - - v , { v z , v , } = I( 1 7 a ){ v , , v , } - -- [ v , , v , } = 0 .

F i n a l l y , t h e f o r m e r p a i r o f c h a r a c t e r is t ic v e c t o r s is p e r p e n d i c u l a r t o t h e l a t t e rk i n d(1 7b ) _{v,, v,} -- {v ,, v,} -= {v ,, v,} = {~s, v,} = O.I t w il l t u r n o u t t h a t a l l t h e o t h e r c a s es i n w h i c h A h a s n o e l e m e n t a r y d i v i s o ra re spee i a l s es o f ( a ) .

" O ,

AFxG. 1 . Pos i t ion of the charac ter i s t i c va lues for the gen era l case a ) in the com plex p lan e .In case b) , k~ and ) ,, co inc ide and are equa l 1 ; in cu e c ) , ), l and ),s co inc ide and are e i th er+ 1 o r - 1 . In c a se d) bo th paire ),z =ffi ~ ffi= I an d ~ - ).ffi~- 4-1 coin cide .( b ) T h e r e i s a p a i r o f co m p l e x ch a r a c t e r i s t i c v a l u e s ), 1 , )~s - - ) ,~ 'z - - ) , ~ ,

),1 ~ ) ,* , [ ) ,1 1 - - [ ),s I - - 1 . N o p a i r w i t h I k~ I ~ 1 , h o w e v e r . T h e n o n ac co u n to f [ 8] , s t i l l ) ~ -- Z * w h i ch g i v e s w i t h I k , ! = 1 , k 8 = + 1 . S i n ce t h e p r o d u c t) , l~ z k ~ ffi 1, o n a c c o u n t o f t h e s e c o n d c o n d i t i o n fo r h o m o g e n e o u s L o r e n t zt r a n s f o r m a t i o n s , a ls o )~ = ~ = 1 . T h e d o u b l e c h a r a c t e r i s t i c v a l u e -4 -1 h a st w o l i n e a r ly i n d e p e n d e n t c h a r a c t e r is t i c v e c t o r s v , a n d v , w h i c h c a n b e a s s u m e dt o b e p e r p e n d i c u l a r t o e a c h o t h e r , { v3 , v,} = 0 . A c c o r d i n g t o [2 ], o n e o f t h ef o u r c h a r a c t e r i s t i c v e c t o r s m u s t b e t i m e - l ik e a n d s i n c e t h o s e o f ),1 a n d ) ,, a r es p a c e -l i k e, t h e t i m e - l ik e o n e m u s t b e l o n g t o + 1 . T h i s m u s t b e p o s i ti v e ,t h e r e f o r e ~ - - ~ = 1 . O u t o f t h e t i m e - l i k e a n d s p a c e - l i k e v e c t o r s [ s a , va} -- - 1a n d { s 4 , v4} ffi 1 , o n e c an b u i l d t w o n u l l v ec t o r s v , -{- t~ an d v , - ~ . D o i n gt h i s , c a s e ( b ) b e c o m e s t h e s p e c i a l c a s e of (a ) i n w h i c h t h e r e a l p o s i t i v e c h a r -a c t e r i s t i c v a l u e s b e c o m e e q u a l ~ = ),7 1 = 1.

( e ) A l l c h a r a c t e r i s t i c v a l u e s a r e r e a l ; t h e r e i s h o w e v e r o n e p a i r X z - - X ~ ,

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E.P Wigner / Un itary representations o f Lor entz group 23

k~ ffi ) ` ~ t , t h e m o d u l u s o f w h i c h is n o t u n i t y . T h e n { ca , ~ } ffi I v 4 , v, | ffi 0an d h > 0 an d o ne ca n con c l u de fo r ),~ an d ) ,ff i, a s be fo re fo r h a n d h t h a t ),~ ffi).2 - - 4 - 1 . T h i s a g a i n i s a s p e c i a l e a s e o f ( a ) ; h e r e t h e t w o c h a r a c t e r i s t i c v a l u e so f m o d u l u s 1 b e c o m e e q u a l.

( d ) A l l c h a r a c t e r i s t i c v a l u e s a r e r ea l a n d o f m o d u l u s 1 . I f a l l o f t h e m a r e + 1 ,w e h a v e t h e u n i t m a t r i x w h i c h c l e a r l y c a n b e c o n s i d e r e d a s a s p e c i a l c a s e o f ( a ) .T he o t he r c a se i s ) `t = )`2 = - 1 , ~a = k4 = -F 1 . T h e cha ra c t e r i s t i c v ec t o r s o fk t a n d ) ,s m u s t b e s p a c e -l ik e , o n a c c o u n t o f t h e t h i r d c o n d i t io n f o r a h o m o g e n e o u sL o r e n t z t r a n s f o r m a t i o n ; t h e y c a n b e a s s u m e d t o b e o r th o g o n a l a n d n o r m a l i z e dt o - 1 . T h i s is t h e n a s p e c ia l c a s e o f ( b ) a n d h e n c e o f ( a ) a l so . T h e c a s e s( a ) , ( b ) , ( c ) , ( d ) a r e i l l u s t r a t e d i n F i g . 1 .

T h e c a s e s r e m a i n t o be c o n s i d e re d in w h i e n A h a s a n e l e m e n t a r y d i v is o r .W e s e t t h e r e f o r e(18) A,v , = ~,v , ; A, to, = ~, to , - t - v. .I t f o l l ow s f ro m [5 ] t h a t e i t h e r I k , I = 1 , o r Iv . , v,} ffi 0 . W e hav e Iv , , to ,} ffi{ / ~ . , A .to.} = I X . ] ff i{v., to .} -F {v , , ~ .}. F r o m th i s equ at io n( 1 9 ) I * . , v . } = 0f o l lo w s f o r l k , l ffi I , s o t h a t ( 1 9 ) h o l d s i n a n y c a s e . I t f o l l o w s t h e n f r o m[6] t h a t k , is r e a l , p o s i t i v e a n d v . , t o . c a n b e a s s u m e d t o b e r e a l a l so . T h e l a s te q u a t io n n o w b e c o m e s {v , , to, } .ffi X, {v , , t 0,} so t h a t e i t h e r ~ , f fi I o r {u , , to , } ffi 0 .F i n a ll y , w e h a v e

= , = x . { t o . , t o .} + 2 X . l t o . , v . } + I v . ,t o . , t o . } { A . t o . A . t o . } 2T h i s e q u a t i o n n o w s h o w s t h a t(19a ) { t o . , v. } - - -- 0e v e n i l k . ffi I . F r o m ( 1 9 ), ( 1 9 a ) i t f o l lo w s tb _ at t o. i s s p a c e - l i k e a n d c a n b en o r m a l i z e d t o(19b ) { t o . , w . } = - -1 .I n s e r t i n g ( 1 9a ) i n t o t h e p r e c e d i n g e q u a t i o n w e fi n a l ly o b t a i n(19e ) ~ . f f i 1 .

[ 9 ] I f A , ha s an e lementary d iv isor , a l l i t s chara cbn ~t ic roote an6 1 .F r o m ( 1 9 c ) w e s e e t h a t t h e r o o t o f t h e e l e m e n t a r y d i v i s o r i s 1 a n d t h i s i s a t

l e a s t a do ub l e roo t . I f A ha d s pa i r o f ch a ra c t e r i s t i c va l ue s ~ ~ 1 , X s ffi X ~ l ,t h e c o r r e s p o n d i n g c h a r a c t e r is t ic v e c t o r s ~ a n d ~ w o u l d b e o r t h o g o n a i t o ~o a n dt he re f o re space - l i ke . O n ac co un t o f [5] , t h en I ) .t I - I ~ I ffi 1 an d { ~ , ~ } ffi 0 .F u r t h e r m o r e , f r o m { to ,, ~ } f fi { A . w . . A . ~ } f fi ) , l { t o . . ~ } + X d ~ . . ~ } a n d f r om{ v ., ~ } = 0 a l so { to ., ~ } ffi 0 fo l l ow s . T h us a l l t h e fou r vec t o r s ~ , ~ , ~ . , t o . w ou l db e m u t ~ ! l y o r t h o g o n a l . T h i s is e x c l u d e d b y [2] a n d ( 1 9) .

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24 ILP Wigner / Unitary representations of Lorentz group

T w o c a s es a r e c on c e iv a b l e n o w . E i t h e r t h e f o u r f o ld c h a r a c t e r i s t i c r o o t h a so n l y o n e c h a r a c t e r i s t i c v e c t o r , o r t h e r e is in a d d i t i o n t o v, (a t l e a s t) a n o t h e rc h a r a c t e r i st i c v ec t o r v l . I ll t h e f o r m e r ca s e f o u r l i n e a rl y i n d e p e n d e n t v e c t o r sv , , w , . z , , x , c o u l d b e f o u n d s u c h t h a t

A ,.v ,. = v , A , w , = w , + v ,h , z , = z , + w , h , x , = x , + z , .

H o w e v e r { v , , x ,} = { h , ~ , , h , x , ] = { v , , z , } + {v , , z , } f r o m w h i c h { v , , z ,} = 0f o ll ow s . O n t h e o t h e r h a n d

I w . , z. } = { A . w . , A ~ . } = { w . , z . } + I w . , w . } + { v . , z . } + { v . , w . } .T h i s g i v e s w i t h ( 1 9 a) a n d ( 1 9b ) { v , , z,} = 1 s o t h a t t h i s c a s e m u s t b e e x c l u d e d .

( e ) T h e r e i s t h u s a v e c t o r v~ s o t h a t i n a d d i t i o n t o ( 1 8)( 1 8 a ) A , v l = v xh o l d s . F r o m { w , , v l} = { A , w , , A,vt}" = { w , , v l } + { v , , v t} fo l lows(19d ) {v , , v ,} = 0 .T i l e e q u a t i o n s ( 1 8) , ( 1 8 a ) w il l r e m a i n u n c h a n g e d i f w e a d d t o w , a n d v t a m u l t i p leo f v , . W e c a n a c h i e v e i n t h is w a y t h a t t h e f o u r t h c o m p o n e n t s o f b o t h t o , a n d~ , ~ 'a n is h . F u r t h e r m o r e , vl c a n b e n o r m a l i z e d t o - 1 a n d a d d e d t o w , a l s o w i t ha n a r b i t r a r y c o ef fi c ie n t, t o m a k e i t o r t h o g o n a l to v l. H e n c e , w e c a n a s s u m e t h a t( 1 9 e ) v . = w . = 0 ; I v , , v ~} = - 1 ; { t o . , v ~} = o .

W e c a n f i n a l l y d e f i n e t h e n u l l v e c t o r z , t o b e o r t h o g o r m l t o w , a n d v~ a n d h a v e as c a l a r p r o d u c t 1 w i t h v ,(1 90 {z , , z ,} = {z , , w~} = { z , , vl} = 0; {z ,, v,} = 1.T h e n t h e n u l l v e c t o r s v , a n d z r e p r es e n t th e m o m e n t a o f t w o l i gh t b e a m s i no p p o s i t e d i re c t i o n s . I f w e s e t A , z , = a v , + b w , + c z , -4- d v t t h e c o n d i t i o n s{ z , v} = { h , z , , A l l g i v e, if w e s e t f o r v t h e v e c t o r s v , , w , , z , , vt t h e c o n d i t i o n sc = 1 ; b = c ; 2 a e - b 2 - d ~ = 0 ; d = 0 . H e n c e

h~ ---- V. A. W. = W. + V~( 2 0 ) h,v~ = v ~ h , z , = z , + w , + v , .A L o r e n t z t r m m f o r m a t i o n w i t h a n e l e m e n t a r y d i vi s o r c a n b e b e s t c h a r a c t e r i z e db y t h e n u l l v e c t o r v w h i c h i s i n v a r i a n t u n d e r i t a n d t h e s p a c e p a r t o f w h i c hf o r m s w i t h t h e t w o o t h e r v e c to r s t o, a n d v l t h r e e m u t u a l l y o r t h o g o n a l v e c t o r s i no r d i n a r y s p a c e . T h e t w o v e c t o r s w , a n d v~ a r e n o r m a l i z e d , 0z i s i n v a r i a n t u n d e rA . w h i l e th e v e c t o r v i s a d d e d t o w u p o n a p p l i c a t i o n o f A . T h e r e s u l t o f t h ea p p l i c a t i o n o f A , t o a v e c t o r w h i c h i s li n e a r l y i n d e p e n d e n t o f v , , t o. a n d v~ i s ,a s w e s a w , a l r e a d y d e t e r m i n e d b y t h e e x p r e s s i o n s f o r h , z , h tO o a n d A , v z .

T h e A . ( 7 ) w h i c h h a v e t h e i n v a r i a n t n u l l v e c t o r v . a n d a l s o to . ( a n d h e n c e a l s o

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E.P. W igner / Unitary representations o f Lorentz group 25v t) i n c o m m o n a n d d i ff e r o n l y b y a d d i n g t o w , d i f f e re n t m u l t ip l e s 7 v , o f t, ,,f o r m a c y c l i c g r o u p w i t h "v - - 0 , th e u n i t t r a n s f o r m a t i o n a s u n i t y :

~ . (v)A. (V ' ) = ACv + ~ ' ) .T h e L o r e n t z t r a n s f o r m a t i o n M ( a ) w h i c h le a v es v t a n d w i n v a r i a n t b u t r e-p l a c e s v , b y ow , ( a n d z . b y a - ~ z, ) h a s t h e p r o p e r t y o f t r a n s f o r m i n g A ,(~ ,) i n t o

m ( , , ) A ( ~, ) M ( ~ ) - t = A . (cry) . ( + )A n e x a m p l e o f A , (~ ) a n d M ( a ) i sI i 1 0 0 i1 ,y 0A , ( 7 ) - - 0 - 7 1 - 7 2 - 75 ;

0 , 5 1 + , ' I ]

M ( a ) = I [0 01 0 00 C a + - - ~ ) ~ ( ~ - - - ~ ) " 0 0 i ( ~ - " - ~ ) ~ ( ~ + " - ~ )

T h e s e L o r e n t z t r a n s f o r m a t i o n s p l a y a n i m p o r t a n t r S le i n th e r e p r e s e n t a ti o n sw i t h s p a c e li k e m o m e n t u m v e c t o rs .A b e h a v i o r l i k e ( + ) i s i m p o s s i b l e f o r f i n it e u n i t a r y m a t r i c e s b e c a u s e t h ec h a r a c t e r i s t i c v a l u e s o f M ( a ) -~ A , ( ~ , )M ( , , ) a n d A , (~ ,) a r e t h e s a m e - - t h o s e o fAo(~a) ---- A ,( -: ) ~ t h e a th pow er s o f t hos e o f A (7 ). Th i s sho ws v e r y s im p ly t h a tt h e L o r e n t z g r o u p h a s n o t r u e u n i t a r y r e p r e s e n ta t i o n i n a f in it e n u m b e r ofd i m e n s i o n s .C . D e c o m p o s i t i o n o f a h o m o g e n e o u s L o r e n t z t r an s f o rm a t i o n i n t o r o t at i on s a n d

a n a c c e l e r a t i o n i n a g i v e n d i r e c t i o nT h e h o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n is , f r o m t h e p o i n t o f v i e w o f t h e

p h y s i ci s t, a t r a n s f o r m a t i o n t o a u n i f o r m l y m o v i n g c o o r d i n a t e s y s t e m , t h e o r ig i no f w h i c h c o i n c i d e d a t t = 0 w i t h t h e o r i g i n o f t h e f i rs t c o o r d i n a t e s y s t e m . O n ec a n , t h e r e f o r e , f i rs t p e r f o r m a r o t a t i o n w h i c h b r i n g s t h e d i r e c t i o n o f m o t i o n o ft h e s e c o n d sy s t e m i n t o a g i v e n d i r e c ti o n m y t h e d i re c t io n o f t h e t h i r d a x i s - -a n d i m p a r t i t a v e l o c i t y i n t h i s d i r e c t i o n , w h i c h w il l b r i n g i t t o r e s t. A f t e rt h i s , t h e t w o c o o r d i n a t e s y s t e m s c a n di ff e r o n l y i n a r o t a t i o n . T h i s m e a n s t h a te v e r y h o m o g e n e o u s L o r e n t s t r a n s f o r m a t i o n c a n b e d e co m p o s e d i n t h e f o l lo w i n gw a y 17( 21 ) A = R Z 8

IT Cf. e .g .L . S i lberatein, T h e T h e o r y o f R d a l i v i q t . Lo nd on 1924, p. 142.

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26 LP. Wigner / Unitary representations of Lorentz group

w h e r e R a n d 8 a r e p u r e r o t a t i o n s , ( i. e. R , =a n d R , = 8 4, = 1, a l s o R ' = R - l , 8 ' = 8 - l )d i r e c t i o n o f t h e t h i r d a x i s , i. e.

Z =

R z, = S , = S ,, = 0 f o r i ~ 4a n d Z i s a n a c c e l e r a t i o n i n t h e

0 0 01 0 0O a b0 b a

w i t h a 2 - b ' = 1 , a > b > 0 . T h e d e c o m p o s i t i o n ( 2 1 ) i s c l e a r l y n o t u n i q u e .I t w i ll b e s h o w n , h o w e v e r , t h a t Z is u n i q u e l y d e t e r m i n e d , i . e. t h e s a m e in e v e r yd e c o m p o s i t i o n o f t h e f o r m ( 2 1 ) .

I n o r d e r t o p ro v e th i s m a t h e m a t i c a l l y , w e c h o s e R s o t h a t i n R - I A = I t h ef i rs t t w o c o m p o n e n t s i n t h e f o u r t h c o l u m n I~ , = I~4 = 0 b e c o m e z e r o : R - 1s h a l l b r i n g t h e v e c t o r w i t h t h e c o m p o n e n t s -~-~4, h 2 ~, A34 i n t o t h e t h i r d a x i s .T h e n w e t a k e I u = ( h ~ , + A~4 + h ~ , ) t a n d I ,~ = A , f o r b a n d a t o f o r m Z ;t h e y s a t i s fy t h e e q u a t i o n I ~ - I~ , = 1. H e n c e , t h e f i rs t t h r e e c o m p o n e n t s o f t h ef o u r t h c o l u m n o f J = Z - I I = Z - t R - I A ~ l l b e c o m e z e ro a n d J ~ = 1, b e c a u s e o fJ ~ , - J1 4 - J ~ - J ~ , = 1. F u r t h e r m o r e , t h e f i r st t h r e e c o m p o n e n t s o f t h ef o u r t h r o w o f J w i l l v a n i s h al s o, o n a c c o u n t o f J l , - J ~ t - J ~ - J ~ s = 1 , i .e .J = 8 = Z - t R - ~ A i s a p u r e r o t a t i o n . T h i s p r o v e s t h e p o s s i b i l i t y o f t h e d e -c o m p o s i t i o n ( 2 i ) .

T h e t r a c e o f A A ' = R Z 2 R - l i s e q u a l t o t h e t r a c e o f Z 2, i .e . e q u a l t o 2 a 2 +2b2 + 2 = 4 a 2 = 4b 2 + 4 w h i c h s h o w s t h a t t h e a a n d b o f Z a r e u n i q u e l y d e t e r -m i n e d . I n p a r t i c u l a r a = 1, b = 0 a n d Z t h e u n i t m a t r i x if h A ' = 1 , i .e . A -ap u r e r o t a t i o n .

I t i s e a s y to s h ow n o w t h a t t h e g r o u p s p a c e o f t h e h o m o g e n e o u s L o r e n t zt r a n s f o r m a t i o n s is o n l y d o u b l y c o n n e c t e d . I f a c o n t i n u o u s s e r ie s A ( t ) o fh o m o g e n e o u s L o r e n t z t r a n s f o r m a t i o n s is g i v en , w h i c h i s u n i t y b o t h f o r t - - 0a n d f o r t = 1 , w e c a n d e c o m p o s e i t a c c o r d i n g t o ( 21 )( 2 1 a ) A ( t ) = R ( t ) Z ( t ) S ( t ) . .

I t i s a l s o c le a r f r o m t h e f o r eg o i n g , t h a t R ( t ) c a n b e a s s u m e d t o b e c o n t i n u o u si n t, e x c e p t f o r v a l u e s o f t , f o r w h i c h h ~ = A24 = A34 = 0 , i .e . f o r w h i c h h is ap u r e r o t a t i o n . S i m i l a r l y , Z ( t ) w i l l b e c o n t i n u o u s i n t a n d t h i s w i l l h o l d e v e nw h e r e A ( t ) i s a p u r e r o t a t i o n . F i n a l l y , S = Z - ~ R - I A w i l l b e c o n t i n u o u s a l s o ,e x c e p t w h e r e h ( t ) i s a p u r e r o t a t i o n .

L e t u s c o n si d e r n o w t h e s er ie s o f L o r e n t z t r a n s f o r m a t i o n s( 2 1 b ) A . ( t ) = R ( t ) z ( t ) ' z ( t )w h e r e t h e b o f Z ( t ) i s s t i m e s t h e b o f Z ( L ) . B y d e c r e a s i n g s f r o m 1 t o 0 w ec o n t i n u o u s l y d e f o r m t h e s e t h i ( t) = A ( t) o f L o r e n t z t r a n s f o r m a t i o n s i n t o a s e t o fr o t a t i o n s l h ( t ) = R ( t ) S ( t ) . B o t h t h e b e g i n n i n g A 0(0 ) = 1 a n d t h e e n d h , ( 1 ) - - 1o f t h e se t r e m a i n t h e u n i t m a t r i x a n d t h e s e t s A 0(t) r e m a i n c o n t i n u o u s i n t f o r

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E.P. Wigner / U nitary representations o f Loren tz group 2 7

all values of s. This last fact is evident for such t for which A(t) is not a rota-tion: for such t all factors of (21b) are continuous. But it is true also for tofor which A(t0) is a rotation, and for which, hence Z(t0) = 1 and A,(t~) = Al(to) ffiA(t0). As Z ( t ) i s everywhere continuous, there ~11 be a neighborhood of toin which Z ( t ) and hence also Z ( t ) ' i s arbitrarily close to the un it matrix. Inthis neighborhood A,(t) = A(0. 8 ( t ) - t Z ( t ) - t Z ( t ) " 8 ( 0 i s arbitrarily close toA(t); and, if the neighborhood is small enough, this is arbitrarily close to

=

Thus (21b) replaces the continuous set A(t) of Lorentz transformations by acontinuous set of rotations. Since these form an only doubly connected mani-fold, the manifold of Lorentz transformations can not be more than doublyconnected. The existence of a two valued representationm shows tha t it isactually doubly and not simply connected.We can form a new group~ from the Lorentz group, the elements of which arethe elements of the Lorentz group, together with a way ACt), connecting A(1)= A with the uni ty A(0) = E. However, two ways which can be continuouslydeformed into each other are not considered different. The product of theelement "A with the way A(t)" with the element " I with the way I ( 0 " i s theelement AI with the way which goes from E along A(0 to A and hence alongAI(t) to AI. Clearly, the Lorentz group is isomorphic with this group and twoelements (corresponding to the two essentially different ways to A) of this groupcorrespond to one element of the Lorentz group. It is well known, M that thisgroup is holomorphic with the group of unimodular complex two dimensionaltransformations.

Every continuous representation of the Lorentz group "up to the sign" is asinglevalued, continuous representation of this group. The tr~for mat ion whichcorresponds to "A with the way A0)" is that d(a) which is obtained by goingover from d(E) = d(A(0)) -- 1 continuously along d( a( 0) to d(A(l) ) ffi d(a) .D. The homogeneous Lorentz group is simple

It will be shown, first, that an invariant subgroup of the homogeneous Lorentsgroup contains a rotation (i.e. a transformation which leaves z, invariant).--We can write an arbitrary element of the invariant subgroup in the form R Z 8of (21). From its presence in the invariant subgroup follows tha t of 8 . R Z 8 . 8 -~= 8 R Z = T Z . If X, is the rotation by T about the first a x i s , X , Z X , ffi g - . land X , T Z X . -~ = X ~ T X . X . Z X ~ = X , T X , Z "~ i s contained in the invariantsubgroup also and thus the transform of this with Z , i.e. Z - - z X , T X , , a ls o. T h eproduct of this with T Z i s T X . T X , which leaves z4 invariant. I f T X , T X , = 1we can take T Y . T Y , . I f this is the unity also, T X , T X , ffi T Y , T Y , a n d Tcommutes with X , Y . , i.e. is a rota tion about the third axis In this c a s e t h e

s ' Cf . H. W eyl , Gruppt, theorie und Qua~m~ .r ~f dk , s t ; ed . Le ipzig 1928, p a g ~ 110-114,2rid ed. Le ipzig 1931, pages 130-133. I t may be interest in g to remark tha t e~ en t ia i ly th esam e isom orph ism has been recogn ized already by L. S i lberetein , I .e . pages 148-15/ .

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28 KI~ Wigner / Unitary representations o f Lorentz groupspace like (complex) characteristic vectors of T Z lie in the plane of the first twoco6rdinate axes. Transforming T Z by an acceleration in the direction of thefirst co6rdinate axis we obtain a new element of the invariant subgroup forwhich the space like characteristic vector will have a not vanishing fourthcomponent. Taking this for RZ,S we can transform it with S again to obtain anew S R Z = T Z . However, since $ leaves z4 invariant, the fourth componentof the space like characteristic vectors of this T Z will not vanish and we canobtain from it by the procedure just described a rotation which must be con-tained in the invariant subgroup.It remains to be shown that an invariant subgroup which contains a rotation,contains the whole homogeneous Lorentz group. Since the three-dimensionalrotation group is simple, all rotations must be contained in the invariant sub-group. Thus the rotation by 7r around the first axis X, and also its transformwith Z and also

Z X , Z - ' . X , = Z . X . Z - I X , = Z 'is contained in the invariant subgroup. However, the general acceleration inthe direction of the third axis can be written in this form. As all rotations arecontained in the invariant subgroup also, (21) shows that this holds for allelements of the homogeneous Lorentz group.It follows from this that the homogeneous Lorentz group has apart from therepresentation with unit matrices only true representations. It follows thenfrom the remark at the end of part B, that these have all infinite dimensions.This holds even for the two-valued representations to which we shall be led inSection 5 equ. (52D), as the group elements to which the positive or negativeunit matrix corresponds must form an invariant subgroup also, and becjtuse theargument at the end of part B holds for two-valued representations also. Oneeasily sees furthermore from the equations (52]3), (52C) that it holds for theinhomogeneous Lorentz group equally well.

5. R E D U C T I O N O F R E P R E S E N T A T I O N S U P T O A FACTORT O ~ w o - V A L U E DREPRESENTATIONS

The reduction will be effected by giving each unitary transformation, which isdefined by the physical content of the theory and the consideration of referenceonly up to a factor of modulus unity, a "phase," which will leave only the signof the representation operators undetermined. The unitary operator cor-responding to the translation a will be denoted by T(a) , that to the homogeneousLorentz transformation A by d(A). To the general inhomogeneous Lorentztransformation then D (a , A) = T (a )d (A ) will correspond. Instead of therelations (12), we shall use the following ones.(22B) T(a )T(b ) = ca(a, b)T (a + b)(22C) d( A )T (a ) = (A, a)T ( A a )d ( A )(22D) d ( A ) d ( I ) = ~,(A, I ) d ( A I ) .

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E.IE Wigner / U nitary representations o f Loren tz group 29T h e ~ a r e n u m b e r s o f m o d u l u s 1. T h e y e n t e r b e c a u s e t h e m u l t i p l ic a t i o n

r u l e s (1 2 ) h o l d fo r t h e r e p r e s e n t a t i v e s o n l y u p t o a f a c to r . O t h e r w i s e , t h er e l a t i o n s ( 22 ) a r e c o n s e q u e n c e s o f (1 2 ) a n d c a n i n t h e i r r e t u r n r e p l a c e ( 1 2) .W e s h a l l r ep l a c e t h e T ( a ) , d ( A ) b y f l ( a ) T ( a ) a n d f i (A ) d ( A ) r e s p e c t i v e l y , f o rw h i c h e q u a t i o n s s i m i l a r t o ( 2 2) h o l d , h o w e v e r W ith(2 2 ') ~(a , b) -- 1; ~0(A, a) -- 1; ,o(A, I) --- :1:1.

A .I t i s n e c e s s a r y , fi rs t, t o s h o w t h a t t h e u n d e t e r m i n e d f a c t o r s in t h e r e p r e s e n t a -

$ ion D ( L ) c a n b e a s s u m e d i n s u c h a w a y t h a t t h e co (a , b ), co (A , a ) , c o ( A , / ) b e c o m ea p a r t f r o m r e g io n s o f l o w e r d i m e n s i o n a l i t y - - c o n t i n u o u s f u n c t i o n s o f t h e i r& r g u n t e nt s . T h i s i s a c o n s e q u e n c e o f t h e c o n t i n u o u s c h a r a c t e r o f t h e r e p r e s e n t a -t i o n a n d s h a l l b e d i s c u s s ~ i f i rs t.( a ) F r o m t h e p o i n t o f v i e w o f t h e p h y s i c i s t, t h e n a t u r a l d e f i n i t i o n o f t h ec o n t i n u i t y o f a r e p r e s e n t a t i o n u p t o a f a c t o r i s a s f o ll ow s . T h e n e i g h b o r h o o d 8o f a L o r e n t z t r a n s f o r m a t i o n / 4 ffi ( /~ , / ) s h a ll c o n t a i n a ll t h e t r a n s f o r m a t i o n sL = ( a , A) f o r w h i ch I a , - b , I < 5 an d [ A~ -- I ~ I < 5 . T he r ep r e se n t a t i onu p t o a f a c t o r D ( L ) i s c o n t i n u o u s i f t h e r e i s t o e v e r y p o s i t i v e n u m b e r , e v e r yn o r m a l i z e d w a v e f u n c t i o n ~o a n d e v e r y L o r e n t z t r a n s f o r m a t i o n L 0 s u c h a n e i g h -b o r h o o d ~ o f Lo t h a t f o r e v e r y L o f t h is n e i g h b o r h o o d o n e c a n f i n d a n f i o fm o d u l u s 1 ( t h e f~ d e p e n d i n g o n L a n d ~o) s u c h t h a t ( u , , u , ) < ~ w h e r e( 2 3 ) u , = ( D ( L , ) -

L e t u s n o w t a k e a p o i n t / 4 i n th e g r o u p sp a c e a n d f i n d a n o r m a l i z e d w a v e[un c t ion ~o for w hic h I (~o, D(Lo)~o) I > 1 /6 . T he re a lw ays exi s t s a ~o w i th th i spro pe r ty , i f I(~o, D(Lo)~0)[ < 1 / 6 th en ~ - - a~o -! - ~D(L0)~o w i th su i t ab ly cho sen- an d ~ wi l l be no r m a l i zed and I ( ~ , D( Lo)~ ,) ] > 1 / 6 . W e cons i de r t h en su chn e i g h b o r h o o d ~ o f / 4 f o r a ll L o f w h i c h I (~0, D(L)~o) J > 1 / 12 . I t i s we l lk n o w n ~' t h a t t h e w h o l e g r o u p s p a c e c a n b e c o v e r e d w i t h s u c h n e i g h b o r h o o d s .W e w a n t t o s h ow n o w t h a t t h e D(L)~o c a n b e m u l t i p l ie d w i t h s u c h p h a s e f a c t o r s( d e p e n d i n g o n L ) o f m o d u l u s u n i t y t h a t i t b e c o m e s s t ro n g l y c o n t in u o u s i n t h er e g i o n ~ .W e s h a l l ch o s e t h a t p h a s e f a c to r s o t h a t (~o, D (L) v,) b e c o m e s r e a l a n d p o s i t i v e .D e n o t i n g t h e n( 2 3 ' ) (D(L~) - D ( L )) ~ o = U , ,t h e ( U , , U , ) c a n b e m a d e a r b i tr a r i ly s m a l l b y l e t t in g L a p p r o a c h s u f fi c ie n tl yn e a r t o L I , i f L t is i n ~ . I n d e e d , o n a c c o u n t o f t h e c o n t in u i t y , a s d e f in e dab ov e , t he re i s an f l ---- e ; ' suc h th a t (u , u ) < ~ i f L is s - ~ c i e n t l y n e a r t o L Iw h e r e

u = -

z~ T h i s c o n d i ti o n i8 t h e " s e p a r a b i l i t y " o f t h e g r o u p . C f . e . g . A . H u n t , A n n . o f M a t h . ,34, 147, 1933.

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30 E.P Wigner / Unitary representations of Lorentz group

T R b i n g t h e a b s o l u t e v a l u e o f th e s c a l a r p r o d u c t o f u w i t h o n e o b t a i n si ( 9 , V ( I a ) ~ ) - c o s , c( , D ( L ) ~ o ) - - i s i n , ~ (~ , D C L ) ~ , ) I - - I ( ~ ' , u ) [ ~_ V / ~ ,

b e c a u s e o f S c h w a r t z 's i n e q u a l i t y . I f o n l y V ~ - < 1 / 1 2 , t h e m u s t b e s m a l l e rt h a n ~ / 2 b e c a u s e t h e a b s o l u t e v a l u e i s c e r t a i n ly g r e a t e r t h a n t h e r e al p a rt , a n db o t h (~ , D ( L ~ ) ) a n d ( , D ( L ) ) a r e r e a l a n d g r e a t e r t h a n 1 / 1 2 .

A s t h e a b s o l u te v a l u e i s a l so g r e a t e r t h a n t h e i m a g i n a r y p a r t, w es i n < 1 2 V ~ .

O n t h e o t h e r h a n d ,

a n d t h u sU , = u - I - ( e ; " - 1 ) D ( L ) ~ ,

( U o , G o ) t _~ ( u , u ) t + [ e " - ] I -~ ~ + 2 s i n , / 2( u o , u o ) _~ 8 2 5 , .

( b ) I t s h a ll b e s h o w n n e x t t h a t i f D(L)~ s s t ron g ly con t inuous in a reg iona n d D ( L ) i s c o n t i n u o u s i n t h e s e n s e d e f in e d a t t h e b e g i n n i n g o f t h i s s ec t io n ,t h e n D ( L ) ~ w i t h a n a r b i t r a r y ~ is ( s t ro n g l y ) c o n t i n u o u s i n t h a t r e g i o n a ls o .W e s h a l l s e e , h e n c e , t h a t t h e D ( L ) , w i t h a n y n o r m a l i z a t i o n w h i c h m a k e s aD ( L ) ~ , s t r o n g l y , c o n t i n u o u s , i s c o n t i n u o u s i n t h e o r d i n a r y s e n s e : T h e r e i s t oe v e r y / , 1 , Ea n d t ~ , r y ~ a 5 s o t h a t ( U , , U ) < ~ w h e r e

U , = ( D ( L t ) - D ( L ) ) ~i f L i s i n t h e n e i g h b o r h o o d ~ o f L x .I t i s s u t t l c i e n t t o s h o w t h e c o n t i n u i t y o f D ( L ) ~ w h e r e ~ i s o r th o g o n a l t o .I n d e e d , e v e ry ~ ' ca n b e d e c o m p o s e d i n to t w o te r m s , ~ ' = a~, + / t ~ t h e o n e ofw h i c h i s p a r a ll e l, t h e o t h e r p e r p e n d i c u l a r t o ~. S i n c e D ( L ) v , i s c o n t i n u o u s ,a c c o r d i n g t o s u p p o s i ti o n , D ( L ) ~ ' = ~ d ) ( L ) ~ - t- t JD ( L ) ~ w i l l b e c o n t i n u o u s a l s o ifD ( L ) ~ , i s c o n t i n u o u s .T h e c o n t i n u i t y o f t h e r e p r e s e n t a t i o n u p t o a f a c t o r r e q u i r e s t h a t i t is p o s si b let o a c h i e v e t h a t ( u , u , ) < ~ a n d ( u + , , u , ~ , ) < ~ w h e r e( 2 3 a) u = ( D ( L ~ ) - f l V ( L ) ) ~ ,( 2 3 b ) u ~ , = ( D ( L , ) - f l ,~ ,D(L)) ( - t - ~o) ,w i t h m f i t a b l y c h o s e n f l 's . A c c o r d i n g t o t h e f o r e g o i n g , i t a l s o is p o s si b l e t oc h o o s e L a n d L~ s o c lo s e t h a t ( U , , U , ) < E.S u b t r a c t i n g ( 2 3 9 a n d ( 2 3a ) fr o m ( 2 3b ) a n d a p p l y i n g D ( L ) -~ o n b o t h s i d es g iv e s

(~ - o , n , ) q , + ( 1 - ~ , ~ , ) ~ , = 9 ( L ) - % , ~ - ~ , - U o )T h e s r ~ l a r p r o d u c t o f t h e r i g h t s i d e w i t h i ts e l f i s l e ss t h a n 9E . H e n c e b o t h] f l - fl~ +~ [ < 3 2 a n d I 1 - f l , ~ , [ < 3 2 o r [ 1 - f t l < 0 E l - B e c a u s e o fU , = u , - (1 - f l , ) D ( L ) ~ , t h e ( U , , U , ) t < ( u , u ) -I- ] 1 - fl i a n d t h u s(U, U) < 49E.

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E.P. W igner Un itary representationsof Lorentz group 31T h i s c o m p l e te s t h e p r o o f o f t h e th e o r e m s t a t e d u n d e r ( b ) . I t a l s o s h o w s t h a t

n o t o n l y t h e c o n t i n u i t y o f D(L)~o h a s b e e n a c h i e v e d i n t h e n e i g h b o r h o o d o f L ob y t h e n o r m a l i z a t i o n u s e d in ( a ) b u t a l s o t h a t o f D ( L ) ~ / w i t h every 41 , i .e . , thec o n t i n u i t y , o f D ( L ) .

I t i s c l e a r a l so t h a t e v e r y f i n it e p a r t o f t h e g r o u p s p a c e c a n b e c o v e r e d b y af in i te n u m b e r o f n e i g h b o r h o o d s in w h i c h D ( L ) c a n b e m a d e c o n t i n u o u s . I t i se a s y t o s e e t h a t t h e w o f ( 2 2) w i l l b e a ls o c o n t i n u o u s i n t h e s e n e i g h b o r h o o d s s ot h a t i s i s p o s s ib l e t o m a k e t h e