GENERAL  ARTICLE
Eugene Paul Wigner – A Tribute*
N Mukunda
N Mukunda is at the
Centre for High Energy
Physics, IISc, Bangalore.
His interests are classical
and quantum mechanics,
theoretical optics and
mathematical physics.
O n e o f o u r la st su rv iv in g lin k s w ith th e p erio d o f th e
crea tio n a n d d ev elo p m en t o f q u a n tu m m ech a n ics w a s
b ro k en w ith th e p a ssin g o f E u g en e W ig n er o n 1 J a n u a ry 1 9 9 5 a t P rin ceto n in U S A . W ig n er w a s rem a rk a b ly ta len ted a n d w id e-ra n g in g in h is in terests, a n d h is
w o rk to u ch ed in n u m era b le a sp ects o f m o d ern p h y sics.
In ev ery a rea th a t h e tu rn ed to , h e d iscov ered n ew a n d
p ro fo u n d in sig h ts a n d in terestin g v iew p o in ts, o ften u n d ersto o d a n d ca rried fu rth er b y o th ers m u ch la ter. H e
w a s a s m u ch a t h o m e in fu n d a m en ta l p ro b lem s o f p h y sics
a n d th eir m a th em a tica l a n a ly sis a s in en g in eerin g a n d
tech n o lo g ica l m a tters. In th is trib u te, I sh a ll ¯ rst d escrib e b rie° y h is life a n d ca reer, th en tu rn to a sk etch o f
h is w o rk , a n d co n clu d e w ith a n a ttem p t to ca p tu re h is
p erso n a lity a n d p h ilo so p h y o f scien ce a n d life.
A B r ie f L ife S k e tc h [1 ]
*Reproduced from Current Science, Vol.69, No.4, 25 August
1995.
Keywords
Symmetry in quantum mechanics, measurement theory in
quantum mechanics, nuclear
physics, relativistic quantum
mechanics, elementary particles.
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E u g en e P a u l (J en o P a l in H u n g a ria n ) W ig n er w a s b o rn
o n 1 7 N ov em b er 1 9 0 2 in B u d a p est, H u n g a ry, to E lisa b eth E in h o rn a n d A n th o n y W ig n er. H e th u s b elo n g ed
to th e sa m e g en era tio n a s W ern er H eisen b erg , E n rico
F erm i a n d P a u l D ira c. L eo S zila rd a n d J o h n v o n N eu m a n n w ere W ig n er's cla ssm a tes a t th e L u th era n H ig h
S ch o o l in B u d a p est { `a t th a t tim e, p erh a p s th e b est
h ig h sch o o l o f H u n g a ry a n d p ro b a b ly a lso o n e o f th e
b est o f th e w o rld ' [2 ]. W ig n er reta in ed g rea t reg a rd fo r
h is m a th em a tics tea ch er L . R a tz, w h o a lso reco g n ized
a n d en co u ra g ed v o n N eu m a n n 's u n u su a l ta len ts.
A fter a y ea r sp en t a t th e T ech n ica l In stitu te in B u d a p est, in 1 9 2 1 W ig n er jo in ed th e T ech n isch e H o ch sch u le
in B erlin to tra in a s a ch em ica l en g in eer. H e co m p leted
h is d o cto ra te in 1 9 2 5 a n d th en w o rk ed fo r a y ea r a n d a
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h a lf a s a lea th er ch em ist. B y th is tim e h e h a d b eco m e v ery m u ch a p a rt o f th e
B erlin p h y sics scen e; h is b rea k ca m e w ith a n a p p o in tm en t a s a ssista n t to R ich a rd
B eck er fo r 1 9 2 6 { 2 7 . T h is w a s fo llow ed in 1 9 2 7 { 2 8 b y a p o sitio n a s a ssista n t to
D av id H ilb ert in G Äo ttin g en , a n d th en a s P riva td o zen t a t G Äo ttin g en d u rin g 1 9 2 8 {
3 0 . A t th is p o in t h e m ov ed to th e U n ited S ta tes, w h ere h e sp en t th e rest o f h is
life.
W ig n er's ca reer in th e U S b eg a n a s a lectu rer in m a th em a tica l p h y sics d u rin g
1 9 3 0 a t P rin ceto n U n iv ersity, q u ick ly eleva ted to a P ro fesso rsh ip fro m 1 9 3 0 to
1 9 3 6 . T h e y ea r 1 9 3 7 { 3 8 w a s sp en t a s a p ro fesso r a t th e U n iv ersity o f W isco n sin
a t M a d iso n . U p o n retu rn to P rin ceto n , h e b eca m e T h o m a s D . J o n es P ro fesso r o f
M a th em a tica l P h y sics in 1 9 3 8 , a p o sitio n h e h eld u n til 1 9 7 1 . T h e a ca d em ic y ea r
1 9 5 7 { 5 8 w a s sp en t a t L eid en in th e N eth erla n d s.
In 1 9 3 7 W ig n er b eca m e a n a tu ra lized citizen o f th e U n ited S ta tes. H e to o k h is
citizen sh ip v ery serio u sly, a n d p lay ed a v ery a ctiv e ro le in p u b lic a ® a irs a n d m a tters o f g ov ern m en t p o licy. A s h is co n trib u tio n to th e w a r e® o rt, h e sp en t 1 9 4 2 { 4 5
a t th e M eta llu rg ica l L a b o ra to ry o f th e U n iv ersity o f C h ica g o , th e la st tw o y ea rs
a s th e h ea d o f th e th eo ry g ro u p th ere. E a rlier h e h a d jo in ed S zila rd a n d F erm i in
p ersu a d in g E in stein to w rite th e fa m o u s A u g u st 1 9 3 9 letter to P resid en t F ra n k lin
R o o sev elt th a t led to th e settin g u p o f th e M a n h a tta n P ro ject. H e w a s p resen t
a t th e U n iv ersity o f C h ica g o 's S ta g g F ield S q u a sh C o u rts o n 2 D ecem b er 1 9 4 2 to
w itn ess th e w o rld 's ¯ rst co n tro lled n u clea r ¯ ssio n rea ctio n set u p u n d er F erm i's
lea d ersh ip . D u rin g 1 9 4 6 { 4 7 h e serv ed a s D irecto r o f w h a t la ter b eca m e th e O a k
R id g e N a tio n a l L a b o ra to ry in T en n essee. In 1 9 5 2 h e w a s fu ll-tim e a d v iser to th e
D u P o n t C o m p a n y to d esig n th e S ava n n a h R iv er h eav y -w a ter p lu to n iu m p ro d u ctio n rea cto rs. S o o n a fter, in 1 9 5 4 h e w a s a p p o in ted to th e G en era l A d v iso ry
C o m m ittee o f th e U n ited S ta tes A to m ic E n erg y C o m m issio n , a n d serv ed a lso o n
m a n y p a n els o f th e S cien ce A d v iso ry C o m m ittee to th e P resid en t o f th e U n ited
S ta tes.
O f th e m a n y aw a rd s th a t ca m e to W ig n er, w e m u st m en tio n th e M ed a l fo r M erit,
th e F ra n k lin M ed a l fo r 1 9 5 0 , th e E n rico F erm i A w a rd o f th e U S A E C fo r 1 9 5 8 ,
th e A to m s fo r P ea ce A w a rd fo r 1 9 6 0 , th e M a x P la n ck M ed a l fo r 1 9 6 1 , a n d th e
1 9 6 3 N o b el P rize in p h y sics (sh a red w ith M a ria M ay er a n d H a n s J en sen ) fo r h is
w id e ra n g e o f co n trib u tio n s to q u a n tu m m ech a n ics.
W ig n er's ¯ rst m a rria g e, to A m elia F ra n ck in 1 9 3 6 , w a s fo llow ed b y a seco n d o n e
in 1 9 4 1 to M a ry A n n ette W h eeler, a p ro fesso r o f p h y sics. H is sister M a rg it B a la sz
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GENERAL  ARTICLE
n ee W ig n er m a rried P a u l D ira c in 1 9 3 7 . It a p p ea rs th a t D ira c w a s so sh y th a t
h e o n ce in tro d u ced h is w ife to a n o ld frien d a s W ig n er's sister. In resp o n se (!)
W ig n er referred to D ira c a s `m y fa m o u s b ro th er-in -law ' [3 ]. T h ere is a ch a rm in g
a cco u n t b y M a rg it o f h er ¯ rst m eetin g w ith D ira c in W ig n er's co m p a n y. A t a
m eetin g in B u d a p est, th e v o n N eu m a n n s h a d in v ited M a rg it to v isit a n d stay
w ith th em in P rin ceto n . A n d th en [4 ]:
`E u g en e in sisted , \ If y o u co m e to P rin ceto n , y o u m u st stay w ith m e. W h a t w o u ld
p eo p le say if y o u d id n o t stay w ith y o u r b ro th er?" I w a s n o t terrib ly th rilled w ith
th e id ea . T h e v o n N eu m a n n s h a d a lov ely h o m e,..., w h ile m y b ro th er lik ed to
a p p ea r, a n d a ct, lik e a p a u p er. W e sa iled in th e fa ll; E u g en e h a d a tw o -b ed ro o m
a p a rtm en t, p ro u d ly b o a stin g th a t h e fu rn ish ed it to th e co st o f u n d er $ 2 5 . It
lo o k ed lik e it ... It w a s so o n a fter m y a rriva l; w e w ere h av in g lu n ch a t o n e
o f th ese resta u ra n ts, w h en a ta ll, slen d er yo u n g m a n en tered th e d in in g ro o m ,
lo o k ed a t E u g en e a n d g reeted h im . H e lo o k ed lo st, a n d sa d . I a sk ed w h o h e
w a s, still sta n d in g u n d ecid ed a n d n o n e to o h a p p y -lo o k in g . I w a s to ld , h e w a s
a n E n g lish p h y sicist, w h o m E u g en e k n ew in G Äo ttin g en , w h ere th ey u sed to h av e
th eir m ea ls to g eth er. \ H e d o es n o t lik e to ea t a lo n e" . \ S o w h y d o n 't y o u a sk
h im to jo in u s?" T h a t w a s h ow I m et P a u l D ira c. T h a t w a s th e fa ll o f 1 9 3 4 . T h e
In stitu te fo r A d va n ced S tu d ies h a d n o b u ild in g o f its ow n a s y et. Its m em b ers,
lik e E in stein , v o n N eu m a n n a n d D ira c a s a v isitin g m em b er, h a d a d jo in in g ro o m s
in a la rg e u n iv ersity b u ild in g , ca lled F in e H a ll. I rem em b er so w ell: to th e left
w a s E in stein 's ro o m , in th e m id d le E u g en e's a n d to th e rig h t o f h im , D ira c's.'
W ig n er, S zila rd a n d v o n N eu m a n n fo rm ed th e fa m o u s H u n g a ria n trio w h o co n trib u ted so d ecisiv ely to in tellectu a l life in th e U n ited S ta tes in th e 1 9 3 0 s a n d
la ter. T h ere is a sto ry th a t d u rin g a m eetin g o f scien tists co n n ected w ith th e
w a r e® o rt th ere w a s so m u ch co n fu sio n d u e to m a n y la n g u a g es b ein g u sed th a t
so m eo n e g o t u p a n d ex cla im ed : `G en tlem en , let u s u se o n e la n g u a g e w e ca n a ll
u n d ersta n d { H u n g a ria n !'
W h en W ig n er d ied h e left b eh in d h is th ird w ife E ileen , a so n a n d tw o d a u g h ters.
C o n trib u tio n s to S c ie n c e a n d E n g in e e r in g
W ig n er's w o rk in p h y sics is ch a ra cterized b y h a rd m a th em a tica l a n a ly sis b a sed
o n sim p le y et p ro fo u n d p h y sica l a ssu m p tion s. W h ile th ere is a d ow n -to -ea rth
p ra ctica l q u a lity to so m e o f h is w o rk , in o th ers h e d ea lt w ith th e m o st fu n d a m en ta l
issu es w ith g rea t re¯ n em en t { h e w a s b o th a n a rtist a n d a n en g in eer, a n d q u a n tu m
m ech a n ics w a s h is m ed iu m . T o q u o te J o h n A . W h eeler [5 ]: `In th e w o rk o f E u g en e
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W ig n er o n e sees th e b a sic h a rm o n y b etw een th e co n cep tu a l fra m ew o rk o f p h y sics
a n d th e stru ctu re o f th e m a th em a tics a sso cia ted w ith th a t p h y sics.' O n th e o th er
h a n d , h is g ra sp o f tech n o lo g y is b est co n v ey ed b y th is p a ssa g e fro m L aw ren ce
D resn er a n d A lv in M . W ein b erg [6 ]: `.... th e fa cility w ith w h ich h e co u ld p a ss b a ck
a n d fo rth b etw een en g in eerin g a n d p h y sics { fro m a d iscu ssio n o f th e p ro b a b le
d istrib u tio n o f en erg y lev els in U 2 3 5 to a critica l ex a m in a tio n o f th e b lu ep rin ts
o f th e co n crete fo u n d a tio n s fo r th e H a n fo rd rea cto rs, o r fro m a g ro u p th eo retica l
a rg u m en t in tra n sp o rt th eo ry to th e d esig n o f a lu m in iu m fu el elem en ts!'
W ig n er's ¯ rst im p o rta n t w o rk in p h y sics, co m p leted d u rin g h is a p p ren ticesh ip
w ith B eck er, w a s a p ow erfu l trea tm en t o f q u a n tu m m a n y -ferm io n sy stem s. A ro u n d
th e tim e o f th e m ov e to G Äo ttin g en , a n d fo llow in g a su g g estio n b y v o n N eu m a n n ,
h e u n d erto o k th e m a jo r ta sk o f in tro d u cin g g ro u p th eo retica l m eth o d s in to q u a n tu m m ech a n ics. B y 1 9 2 8 h e h a d p u b lish ed six la n d m a rk p a p ers o n th e su b ject; h e
sh a res w ith H erm a n n W ey l th e cred it fo r m a k in g th is a n essen tia l a n d ch a ra cteristic co m p o n en t o f q u a n tu m p h y sics w h ich p erva d es a ll its a p p lica tio n s. D u rin g
th e 1 9 3 0 s h e w o rk ed in so lid -sta te p h y sics a n d a t th e fro n tiers o f th e d ev elo p in g su b ject o f n u clea r p h y sics, m a k in g a m a jo r e® o rt to u n d ersta n d th e fo rces
b etw een n u cleo n s, a n d d ev elo p in g th e co m p o u n d n u cleu s m o d el to ex p la in reso n a n ce p h en o m en a in n eu tro n -in d u ced n u clea r rea ctio n s. H is d ev elo p m en t la ter o f
th e R -m a trix th eo ry o f n u clea r rea ctio n s w a s a resp o n se to a co m m en t b y F erm i
th a t th e co m p o u n d n u cleu s m o d el n eed ed a ¯ rm th eo retica l fo u n d a tio n . P ro b a b ly h is m o st rem a rka b le w o rk in m a th em a tica l p h y sics { th e stu d y o f th e u n ita ry
rep resen ta tio n s o f th e in h o m o g en eo u s L o ren tz g ro u p { g rew o u t o f a su g g estio n
m a d e to h im b y D ira c in 1 9 2 8 . T h is w a s co m p leted in M a d iso n in 1 9 3 7 , a n d su b seq u en tly b eca m e th e b a sic fra m ew o rk fo r a ll rela tiv istic q u a n tu m th eo ries. H e
ca m e b a ck to p ro b lem s o f n u clea r stru ctu re in h is su p erm u ltip let th eo ry o f 1 9 3 7 ,
a n d la ter in h is sta tistica l trea tm en t o f n u clea r sp ectro sco p y b a sed o n ra n d o m
m a trices.
In th e m id st o f a ll th is, in th e 1 9 4 0 s h e w o rk ed o n th e th eo ry o f n eu tro n ch a in
rea cto rs a n d th e d esig n o f p lu to n iu m b reed er rea cto rs.
W ig n er's co n cern w ith th e stru ctu re o f q u a n tu m m ech a n ics h a s led to a series o f
in cisiv e in sig h ts ov er m a n y y ea rs. In th e ea rly 1 9 6 0 s h e tu rn ed to p ro b lem s o f
in terp reta tio n a n d ep istem o lo g y ra ised b y th e sta n d a rd in terp reta tio n o f q u a n tu m
m ech a n ics. A t th is p o in t it is co n v en ien t to p resen t b rie° y a n d selectiv ely sk etch es
o f W ig n er's w o rk u n d er sev era l b ro a d a rea s. T h is is a d m itted ly a n in a d eq u a te,
in co m p lete a n d p o ssib ly su p er¯ cia l w ay to su rv ey h is w o rk , y et it m ay su cceed
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in co n v ey in g so m e id ea o f th e ra n g e a n d m a g n itu d e o f h is a ch iev em en ts. B efo re
em b a rk in g o n th is, w e m ay reca ll th e fo llow in g im p o rta n t b o o k s p u b lish ed b y
W ig n er: (1 ) G rou p T heory an d its A pplication to the Q u an tu m M echan ics of
A tom ic S pectra (A ca d em ic P ress, N ew Y o rk , 1 9 5 9 ; th e o rig in a l G erm a n v ersio n
p u b lish ed b y F ried rich V iew eg , B ra u n sch w eig , 1 9 3 1 ); (2 ) N u clear S tru ctu re, w ith
L eo n a rd E isen b u d (P rin ceto n U n iv ersity P ress, 1 9 5 8 ); (3 ) T he P hysical T heory of
N eu tron C hain R eactors, w ith A lv in M . W ein b erg (U n iv ersity o f C h ica g o P ress,
1 9 5 8 ); a n d (4 ) S ym m etries an d R e° ection s { S cien ti¯ c E ssays (In d ia n a U n iv ersity
P ress, 1 9 6 7 ). W e m ay a lso m en tio n th a t th e O cto b er 1 9 6 2 issu e o f th e R eview s of
M odern P hysics, p u b lish ed o n th e o cca sio n o f h is 6 0 th b irth d ay, co n ta in s m a n y
a rticles su rv ey in g W ig n er's w o rk in sev era l a rea s.
S tr u c tu re a n d C o n te n t o f Q u a n tu m M ec h a n ic s
A n y serio u s u ser o f q u a n tu m m ech a n ics is su re to ¯ n d h erself em p loy in g rep ea ted ly, eith er ex p licitly o r im p licitly, o n e o r a n o th er o f th e m a n y b a sic co n cep ts
a n d m eth o d s in v en ted b y W ig n er. O n e o f th e ea rliest is th e co n cep t o f p a rity [7 ].
In cla ssica l p h y sics, sp a ce in v ersio n is m erely a g eo m etrica l o p era tio n o r tra n sfo rm a tio n , a ru le to m a p ea ch p o in t in sp a ce to its im a g e b y in v ersio n th ro u g h
a ch o sen o rig in . T h e tim e is left u n a ® ected . A p a rticle tra jecto ry, fo r ex a m p le,
w o u ld b e m a p p ed o n to a n o th er p o ssib le tra jecto ry.
C lassical space in version :
P : x ! ¡ x ;t ! t;
x (t) ! ¡ x (t):
W ig n er sh ow ed th a t in q u a n tu m m ech a n ics, p a rity is m o re th a n a tra n sfo rm a tio n ,
it is a p h y sica l o b serv a b le w h o se va lu e ca n b e ex p erim en ta lly m ea su red . T h e
p o ssib le resu lts o f m ea su rem en t a re § 1 , a n d th e co rresp o n d in g q u a n tu m sta tes
a re sa id to p o ssess ev en o r o d d p a rity, resp ectiv ely.
Q u an tu m space in version :
P Ã (x ;t) = Ã (¡ x ;t);
à (¡ x ;t) = § à (x ;t) ) P = § 1 ;
ev en = o d d sta tes:
It w a s th is ro le o f p a rity in q u a n tu m m ech a n ics th a t w a s sh ow n b y W ig n er to b e
th e ex p la n a tio n fo r L a p o rte's selectio n ru le in a to m ic sp ectro sco p y [8 ]: th e m a trix
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elem en ts o f th e electric d ip o le m o m en t o p era to r, a n d h en ce th e co rresp o n d in g
tra n sitio n s, va n ish u n less th e tw o co n cern ed sta tes h av e o p p o site p a rities.
T h e d eep co n n ectio n b etw een in v a ria n ce p rin cip les a n d co n serva tio n law s, b o th
in cla ssica l p h y sics a n d in th e q u a n tu m d om a in w ith sp eci¯ ca lly n ew a n d su b tle fea tu res, rem a in ed a lifelo n g co n cern fo r W ig n er, so m eth in g h e ca m e b a ck
to tim e a n d a g a in . In th e p a rticu la r ca se o f ro ta tio n a l sy m m etry, th e g en era l
p ro g ra m m e o f in co rp o ra tin g g ro u p th eo retica l m eth o d s in to q u a n tu m m ech a n ics led to W ig n er's im p ressiv e b o d y o f resu lts co n cern in g a n g u la r m o m en tu m in
q u a n tu m m ech a n ics [9 ]. T h e d eta iled rep resen ta tio n th eo ry o f th e ro ta tio n g ro u p
S O (3 ) a n d its cov erin g g ro u p S U (2 ), w h ich is b a sic to q u a n tu m m ech a n ics, w a s
d ev elo p ed b y h im in a fo rm su ited to p ractica l a p p lica tio n . T h e a n g u la r m o m en tu m a d d itio n th eo rem , th e co n cep t o f ten so r o p era to rs, th e W ig n er{ E cka rt
th eo rem fo r th eir m a trix elem en ts, ex p licit ex p ressio n s fo r th e C leb sch { G o rd a n
co u p lin g co e± cien ts (a lso ca lled th e W ig n er 3 j sy m b o ls), lea d in g o n to th e in trica te R a ca h { W ig n er ca lcu lu s fo r co u p lin g o f ten so r o p era to rs a n d co m p u tin g
th e resu ltin g m a trix elem en ts, th e g en era liza tio n s to o th er sy m m etry g ro u p s {
a ll th ese o h -so fa m ilia r to o ls o f th e tra d e in a to m ic, n u clea r a n d p a rticle p h y sics
o rig in a te fro m h is w o rk .
In h is b o o k o n g ro u p th eo ry, W ig n er fo rm u la ted a n d p rov ed a fu n d a m en ta l th eo rem co n cern in g th e rep resen ta tio n o f sy m m etry o p era tio n s in q u a n tu m m ech a n ics
[1 0 ]. T h is is a v ery d eep a n d su b tle resu lt, a n d a b rief ex p la n a tio n w o u ld n o t b e
o u t o f p la ce. T h e rela tio n b etw een p h y sica l sta tes a n d w av e fu n ctio n s (o r H ilb ert
sp a ce sta te v ecto rs) in q u a n tu m m ech a n ics is o n e-to -m a n y. T h is is b eca u se a
ch a n g e in th e ov era ll p h a se o f a w av e fu n ctio n is p h y sica lly u n o b serva b le:
v ecto rs à ;e i® à ;e i¯ à ;::: ! sa m e p h y sica l sta te;
v ecto rs
M a n y -to -o n e
$
p h y sica l sta tes:
T h erefo re, w h a t p h y sica l sta tes co rresp o n d to in a o n e-to -o n e m a n n er a re n o t
v ecto rs b u t ray s: a ray is a n eq u iva len ce cla ss o f v ecto rs, tw o v ecto rs b ein g
d ecla red eq u iva len t if th ey d i® er o n ly b y a p h a se. T h e ray to w h ich a v ecto r Ã
b elo n g s ca n b e u n a m b ig u o u sly d escrib ed b y th e co rresp o n d in g p ro jectio n o p era to r
o r d en sity m a trix ½ Ã :
v ecto r à ! ray ½ à = à à + ;
o n e-to -o n e
$
ray s
p h y sica l sta tes.
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GENERAL  ARTICLE
R ay s d o n o t fo rm a v ecto r sp a ce, so th eir g eo m etry is so m ew h a t h a rd er to v isu a lize th a n th a t o f v ecto rs à . W ig n er's th eo rem th en sh ow s th a t a n y m a p p in g ¡ o f
ray s (i.e., p h y sica l sta tes) o n to th em selv es p reserv in g q u a n tu m -m ech a n ica l p ro b a b ilities { a n d a n y sy m m etry o p era tio n m u st b e o f su ch a n a tu re { ca n b e `lifted '
to eith er a lin ea r u n ita ry o r a n a n tilin ea r u n ita ry (a n tiu n ita ry ) tra n sfo rm a tio n T
o n v ecto rs.
U n ita ry -A n tiu n ita ry T h eo re m :
S y m m etry o p era tio n ¡ ,
½Ã = à Ã
y
! ¡ (½ Ã ) = ½ Ã
0
= Ã
0
Ã
0y
0
0
½ ' = ' ' y ! ¡ (½ ' ) = ½ ' 0 = ' ' y
0
0
à ;' ;:::d eterm in ed u p to p h a ses,
0
0
j(' ;Ã )j = j(' ;Ã )j )
eith e r
0
à = T à ;' 0= T ' ;:::;
T lin ea r u n ita ry,
0
0
(' ;Ã ) = (' ;Ã ) ! u n ita ry a ltern a tiv e;
or
0
0
à = T à ;' = T ' ;:::;
T a n tilin ea r u n ita ry,
0
0
(' ;Ã ) = (' ;Ã )¤
= (Ã ;' ) ! a n tiu n ita ry a ltern a tiv e.
(H ere th e in n er p ro d u ct o f th e H ilb ert sp a ce v ecto rs ' ;Ã is d en o ted b y (' ;Ã )).
T h is rem a rka b le th eo rem h a s b een ex ten d ed a n d p rov ed u n d er d i® eren t co n d itio n s
b y o th ers ov er th e d eca d es.
M o st sy m m etries in q u a n tu m m ech a n ics tu rn o u t to b e o f th e u n ita ry ty p e; tim e
rev ersa l is o n e ex a m p le w h ere th e a n tiu n ita ry a ltern a tiv e is rea lized . T h e a n a ly sis
o f th is tra n sfo rm a tio n in q u a n tu m m ech a n ics w a s g iv en b y W ig n er [1 1 ] in 1 9 3 2 .
In S ch rÄo d in g er's q u a n tu m m ech a n ics, tim e rev ersa l a cts o n w av e fu n ctio n s th u s:
T Ã (x ;t) = Ã (x ;¡ t)¤:
U n lik e p a rity, h ow ev er, th is o p era tio n d o es n o t h av e th e sta tu s o f a p h y sica l
o b serva b le in q u a n tu m m ech a n ics, a n d its eig en v a lu es a re n o t in va ria n tly d e¯ n ed
a n d a re n o t ex p erim en ta lly m ea su ra b le.
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GENERAL  ARTICLE
C o n tin u in g w ith th e th em e o f sy m m etry in q u a n tu m m ech a n ics, W ig n er a n d v o n
N eu m a n n p rov ed a v ery in terestin g resu lt in 1 9 2 9 , w h ich is o f g rea t im p o rta n ce
esp ecia lly in m o lecu la r p h y sics [1 2 ]: if th e electro n ic sta tes in a m o lecu le a re cla ssi¯ ed a cco rd in g to th eir sy m m etry, i.e., th e rep resen ta tio n o f th e fu ll g ro u p o f
sy m m etry o f th e releva n t m o lecu le, a n d if w e h av e tw o d istin ct eig en va lu es a n d
eig en sta tes sh a rin g th e sa m e sy m m etry (tw o electro n term s), th ese eig en va lu es
w ill n o t cro ss (b eco m e a ccid en ta lly eq u a l) a s o n e va ries th e in tern u clea r d ista n ces
in th e m o lecu le. O n th e o th er h a n d , electro n term s o f d istin ct sy m m etry ca n
cro ss. T h is is a g en era l th eo rem o f q u a n tu m m ech a n ics, a p p lica b le to a g en eric
h a m ilto n ia n p o ssessin g so m e sy m m etries a n d d ep en d en t o n a co n tin u o u s p a ra m eter: a s th e p a ra m eter is v a ried , d istin ct eig en va lu es `o f th e sa m e sy m m etry '
w ill n o t a ccid en ta lly cro ss b u t w ill rep el each o th er.
M a n y y ea rs la ter, W ick , W ig h tm a n a n d W ig n er [1 3 ] b ro u g h t to lig h t a n o th er
a sp ect o f sy m m etry in q u a n tu m m ech a n ics, n a m ely th e ex isten ce o f su p erselectio n
ru les. T h is a m o u n ts to a restrictio n o n th e a p p lica b ility o f th e su p erp o sitio n
p rin cip le in q u a n tu m m ech a n ics. In g en era l, th e H ilb ert sp a ce o f sta tes o f a
q u a n tu m sy stem b rea k s u p in to secto rs, an d th e fo rm a tio n o f co m p lex lin ea r
co m b in a tio n s to p ro d u ce n ew sta tes fro m old is lim ited to o n e secto r a t a tim e,
n o t cu ttin g a cro ss secto rs. T h is is th e rea son w h y th e p h a se o f a sp in o r ¯ eld { a
¯ eld w ith h a lf-o d d in teg er sp in { is n o n o b serva b le. S o , fo r in sta n ce, a n o n triv ia l
lin ea r co m b in a tio n o f sta tes w ith in teg er a n d h a lf-o d d in teg er a n g u la r m o m en ta
ca n n o t b e p rep a red . A s a n o th er ex a m p le, on e ¯ n d s th a t lin ea r su p erp o sitio n s o f
sta tes o f d istin ct electric ch a rg e a re u n p h y sica l. It is su sp ected th a t th ese resu lts
h a d lo n g b een k n ow n to W ig n er, a n d h e w a s p ersu a d ed b y h is co a u th o rs to jo in
th em a n d say so in p rin t.
In th e p refa ce to h is b o o k o n g ro u p th eo ry, W ig n er rela tes a co n v ersa tio n w ith
v o n L a u e o n th e u se o f g ro u p th eo ry a s th e n a tu ra l to o l w ith w h ich to ta ck le
p ro b lem s in q u a n tu m m ech a n ics [1 4 ]. H e say s: `I lik e to reca ll h is q u estio n
a s to w h ich resu lts ... I co n sid ered m o st im p o rta n t. M y a n sw er w a s th a t th e
ex p la n a tio n o f L a p o rte's ru le (th e co n cep t o f p a rity ) a n d th e q u a n tu m th eo ry o f
th e v ecto r a d d itio n m o d el a p p ea red to m e m o st sig n i¯ ca n t. S in ce th a t tim e, I
h av e co m e to a g ree w ith h is a n sw er th a t th e reco g n itio n th a t a lm o st a ll ru les o f
sp ectro sco p y fo llow fro m th e sy m m etry o f th e p ro b lem is th e m o st rem a rka b le
resu lt'.
T h e ex p o n en tia l d ecay law fo r u n sta b le sta tes h a s b een w ell k n ow n sin ce th e
d ay s o f R u th erfo rd 's ex p erim en ts o n ra d io a ctiv ity. T h e ¯ rst p ro p erly q u a n tu m -
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GENERAL  ARTICLE
m ech a n ica l d iscu ssio n a n d d eriva tio n o f th is law is d u e to W eissk o p f a n d W ig n er
[1 5 ]. T h ey w ere a b le to p rov id e th e b a sic th eo ry fo r th e n a tu ra l lin ew id th s a n d
lifetim es o f a to m ic sta tes d ecay in g v ia tra n sitio n s to o th er sta tes w ith em issio n
o f ra d ia tio n . T h eir u se o f seco n d -o rd er p ertu rb a tio n th eo ry a lo n g w ith ju d icio u s
a n d d elica te a ssu m p tio n s a lso d isclo sed th a t th e ex p o n en tia l d ecay law is o n ly
a n a p p rox im a te, n o t a n ex a ct, co n seq u en ce o f q u a n tu m m ech a n ics; so d ep a rtu res
fro m it fo r b o th v ery sh o rt a n d v ery lo n g tim es a re to b e ex p ected .
T h e lin ea r su p erp o sitio n p rin cip le o f q u a n tu m m ech a n ics, a lrea d y referred to
a b ov e, ¯ n d s its m o st n a tu ra l ex p ressio n a t th e lev el o f sta te v ecto rs in H ilb ert
sp a ce. T h e ray sp a ce o r d en sity m a trix d escrip tio n o f p h y sica l sta tes, w h ich is
clo ser to a cla ssica l d escrip tio n , o b scu res th is p rin cip le so m ew h a t { it is p resen t
b u t n o t m a n ifest. In 1 9 3 2 , w h ile stu d y in g th erm o d y n a m ic eq u ilib riu m in q u a n tu m
m ech a n ics, W ig n er in tro d u ced a n o th er d escrip tio n o f sta tes fo r q u a n tu m sy stem s
p o ssessin g cla ssica l ca n o n ica l a n a lo g u es [1 6 ]. T h u s, ea ch q u a n tu m sta te is d escrib a b le b y a certa in rea l d istrib u tio n o r fu n ctio n o n th e cla ssica l p h a se sp a ce.
In o n e d im en sio n w ith cla ssica l p h a se sp a ce va ria b les x a n d p , th e co n stru ctio n
is a s fo llow s:
1
à (x ) ! W (x ;p ) =
h
Z1
¡1
0
dx Ã
µ
¶
1 0
x ¡ x Ã
2
µ
1 0
x + x
2
¶¤
0
ex p (ix p = ¹h ) :
T h ese d istrib u tio n s { n a m ed a fter W ig n er { a re a t th e lev el o f d en sity m a trices,
n o t sta te v ecto rs. T h ey a re su g g estiv ely lik e cla ssica l p ro b a b ility d istrib u tio n s o n
p h a se sp a ce, su ch a s o n e u ses in cla ssica l sta tistica l m ech a n ics. H ow ev er, sin ce
in g en era l W (x ;p ) ca n b eco m e n eg a tiv e fo r so m e a rg u m en ts, w e d o n o t h av e a
cla ssica l sta tistica l p ictu re w ith w ell-d e¯ n ed p ro b a b ilities. T h is is a s it sh o u ld b e,
sin ce q u a n tu m fea tu res m u st b e p reserv ed . T h is d escrip tio n o f sta tes in q u a n tu m
m ech a n ics tu rn ed o u t to b e th e co u n terp a rt, o r co m p a n io n , to a ru le o r co n v en tio n
g iv en b y W ey l fo r a sso cia tin g a q u a n tu m -m ech a n ica l o p era to r w ith ev ery cla ssica l
d y n a m ica l va ria b le; a n d th ese id ea s w ere fu rth er ex ten d ed , p a rticu la rly b y M oy a l
[1 7 ].
W ig n er co n trib u ted a g rea t d ea l to th e fo rm a l d escrip tio n o f sca tterin g a n d rea ctio n p ro cesses in q u a n tu m m ech a n ics, esp ecia lly in th e co n tex t o f n u clea r p h y sics.
O n e o f h is resu lts co n cern s th e p h y sica l m ea n in g o f p h a se sh ifts. In g en era l,
sca tterin g cro ss-sectio n s a re d eterm in ed b y th e sq u a red m a g n itu d es o f S -m a trix
elem en ts, a n d in th ese th e p h a ses g et w a sh ed o u t. O n th e o th er h a n d , th e sp a tio tem p o ra l d ev elo p m en t o f a sca tterin g p ro cess d escrib ed w ith in th e lim its set
b y q u a n tu m m ech a n ics, in v o lv es th ese p h a ses. T h e b ea u tifu l co n n ectio n fo u n d
952
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b y W ig n er is th e ex p ressio n fo r tim e d elay ca u sed b y in tera ctio n a n d its rela tio n
to th e en erg y d ep en d en ce o f th e sca tterin g p h a se sh ift [1 8 ]:
¢ T (E ) = 2
d
± (E ) :
dE
H ere ± (E ) is th e p h a se sh ift a t en erg y E ; th u s, a n a ttra ctiv e (rep u lsiv e) in tera ctio n lea d s to ± (E ) in crea sin g (d ecrea sin g ) w ith en erg y, h en ce to a d elay (a d va n ce)
in th e a p p ea ra n ce o f th e ¯ n a l-sta te p ro d u cts o f a co llisio n a fter u n d erg o in g in tera ctio n .
W e co n clu d e th is a cco u n t w ith a co u p le o f `cu rio s'. C la ssica lly, o n e ex p ects th e
p o ssib le sta tes o f a sy stem o f in tera ctin g p a rticles { esp ecia lly, a tw o -b o d y sy stem
{ to sep a ra te in to tw o ty p es: u n b o u n d ed o r sca tterin g sta tes, h a v in g p o sitiv e
en erg y, a n d b o u n d sta tes, h av in g n eg a tiv e en erg y. In q u a n tu m m ech a n ics w e
ex p ect th e en erg y eig en va lu es to b eh av e a n a lo g o u sly : a co n tin u u m o f u n b o u n d ,
p o sitiv e-en erg y sca tterin g sta tes sittin g o n to p o f a set o f d iscrete n eg a tiv e-en erg y,
b o u n d sta tes. O n ly th e la tter h av e n o rm a liza b le w av e fu n ctio n s. In a rem a rka b le
p a p er in 1 9 2 9 , W ig n er a n d v o n N eu m a n n p ro d u ced a n ex a m p le o f a tw o -b o d y
p o ten tia l w h ich p o ssesses a b o u n d sta te em b ed d ed in th e co n tin u u m [1 9 ]! T h is
is a n u n ex p ected a n d essen tia lly q u a n tu m -m ech a n ica l resu lt. T h e p o ten tia l is
`a rti¯ cia l' in th a t it h a s to b e ca refu lly en g in eered to p ro d u ce th e d esired resu lt,
a n d th e sta te in v o lv ed is u n sta b le ev en u n d er sm a ll p ertu rb a tio n s.
T h e p a ssa g e fro m cla ssica l to q u a n tu m m ech a n ics resu lts, a t th e lev el o f d y n a m ica l va ria b les, in th e lo ss o f com m u tativity in m u ltip lica tio n . T h u s, fo r tw o p h y sica l
q u a n tities rep resen ted b y o p era to rs A a n d B , in g en era l A B 6= B A . H ow ev er,
th is d ep a rtu re fro m th e cla ssica l is lim ited in th e sen se th a t associativity is m a in ta in ed : fo r th ree (o r m o re) q u a n tities m u ltip lied in a g iv en seq u en ce th e p ro d u ct
is u n a m b ig u o u s: (A B )C = A (B C ) = A B C . O n e ca n a sk h ow q u a n tu m m ech a n ics m ig h t b e m o d i¯ ed if o n e ta k es th e n o n cla ssica l p a th o n e step fu rth er a n d ,
a lo n g w ith co m m u ta tiv ity, o n e g iv es u p a sso cia tiv ity a s w ell. T h is w a s ex a m in ed
b y J o rd a n , v o n N eu m a n n a n d W ig n er [2 0 ] in 1 9 3 4 . It d id n o t, h ow ev er, lea d
to a n y a ltern a tiv es w ith su ± cien tly in terestin g a n d ° ex ib le p ro p erties to g iv e a
fu rth er ex ten sio n o f q u a n tu m m ech a n ics.
G o in g ov er th is rich list o f co n trib u tio n s, o n e is tem p ted to say th a t W ig n er to o k
h is rev en g e fo r n o t h av in g b een in v o lv ed in th e d iscov ery o f q u a n tu m m ech a n ics,
a n d co m p en sa ted fo r it a cco rd in g ly !
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GENERAL  ARTICLE
N u c lea r F o rce s, S tru c tu re a n d R ea c tio n s
F o llow in g th e d iscov ery o f th e n eu tro n b y C h a d w ick in 1 9 3 2 , th ere w a s a g rea t
d ea l o f w o rk ex p lo rin g th e n a tu re o f th e stro n g n u clea r fo rces b etw een n eu tro n s
a n d p ro to n s. It w a s rea lized th a t th ese w o u ld b e strik in g ly d i® eren t fro m th e
fa m ilia r C o u lo m b fo rces b etw een p ro to n s, o f v ery sh o rt ra n g e, a n d w ith co m p lica ted d ista n ce d ep en d en ces. F u rth er d ep en d en ces o n sp in a n d sp a ce ex ch a n g e
w ere a lso a n ticip a ted . W ig n er w a s o n e o f th e ea rliest co n trib u to rs to th is ¯ eld ,
a n d h is n a m e is a sso cia ted w ith o n e o f th e fo u r b a sic ty p es o f term s in th e p o ten tia l en erg y ex p ressio n [2 1 ]:
p o ten tia l en erg y b etw een p ro to n a n d n eu tro n =
p u rely d ista n ce-d ep en d en t W ig n er term +
sp in ex ch a n g e B a rtlett term +
sp a ce ex ch a n g e M a jo ra n a term +
sp in a n d sp a ce ex ch a n g e H eisen b erg term .
T h u s, th e W ig n er fo rce is th e sim p lest o f a ll; th e o th ers eith er d istin g u ish b etw een
sin g let a n d trip let sp in sta tes, o r b etw een ev en a n d o d d o rb ita l a n g u la r m o m en ta ,
o r b o th . S u ch p h en o m en o lo g ica l p o ten tia ls a re u sefu l in a n a ly sin g low -en erg y
n u clea r b o u n d sta tes, sca tterin g p ro cesses, etc.
T h e low -en erg y (in th e k eV to few M eV ra n g e) sca tterin g cro ss-sectio n s o f n eu tro n s o ® va rio u s n u clei w ere ex p erim en ta lly stu d ied b y F erm i a n d h is co lla b o ra to rs, a n d m a n y o th er g ro u p s, a ro u n d 1 9 3 6 . T h ey fo u n d strik in g reso n a n ce
stru ctu res in th ese cro ss-sectio n s, w ith sh a rp m a x im a sep a ra ted b y v ery sm a ll
va lu es in b etw een . S o o n a fter, a th eo retica l ex p la n a tio n w a s o ® ered in d ep en d en tly b y N iels B o h r o n th e o n e h a n d , a n d b y G reg o ry B reit a n d W ig n er [2 2 ] o n
th e o th er. T h is is th e so -ca lled co m p o u n d n u cleu s m o d el. It p ictu res th e sca tterin g a n d rea ctio n p ro cesses a s ta k in g p la ce in tw o step s. A t ¯ rst th e in co m in g
low -en erg y p ro jectile (w h ich co u ld b e so m e lig h t n u cleu s ra th er th a n a n eu tro n )
a n d th e ta rg et co m b in e to p ro d u ce a co m p o u n d n u cleu s in o n e o f sev era l p o ssib le
m eta sta b le sta tes. In th is p ro cess th e p ro jectile en erg y is q u ick ly sh a red w ith a ll
th e n u cleo n s in th e co m p o u n d stru ctu re, a n d th en th e m o d e o f fo rm a tio n o f th is
stru ctu re is `fo rg o tten '. In th e seco n d step , th e d ecay o f th e co m p o u n d n u clea r
sta te in to va rio u s en erg etica lly a llow ed ch a n n els is g ov ern ed b y p ro b a b ility law s.
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It is th e p ro b a b ility o f o ccu rren ce o f th e ¯ rst step th a t sh ow s a n ex trem ely sen sitiv e en erg y d ep en d en ce a n d g iv es rise to th e o b serv ed reso n a n ces. In th eir w o rk
B reit a n d W ig n er d eriv ed th e fa m o u s b ell-sh a p ed sin g le-lev el reso n a n ce fo rm u la
k n ow n a fter th eir n a m es:
P ro b a b ility o f fo rm a tio n o f co m p o u n d n u cleu s:
® ¡ ¸ = f (E ¡ E ¸ )2 +
1 2
¡ g;
4 ¸
E = to ta l in itia l en erg y,
E ¸ ;¡ ¸ = av era g e en erg y, w id th , o f co m p o u n d n u clea r sta te ¸ .
T h e p a rtia l cro ss-sectio n s fo r su b seq u en t d ecay s in to ea ch o f th e sev era l ava ila b le
¯ n a l ch a n n els reta in th is ch a ra cteristic en erg y d ep en d en ce.
S o m etim e a fter th is, a ro u n d 1 9 4 4 , F erm i rem a rk ed to W ig n er (a s w a s m en tio n ed
ea rlier) th a t a g o o d th eo retica l b a sis fo r th e co m p o u n d n u cleu s m o d el w a s la ck in g . T h ereu p o n W ig n er set a b o u t fo rm u la tin g o n e. T h is w a s th e sta rtin g p o in t o f
th e R -m a trix th eo ry o f n u clea r rea ctio n s, d ev elo p ed b y h im la rg ely in co lla b o ra tio n w ith E isen b u d [2 3 ]. T h e b a sic id ea is to sep a ra te th e to ta l m u ltid im en sio n a l
co n ¯ g u ra tio n sp a ce o f a ll th e n u cleo n s in th e co m p o u n d n u cleu s (i.e., th e p ro jectile n u cleo n s p lu s th e ta rg et n u cleo n s) in to tw o p a rts: a n in terio r reg io n w h ere
th ey a re all w ith in th e ra n g e o f n u clea r fo rces a ctin g b etw een ev ery p a ir, a n d
a n ex terio r reg io n w h ere th is is n o t so . In th e la tter reg io n , o n e th en d e¯ n es o r
p ick s o u t essen tia lly n o n ov erla p p in g su b reg io n s, o n e fo r ea ch p o ssib le (tw o -b o d y )
¯ n a l ch a n n el in to w h ich th e co m p o u n d n u cleu s ca n d ecay. In stea d o f p o sin g
a m u ltich a n n el h a m ilto n ia n eig en fu n ctio n a n d eig en va lu e p ro b lem , a series o f
m a tch in g co n d itio n s co n n ectin g th e in terio r a n d ex terio r ch a n n el w av e fu n ctio n s
a n d th eir ra d ia l d eriva tiv es, a cro ss th e b o rd ers b etw een th e in terio r a n d ea ch ex terio r reg io n , a re set u p . T h e R -m a trix elem en ts a re q u a n tities th a t en ter th ese
rela tio n s, th ey a re a m u ltich a n n el g en era liza tio n o f th e lo g a rith m ic d eriva tiv e o f
a w av e fu n ctio n in a o n e-d im en sio n a l ra d ia l p ro b lem . T h e p a ra m eters en terin g
th e R -m a trix a re th e en erg y va lu es a n d th e va rio u s p a rtia l d ecay w id th s o f a ll
p o ssib le co m p o u n d n u cleu s lev els. T h u s, th e R -m a trix b eca m e sim u lta n eo u sly
a co n v en ien t m eth o d fo r p a ra m etriza tio n o f sca tterin g a n d rea ctio n a m p litu d es
u sin g p h en o m en o lo g ica lly a ccessib le co m p o u n d n u clea r sta te en erg ies a n d w id th s,
a n d w ith fu rth er d ev elo p m en ts a w ay to em b o d y g en era l p h y sica l p rin cip les, su ch
a s u n ita rity a n d ca u sa lity, g ov ern in g rea ctio n p ro cesses. In ter alia th is led to a
m u lti-lev el g en era liza tio n o f th e B reit{ W ig n er reso n a n ce ex p ressio n g iv en a b ov e,
a n d to a criterio n fo r th e va lid ity o f th e sin g le-lev el fo rm u la .
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R etu rn in g to th e p ro b lem o f n u clea r fo rces a n d stru ctu re, in 1 9 3 7 W ig n er ca m e u p
w ith th e S U (4 ) su p erm u ltip let th eo ry to sy stem a tize th e low -ly in g en erg y lev els
o f lig h t n u clei [2 4 ]. T h e id ea w a s th a t th e in tera ctio n s a m o n g p ro to n s a n d n eu tro n s, reg a rd ed a s n u cleo n s p o ssessin g th e iso sp in d eg ree o f freed o m in tro d u ced
b y H eisen b erg [2 5 ] a s ea rly a s 1 9 3 2 , m ig h t to a g o o d a p p rox im a tio n b e b o th sp in a n d iso sp in -in d ep en d en t. M o re g en era lly, it m ig h t b e in va ria n t u n d er a ll fo u rd im en sio n a l u n ita ry tra n sfo rm a tio n s m ix in g u p th e fo u r in d ep en d en t sp in -iso sp in
sta tes o f a n u cleo n . (T h is a ssu m p tio n a ctu a lly lea d s to sp eci¯ c sp in a n d iso sp in
d ep en d en ces in th e in tera ctio n .) It w o u ld th en b e p o ssib le to a rra n g e th e en erg y
lev els o f `n eig h b o u rin g ' n u clei w ith a co m m o n m a ss n u m b er in to va rio u s u n ita ry
irred u cib le rep resen ta tio n s (U IR s) o f S U (4 ), co n sid er sy stem a tica lly th e b rea k in g
o f th is sy m m etry, etc. E a ch U IR o f S U (4 ) is m a d e u p o f sev era l sp in -iso sp in m u ltip lets in a d e¯ n ite w ay. W h ile th e id ea w a s p h y sica lly w ell m o tiva ted a s a u sefu l
¯ rst a p p rox im a tio n , it w a s p u rsu ed o n ly to a lim ited ex ten t. H ow ev er, m a n y
y ea rs la ter, in 1 9 6 4 , W ig n er's th eo ry p rov id ed th e in sp ira tio n fo r a sim ila r S U (6 )
in va ria n t th eo ry o f b a ry o n s a n d m eso n s in th e fra m ew o rk o f th e q u a rk m o d el [2 6 ].
A t th e o th er en d o f th e sca le fro m low -ly in g w ell-sep a ra ted en erg y lev els o f lig h t
n u clei, w e h a v e th e rela tiv ely h ig h ly ex cited a n d clo sely sp a ced lev els o f h eav y
n u clei w ith m a n y d eg rees o f freed o m . H ere W ig n er p ro p o sed a co m p letely d i® eren t p h y sica l a p p ro a ch , o n e w h ich h a s stim u la ted w o rk b y m a n y o th ers a n d led
to co n n ectio n s w ith sev era l o th er p ro b lem s [2 7 ]. T h e p h y sica l id ea s m ay b e m o tiva ted a s fo llow s. A s th e ex cita tio n en erg y (o f a co m p lica ted n u cleu s) in crea ses,
o n e ex p ects th e en erg y lev els to g et clo ser a n d clo ser, a n d o n e a lso lo ses h o p e
o f b ein g a b le to o b ta in th em in d iv id u a lly fro m a ¯ rst-p rin cip les H a m ilto n ia n .
In stea d , w h a t w o u ld b e m o re a ccessib le a n d p h y sica lly in terestin g a re va rio u s
sta tistica l p ro p erties o f th e lev els; th e p ro b a b ility d istrib u tio n s fo r su ccessiv e lev els to o ccu r a t va rio u s en erg ies, fo r th e sp a cin g b etw een n eig h b o u rin g lev els to
h av e d i® eren t va lu es, a n d so o n . T o o b ta in th ese sta tistica l fea tu res, a n d a t th e
sa m e tim e to re° ect th e fa ct th a t o n e is d ea lin g w ith a v ery co m p lex sy stem w ith
m a n y d eg rees o f freed o m , W ig n er p ro p o sed th a t th e b a sic H a m ilto n ia n (a fter
tru n ca tio n to a la rg e b u t ¯ n ite d im en sio n ) b e itself reg a rd ed a s a ra n d o m m a trix , b elo n g in g to a n en sem b le w ith sp eci¯ ed p ro p erties. O n ce o n e sp eci¯ es th e
n a tu re o f th is en sem b le, reg a rd ed a s a p rim a ry in p u t, th e sta tistica l p ro p erties
o f th e eig en va lu es o f th e H a m ilto n ia n , th e sp a cin g d istrib u tio n , etc., ca n a ll b e
d eriv ed , in p rin cip le, a s seco n d a ry co n seq u en ces. It tu rn s o u t th a t in u sin g th is
a p p ro a ch o n e m u st d ea l w ith o n e `sim p le seq u en ce' o f n u clea r lev els a t a tim e;
th is is a set o f lev els p o ssessin g th e sa m e ex a ctly co n serv ed q u a n tu m n u m b ers
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{ `b elo n g in g to th e sa m e sy m m etry ' { su ch a s th e to ta l a n g u la r m o m en tu m a n d
p a rity. P ro p erties o f d i® eren t sim p le seq u en ces a re in d ep en d en t. T h u s, W ig n er's
h y p o th esis w a s th a t th e lo ca l sta tistica l b eh av io u r o f th e lev els in a sim p le seq u en ce is g iv en b y th e p ro p erties o f th e eig en va lu e sp ectru m o f a ra n d o m m a trix
d raw n fro m a su ita b le en sem b le. T h e ty p e o f en sem b le to b e u sed d ep en d s o n
th e in teg er o r h a lf-o d d in teg er n a tu re o f th e to ta l a n g u la r m o m en tu m , b eh av io u r
u n d er tim e rev ersa l, a n d p resen ce o r a b sen ce o f ro ta tio n a l sy m m etry. L a ter w o rk
h a s sh ow n th a t th ere a re th ree n a tu ra l ty p es o f en sem b les, in co rresp o n d en ce w ith
th e th ree g rea t fa m ilies o f cla ssica l co m p a ct sim p le L ie g ro u p s: th e G a u ssia n rea l
o rth o g o n a l, th e G a u ssia n co m p lex u n ita ry, a n d th e G a u ssia n sy m p lectic en sem b les. T h ese en sem b les co n sist resp ectiv ely of rea l sy m m etric, co m p lex h erm itia n
a n d rea l q u a tern io n ic m a trices (o f su ita b le d im en sio n s, ev en in th e la st ca se). T h e
p ro b a b ility d istrib u tio n d e¯ n in g th e en sem b le is in v a ria n t u n d er a rea l o rth o g o n a l, co m p lex u n ita ry o r u n ita ry sy m p lectic g ro u p o f tra n sfo rm a tio n s a p p lied to
its elem en ts; m o reov er, th e m a trix elem en ts o f th e H a m ilto n ia n a re a ssu m ed to
b e in d ep en d en t ra n d o m va ria b les. It is th e co m b in a tio n o f th ese tw o p ro p erties
th a t m a k es th ese en sem b les G a u ssia n .
A g rea t d ea l o f so p h istica ted m a th em a tica l a n a ly sis h a s g o n e in to th ese o b jects,
a n d th is a ctiv ity co n tin u es [2 8 ]. O n e v ery in terestin g fea tu re th a t w a s reco g n ized
v ery ea rly w a s th a t th e sp a cin g d istrib u tion va n ish es a s a p ow er o f th e sp a cin g
a s th e sp a cin g ten d s to zero . T h e ra te o f th is va n ish in g , th e p ow er in v o lv ed , is
ch a ra cteristic fo r ea ch o f th e th ree fa m ilies o f en sem b les. T h e p h y sica l m ea n in g
o f th is resu lt { b o rn e o u t b y ex p erim en ts a n d rem in d in g u s o f th e n o -cro ssin g
th eo rem o f W ig n er a n d v o n N eu m a n n fo r electro n term s o f th e sa m e sy m m etry
{ is th a t w ith in a sim p le seq u en ce n eig h b ou rin g lev els d o n o t lik e to co m e v ery
clo se to o n e a n o th er. H a d w e im a g in ed th a t th e en erg y lev els th em selv es w ere
in d ep en d en tly sta tistica lly d istrib u ted , th ere w o u ld h av e b een n o ca u se fo r su ch
lev el rep u lsio n . T h is o n ly em p h a sizes W ign er's id ea th a t th e p ro p erties o f th e
en sem b le o f H a m ilto n ia n s m u st b e ch o sen ¯ rst, a n d o th er p ro p erties th en o b ta in ed
a s co n seq u en ces.
Q u a n tu m F ie ld T h eo ry , R e la tiv istic C la ssica l a n d Q u a n tu m M ec h a n ic s
T h e ru les fo r ca n o n ica l q u a n tiza tio n { crea tin g a q u a n tu m th eo ry fro m a cla ssica l
o n e { w ere o rig in a lly in v en ted in th e co n tex t o f n o n rela tiv istic p a rticle q u a n tu m
m ech a n ics. T h e ¯ rst su ccessfu l a p p lica tio n o f th ese ru les to a cla ssica l ¯ eld th eo ry ca m e w ith D ira c's q u a n tiza tio n o f th e electro m a g n etic ¯ eld . T h is led to h is
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cla ssic 1 9 2 7 p a p er in w h ich h e trea ted th e p ro cesses o f em issio n a n d a b so rp tio n o f
ra d ia tio n b y m a tter, u sin g q u a n tu m p rin cip les a n d th e p h o to n co n cep t [2 9 ]. T h e
q u a n tized ¯ eld led to a sy n th esis o f co m p lem en ta ry cla ssica l p a rticle a n d ¯ eld
la n g u a g es, a n d co u ld d escrib e sta tes w ith va ria b le n u m b ers o f id en tica l p a rticles.
T h e ca n o n ica l q u a n tiza tio n m eth o d led to co m m u ta tio n rela tio n s o f th e fo rm
a r a ys ¡ a ys a r = ± r s ;
a r a s ¡ a s a r = a yr a ys ¡ a ys a yr = 0 :
H ere a r (a yr ) a re th e a n n ih ila tio n (crea tio n ) o p era to rs fo r p h o to n s in va rio u s sta tes
in d ex ed b y r . T h ese sta tes a re a n in d ep en d en t, o rth o g o n a l a n d co m p lete set o f
m o d es o f th e electro m a g n etic ¯ eld . T h e o p era to rs a r ;a yr a re q u a n tu m a n a lo g u es
o f th e cla ssica l co m p lex co e± cien ts in a n ex p a n sio n o f th e cla ssica l ¯ eld in th ese
m o d es. In th is ca se th e a p p ea ra n ce o f co m m u ta tio n rela tio n s led n a tu ra lly to
B o se{ E in stein sta tistics fo r p h o to n s. V ery so o n a fter D ira c's p a p er, J o rd a n a n d
W ig n er sh ow ed th a t to d escrib e ferm io n s (su ch a s electro n s) o b ey in g P a u li's ex clu sio n p rin cip le a n d F erm i{ D ira c sta tistics, th e p a rticle a n n ih ila tio n a n d crea tio n
o p era to rs m u st o b ey a n tico m m u ta tio n rela tio n s [3 0 ]:
a r a ys + a ys a r = ± r s ;
a r a s + a s a r = a yr a ys + a ys a yr = 0 :
F o r a ¯ n ite n u m b er o f m o d es, th ey p rov ed th a t u p to eq u iva len ce th ere is o n ly
o n e irred u cib le rep resen ta tio n o f th ese rela tio n s, a n d it is ¯ n ite-d im en sio n a l. T h is
u n iq u en ess is sim ila r to a co rresp o n d in g resu lt in th e ca se o f co m m u ta tio n rela tio n s. T h e m a jo r d i® eren ce is th a t fro m a m a th em a tica l p o in t o f v iew sy stem s
o f o p era to rs o b ey in g th e a n tico m m u ta tio n rela tio n s a re q u ite `h a rm less', w h ile in
th e ca se o f co m m u ta tio n rela tio n s th ey a re u n b o u n d ed a n d th e sp a ce is in ¯ n ite
d im en sio n a l { ev en fo r a ¯ n ite n u m b er o f m o d es. O f co u rse, in th e J o rd a n { W ig n er
ca se th ere is n o sen sib le cla ssica l lim it.
It is in terestin g to n o te th a t D ira c's in itia l rea ctio n to th is w o rk o f J o rd a n a n d
W ig n er w a s d ecid ed ly n eg a tiv e [3 1 ]. W ig n er la ter a ttrib u ted th is to D ira c's b ein g
v ery co m m itted to th e H a m ilto n ia n p o in t o f v iew in d y n a m ics { `a ca p tiv e o f th e
H a m ilto n ia n fo rm a lism '. H ow ev er, it b eca m e clea r v ery so o n th a t th is w a s th e
co rrect w ay to set u p q u a n tu m ¯ eld th eo ry fo r ferm io n s, a n d it b eca m e p a rt o f
th e fo u n d a tio n s o f th e su b ject.
T h e ¯ rst a ttem p ts a t u n itin g q u a n tu m m ech a n ics a n d sp ecia l rela tiv ity w ere d u e
to K lein a n d G o rd o n . T h is resu lted in th e w av e eq u a tio n n a m ed a fter th em , b u t it
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fa ced p ro b lem s o f in terp reta tio n a t th e o n e-p a rticle lev el. T h e n ex t, sp ecta cu la rly
su ccessfu l, a ttem p t w a s D ira c's w o rk in 1 9 2 8 th a t led to h is w av e eq u a tio n fo r th e
electro n a n d its series o f a m a zin g co n seq u en ces [3 2 ]. P ro b a b ly so o n a fter, in 1 9 2 8
itself, D ira c su g g ested to W ig n er a co m p reh en siv e stu d y o f a ll p o ssib le u n ita ry
irred u cib le rep resen ta tio n s o f th e in h o m o g en eo u s L o ren tz g ro u p (IH L G ), i.e., o f
th e h o m o g en eo u s L o ren tz g ro u p (H L G ) su p p lem en ted b y sp a ce{ tim e tra n sla tio n s.
B y a b o u t 1 9 3 2 , M a jo ra n a h a d co n stru cted m a n y o f th ese U IR s, a n d la ter th ese
w ere sim p li¯ ed b y D ira c a n d P ro ca [3 3 ]. T h e so lu tio n o f th is p ro b lem p o sed b y
D ira c to W ig n er b eca m e a h ercu lea n e® o rt, b ein g co m p leted o n ly in 1 9 3 7 . T h e
resu lt w a s a n a ll-tim e cla ssic p a p er in m a th em a tica l p h y sics [3 4 ]. In it, W ig n er
a ck n ow led g es th e h elp a n d g u id a n ce h e receiv ed n o t o n ly fro m D ira c b u t a lso o n
m a th em a tica l a sp ects fro m v o n N eu m a n n . A t so m e sta g e D ira c a d v ised W ig n er
to b e ca refu l, a n d th e la tter rep lied [3 5 ]: `Y o u p o in t o u t th a t ca re is n eed ed in
th e a n a ly sis o f th e rep resen ta tio n s o f th e L o ren tz g ro u p ; I p ro m ise y o u th a t I w ill
b e ca refu l'.
W ig n er's p a p er co n ta in s a d eta iled a n a ly sis o f th e stru ctu re o f th e H L G a n d th e
IH L G , a n d o f g en era l u n ita ry rep resen ta tio n s (U R s) o f th e IH L G in th e co n tex t
o f q u a n tu m m ech a n ics; it th en tu rn s to a stu d y o f th e U IR s. T h e resu lt w a s
th a t th ese co u ld b e cla ssi¯ ed in to fo u r b ro a d ty p es, d ep en d in g u p o n th e n a tu re
o f th e p o ssib le va lu es o f en erg y { m o m en tu m p ¹ o ccu rrin g w ith in th e U IR , a n d
th e a llow ed `sta tes o f p o la riza tio n ' fo r ea ch en erg y { m o m en tu m . T h e h elicity ¸ is
d e¯ n ed a s th e co m p o n en t o f a n g u la r m o m en tu m in th e d irectio n o f m o m en tu m .
F o r ea ch k in d o f p ¹ (p rov id ed it is n o t id en tica lly va n ish in g ) th e a llow ed va lu es
o f ¸ a re d eterm in ed b y so m e U IR o f a co rresp o n d in g su b g ro u p o f th e H L G , th e
so -ca lled `little g ro u p ' fo r th a t p ¹ ; it co n sists o f a ll elem en ts o f th e H L G w h ich
leav e p ¹ in va ria n t. T h e p a ttern o f U IR s o f th e IH L G is d isp lay ed in T able 1 ¤ (h ere
sp a ce in v ersio n o r p a rity h a s b een in clu d ed in th e H L G , ex cep t th a t fo r n eu trin o s
th is o p era tio n is u n d e¯ n ed ) [3 6 ].
W h ile m a n y o f th ese U IR s w ere k n ow n ea rlier to M a jo ra n a a n d D ira c, th e so ca lled in ¯ n ite-sp in o r co n tin u o u s-sp in rep resen ta tio n s in ca ses (b ) a n d (c) w ere
g en u in ely n ew . In h is w o rk , W ig n er d id n o t ca rry th e in v estig a tio n o f th ese, o r
o f ca se (d ), to co m p letio n . H e m en tio n ed th eir ex isten ce, a n d o n ly rem a rk ed :
`... th e la st ca se m ay b e th e m o st in terestin g fro m th e m a th em a tica l p o in t o f
v iew . I h o p e to retu rn to it in a n o th er p a p er. I d id n o t su cceed so fa r in g iv in g a co m p lete d iscu ssio n o f th e 3 rd cla ss.' W ig n er's `la st ca se' a n d `3 rd cla ss'
*
At the time this article was written it was generally believed, as indicated in Table 1, that neutrinos are massless.
Over the past decade this situation has changed, some of them definitely have (very small) nonzero masses.
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Nature of p ¹
Little group within
HLG(SL(2,C) )
Number of polarization
states, spectrum of ¸
Remarks
(a) Time-like
(positive or
negative)
SO(3) (SU(2))
2s + 1 for s = 0;1= 2;1:::
¸ = s ;s ¡ 1;:::;¡ s
Massive particles
with zero or ¯nite
spin, s = 0 for ¼ meson
s = 1= 2 for electron
(b) Light-like
(positive or
negative)
E(2), twodimensional
Euclidean
group
One: ¸ = 0
Two: ¸ = § s;s = 1= 2;1;:::
In¯nite: ¸ = 0;§ 1;§ 2::::
or
¸ = § 1= 2;§ 3= 2;:::;
No known particles
s = 1 for photons
s = 1= 2;¸ = ¡ 1= 2 for
neutrinos
No known particles
(c) Space-like
SO(2,1) (SL(2,R))
One: ¸ = 0
In¯nite: ¸ = s ;s + 1;::: or
¡ s;¡ s ¡ 1;:::; s = 1= 2;1;::: or
¸ = 0;§ 1;§ 2;::: or
¸ = § 1= 2;§ 3= 2 :::
Imaginary mass,
unphysical
(d) Vanishing
HLG(SL(2,C) )
{
{
Table 1.
co rresp o n d resp ectiv ely to (c) a n d (d ) in o u r ta b le. W e a lso see th a t n o t ev ery
m a th em a tica lly a ccep ta b le U IR o f th e IH L G is a ccep ta b le o n p h y sica l g ro u n d s.
R ela tiv istic q u a n tu m sy stem s d escrib ed b y a n y U IR o f th e IH L G a re ca lled `elem en ta ry sy stem s'. T ru ly elem en ta ry p a rticles, a b le to ex ist in iso la tio n , a re
d escrib ed u sin g th em . E x a m p les a re p h o to n s, n eu trin o s, electro n s a n d m u o n s.
T h e p h ra se `elem en ta ry sy stem s' co n v ey s th e m ea n in g th a t a ll th eir p ro p erties
a re rev ea led b y stu d y in g th eir b eh av io u r u n d er a ll elem en ts o f th e IH L G { th ere
is n o in tern a l stru ctu re in v o lv ed . In th e a b ov e listin g , o n ly ca ses (a ) a n d (b ) fo r
¯ n ite h elicity a re rea lized in n a tu re.
T h e U IR s o f ca se (d ) a re a ctu a lly U IR s o f th e H L G S O (3 ,1 ) (o r o f th e clo sely
rela ted g ro u p S L (2 , C )). It rem a in ed fo r H a rish -C h a n d ra a n d fo r G el'fa n d a n d
N a im a rk to d eterm in e th em in d ep en d en tly [3 7 ]. T h e in p u ts n eed ed to co n stru ct
th e U IR s o f ca se (c) fo r in ¯ n ite sp in w ere p rov id ed b y B a rg m a n n th ro u g h h is
co n stru ctio n o f th e U IR s o f S O (2 ,1 ) a n d S L (2 ,R )[3 8 ].
In h is co n trib u tio n to R M P , D ira c m a d e th e fo llow in g co m m en ts o n W ig n er's
w o rk [3 9 ]. `T h e p ro b lem o f w o rk in g o u t a ll u n ita ry rep resen ta tio n s o f th e IH L G
h a s b een d ea lt w ith b y W ig n er, ta k in g th e m a th em a tica l p o in t o f v iew th a t tw o
rep resen ta tio n s a re eq u iva len t if th ey a re co n n ected b y a u n ita ry tra n sfo rm a tio n .
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H e d eco m p o ses th e rep resen ta tio n s in to th eir irred u cib le co n stitu en ts a n d ¯ n d s
th a t th e irred u cib le co n stitu en ts p rov id e th eo ries o f elem en ta ry p a rticles w ith
va rio u s sp in s. T h is w o rk d o es n o t lea d to an y in tera ctio n b etw een p a rticles. T o
b rin g in in tera ctio n , o n e m u st d ep a rt fro m th e p o in t o f v iew o f lo o k in g a t tw o
rep resen ta tio n s a s eq u iva len t if th ey a re co n n ected b y a u n ita ry tra n sfo rm a tio n ,
a p o in t o f v iew w h ich in v o lv es lo o k in g u p o n a ll u n ita ry tra n sfo rm a tio n s a s triv ia l. T o a p h y sicist, so m e u n ita ry tra n sfo rm a tio n s a re triv ia l, w h erea s o th ers (fo r
ex a m p le, th e S m a trix ) a re fa r fro m triv ia l, so a p h y sicist ca n n o t lo o k u p o n tw o
rep resen ta tio n s co n n ected b y a u n ita ry tra n sfo rm a tio n a s n ecessa rily eq u iva len t.'
T h e p o in t is th a t fo r a n y rea lly in terestin g rela tiv istic q u a n tu m sy stem , su ch a s
a rela tiv istic q u a n tu m ¯ eld th eo ry, it is n o t o n ly im p o rta n t to k n ow w h ich U IR s
o f th e IH L G a re p resen t, b u t a lso h ow th ey a re p u t to g eth er. H ow ev er it m u st
b e p o in ted o u t th a t a s ea rly a s 1 9 4 9 W ig n er h im self h a d d raw n a tten tio n to th is
situ a tio n [4 0 ]: `T h e elem en ta ry sy stem s co rresp o n d m a th em a tica lly to irred u cib le
rep resen ta tio n s o f th e L o ren tz g ro u p a n d a s su ch ca n b e en u m era ted ... H ow ev er,
in th e d escrip tio n b y irred u cib le sta tes, th e fo rm o f a lm o st a ll p h y sica lly im p o rta n t o p era to rs rem a in s u n k n ow n a n d , in fact, d ep en d s o n th e sy stem , th e ty p es
o f in tera ctio n s, etc. T h is lea d s to a ra th er stra n g e d ilem m a : in th e cu sto m a ry
d escrip tio n th e fo rm o f th e p h y sica lly im p o rta n t o p era to rs is k n ow n b u t th e tim e
d ep en d en ce o f th e sta tes is u n p red icta b le o r d i± cu lt to ca lcu la te. In th e d escrip tio n ju st m en tio n ed , th e situ a tio n is o p p o site: th e tim e d ep en d en ce o f th e sta tes
fo llow s fro m th e in va ria n ce p ro p erties, b u t th e fo rm o f th e p h y sica lly im p o rta n t
o p era to rs is h a rd to esta b lish .'
W ig n er retu rn ed o n m a n y o cca sio n s to a d escrip tio n o f th e resu lts o f h is cla ssic
w o rk . H e a lso co n stru cted w ith B a rg m a n n a u n i¯ ed set o f w av e eq u a tio n s w h o se
so lu tio n s w o u ld lea d to U IR s o f ty p es (a ) a n d (b ) in o u r ta b le [4 1 ]. H is w o rk w ith
N ew to n o n th e p ro b lem o f p o sitio n is p a rticu la rly in terestin g , so I d escrib e it in
a little d eta il [4 2 ].
T h e sta rtin g p o in t o f n o n rela tiv istic p a rticle q u a n tu m m ech a n ics is th e set o f
p o sitio n s a n d m o m en ta a s p rim a ry d y n a m ica l va ria b les, o u t o f w h ich a ll o th er
va ria b les a re b u ilt u p . (L a ter w o rk h a s sh ow n th a t th ese p o sitio n s a n d m o m en ta
ca n b e d eriv ed a s seco n d a ry o b jects sta rtin g fro m su ita b le q u a n tu m -m ech a n ica l
rep resen ta tio n s o f th e G a lilei g ro u p ). N ow , fro m W ig n er's p o in t o f v iew in th e
rela tiv istic co n tex t, th e p rim a ry th in g s a re th e U IR s o f th e IH L G . A fter h av in g
set th em u p , o n e m u st ex a m in e w ith in w h ich U IR s o n e ca n co n stru ct p o sitio n
o p era to rs w ith p h y sica lly d esira b le p ro p erties. S u ch a n a n a ly sis w a s ¯ rst u n d erta k en b y N ew to n a n d W ig n er. T h ey w ere ab le to sh ow th a t in ev ery ¯ n ite m a ss
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a n d ¯ n ite sp in U IR (ca se (a )) a u n iq u e set o f p o sitio n o p era to rs p o ssessin g sev era l
p h y sica lly rea so n a b le p ro p erties d o es in d eed ex ist. H ow ev er, co n tra ry to n a iv e
ex p ecta tio n , th ey d o n o t fo rm th e sp a ce co m p o n en ts o f a rela tiv istic fo u r-v ecto r.
T h is illu stra tes th e fa ct th a t in q u a n tu m th eo ry th e u n ita ry tra n sfo rm a tio n law
is m o re b a sic th a n th e g eo m etric o n e o r m a n ifest cova ria n ce. In th e m a ssless
ca se w ith ¯ n ite n o n zero h elicity ev en th is m u ch ca n n o t b e d o n e. T h u s, n eith er
p h o to n s n o r n eu trin o s ca n b e lo ca lized in sp a ce.
In o th er rela ted w o rk w e m en tio n th e stu d y b y In o n u a n d W ig n er o f th e p ro cess
o f `g ro u p co n tra ctio n ' b y w h ich th e IH L G g o es ov er in th e n o n rela tiv istic lim it to
th e G a lilei g ro u p [4 3 ]; S a leck er a n d W ig n er's a n a ly sis o f d eep co n cep tu a l p ro b lem s
in b rin g in g to g eth er q u a n tu m m ech a n ics a n d g en era l rela tiv ity, ca u sed b y q u a n tu m lim ita tio n s o n p o sitio n m ea su rem en ts [4 4 ]; a n d va n D a m a n d W ig n er's co n stru ctio n o f cla ssica l rela tiv istic d irect-in tera ctio n th eo ries restin g u p o n in teg ro d i® eren tia l eq u a tio n s fo r p a rticle tra jecto ries [4 5 ]. O n e o f W ig n er's co n clu sio n s
w a s th a t w h ile sp ecia l rela tiv ity a n d q u a n tu m m ech a n ics co u ld a t lea st co n cep tu a lly b e co m b in ed , w ith g en era l rela tiv ity a n d q u a n tu m m ech a n ics th ere w a s n o
co m m o n g ro u n d a t a ll.
In te r p re ta tio n o f Q u a n tu m M ec h a n ic s
In th e ea rly 1 9 6 0 s W ig n er tu rn ed to a serio u s ex a m in a tio n o f th e p ro b lem s o f
in terp reta tio n o f q u a n tu m m ech a n ics, a n d a clea r ex p ressio n o f th e o rth o d ox p o sitio n w h ich essen tia lly co in cid ed w ith h is ow n [4 6 ]. A s ev id en ce fo r th e la tter,
h ere is h is ow n sta tem en t: `T h e o rth o d ox v iew is v ery sp eci¯ c in its ep istem o lo g ica l im p lica tio n s ...A la rg e g ro u p o f p h y sicists ¯ n d s it d i± cu lt to a ccep t th ese
co n clu sio n s a n d , ev en th o u g h th is d o es n o t a p p ly to th e p resen t w riter, h e a d m its
th a t th e fa r-rea ch in g n a tu re o f th e ep istem o lo g ica l co n clu sio n s m a k es o n e u n ea sy.'
H e a lso o ften sa id th a t h e w a s a d d in g h a rd ly a n y th in g n ew to L o n d o n a n d B a u er's
cla ssic 1 9 3 9 ex p o sitio n [4 7 ]. H e a ccep ted th e trea tm en t o f m ea su rem en t th eo ry
th a t h a d b een a rticu la ted b y h is frien d v o n N eu m a n n [4 8 ] a s ea rly a s 1 9 3 2 , a n d
w a n ted to resta te it fo r a n ew g en era tio n a n d ex tra ct its u ltim a te co n seq u en ces
fo r ep istem o lo g y.
W ig n er em p h a sized th a t th e sta te v ecto r o f a q u a n tu m sy stem ch a n g es in tw o
m u tu a lly ex clu siv e w ay s { co n tin u o u s, d eterm in istic S ch rÄo d in g er ev o lu tio n w h en
n o t su b ject to o b serva tio n , a n d d isco n tin u o u s, p ro b a b ilistic, co lla p se w h en m ea su rem en ts a re m a d e. H e w en t to m u ch len g th to sh ow th a t th e lin ea r S ch rÄo d in g er
eq u a tio n { ev en in clu d in g th e a p p a ra tu s a n d th e sy stem 's co u p lin g to it { ca n
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n ev er p ro d u ce th e m a cro sco p ica lly d esired co lla p se p h en o m en o n , a n d stressed rep ea ted ly th a t p u re sta tes a n d m ix tu res h av e v ery d i® eren t p h y sica l p ro p erties. H e
a lso p resen ted a p ra g m a tic a n sw er to th e q u estio n `W h a t is th e sta te v ecto r?'. It
w a s th a t it co d i¯ es in a co m p a ct w ay a ll p ast in fo rm a tio n a b o u t a sy stem , o n th e
b a sis o f w h ich w e ca n sta te th e p ro b a b ilistic co n n ectio n s th a t q u a n tu m m ech a n ics
g iv es a m o n g a series o f m ea su rem en ts ca rried o u t su b seq u en tly a n d seq u en tia lly
in tim e; a ll th e co n seq u en ces o f q u a n tu m m ech a n ics a re ju st su ch sta tem en ts. S o ,
a s th e o rth o d ox v iew cla im s, `th e law s o f q u a n tu m m ech a n ics ca n b e ex p ressed
o n ly in term s o f p ro b a b ility co n n ectio n s', an d ca n n o t b e fo rm u la ted in term s o f
o b jectiv e rea lity.
P u rsu in g th is a n a ly sis fu rth er, W ig n er ca m e to th e co n clu sio n th a t h u m a n co n scio u sn ess is a n essen tia l ex tern a l in g red ien t n eed ed to m a k e co m p lete sen se o f
q u a n tu m m ech a n ics. T h e co lla p se o f th e state v ecto r o ccu rs w h en a n d o n ly w h en
a n o b serva tio n is reg istered in so m e in d iv id u a l co n scio u sn ess: `It is a t th is p o in t
th a t th e co n scio u sn ess en ters th e th eo ry u n av o id a b ly a n d u n a ltera b ly. If o n e
sp ea k s in term s o f th e w av e fu n ctio n , its ch an g es a re co u p led w ith th e en terin g o f
im p ressio n s in to o u r co n scio u sn ess'. A n d a g a in : `... it w a s n o t p o ssib le to fo rm u la te th e law s o f q u a n tu m m ech a n ics in a fu lly co n sisten t w ay w ith o u t referen ce to
th e co n scio u sn ess'. In su p p o rt o f th is d ecla ra tio n , W ig n er a p p ea ls to H eisen b erg
a n d say s: `W . H eisen b erg ex p ressed th is m o st p o ig n a n tly (D aedalu s, 1 9 5 8 , 8 7 ,
9 9 ): \ T h e law s o f n a tu re w h ich w e fo rm u la te m a th em a tica lly in q u a n tu m th eo ry
d ea l n o lo n g er w ith th e p a rticles th em selv es b u t w ith o u r k n ow led g e o f th e elem en ta ry p a rticles ... T h e co n cep tio n o f o b jectiv e rea lity... eva p o ra ted in to th e...
m a th em a tics th a t rep resen ts n o lo n g er th e b eh av io u r o f elem en ta ry p a rticles b u t
ra th er o u r k n ow led g e o f th is b eh av io u r" '.
A s o n e ca n im a g in e, th is lin e o f th in k in g led W ig n er in ex o ra b ly to a k in d o f so lip sism , a n d to th e d elin ea tio n o f tw o k in d s o f rea lity { th e co n ten t o f o n e's ow n
co n scio u sn ess, th e o n ly a b so lu tely rea l, a n d ev ery th in g else ex tern a l to o n eself,
in clu d in g ev ery o th er p erso n 's co n scio u sn ess. T o su p p o rt th e fo rm er h e tu rn ed
to S ch rÄo d in g er; `... th e m o st elo q u en t sta tem en t o f th e p rim e n a tu re o f th e co n scio u sn ess w ith w h ich th is w riter is fa m ilia r a n d w h ich is o f recen t d a te is o n p a g e
2 o f S ch rÄo d in g er's M in d an d M atter: \ W o u ld it (th e w o rld ) o th erw ise (w ith o u t
co n scio u sn ess) h av e rem a in ed a p lay b efo re em p ty b en ch es, n o t ex istin g fo r a n y b o d y, th u s q u ite p ro p erly n o t ex istin g ?" ' B u t th ere w a s a sig n o f a sy m m etry {
th e o n ly a b so lu tely rea l, o n e's ow n co n scio u sn ess, d o es d ep en d o n fo o d , a ir a n d
w a ter fo r its ow n su rv iva l a n d fu n ctio n in g , a s w e a re p a in fu lly aw a re; so h e m a d e
a ca se fo r d ev isin g ex p erim en ts w h ich m ig h t sh ow u p th e e® ects o f co n scio u sn ess
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o n m a tter. In ta lk in g o f th e ¯ rst k in d o f rea lity, W ig n er a lso rea lized a n d sta ted
its o b v io u s lim ita tio n s { its aw a k en in g w ith b irth a n d in fa n t g row th , its ex tin ctio n
a t d ea th . S o h e a rg u ed fo r a d eep stu d y o f th e fo rm er p h a se, to u n d ersta n d th e
n a tu re o f co n scio u sn ess.
W ig n er felt th a t th e d ev elo p m en t o f q u a n tu m m ech a n ics h a d w id en ed th e o u tlo o k o f m o st p h y sicists, a n d a lso in a sen se m a d e th em in w a rd -lo o k in g : `U n til n o t
m a n y y ea rs a g o , th e \ ex isten ce" o f a m in d o r so u l w o u ld h av e b een p a ssio n a tely
d en ied b y m o st p h y sica l scien tists ... E v en to d ay, th ere a re a d h eren ts to th is v iew
th o u g h few er a m o n g th e p h y sicists th a n { iro n ica lly en o u g h { a m o n g b io ch em ists'.
H e a lso saw th a t q u a n tu m m ech a n ics rein fo rces th e circu m sta n ce th a t a n y o b serva tio n a n d in terp reta tio n o f m ea su rem en t rests o n p rev io u sly co n stru cted a n d
u n d ersto o d th eo ry. T h u s, w e a re lin k ed in a ch a in to th e v ery b eg in n in g s o f
o u r a cq u isitio n o f k n ow led g e o f o u r su rro u n d in g s a n d its reg u la rities { in d eed to
p h y lo g en esis a n d o n to g en esis.
T o d ay it m ay seem th a t th ese co n clu sio n s o f W ig n er w ere p rem a tu re. C erta in ly,
e® o rts a re a p len ty to ¯ n d m o re `a ccep ta b le' in terp reta tio n s o f q u a n tu m m ech a n ics, w ith o u t a p p ea l to o u rselv es a s essen tia l p rereq u isites. W a s W ig n er th en `a
v ictim o f h is g en era tio n '? S h o u ld w e sm ile a t th ese co n clu sio n s w h ich h e fo u n d
in esca p a b le? O r w a s h e o n ly b ein g ru th lessly h o n est a n d ex p ressin g clea rly w h a t
o th ers h esita ted to p u t in to w o rd s?
S o lid -S ta te P h y sic s, R ea c to r T h eo ry a n d T ec h n o lo g y
I w ill to u ch u p o n th ese a rea s o n ly b rie° y. W ig n er's in terest in p ro b lem s o f so lid sta te p h y sics a n d m a teria ls scien ce stem m ed fro m a v ery ea rly d a te. T h ere m u st
h av e b een lin k s to h is o rig in a l tra in in g a s a ch em ica l en g in eer; la ter o n h is d eta iled
k n ow led g e o f p ro p erties o f m a teria ls p lay ed a k ey ro le in h is w o rk o n rea cto rs.
A m o n g h is g ifted stu d en ts in so lid -sta te scien ce in th e 1 9 3 0 s w e m ay m en tio n J o h n
B a rd een , G reg o ry W a n n ier a n d F red erick S eitz. It w a s W ig n er w h o su g g ested to
W a n n ier [4 9 ] `th a t th ere o u g h t to b e a w ay to reco n cile th e lo ca l a n d th e b a n d
co n cep t fo r electro n s, a n d th a t su ch a reco n cilia tio n w o u ld p ro b a b ly b e u sefu l
in u n d ersta n d in g th e sp ectra o f in su la to rs'. W ig n er a lso w o rk ed o n ra d ia tio n
d a m a g e in so lid s { th e d eta iled m icro sco p ic p ictu re o f la ttice d efects o ccu rrin g
w h en m a teria ls a re irra d ia ted w ith n eu tro n s, th e resu ltin g ch a n g es in h ea t a n d
electrica l co n d u ctiv ity a n d d u ctility, a n d a lso th e w ay s in w h ich th e m a teria l
seem s to recov er fro m th e d a m a g e a s tim e g o es o n [5 0 ].
W ig n er w a s th e so u rce o f m u ch o f th e th eo ry a n d th e m a jo r tech n o lo g ica l
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d ev elo p m en ts co n n ected w ith n u clea r rea cto rs. H is co n trib u tio n s b eg a n in 1 9 4 0 .
A s b rie° y m en tio n ed ea rlier, h e w a s a lea d er a t th e U n iv ersity o f C h ica g o M eta llu rg ica l L a b o ra to ry d u rin g 1 9 4 3 { 4 5 . H e co n trib u ted to th e d ev elo p m en t o f
resea rch rea cto rs, p ow er rea cto rs a n d p lu to n iu m p ro d u ctio n rea cto rs. O n th e
th eo retica l fro n t h e m a d e m a jo r co n trib u tion s to th e sp ectru m o f th e B o ltzm a n n
eq u a tio n , n eu tro n th erm a liza tio n , th erm a l u tiliza tio n a n d reso n a n ce a b so rp tio n .
A ll v ery p ra ctica l co n trib u tio n s `w h ich o n e w o u ld h a rd ly, a priori, h av e a sso cia ted
w ith th e sa m e m a n w h o in tro d u ced g ro u p th eo ry in to q u a n tu m m ech a n ics' [5 1 ].
V ie w s o n S c ie n c e , P h ilo so p h y a n d L ife
W ig n er w a s a g ifted a n d a rticu la te ex p o sito r o f scien ce a n d its p rin cip les to g en era l
a u d ien ces. H ow ev er, h e freq u en tly in d u lg ed in a k in d o f m o ck h u m ility { a s h is
P rin ceto n co llea g u es ex p la in ed h is la n g u a ge [5 2 ], `A p iece o f w o rk is \ a m u sin g "
if it is co rrect a n d b ea u tifu l; it is \ in terestin g " if it is w ro n g a n d m essy.' A n d
in d escrib in g th e ep istem o lo g y o f q u a n tu m m ech a n ics to a n a u d ien ce o f n o n p h y sicists, h e sa id o f h im self th e w riter [5 3]: `H e rea lizes th e p ro fu n d ity o f h is
ig n o ra n ce o f th e th in k in g o f so m e o f th e grea test p h ilo so p h ers a n d is u n d er n o
illu sio n th a t th e v iew s to b e p resen ted w ill b e v ery n ov el. H is h o p e is th a t th ey
w ill a p p ea r sen sib le.' H e co u ld co n v ey sh a rp id ea s p ith ily : `S o m eo n e o n ce sa id
th a t p h ilo so p h y is th e m isu se o f a term in olo g y w h ich w a s in v en ted ju st fo r th is
p u rp o se'.
T h ese a p a rt, h is g ra sp o f a n d co n cern fo r th e g ra n d p rin cip les o f scien ce w ere
v ery d eep . T h e ro le o f in va ria n ce p rin cip les a n d th eir a sso cia ted co n serva tio n
law s ca p tiva ted h im { h e d w elt u p o n th em a t len g th o n m a n y o cca sio n s [5 4 ],
a n d sa id : `A la rg e p a rt o f m y scien ti¯ c w o rk h a s b een d ev o ted to th e stu d y
o f sy m m etry p rin cip les in p h y sics....' H e titled h is N o b el lectu re `E v en ts, law s
o f n a tu re, a n d in va ria n ce p rin cip les'. H e often d escrib ed a s a m ira cle th e fa ct
th a t h u m a n u n d ersta n d in g co u ld u n cov er law s o f n a tu re, a n d sep a ra te th em fro m
th e a ccid en ts o f in itia l co n d itio n s. T h e law s p rov id e stru ctu re a n d co h eren ce to
ev en ts, a n d , in tu rn , th e sy m m etry p rin cip les p rov id e th ese q u a lities to law s; th u s
o n e h a s th e a scen d in g p ro g ressio n : ev en ts to law s to sy m m etry p rin cip les.
T u rn in g to th e ro le o f m a th em a tics in n a tu ra l scien ce, h e ex p ressed w o n d er a t
th e w ay in w h ich m a th em a tica l co n cep ts a n d co n n ectio n s sh ow u p in u n ex p ected
w ay s a n d p la ces, a n d a lso a t th e fa ct th a t ten ta tiv e th eo ries tu rn o u t u p o n fu rth er
d ev elo p m en t to b e fa r m o re a ccu ra te th a n co u ld rea so n a b ly h av e b een ex p ected
a t th e o u tset. T h is led h im to co n clu d e th a t, sin ce w e d o n o t q u ite k n ow w h y
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GENERAL  ARTICLE
w e su cceed so w ell so o ften , w e m u st b e ca u tio u s a n d n o t im m ed ia tely reg a rd a
su ccessfu l ex p la n a tio n a s th e tru th !
P o n d erin g o n th e lik ely fu tu re o f scien ce, W ig n er w o n d ered w h eth er it m ig h t n o t
w in d d ow n u n d er its ow n w eig h t, a n d lo se its a ttra ctiv en ess to th e y o u n g . T h e
in crea sin g ex ten t o f scien ce m a k es it g o b ey o n d th e rea ch o f a n y o n e in d iv id u a l.
B u t th e resp o n se to th is ca n n o t ju st b e a n in crea se in tea m e® o rts, b eca u se th is
ca n n ev er ca p tu re tru e crea tiv e th in k in g in th e in d iv id u a l su b co n scio u s. T h ere is
a n eed h ere to ¯ n d d eep er w ay s o f sh a rin g in fo rm a tio n a n d in sig h t, o f h a rm o n izin g
th e co llectiv e co n scio u s w ith th e su b co n scio u s in ea ch in d iv id u a l.
C o n tin u in g o n th e th em e o f th e g row th o f scien ce a n d th e em erg en ce o f la rg e
co lla b o ra tiv e e® o rts, h e a rg u ed fo r p ro tectin g th e in d iv id u a l a n d g iv in g va lu e
a n d esteem to little scien ce: `O n e d o es n o t h av e th e sa tisfa ctio n w h ich crea tiv e
w o rk , a s w e k n ow it to d a y, p rov id es, if o n e's a ctiv ities a re to o clo sely d irected b y
o th ers'. A b o u t th e em erg en ce o f d eep in sig h ts, `It is h a rd to im a g in e h ow th ey
ca n b e d ev elo p ed o th er th a n in co m p a ra tiv e so litu d e'. A n d a s fo r th e p lea su res
o f p u rsu in g scien ce: `It h a s b een sa id th a t th e o n ly o ccu p a tio n s w h ich b rin g tru e
joy a n d sa tisfa ctio n a re th o se o f p o ets, a rtists, a n d scien tists, a n d , o f th ese, th e
scien tists a re a p p a ren tly th e h a p p iest.'
T h ro u g h th e d escrip tio n o f h is w o rk I h av e tried to co n v ey th e fa ct th a t W ig n er
a ck n ow led g ed v ery g ra cio u sly h is d eb t to so m e o f h is m o st g ifted co n tem p o ra ries.
H e w a s a lso g en ero u s in h is a ssessm en t o f th em . O f v o n N eu m a n n h e w ro te: `...
w h en ev er I ta lk ed w ith th e sh a rp est in tellect w h o m I h av e k n ow n { w ith v o n
N eu m a n n { I a lw ay s h a d th e im p ressio n th a t o n ly h e w a s fu lly aw a k e, th a t I w a s
h a lfw ay in a d rea m .' A n d a b o u t R ich a rd F ey n m a n : `H e is a seco n d D ira c, o n ly
th is tim e m o re h u m a n '.
T w o p erso n s th a t W ig n er h a d b een v ery clo se to { E n rico F erm i a n d v o n N eu m a n n
{ b o th d ied in th eir ¯ fties. W ig n er d escrib ed a n d co n tra sted th eir a ttitu d es to th e
in ev ita b le. W ith F erm i, `O n a h ero ic sca le w a s h is a ccep ta n ce o f d ea th ...H e w a s
so co m p letely co m p o sed th a t it a p p ea red su p erh u m a n '. B u t w ith v o n N eu m a n n
it w a s v ery d i® eren t: 'It w a s h ea rtb rea k in g to w a tch th e fru stra tio n o f h is m in d ,
w h en a ll h o p e w a s g o n e, in its stru g g le w ith th e fa te w h ich a p p ea red to h im
u n av o id a b le b u t u n a ccep ta b le'. T h ese ex p erien ces m u st h av e a ® ected W ig n er
d eep ly ; a t a co n v o ca tio n a d d ress to a n a u d ien ce o f y o u n g stu d en ts so o n a fter, h e
sa id : `O u r cu ltu re is co m m ittin g a sin b y cov erin g o u r ey es a g a in st th e rea liza tio n
th a t n o n e o f u s w ill b e h ere a lw ay s'. A n d to a g en era l a u d ien ce so m e tim e la ter:
966
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GENERAL  ARTICLE
`T h e reco g n itio n th a t p h y sica l o b jects a n d sp iritu a l va lu es h av e a v ery sim ila r
k in d o f rea lity h a s co n trib u ted in so m e m ea su re to m y m en ta l p ea ce'. T h ese
va rio u s ex p ressio n s seem rela ted .
W ig n er w a s a p h y sicist w h o a ch iev ed ra re ra n g e a n d d ep th in h is life a n d w o rk . H e
w a s a p ro d u ct o f th e o ld w o rld w h o ° ow ered d u rin g th e g o ld en a g e o f th eo retica l
p h y sics, a n d ca rried th e fra g ra n ce o f h is su b ject to th e n ew w o rld . T h e tim e
seem s p a st w h en su ch a n o th er ca n a p p ea r.
Suggested Reading
[1]
In the following, frequent references will be made to the following three sources: (a) E P Wigner,
Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New
York, 1959; (b) Reviews of Modern Physics, Vol.34, No.4, 1962 (c) E P Wigner, Symmetries and
Reflections – Scientific Essays, Indiana University Press, 1967. For brevity we shall refer to these
as GT, RMP and SR, respectively.
[2]
SR, p. 257.
[3]
E P Wigner, Remembering Paul Dirac, in Reminiscences about a Great Physicist: Paul Adrien Maurice Dirac,
(eds. B N Kursunoglu and E P Wigner), Cambridge University Press, Cambridge, 1987.
[4]
Margit Dirac, Thinking of my darling Paul, in Reminiscences about a Great Physicist: Paul Adrien Maurice
Dirac, (eds. B N Kursunoglu and E P Wigner), Cambridge University Press, Cambridge, 1987.
[5]
[6]
J A Wheeler, in RMP, p.873.
L Dresner and A M Weinberg in RMP, p.747.
[7]
E P Wigner, Über die Erhaltungssatze in der Quanten mechanik, Nachr. Ges. Wiss. Göttingen, Math
– Physik, K1. p.375, 1927; SR, p.61.
[8]
O Laporte, Z. Phys., Vol.23, p.135, 1924; GT, Chap.18.
[9]
G T, Chaps. 15 and 16; V Bargmann in RMP, p.829; L C Biedenharn and J D Louck, Angular
Momentum in Quantum Physics – Theory and Application, Encyclopedia of Mathematics and its
Applications (ed. G C Rota), Addison–Wesley, Reading MA, Vol.8, 1981.
[10] GT, Appendix to Chap.20; V Bargmann, J Math, Phys., Vol.5, p.862; 1964; L C Biedenharn and
J D Louck, The Racah–Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its
Applications, (ed. G C Rota), Addison–Wesley, Reading, MA, Vol.9, Chap.5, 1981.
[11] E P Wigner, Über die Operation der Zeitumkehr in der Quanten mechanik, Nachr. Ges. Wiss.
Gottingen, Math-Physik, K1., p.546, 1932; GT, Chap. 26.
[12] J von Neumann and E P Wigner, Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys.
Z., Vol.30, p.467, 1929; see also L D Landau, and E M Lifshitz, Quantum Mechanics, Pergamon Press,
Oxford, p. 279, 1965.
[13] G C Wick, A S Wightman and E P Wigner, The intrinsic parity of elementary particles, Phys.
Rev., Vol.88, p.101, 1952; see also R F Streater and A S Wightman, PCT, Spin and Statistics. and
all that, W A Benjamin, New York, Chap,1, 1964.
[14]
GT, p. v.
[15] V Weisskopf and E P Wigner, Berechnung der naturlichen Linienbreite auf Grund der
Diracschen Licht theorie. Z.Phys., Vol.63, p.54, 1930; Über die naturliche Linien breite in der
Strahlung des harmonischen Oszillators, ibid., Vol.65, p.18, 1930; see also M L Goldberger and
K M Watson, Collision Theory, Wiley, New York, Chap.8, 1964; E Merzbacher, Quantum Mechanics,
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GENERAL  ARTICLE
Wiley, New York, 2nd edn., Chap.18, 1970.
[16]
E P Wigner, On the quantum correction for thermodynamic equilibrium, Phys.. Rev., Vol.40, p.749, 1932;
see also M Hillery, R F O’Connell, M O Scully and E P Wigner, Distribution functions in physics:
Fundamentals, Phys. Rep., Vol.106, p.121, 1984.
[17]
H Weyl, The Theory of Groups and and Quantum Mechanics, Dover, New York, p.274, 1931; J E Moyal, Proc.
Camb, Phil. Soc., Vol.45, p.99, 1949.
[18] E P Wigner, Lower limit for the energy derivative of the scattering phase shift, Phys. Rev, Vol.98,
p.145, 1955; see also M L Goldberger and K M Watson in ref. 15, Chap.8, p.492; W Brenig and
R Haag, General quantum theory of collision processes, in Quantum Scattering Theory (ed. Marc
Ross.), Indiana University Press, Bloomington, Section 3, 1963.
[19]
J von Neumann and E P Wigner, Über merkwurdige diskrete Eigenworte, Phys. Z., Vol.30, p.465, 1929; see
also L E Ballentine, Quantum Mechanics, Prentice-Hall, Englewood Cliffs, NJ, p.205, 1990.
[20]
P Jordan, J von Neumann and E P Wigner, On an algebraic generalization of the quantum mechanic formalism, Ann. Math., Vol.35, p.29, 1934; see also L C Biedenharn and J D Louck, in ref.10, Chap.5.
[21] E P Wigner, On the mass defect of helium, Phys. Rev., Vol.43, p.252, 1933; Über diestreuung von
neutronen an protonen, Z. Phys., Vol.83, p.253, 1933; J H Bartlett Jr., Phys. Rev., Vol.49, p.102,
1936; E Majorana, Z. Phys., Vol.82, p.137, 1933; W Heisenberg, Z. Phys.,Vol.77, p.1, 1932.
[22]
N Bohr, Nature, Vol.137, p.344, 1936; G Breit and E P Wigner, Capture of slow neutrons, Phys. Rev., Vol.49,
p.519, 1936.
[23] E P Wigner, Resonance reactions and anomalous scattering, Phys. Rev., Vol.70, p.15, 1946;
Resonance reactions, Phys. Rev. Vol.70, p.606, 1946; L Eisenbud and E P Wigner, Higher angular
momenta and low-range interaction in resonance reactions, Phys. Rev., Vol.72, p.29, 1947; SR,
p.93; E Vogt, in RMP, p.723.
[24] E P Wigner, On the consequences of the symmetry of the nuclear Hamiltonian on the
spectroscopy of nuclei, Phys. Rev., Vol.51, p.106, 1937; reprinted in F J Dyson, Symmetry Groups
in Nuclear and Particle Physics, W A Benjamin, New York, 1966.
[25] W Heisenberg,
[26]
Z. Phys., Vol.77, p.1, 1932.
See, for instance, the reprint group 3 in F J Dyson, ref. 24.
[27]
E P Wigner, Gatlinberg Conf. on Neutron Physics, Oak Ridge Natl. Lab. Rept. No.ORNL-2309, p.59; On
the statistical distribution of thc widths and spacings of nuclear resonance levels, Proc. Camb.
Phil. Soc., Vol.47, p.790, 1951.
[28] F J Dyson, J. Math. Phys., Vol.3, No.140, p.1191, 1962; F J Dyson and M L Mehta, J. Math. Phys.,
Vol.4, p.701, 1963; M L Mehta, Random Matrices, Academic Press, New York, 1967.
[29] P A M Dirac, Proc. R. Soc. London, Vol.A114, p.243, 1927.
[30] P Jordan, and E Wigner, Über das Paulische Aquivalenzverbot, Z. Phys., Vol.47, p.631, 1928.
[31] H S Kragh, Dirac – A Scientific Biography, Cambridge University Press, Cambridge, pp.128–129,
p.289, 338, 1990.
[32] P A M Dirac, Proc. R. Soc. London, Vol.A117, p.610, 1928; Vol.A118, p.351.
[33] E Majorana, Nuovo Cim., Vol.9, p.335, 1932; P A M Dirac, Proc. Roy. Soc. London, Vol.A155, p.447,
1936; Al. Proca, J. Phys. Road., Vol.7, p.347, 1936.
[34] E Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math., Vol.40, p.149,
1939.
[35]
Quoted in ref. 13.
[36]
For quantum-mechanical purposes, one has to deal with the group SL(2,C), the universal covering group of
the HLG, and its subgroups.
[37] Harish-Chandra, Proc. Roy. Soc. London, Vol.A189, p.372, 1947; I M Gel’fand and M A Naimark,
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Izv. Akad. Nauk. SSSR, Vol.11, p.411, 1947.
[38] V Bargmann, Ann. Math., Vol.48, p.568, 1947.
[39] P A M Dirac, in RMP, p.592.
[40] E P Wigner, Invariance in physical theory, Proc. Am. Phil. Soc., Vol.93, p.521, 1949; reprinted in
SR, pp.8–9.
[41]
V Bargmann, and E P Wigner, Group theoretic discussion of relativistic wave equations, Proc. Natl. Acad.
Sci., USA, Vol.34, p.211, 1948.
[42] T D Newton, and E P Wigner, Localized states for elementary systems, Rev. Mod. Phys., Vol.21,
p.400, 1949.
[43] E Inonu and E P Wigner, On the contraction of groups and their representations, Proc. Natl. Acad.
Sci. USA, Vol.39, p.510, 1953.
[44] H Salecker and E P Wigner, Quantum limitations of the measurement of space-time distances,
Phys. Rev., Vol.109, p.571, 1958; SR, pp.62 ff.
[45] E P Wigner and H van Dam, Classical relativistic mechanics of interacting point particles, Phys.
Rev., Vol.B138, p.1576, 1965.
[46]
E P Wigner, The problem of measurement, Am. J. Phys., Vol.31, p.6, 1963; Remarks on the mindbody question, in The Scientist Speculates (ed. I J Good), William Heinemann, London, 1961; Two
kinds of reality, The Monist, Vol.48, No.2, 1964; all reprinted in SR, pp.153, 171, 185.
[47] F London, and E Bauer, La Theorie de I’Observation en Mecanique Quantique, Hermann et Cie, Paris,
1939.
[48] J von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press,
Princeton, New Jersey, 1955.
[49] G Wannier, in RMP, p.645.
[50]
SR, p.126.
[51]
Dresner Lawrence and A M Weinberg, in RMP, p.747.
[52] V Bargmann et al., in RMP, p.588.
[53]
SR, p. 186.
[54]
In addition to the articles included in SR, see also R M F Houtappel, H van Dam, and E P Wigner, The
conceptual basis and use of the geometric invariance principles, Rev. Mod. Phys., Vol.37, p.595, 1965.
Address for Correspondence: N Mukunda, Centre for High Energy Physics, Indian Institute of Science, Bangalore
560 012, India. Email: nmukunda@cts.iisc.ernet.in
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