8 .2 2 5 / S T S .0 4 2 , Ph y s ic s in the 2 0 th C e ntur y Pr of e s s or Da v id K a is e r , 1 6 No v e mbe r 2 0 2 0

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1. Q u a nt u m F ie ld s a nd Infinit ie s

2. Q u a nt u m E le c t r od y na m ic s a ft e r t he W a r : N e w E x pe r im e nt s

3 . J u lia n Sc hw ing e r a nd R e nor m a liz a t ion

N e a r ly a s so o n a s p h y sic ist s h a d c o b b le d t o g e t h e r q u a n t u m me c ha ni c s i n 1 9 2 5 - 2 6 , s e v e r a l be g a n t r y i ng t o a ppl y t he ne w for ma l i s m t o r a d ia t ion . A f t e r a ll, E in st e in h a d su g g e st e d t h a t lig h t c o n sist s o f qua nt a b ac k i n 1 9 0 5 . P au l D i r ac pr opos e d a q u a n t ize d d e sc r ip t io n o f M a x w e llia n e le c t r o m a g n e t ic f ie ld s in 1 9 2 7 , t he r e b y he l pi ng t o l a u nc h qua nt um f ie l d t h e o r y .

Us i ng F o uri e r a na l y s i s , w e m a y r e p r es en t a n a r b i tr a r y cur v e a s a

s up e r p o s it io n o f w a v e s w it h sp e c if ic w a v e n u m b e r s | k | = 2 π /  :

C o urtesy of Evan Bianco, AgileScientific. Used under CC BY.

D ir a c : g e n e r a lize for fi e lds t o

P au l D i r ac ’ s f i r s t pu bl i c a t i on on a q u a nt u m - t he or e t i c a l t r e a t m e nt of t he e l e c t r om a g ne t i c f i e l d , 1 9 2 7 .

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T h e n t h e st a t e o f t h e f ie ld ca n b e r e p r es en ted a s a c o lle c t io n o f qua nt a in v a r io u s st a t e s ( E p , p ) . Cha ngi ng t he fi e l d c or r e s pond s t o e mit t ing or a b s o rb ing q u a n t a in sp e c if ic s ta tes .

But t h e u n c e r t a in t y p r in c ip le a llo w s f o r “ v ir t u a l” p r o c e sse s t h a t t e mp o ra ri l y v i ol a t e t he c ons e r v a t i on of e ne r g y :

H en ce th e e x a c t s ta te o f th e f i el d a t a n y g i v en ti m e ca n n ev er b e st ip u la t e d ; t h e r e w ill a lw a y s e x ist qua nt um f l uc t ua t io ns .

Cons i d e r a n e l e c t r on t r a v e l i ng fr om l oc a t i on x 1 t o x 2 w i t h s ome mome nt u m p :

T he v i r t u a l phot on c ou l d “bor r o w ” a ny a mou nt of e ne r g y  E = c h k , w it h k = | k | an d 0 " k " # , a s l ong a s  t ~ h /  E w a s c or r e s pond i ngl y br i e f . T he p r o b a b i l i t y am p l i t u d e ~ 1/ k t o e m it a n d r e - a bs or b a v i r t u a l phot on of mome nt u m k ( hi ghe r - m o m e n t u m st a t e s le ss lik e ly ) . B u t sin c e t h e v ir t u a l p h o t o n co ul d h a v e a ny v a lue | k | , o n e m u st su m u p a ll t h e p o ssib ilit ie s:

T h e e l e c tro n ’ s “ s e l f - e n e rg y ” di v er ges !

A f ir st id e a , w h ic h se v e r a l p h y sic ist s e x p lo r e d d u r in g t h e 1 9 3 0 s , w as s i m p l y t o c u t of f t h e in t e g r a ls , r a t h e r t h a n in t e g r a t e u p t o | k | ! " .

But in se r t in g a c u t - off k m ax m e a n t t h a t t h e e q u a t io n s w o u ld n o lo n g e r r e sp e c t t h e sy m m e t r ie s o f sp e c ia l r e la t i v it y — ev en t h o u g h t h is w a s s up p o s e d t o d e sc r ib e ( a q u a n t ize d v e r sio n o f ) M a x w e ll ’ s e q u a t io n s , w h ic h d o o b e y t h e r e la t i v ist ic s ym m etr i es .

A p r a c t ic a l c o n se q u e n c e : if o n e sim p ly in se r t e d k m ax b y h a n d , t h e n t h e r e su lt s o f a n y c a lc u la t io n w o u ld r e m a in a mb ig uo us : d if f e r e n t in e r t ia l o b se r v e r s w o u ld n o lo n g e r a g r e e o n o b se r v a b le q u a n t it ie s .

V ir t u a l P a r t ic le s : V a c c u m P o la riz a t io n

S im ila r v ir t u a l p r o c e sse s a f f e c t e d e l e c t ric c h a r g e . A phot on c ou l d sp o n t a n e o u sly e m it a n d la t e r r e a b so r b a v ir t u a l e le c t r o n - pos i t r on pa i r .

H e n c e o n e c o u ld a sk a b o u t a n e le c t r o n ’ s in t e r a c t io n w it h i t s o w n e l e c t r o ma g ne t ic f ie l d . A f t e r a ll, t h e E a n d B fi e l d s c ons i s t of phot ons , a ny o f w h ic h c o u ld sp o n t a n e o u sly u n d e r g o t h e se v ir t u a l p r o c e sse s . I n t h e v ic in it y o f a n e le c t r o n , e mp t y s p a c e it s e l f w ou l d be c ome po lar i z e d.

“ clo u d ” o f v ir t u a l e l e c t r o n - pos i t r on pa i r s

- + + -

T he or i gi na l c ha r g e w ou l d be s c r e e ne d . By h o w m uc h ?

E a c h v ir t u a l p a ir c o u ld b o r r o w u p t o | k | ! " , so

T h e e ffe c ti v e c h arg e o f an e l e c tro n di v er ges !

V ir t u a l P a r t ic le s : In t e rw a r R e s p o n s e s

S e v e r a l le a d in g p h y sic ist s in E u r o p e , su c h a s N ie l s Bo h r a n d W e r ne r H ei sen b er g , a r g u e d t h a t t h e in f in it ie s in q u a n t u m e le c t r o d y n a m ic s p o in t e d t o t h e n e e d f o r ( y e t a n o t h e r ) r a d ic a l c o n c e p t u a l r e v o lu t io n , a k in t o r e la t i v it y a n d q u a n t u m m e c h a n ic s .

H ei sen b er g : m a yb e th er e ex i s ts a f und a me nt a l ( s h o r t e s t ) l e ng t h ,  0 , an d he nc e s ome l a r g e s t pos s i bl e mome nt u m h k m ax = h /  0 < # . T h a t w o u ld r e m o v e t h e in f in it ie s b ut r e q u ir e a r a d ic a l r e t h in k in g o f sp a c e a n d t im e , b e y o n d r e la t i v it y .

O th er s : m a yb e th e i d ea o f vi r t u al par t i c le s is a t f a u lt , in w h ic h c a se t h e unc e r t a int y p rinic p l e a t t h e h e a r t o f q u a n t u m m e c h a n ic s w o u ld n e e d t o b e r e v ise d .

Q u e s tion s ?

Q E D a f t e r t h e W a r : N e w E xp e rim e n t s

Du r i ng t he S e c ond W or l d W a r , r e s e a r c he r s a t t he M IT - b a s ed R adi at i o n L abo r at o r y h a d d ev el o p ed a d v a n ced c a p a b ilit ie s t o p r o d u c e a n d m e a su r e c e n t im e t e r - w a v e le n g t h ( m ic r o w a v e ) e le c t r o m a g n e t ic r a d ia t io n , f o r th e r a d a r p r o j ect.

Af ter th e w a r , s ev er a l R a d L a b r es ea r c h er s w er e a b l e t o t a k e s ur p l us e quip me nt b a c k t o t h e ir u n i v e r sit ie s , w h ic h e na bl e d t he m t o c ond u c t ne w e x pe r i me nt s w i t h hi ghl y se n sit i v e e le c t r o n ic s .

M I T R a d L a b “ R e c e i v e r L a b o r a t o r y , ” B u i l di n g 4, 1941

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T w o of t he s e e x pe r i me nt s t ook pl a c e a t Col u mbi a U n i v e r sit y in t h e la t e 1 9 4 0 s b y Ra d L a b a lu m n i, u sin g su r p lu s Ra d L a b e q u ip m e n t : t h e L am b s h i ft a nd t he a no ma l o us ma g ne t ic mo me nt .

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A c c or d i ng t o t he B ohr mod e l , S c hr öd i ng e r ’ s e q u a t i on, a nd e v e n D ir a c ’ s r e la t i v ist ic e q u a t io n f o r t h e e le c t r o n , t h e e n e r g y of a n e l e c t r on i n t he 2 S st a t e o f h y d r o g e n sh o u ld b e id e n t ic a l to th e en er g y o f th e 2 P s ta te: E n ~ 1 / n 2 .

In s tea d , W il l is L a mb a n d h is st u d e n t Ro b e r t Re t h e rf o rd w e r e a b le t o u se Ra d L a b e q u ip m e n t a n d t e c h n iq u e s t o m e a su r e a t in y d i f f er en ce b etw een th e en er g i es o f th e 2 S a n d 2 P st a t e s in 1 9 4 7 :  E / E ~ 1 0 - 6 . In p a r ti cul a r , th e en er g y o f th e 2 S s ta te w a s slig h t ly g r e a t e r t h a n p r e d ic t e d b y o r d in a r y q u a n t u m me c ha ni c s .

T h is le d se v e r a l t h e o r ist s t o w o n d e r : p e r h a p s t h e e n e r g y sh if t w a s d u e t o e f f e c t s f r o m vi r t u al par t i c le s . L a mb pr e s e nt e d t h e p r e lim in a r y r e su lt s a t a p r i v a t e p h y sic s m e e t in g o n S hel t er I s lan d in J u n e 1 9 4 7 . O n h is t r a in r id e b a c k t o u p st a t e N e w Y o r k , H a n s B et he w or k e d ou t a r ou gh c a l c u l a t i on of t he e ffe c t .

B e t h e ’ s b a sic id e a w a s t h a t v ir t u a l p a r t ic le s w o u ld c o n st a n t ly

V bomba r d t he e l e c t r on, s me a r i ng ou t i t s a v e r a g e pos i t i on w i t hi n s ome r a ng e  r . T h e p o ten ti a l en er g y V ( r ) w o u ld

r th er ef o r e b eco m e p e r t urb e d :

T h e r a n d o m p er tu r b a tio n s w o u ld a v er a g e o u t, s o (  r ⟩ = 0 , e v e n t hou g h ( (  r ) 2 ⟩ $ 0 .

2 S 2 P T he w a v e fu nc t i on for a n e l e c t r on i n t he 2 S s ta te i s s p h e ric a l l y s y mme t ric a n d no nv a nis h ing a t t h e o r ig in ,  ( r = 0 ) # 0 , so it s e n e r g y is

T h e p o ten tia l n ea r th e o r ig in is t r u n c a t e d l if t e d b y t he mod i fi e d be ha v i or of t he e ffe c t i v e pot e nt i a l , w he r e a s

c om pa r e d t o t he “ ba r e ” pot e nt i a l t he w a v e fu nc t i on for t he 2 P s ta te v a n i shes a t t h e o r ig in ,  ( r = 0 ) = 0 ,

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https://ocw.mit.edu/help/faq-fair-use/

so it s e n e r g y is u n a f f e c t e d b y t h e c h a n g e .

V bomba r d t he e l e c t r on, s me a r i ng ou t i t s a v e r a g e pos i t i on w i t hi n s ome r a ng e  r . T h e p o ten ti a l en er g y V ( r ) w o u ld

r th er ef o r e b eco m e p e r t urb e d :

T h e r a n d o m p er tu r b a tio n s w o u ld a v er a g e o u t, s o (  r ⟩ = 0 , e v e n t hou g h ( (  r ) 2 ⟩ $ 0 .

2 S 2 P

T h e p o ten tia l n ea r th e o r ig in is t r u n c a t e d

c om pa r e d t o t he “ ba r e ” pot e nt i a l

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https://ocw.mit.edu/help/faq-fair-use/

T he w a v e fu nc t i on for a n e l e c t r on i n t he 2 S s ta te i s s p h e ric a l l y s y mme t ric a n d no nv a nis h ing a t t h e o r ig in ,  ( r = 0 ) # 0 , so it s e n e r g y is l if t e d b y t he mod i fi e d be ha v i or of t he e ffe c t i v e pot e nt i a l , w he r e a s t he w a v e fu nc t i on for t he 2 P s ta te v a n i shes a t t h e o r ig in ,  ( r = 0 ) = 0 , so it s e n e r g y is u n a f f e c t e d b y t h e c h a n g e .

J u s t d o w n t he ha l l w a y fr om L a mb a nd R e t he r for d , I . I . R abi a n d co l l ea g ues w er e als o u sin g su r p lu s Ra d L a b e q u ip m e n t t o p e r f o r m s e ns i t i v e me a s u r e me nt s of e l e c t r ons i n h y d r og e n a t oms . I n s pr i ng 1 9 4 7 , th ey m ea s ur ed th e el ectr o n ’ s ma g ne t ic mo me nt a nd fou nd a t i ny b u t m e a su r a b le d e v ia t io n f r o m t h e o r e t ic a l p r e d ic t io n s .

R e c a ll t h e q u a n t u m - me c ha ni c a l pr e d i c t i on: a n e l e c t r on ha s c ha r g e e a n d in t r in sic a n g u la r m o m e n t u m ( sp in ) S , s o i t s ma gne t i c mome nt sh o u ld b e :

“ B ohr m a g ne t on”

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B u t u sin g t h e se n sit i v e m ic r o w a v e e le c t r o n ic s , N a f e , N e lso n , a n d Ra b i m e a su r e d a s li g h t ly lar g e r v a lue :   /  B ~ 1 0 - 3 .

L ik e L a m b , Ra b i p r e se n t e d h is g r o u p ’ s r e su lt s a t t h e

“ clo u d ” o f v ir t u a l S h e l t e r I s l a nd c onfe r e nc e i n J u ne 1 9 4 7 . A g a i n, t he or i s t s

w o n d er ed i f th e m ea s ur ed d i s cr e p a n cy m i g h t b e d ue

- +

e l e c t r o n - pos i t r on pa i r s

+ -

t o t h e p h y s ic a l e f f e c t s of vi r t u al par t i c le s .

J u st lik e c o m p a s n e e d le s a lig n in g in t h e E a r t h ’ s ma gne t i c fi e l d , pe r ha ps t he s pi ns of e a c h of t he vi r t u al par t i c le s s u r r ou nd i ng a gi v e n e l e c t r on w ou l d be c ome a l i gne d i n a n e x t e r na l ma gne t i c fi e l d . T he n t he c ont r i bu t i ons fr om t he vi r t u al c lo u d w o u ld y ie ld s o m e l a r g er e f f e c t i v e sp in f o r t h e sy st e m , w h ic h c o u ld sh if t

Q u e s tion s ?

T h e o ris t s ’ R e s p o n s e s a f t e r t h e W a r

J . Ro b e r t O p p e nh e ime r or g a ni z e d t he i nfor ma l , pr i v a t e w o r k sh o p o n S h e lt e r I sla n d f o r 2 4 p h y sic ist s in J u n e 1 9 4 7 , e x p r e ssly t o h e lp ju m p - st a r t r e se a r c h in t h e o r e t ic a l p h y sic s a f t e r t h e d isr u p t io n s o f t h e w a r . T h e N a t io n a l A c a d e m y o f S c ie n c e s pa i d for t he me e t i ng, w hos e s t a t e d g oa l w a s “t o g a t he r a g r ou p c ons i s t i ng of t he y ou ng e r me n, w ho w ou l d u nd e r s t a nd e a c h ot he r ’ s j a r g on. ”

Ph y si c i st s di sc u ss Q E D a t t h e J u n e 1 9 4 7 S h e l t e r I sl a n d c o n f e r e n c e . S ta n d in g , le f t to r ig h t: W illis L a m b , J o h n W he e l e r . Se a t e d , l e f t t o r i g ht : A br a ha m P a i s , R i c ha r d F e y nm a n, H e r m a n F e s hba c h, a nd J u l i a n Sc hw i ng e r .

N e a r ly a ll o f t h e p a r t ic ip a n t s h a d w o r k e d o n w a r t im e pr oj e c t s l i k e r a d a r a nd t he Ma nha t t a n P r oj e c t a nd k ne w e a c h ot he r .

U n lik e t h e o r ist s’ r e sp o n se s in t h e 1 9 3 0 s t o t h e in f in it ie s o f Q E D , f e w o f t h e U S - b a s ed t h e o r ist s a f t e r t h e w a r t a lk e d o f g r a n d c o n c e p t u a l r e v o lu t io n s . Ra t h e r , lik e t h e ir e f f o r t s d u r in g t he w a r , ma ny a i me d t o fi nd pr a c t i c a l me a ns of “g e t t i ng t he n u mbe r s ou t ” t o c ompa r e w i t h m e a su r e m e n t s , r a t h e r t h a n g e t t in g st u c k w it h f o r m a l, f ir st - p r in c ip le s d e r i v a t io n s .

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J ul ia n S c h w ing e r w a s a mong t he S he l t e r I s l a nd pa r t i c i pa nt s . H e ha d w o r k e d a t t h e Ra d L a b d u r in g t h e w a r , a n d w a s ( a t t h e t im e ) t h e y o u n g e st p e r so n e v e r t o g e t t e n u r e a t H a r v a r d ( a t a g e 2 9 ! ) .

J u l i an S c h w i n g e r at t h e M I T R ad L ab , 1940s

A t t he R a d L a b , he ha d l e a r ne d fr om t he e ngi ne e r s t o foc u s on e f f e c t i v e c ir c uit s a nd i npu t - o u t p u t r e la t io n sh ip s .

Upon he a r i ng fr om L a mb a n d R abi a t S h e lt e r I sla n d a b o u t t h e ir la t e st m e a su r e m e n t s , S c h w ing e r be g a n t hi nk i ng a bou t vi r t u al par t i c le s in a sim ila r w a y .

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