13. The Optics Of Particles

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Snippets of Physics 13. The Optics of Particles T Padmanabhan

T Padmanabhan works at IUCAA, Pune and is interested in all areas of theoretical physics, especially those which have something to do with gravity.

T h e p ro b a b ility a m p litu d e fo r a p a rtic le to p ro p a g a te fro m e v e n t to e v e n t in sp a c e tim e sh o w s so m e n ic e sim ila ritie s w ith th e c o rre sp o n d in g p r o p a g a to r fo r th e e le c tro m a g n e tic w a v e a m p litu d e d isc u sse d in th e la st in sta llm e n t. In fa c t, th is a n a lo g y p ro v id e s a n in te r e stin g in sig h t in to th e tr a n sitio n fr o m q u a n tu m ¯ e ld th e o r y to q u a n tu m m e c h a n ic s! In cla ssica l m ech a n ics, th e m o tio n o f a p a rticle w ith p o sitio n x (t) u n d er th e a ctio n o f a (tim e in d ep en d en t) p o ten tia l V (x ) is d eterm in ed th ro u g h th e N ew to n 's law o f m o tio n m xÄ = ¡ r V . G iv en th e in itia l p o sitio n x (0 ) a n d v elo city v (0 ) a t t = 0 w e ca n in teg ra te th is eq u a tio n to d eterm in e th e tra jecto ry. A ll o th er p h y sica l o b serv a b les in cla ssica l m ech a n ics ca n b e o b ta in ed fro m th e tra jecto ry x (t). W h a t is th e co rresp o n d in g situ a tio n in q u a n tu m m ech a n ics? H ere th e w av efu n ctio n o f th e p a rticle à (t;x ) co n ta in s co m p lete in fo rm a tio n a b o u t th e sta te o f th e sy stem a n d sa tis¯ es th e S ch rÄo d in g er eq u a tio n @à ~2 2 i~ r à + V à ´ H à : = ¡ @t 2m

Keywords Optics, waves.

. 8

(1 )

G iv en th e w av efu n ctio n à (0 ;x ) a t t = 0 w e ca n in teg ra te th is eq u a tio n a n d o b ta in th e w av efu n ctio n a t a n y la ter tim e. B u t, u n lik e in th e ca se o f cla ssica l m ech a n ics, th ere is a n ice w ay o f sep a ra tin g th e d y n a m ica l ev o lu tio n fro m th e in itia l co n d itio n in th e ca se o f q u a n tu m m ech a n ics w h ich w e sh a ll ¯ rst d escrib e. T o k eep th e d iscu ssio n so m ew h a t g en era l w e w ill a ssu m e th a t th e

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sp a ce h a s D d im en sio n s w ith th e u su a l ch o ices b ein g D = 1 ;2 ;3 . W e k n ow th a t w h en th e p o ten tia l is in d ep en d en t o f tim e, eq u a tio n (1 ) h a s en erg y eig en sta tes w h ich sa tisfy th e eig en va lu e eq u a tio n H Á n (x ) = E n Á n (x ). U sin g th ese eig en fu n ctio n s w e ca n ex p a n d th e in itia l w av efu n ctio n à (0 ;x ) in term s o f th e en erg y eig en fu n ctio n s a s Z X à (0 ;x ) = c n Á n (x ); c n = d y à (0 ;y )Á ¤n (y ); (2 ) n

w h ere th e ex p ressio n fo r c n fo llow s fro m th e o rth o n o rm a lity o f th e en erg y eig en fu n ctio n s a n d th e sp a tia l in teg ra tio n s a re ov er th e D -d im en sio n a l sp a ce. S in ce th e en erg y eig en fu n ctio n ev o lv es in tim e w ith a p h a se fa cto r ex p (¡ iE n t= ~), it fo llow s th a t th e w av efu n ctio n a t tim e t is g iv en b y X Ã (t;x ) = c n Á n (x )e ¡ iE n t= ~ ; (3 ) n

w h ich , in p rin cip le, so lv es th e p ro b lem . A ctu a lly, w e ca n d o b etter b y ex p ressin g th e c n s in (3 ) in term s o f à (0 ;x ) u sin g th e seco n d rela tio n in (2). T h is g iv es Z X à (t;x ) = d y à (0 ;y ) Á n (x )Á ¤n (y )e ¡ iE n t= ~ ´

Z

n

d y K (t;x ;0;y )Ã (0 ;y );

(4 )

w h ere w e h av e d e¯ n ed th e fu n ctio n { u su a lly ca lled th e p ro p a g a to r o r k ern el { b y X K (t;x ;0 ;y ) = Á ¤n (y )Á n (x )e ¡iE n t= ~ : (5 ) n

It is o b v io u s th a t th e p ro p a g a to r K (t;x ;0 ;y ) ca rries co m p lete in fo rm a tio n a b o u t th e d y n a m ica l ev o lu tio n o f th e sy stem a n d ca n b e co m p u ted o n ce th e H a m ilto n ia n

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It is obvious that the propagator K(t,x;0,y) carries complete information about the dynamical evolution of the system.

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Curiously enough, such a separation has no direct analog in the case of classical mechanics.

H is k n ow n , in term s o f its eig en fu n ctio n s a n d th e eig en va lu es. W h a t is m o re, eq u a tio n (4 ) n icely sep a ra tes th e d y n a m ics, en co d ed in K (t;x ;0 ;y ), fro m th e in itia l co n d itio n en co d ed in à (0 ;y ). C u rio u sly en o u g h , su ch a sep a ra tio n h a s n o d irect a n a lo g in th e ca se o f cla ssica l m ech a n ics. S in ce à (0 ;y ) g iv es th e a m p litu d e to ¯ n d th e p a rticle a ro u n d y a t t = 0 , it fo llow s th a t K (t;x ;0 ;y ) ca n b e th o u g h t o f a s th e p ro b a b ility a m p litu d e fo r a q u a n tu m p a rticle to p ro p a g a te fro m th e ev en t (0 ;y ) to th e ev en t (t;x ). O f co u rse, sin ce th e p o ten tia l is in d ep en d en t o f tim e, w e ca n u se tim e tra n sla tio n in va ria n ce to w rite a n ex p ressio n fo r K (t2 ;x 2 ;t1 ;x 1 ) b y rep la cin g E n t= ~ in (5 ) b y (E n = ~)(t2 ¡ t1 ). F o r th is in terp reta tio n to b e va lid th e p ro p a g a to r m u st sa tisfy th e in teg ra l co n d itio n : Z K (t3 ;x 3 ;t1 ;x 1 ) = d x 2 K (t3 ;x 3 ;t2 ;x 2 )K (t2 ;x 2 ;t1 ;x 1 ): (6 )

U sin g th e d e¯ n itio n in (5 ) a n d th e o rth o n o rm a lity o f eig en fu n ctio n s, y o u ca n p rov e th a t th is is in d eed tru e. N o te th a t th is is a n o n triv ia l co n d itio n : A t th e in term ed ia te ev en t w e in teg ra te o n ly ov er x 2 leav in g t2 a lo n e; n ev erth eless, th e ¯ n a l resu lt { a n d th e left h a n d sid e { is in d ep en d en t o f t2 . S o th e p ro p a g a to r a ctu a lly p ro p a g a tes th e p a rticle fro m ev en t to ev en t. Note that equation (6) is a nontrivial condition: At the intermediate event we integrate only over x2 leaving t2 alone; nevertheless, the final result – and the left hand side – is independent of t2.

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S in ce Á n s a re en erg y eig en fu n ctio n s, it is a lso stra ig h tfo rw a rd to v erify th a t th e p ro p a g a to r sa tis¯ es th e S ch rÄo d in g er eq u a tio n µ ¶ @ i~ ¡ H K (t;x ;0 ;y ) = 0 ; (7 ) @t w ith th e sp ecia l in itia l co n d itio n lim K (t;x ;0 ;y ) = ± D (x ¡ y ): t! 0

(8 )

T h is co n d itio n ca n a lso b e o b ta in ed ea sily fro m (4 ) b y ta k in g th e lim it o f t ! 0 . RESONANCE  January 2009

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A fter th is p rea m b le a b o u t th e p ro p a g a to r, w e w ill tu rn to th e k ey to p ic o f th is in sta llm en t. T o in tro d u ce it, let u s w o rk o u t th e p ro p a g a to r for a free p a rticle (V = 0 ) u sin g th e ex p ressio n in (5 ). In th e ca se o f a free p a rticle th e en erg y eig en fu n ctio n s a n d eig en va lu es ca n b e ta k en to b e la b elled b y a w av en u m b er p in stea d o f a d iscrete in d ex n w ith Á p (x ) =

1 (2 ¼ )D

=2

ex p i(p ¢ x );

E

p

=

~2 p 2 : 2m

(9 )

T h e n o rm a liza tio n o f Á p (x ) is so m ew h a t a rb itra ry b u t w e u se th e co n v en tio n th a t m o m en tu m sp a ce in teg ra ls co m e w ith a m ea su re d p so th at th e o rth o n o rm a lity co n d itio n rea d s a s Z d p Á p (x )Á ¤p (y ) = ± (x ¡ y ): (1 0 ) T h e p ro p a g a to r is n ow g iv en b y a n ex p ressio n sim ila r to (5 ) b u t w ith a n in teg ra l ov er p ra th er th a n a su m ov er th e d iscrete in d ex n . H en ce w e g et Z d p ip ¢(x ¡y )= ~ ¡ip 2 ~t= 2 m K (t;x ;0 ;y ) = e e (2 ¼ )D · ¸ ³ m ´D = 2 im (x ¡ y )2 = ex p (;1 1 ) ~ 2 ¼ i~t 2t w h ere D is th e d im en sio n o f sp a ce (1 , 2 o r 3 ) in w h ich th e p a rticle is m ov in g . (T h e in teg ra l is ju st th e D d im en sio n a l F o u rier tra n sfo rm o f a G a u ssia n w h ich sep a ra tes o u t in ea ch o f th e d im en sio n s.) W e ca n v erify d irectly th a t K (t;x ;0 ;y ) sa tis¯ es (7 ) a n d (8 ). It is a lso o b v io u s th a t it sa tis¯ es th e \ n o rm a liza tio n co n d itio n " Z d x K (t;x ;0 ;y ) = 1 : (1 2 ) U su a lly in q u a n tu m m ech a n ics w e n o rm a lize th e p ro b a b ilities a n d n o t th e p ro b a b ility a m p litu d es. B u t th is is a n ex cep tio n to th e ru le in w h ich p ro b a b ility a m p litu d e

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Equation (12) is an exception to the rule in which probability amplitude for propagation from event to event comes out normalized to unity.

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fo r p ro p a g a tio n fro m ev en t to ev en t co m es o u t n o rm a lized to u n ity. T h ere is a co n v en tio n a l in terp reta tio n o f th e p h a se o f th e p ro p a g a to r in (1 1 ) w h ich w e w ill n ow d escrib e. T o d o th is, let u s co n sid er th e a ctio n fo r th e free p a rticle in cla ssica l m ech a n ics g iv en b y Zt m A (t;x ;0 ;y ) = x j2 ; d t j_ (1 3 ) 2 0 w h ich is d e¯ n ed fo r a n y tra jectory x (t) th a t co n n ects th e tw o en d p o in ts. In p a rticu la r, w e ca n d eterm in e th e cla ssica l tra jecto ry x c (t) fro m ex trem isin g th e a ctio n a n d eva lu a te th e cla ssica l va lu e o f th e a ctio n A c (t;x ;0 ;y ) fo r th is p a rticu la r cla ssica l tra jecto ry. F o r th e free p a rticle, th is is triv ia l to eva lu a te a n d is g iv en b y A c (t;x ;0 ;y ) =

m (x ¡ y )2 ; t 2

(1 4 )

so th a t th e p ro p a g a to r in (1 1 ) ca n b e ex p ressed in th e fo rm · ¸ i K (t;x ;0 ;y ) = N (t) ex p A c (t;x ;0 ;y ) : (1 5 ) ~ W e see th a t th e p h a se o f th e p ro p a g a to r ca n b e in terp reted a s ju st th e cla ssica l va lu e o f th e a ctio n d iv id ed b y ~.

We see that the phase of the propagator can be interpreted as just the classical value of the action divided by ~).

12

T h e situ a tio n is a ctu a lly b etter th a n th is. L et u s co n sid er, in stea d o f th e cla ssica l p a th , a n y a rb itra ry p a th co n n ectin g th e tw o ev en ts (la b elled 1 a n d 2 ) w e a re in terested in . S in ce o n e ca n n o t m ea su re p o sitio n a n d v elo city o f a p a rticle sim u lta n eo u sly in q u a n tu m m ech a n ics, it d o es n o t m a k e sen se to say th a t th e p a rticle w en t fro m o n e p o in t to a n o th er a lo n g a p a rticu la r tra jecto ry. T h e b est w e ca n say is th a t th ere is so m e a m p litu d e P (2 ;1 jx (t)) ´ P (t2 ;x 2 ;t1 ;x 1 jx (t)) fo r th e p a rticle to ch o o se a p a rticu la r p a th x (t). W e n ow p o stu la te, fo llo w in g D ira c a n d F ey n m a n , th a t th is a m p litu d e is g iv en b y

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P (2 ;1 jx (t)) = ex p (iA [2 ;1 jx (t)]= ~). T h en th e to ta l a m p litu d e fo r p ro p a g a tio n fro m ev en t 1 to ev en t 2 (w h ich , o f co u rse, is o u r p ro p a g a to r) m u st b e g iv en b y X X K (2 ;1 ) = P (2 ;1 jx (t)) = ex p (iA [2 ;1 jx (t)]= ~); p a th s

x (t)

(1 6 )

w h ere th e su m m a tio n sy m b o l in d ica tes th a t w e h av e to su m ov er a ll p a th s x (t) co n n ectin g th e tw o ev en ts. In g en era l, it is n o t ea sy to d e¯ n e a n d eva lu a te th is su m b u t so m eth in g in terestin g h a p p en s if th e a ctio n co n ta in s n o term s w h ich a re m o re th a n q u a d ra tic in v elo city o r p o sitio n . In th ese ca ses, w e b eg in b y w ritin g a n y a rb itra ry p a th { ov er w h ich w e h av e to su m { in term s o f th e cla ssica l p a th x c (t) p lu s a d ev ia tio n fro m it: th a t is, w e w rite x (t) = x c (t) + r(t). S u m m in g ov er a ll x (t) is th e sa m e a s su m m in g ov er a ll r (t) b u t th e `b o u n d a ry co n d itio n s' o n r (t) a re ea sier to h a n d le. S in ce b o th th e cla ssica l p a th x c (t) a n d a n y a rb itra ry p a th x (t) co n n ect th e sa m e en d p o in ts, th e r(t) m u st va n ish a t th e en d p o in ts. F u rth er, if th e a ctio n h a s o n ly u p to q u a d ra tic term s in x (t), th en it sp lits u p a s th e su m : A [x (t)] = A [x c (t) + r(t)] = A [x c (t)] + A lin [x c (t);r (t)] + A

q u a d [r(t)];

In general, it is not easy to define and evaluate the sum in (16) but something interesting happens if the action contains no terms which are more than quadratic in velocity or position.

(1 7 )

w h ere A lin is lin ea r in x (t) a n d r(t) w h ile A q u a d is q u a d ra tic in r(t). (T h is is essen tially p a ra p h ra sin g th e fo rm u la fo r (a + b)2 !). B u t reca ll th a t th e cla ssica l p a th is a n ex trem u m o f th e a ctio n ; so th e ch a n g e in a ctio n w h en th e p a th ch a n g es b y a d ev ia tio n r(t) h a s to b e to q u a d ra tic o rd er in r . T h erefo re, A lin = 0 a n d th e su m ov er p a th s in (1 6 ) ca n b e w ritten a s: X K (2 ;1 ) = ex p (iA [2 ;1 jx (t)]= ~) x (t)

=

ex p (iA [2 ;1 jx c (t)]= ~)

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X

ex p (iA

q u a d [r(t)]= ~):(1 8 )

r(t)

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The sum over r(t) in (18) (which we haven’t even defined properly!) can only be a function of t2,t1. We get the neat result that for all actions which have only up to quadratic terms in their variables, the propagator has the form in (15).

1

See equation (7) in Resonance, Vol.13, p.1100, December 2008.

T h e cru cia l p o in t is th a t th e su m ov er r (t) is d o n e w ith th e co n d itio n r(t2 ) = r (t1 ) = 0 a n d h en ce it d o es n o t d ep en d o n x 1 ;x 2 . S o th e su m ov er r (t) in (1 8 ) (w h ich w e h av en 't ev en d e¯ n ed p ro p erly !) ca n o n ly b e a fu n ctio n o f t2 ;t1 . W e g et th e n ea t resu lt th a t fo r a ll a ctio n s w h ich h av e o n ly u p to q u a d ra tic term s in th eir va ria b les, th e p ro p a g a to r h a s th e fo rm in (1 5 ). (In fa ct, o n e ca n a lso d eterm in e N (t2 ;t1 ) u sin g th e co n d itio n (6 ) b u t w e w o n 't g o in to th a t.). F o r a ll th ese ca ses, o f w h ich th e free p a rticle is a sp ecia l ca se, th e p h a se o f th e p ro p a g a to r is ju st th e cla ssica l a ctio n . T h is is a ll n ice b u t in n o n -rela tiv istic m ech a n ics th e a ctio n fu n ctio n a l in (1 3 ) h a s n o sim p le g eo m etrica l in terp reta tio n . W e w ill n ow p rov id e a n a ltern a tiv e p ersp ectiv e o n th e p h a se o f th e p ro p a g a to r w h ich w ill lea d to a g eo m etric in sig h t. T h e m o st rem a rka b le fea tu re a b o u t th e p ro p a g a to r in (1 1 ), fro m th is a ltern a tiv e p ersp ectiv e, is th a t w e h av e a lrea d y seen th is ex p ressio n in th e la st in sta llm en t in co n n ectio n w ith th e p ro p a g a tio n o f electro m a g n etic w a v es a lo n g th e z -d irectio n ! T h ere w e h a d th e ex p ressio n 1 fo r a p ro p a g a to r w h ich is rep ro d u ced h ere fo r y o u r co n v en ien ce: G (z ¡ z 0;x ? ¡ x 0? ) = ³ ! ´ 1 ex p 2 ¼ ic jz ¡ z 0j

" # i! (x ? ¡ x 0? )2 : 2 c (z ¡ z 0)

(1 9 )

C o m p a rin g (1 9 ) w ith (1 1 ) w e see th e fo llow in g co rresp o n d en ce. T h e (z ¡ z 0)= c, w h ich is th e tim e o f lig h t trav el a lo n g th e z -a x is (a lo n g w h ich th e w av e is p ro p a g a tin g ) is a n a lo g o u s to tim e t in q u a n tu m m ech a n ics. T h e tw o tra n sv erse sp a tia l d irectio n s in th e ca se o f electro m a g n etic w av e p ro p a g a tio n a re a n a lo g o u s to th e sp a tia l co o rd in a tes in q u a n tu m m ech a n ics in 2 -d im en sio n s; so w e ca n set D = 2 in (1 1 ). T h e freq u en cy sh o u ld g et m a p p ed to th e rela tio n ~! = m c 2 w h ich is essen tia lly

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th e freq u en cy a sso cia ted w ith th e C o m p to n w av elen g th o f th e p a rticle. T h is w ill m a k e th e p ro p a g a to rs id en tica l! O b v io u sly, th is d eserv es fu rth er p ro b in g esp ecia lly sin ce th e co rresp o n d en ce b rin g s in a c fa cto r w h en w e th o u g h t w e a re d o in g n o n rela tiv istic q u a n tu m m ech a n ics.

Obviously, this

In th e ca se o f th e p ro p a g a tio n o f electro m a g n etic w av e a m p litu d e, w e w ere p ro p a g a tin g it a lo n g th e p o sitiv e z -d irectio n w ith x a n d y a ctin g a s tw o tra n sv erse d irectio n s. In th e ca se o f q u an tu m m ech a n ics, w e a re p ro p a g a tin g th e a m p litu d e fo r a p a rticle a lo n g th e p o sitiv e t-d irectio n w ith a ll th e sp a tia l co o rd in a tes a ctin g a s `tra n sv erse d irectio n s'. In th e la n g u a g e o f p a ra x ia l o p tics, th e sp ecia l a x is is a lo n g th e tim e d irectio n in q u a n tu m m ech a n ics.

brings in a c factor

deserves further probing especially since the correspondence when we thought we are doing nonrelativistic quantum mechanics.

B u t w e k n ow th a t p a ra x ia l op tics is ju st a n a p p rox im a tio n to a m o re ex a ct p ro p a g a tio n in term s o f th e w av e eq u a tio n . In th e w av e eq u a tio n fo r th e electro m a g n etic w av e, th e th ree co o rd in ates (x ;y ;z ) a p p ea r q u ite sy m m etrica lly a n d to o b ta in p a ra x ia l lim it, w e ch o o se o n e a x is (z -a x is) a s sp ecia l a n d p ro p a g a te th e a m p litu d e a lo n g th e p o sitiv e d irectio n . T h is is w h y th e p ro p a g a to r in (1 9 ) h a s th e x ;y co o rd in a tes a p p ea rin g d i® eren tly co m p a red to th e z -a x is. D o in g a b it o f rev erse en g in eerin g w e ca n a sk th e q u estio n : If th e q u a n tu m m ech a n ica l p ro p a g a to r is so m e k in d o f p a ra x ia l o p tics lim it o f a m o re ex a ct th eo ry, w h a t w ill it b e? A n o b v io u s w ay to ex p lo re th e situ a tio n is to resto re th e sy m m etry b etw een z a n d x ;y in o p tics a n d , sim ila rly, resto re th e sy m m etry b etw een t a n d x in q u a n tu m m ech a n ics. W e ca n d o th is if w e reca ll th e in terp reta tio n o f th e p h a se a s d u e to th e p a th d i® eren ce in th e ca se o f electro m a g n etic w av e. T h e releva n t eq u a tio n 2 is a g a in rep ro d u ced b elow : ·q ¸ ! 2 0 0 2 0 k¢ s = (x ? ¡ x ? ) + (z ¡ z ) ¡ (z ¡ z ) c

RESONANCE  January 2009

2 See equation (9) in Resonance, Vol.13, p.1103, December 2008.

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»= ! c

" # 1 (x ? ¡ x 0? )2 : 2 (z ¡ z 0)

(2 0 )

W e u se th e fa ct th a t a p a th d i® eren ce ¢ s b etw een tw o p o in ts in sp a ce w ill in tro d u ce a p h a se d i® eren ce o f k ¢ s in a p ro p a g a tin g w av e. T h e p a ra x ia l o p tics resu lts w h en th e tra n sv erse d isp la cem en ts a re sm a ll co m p a red to th e lo n g itu d in a l d ista n ce. T a k in g a cu e fro m th is, let u s co n stru ct th e q u a n tity o ¤ l(t;x ;0 ;y ) ¡ ct m c n£ 2 2 2 1=2 ´ c t ¡ (x ¡ y ) ¡ ct ; ¸ ~ (2 1 ) w h ere l(t;x ;0 ;y ) is th e sp ecia l rela tiv istic sp a cetim e in terva l b etw een th e tw o ev en ts. W e a re su b tra ctin g fro m it th e `p a ra x ia l d ista n ce' ct a lo n g th e tim e d irectio n a n d d iv id in g b y ¸ ´ (~= m c) w h ich is th e C o m p to n w av elen g th o f th e p a rticle. T h is is ex a ctly th e co n stru ctio n su g g ested b y th e co rresp o n d en ce b etw een (1 9 ) a n d (1 1 ), d iscu ssed p rev io u sly, ex cep t fo r u sin g th e sp ecia l rela tiv istic lin e in terva l, w ith a m in u s sig n b etw een sp a ce a n d tim e. T h e p a ra x ia l lim it n ow a rises a s th e n o n rela tiv istic lim it o f th is ex p ressio n in (2 1 ) w h en c ! 1 ; th is is g iv en b y

We are subtracting from l(t, x; 0, y) the ‘paraxial distance’ ct along the time direction and dividing by

 (~ / mc) which is the Compton wavelength of the particle.

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l ¡ ct » m (x ¡ y )2 ; = ¡ ¸ ~t 2

(2 2 )

w h ich is p recisely th e p h a se o f th e p ro p a g a to r in (1 1 ) ex cep t fo r a sig n . S o th e p ro p a g a to r ca n b e th o u g h t o f a s th e n o n rela tiv istic lim it o f th e fu n ctio n µ · ¸¶ l(t;x ;0 ;y ) i(m c 2 = ~)t K (t;x ;0 ;y ) = N (t)e : ex p ¡ i ¸ (2 3 ) S o th e p h a se o f th e p ro p a g a to r is ju st th e p ro p er d ista n ce b etw een th e tw o ev en ts, in u n its o f th e C o m p to n

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SERIES  ARTICLE

w av elen g th , ju st a s th e p h a se in th e ca se o f th e electro m a g n etic w av e p ro p a g a to r is th e p a th len g th in u n its o f th e w av elen g th . (T h e ex tra fa cto r (m c 2 = ~)t d o es n o t co n trib u te to th e p ro p a g a tio n in teg ra l in (4 ) a n d g o es fo r a rid e; w e ca n ig n o re it b u t if y o u w a n t y o u ca n a lso th in k o f it a s a risin g fro m th e en erg y b ein g sh ifted b y m c 2 .). W e ca n th in k o f th e p a th d i® eren ce b etw een a stra ig h t p a th a lo n g th e tim e d irectio n (w ith x = y ) a n d a n o th er sp eci¯ ed p a th a s co n trib u tin g a p h a se l= ¸ to th e p ro p a g a to r. T h is g eo m etric in terp reta tio n is lo st fo r th e p h a se in th e p a ra x ia l lim it (in th e ca se o f electro m a g n etic th eo ry ) a n d in th e n o n rela tiv istic lim it (in th e ca se o f a p a rticle).

This geometric interpretation is lost for the phase in the paraxial limit (in the case of electromagnetic theory) and in the nonrelativistic limit (in the case of a particle).

T h is ex ten sio n su g g ests th a t th e p h a se in th e rela tiv istic ca se ca n b e rela ted to th e co rresp o n d in g a ctio n . T h e a ctio n fo r a free p a rticle in sp ecia l rela tiv ity is g iv en b y ¶1 = 2 Zt µ v2 2 A R (t;x ;0 ;y ) = ¡ m c : dt 1 ¡ 2 (2 4 ) c 0 O n ce a g a in , ev a lu a tin g th is fo r a rela tiv istic cla ssica l tra jecto ry w e g et · ¸1 = 2 (x ¡ y )2 c 2 A R (t;x ;0 ;y ) = ¡ m c t 1 ¡ c 2 t2 £ ¤1 = 2 ; (2 5 ) = ¡ m c c 2 t2 ¡ (x ¡ y )2 w h ich is essen tia lly th e in terva l b etw een th e tw o ev en ts in th e sp a cetim e. T h is su g g ests ex p ressin g th e p ro p a g a to r fo r th e rela tiv istic free p a rticle in th e fo rm µ c ¶ iA R im c 2 t K (t;x ;0 ;y ) = N (t) ex p : + (2 6 ) ~ ~ T h is resu lt is tru e b u t o n ly in a n a p p rox im a te sen se to th e lea d in g o rd er; th e a ctu a l p ro p a g a to r fo r a p a rticle in rela tiv istic q u a n tu m th eo ry tu rn s o u t to b e m o re co m p lica ted . T h is is b eca u se th e a ctio n in (2 4 ) fo r th e rela tiv istic p a rticle is n o t q u a d ra tic a n d o u r p rev io u s resu lt

RESONANCE  January 2009

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SERIES  ARTICLE

It is the second interpretation which makes contact with optics so clear and is lacking when we do non-relativistic quantum mechanics.

Address for Correspondence T Padmanabhan IUCAA, Post Bag 4 Pune University Campus Ganeshkhind Pune 411 007, India. Email: [email protected] [email protected]

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in (1 8 ) d o es n o t h o ld . B u t, to th e lea d in g o rd er, a ll o f it h a n g s to g eth er v ery n icely. T h e p h a se o f th e p ro p a g a to r is in d eed th e va lu e o f th e cla ssica l a ctio n d iv id ed b y ~ a n d it is a lso g iv en b y th e ra tio o f th e sp a cetim e in terva l b etw een th e ev en ts a n d th e C o m p to n w av elen g th . It is th e seco n d in terp reta tio n w h ich m a k es co n ta ct w ith o p tics so clea r a n d is la ck in g w h en w e d o n o n -rela tiv istic q u a n tu m m ech a n ics. T h ere is a ctu a lly a va lid m a th em a tica l rea so n fo r th is to h a p p en w h ich ca n b e d escrib ed q u a lita tiv ely a s fo llow s: T h e S ch rÄo d in g er eq u a tio n d escrib in g th e n o n -rela tiv istic p a rticle in v o lv es ¯ rst d eriva tive w ith resp ect to tim e b u t seco n d d eriva tiv e w ith resp ect to sp a tia l co o rd in a tes. T h is w o rk s in n o n -rela tiv istic m ech a n ics in w h ich tim e is sp ecia l a n d a b so lu te. In co n tra st, in rela tiv istic th eo ries, w e trea t tim e a n d sp a ce a t a m o re sy m m etric fo o tin g a n d u se a w av e eq u a tio n in w h ich seco n d d eriva tiv es w ith resp ect to tim e a lso a p p ea r. T h e so lu tio n s to su ch a n eq u a tio n w ith a llow p ro p a g a tio n o f a m p litu d es b o th fo rw a rd a n d b a ck w a rd in tim e co o rd in a te ju st a s it a llo w s p ro p a g a tio n b o th fo rw a rd s a n d b a ck w a rd s in sp a tia l co o rd in a tes. W hen on e takes the n on -relativistic lim it of the ¯ eld theory, w e select ou t the m odes w hich on ly propagate forw ard in tim e. T h is is ex a ctly in a n a lo g y w ith p a ra x ia l o p tics w e stu d ied in th e la st in sta llm en t. T h e b a sic eq u a tio n fo r electro m a g n etic w av e w ill a llow p ro p a g a tio n in b o th p o sitiv e z -d irectio n a s w ell a s n eg a tiv e z -d irectio n . B u t, w h en w e co n sid er a sp eci¯ c co n tex t o f p a ra x ia l o p tics (fo r ex a m p le, a b ea m o f lig h t h ittin g a co u p le o f slits in a screen a n d fo rm in g a n in terferen ce p a ttern o r lig h t p ro p a g a tin g th ro u g h a len s a n d g ettin g fo cu sed ), w e select o u t th e m o d es w h ich a re p ro p a g a tin g in th e p o sitiv e z -d irectio n . It is th erefo re n o w o n d er th a t th e p ro p a g a to r in n o n -rela tiv istic q u a n tu m m ech a n ics is m a th em a tica lly id en tica l to th a t in p a ra x ia l o p tics!

RESONANCE  January 2009