THE QUANTUM-MECHANICAL DIVERGENCES OF A SIMPLE FIELD

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U53907 S ,37 Sollfrey, William, 19251950 The quantum-mechanical divergence® oSt> of a simple fields Hew York, 1950. iii,96 typewritten leaves# diagrs'* 29 cm» Thesis {PhoD.) - New York Universit „ Graduate School, 1950, Bibliography: p„93-9°» C57607

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Xerox University Microfilms, Ann Arbor, Michigan 48106

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

I library op

HEW YOU UIYSRSITT OHIV1RSITT HEIGHTS

THE QUANTUM-MECHANICAL DIVERGENCES OP A SIMPLE FIELD

WILLIAM SOLLFREY

S u b m itte d I n

p a r t i a l f u l f i l l m e n t o f th e

re q u ire m e n ts f o r t h e d e g re e o f D o c to r o f P h ilo s o p h y i n t h e G ra d u ate S c h o o l o f A r ts and S c ie n c e

o f New York U n i v e r s i ty

1950

ABSTRACT

I n o r d e r t o s tu d y t h e s o u rc e s o f q u an tu m -m e c h an ica l d iv e r g e n c e s , a n e le m e n ta ry m o d el, c o n s i s t i n g o f a v i b r a t i n g s t r i n g e l a s t i c a l l y c o u p le d t o a harm onic o s c i l l a t o r , i s a n a ly z e d i n d e t a i l .

F o llo w in g a p r e l i m i n a r y d i s c u s s i o n o f

th e c l a s s i c a l e ig e n f u n c tio n s o f b o th t h e u n c o u p le d and c o u p le d s y s te m s , t h e m odel i s q u a n tiz e d by t h e s ta n d a r d m ethods o f b o so n f i e l d t h e o r y .

The o p e r a t o r s i n t h e H a m ilto n ia n a r e so

o rd e r e d t h a t t h e e x p e c ta ti o n v a lu e o f t h e e n e rg y v a n is h e s i n th e s t a t e i n w hich no c o u p le d modes a r e e x c i t e d .

The e n e rg y

o f t h e s t a t e w ith no u n c o u p le d modes e x c i t e d i s fo u n d .

It

i s shown t o hav e a lo w -fre q u e n c y d iv e r g e n c e and t o in v o lv e th e lo g a r ith m o f t h e c o u p lin g c o n s t a n t .

The mean s q u a r e e n e rg y i n

t h i s s t r i n g i s shown t o d iv e r g e l o g a r i t h m i c a l l y a t b o th h ig h and low f r e q u e n c i e s , im p ly in g t h e s t a t e h a s an i n f i n i t e r a t e o f change and t h e r e f o r e c a n n o t be p r e p a r e d .

E x c ite d s t a t e s a r e

t r e a t e d , i n v e s t i g a t i n g t h e i r d e c a y , e m is s io n s p e c t r a , s e l f e n e r g i e s , and d iv e r g e n c e s i n t h e i r b e h a v io r .

M ethods o f c u t t i n g

o f f t h e sy ste m a t b o th h ig h an d low f r e q u e n c ie s by m o d ify in g th e r e q u ire m e n ts on t h e m e a su rin g a p p a r a tu s a r e i n v e s t i g a t e d . The g e n e r a l v ie w p o in t i s ta k e n th ro u g h o u t t h a t some q u a n tu m -m e c h an ica l d iv e r g e n c e s a r e o f m a th e m a tic a l o r i g i n , and may be e lim in a te d by m o d if i c a t io n s o f t h e p e r t u r b a t i o n th e o r y t e c h n iq u e , w h ile o t h e r s a r e p h y s i c a l and a r i s e from p l a c i n g i n p o s s i b l e re q u ire m e n ts on th e a p p a r a tu s u s e d t o m easu re t h e systemfe p r o p e r t i e s .

I f t h e sy ste m i s s e n s i b l e , th e n s e n s i b l e

q u e s t io n s s h o u ld y i e l d s e n s i b l e a n s w e rs .

TABLE OP CONTENTS C h a p te r

Page A b s tr a c t

..............................................................

11

I

I n t r o d u c t i o n ..................................................................................

1

II

The F i n i t e S t r i n g - C l a s s i c a l T h e o r y ............. E q u a tio n s o f M otion - U ncoupled System E ig e n ­ f u n c t i o n s - C oupled System E ig e n f u n c tio n s I n i t i a l V alue P roblem - T r a n s f o r m a tio n betw een C oupled and U ncoupled System V a r i a b l e s .

11

Ill

The F i n i t e S t r i n g - Quantum T h e o r y ............................... F o rm u la tio n - Vacuum S t a t e E n e rg y - F l u c t u a ­ t i o n s o f Vacuum S t a t e E n e rg y .

25

IV

The I n f i n i t e S t r i n g .................................................................. F o rm u la tio n - D ia g o n a liz in g T ra n s f o rm a tio n D is c u s s io n o f F ( a ) “ Q u a n tiz a tio n - Vacuum S t a t e E n erg y - E x c ite d S t a t e s : l ) Decay o f E x c ite d S t a t e ; 2) E m iss io n S p e c tru m ; 3) E n erg y o f E x c ite d S t a t e .

4l

C u to f f C o n s i d e r a t i o n s ................. H ig h -F re q u e n c y C u to f f - L ow -Frequency C u to f f .

71

R e s u l ts and C o n c lu s io n s . • • • • ................

8l

A ppendix A .........................................................

90

R e fe re n c e s and B i b l i o g r a p h y

93

V VI

ill

...............................

CHAPTER I INTRODUCTION

The f a c t t h a t c e r t a i n q u a n t i t i e s d iv e r g e when c a l ­ c u l a t e d q u a n tu m -m e c h a n ic a lly h a s been known f o r many y e a rs* The b e s t known exam ple i s t h e s e l f - e n e r g y o f th e e l e c t r o n due t o i t s i n t e r a c t i o n w ith i t s own e le c tr o m a g n e tic f i e l d * C a l c u l a t i o n s , 1* u s in g t h e D ira c o n e - e l e c t r o n t h e o r y , l e d t o t h e r e s u l t t h a t t o t h e se c o n d a p p ro x im a tio n i n t h e e x p a n sio n o f th e e n e rg y i n pow ers o f th e e l e c t r o n c h a rg e t h e e n e rg y i s in fin ite *

I n te rm s o f e m is s io n and a b s o r p t i o n o f a v i r t u a l

quantum o f e n e rg y h k , t h e e n e rg y i s o f o r d e r k , and s in c e q u a n ta o f a l l e n e r g i e s may be e m itte d and th e n r e a b s o rb e d by v i r t u a l p r o c e s s e s , t h e t o t a l e n e rg y i s o f o r d e r K2 , w here K o i s t h e maximum quantum energy* L a t e r c a l c u l a t i o n s , u s in g h o le t h e o r y , showed t h a t t h i s d iv e rg e n c e i s o n ly o f o r d e r lo g K, an d t h a t t h i s ty p e o f d iv e rg e n c e p e r s i s t s t o a l l o r d e r s i n t h e e x p a n s io n i n pow ers o f e . S e v e r a l a tte m p ts w ere made t o a v o id t h i s d i f f i c u l t y * Among t h e s e w ere: 1) M o d if ic a tio n o f t h e e q u a tio n s o f m o tio n i n t h a t th e y become n o n - l i n e a r a t s m a ll d i s t a n c e s from t h e p o i n t c h a r g e ;^ 2 ) The i n t r o d u c t i o n o f h ig h e r d e r i v a t i v e s th a n k t h e f i r s t i n t o t h e L a g ra n g ia n o f th e s y s te m ;^ ♦ A ll r e f e r e n c e s a r e t o be fo u n d i n t h e B ib lio g ra p h y *

2

3) I n t r o d u c t i o n o f a d v an c ed a s w e ll a s r e ­ t a r d e d p o t e n t i a l s t o change t h e f i e l d a c t i n g on an e le c tro n ;5

4) D evelopm ent o f t h e quantum th e o r y o f r a d i a t i o n d am p in g .^ I n a d d i t i o n t o t h e s e new p h y s i c a l t h e o r i e s , some new m a th e m a tic a l d e v e lo p m e n ts w ith r e s p e c t t o p e r t u r b a t i o n th e o r y were worked o u t .

T hese I n c lu d e d

1) The h i g h e r a p p ro x im a tio n s to t r a n s i t i o n 7 p ro b a b ilitie s ; 2) A re g ro u p in g p r o c e d u r e d e s ig n e d t o p u t th e g

d iv e r g e n c e s I n t o h a rm le s s p o s i t i o n s ;

3) A p p l i c a ti o n o f t h e m ethods o f a n a l y t i c c o n tin u a tio n ;^

4) Use o f t h e s o - c a l l e d

X - l i m i t i n g p r o c e s s . 10

T hese m a th e m a tic a l an d p h y s i c a l m odels w ere a l l d e v e lo p e d t o e l i m i n a t e t h e d iv e r g e n c e s a t th e h ig h - f re q u e n c y end o f t h e sp e c tru m . t h e lo w -fre q u e n c y e n d .

I n a d d i t i o n , t h e r e were d i f f i c u l t i e s a t The m a jo r e f f e c t s w ere lo w -fre q u e n c y

d iv e r g e n c e s I n t h e t h e o r y o f e l e c t r o n s c a t t e r i n g and Compton e ffe c t.

T h ese w ere t r e a t e d by c o n s i d e r i n g t h e e l e c t r o n a s

a lw a y s b e in g a s s o c i a t e d w ith a n a sse m b la g e o f lo w -fre q u e n c y p h o t o n s . 11 Most i n v e s t i g a t o r s b e li e v e d t h a t t h e s e d iv e rg e n c e s had a d e e p - s e a t e d p h y s i c a l m ea n in g , a n d w ere d i r e c t l y in v o lv e d i n t h e a tte m p t t o u n i f y r e l a t i v i t y a n d quantum m e c h a n ic s . H ow ever, t h e r e were sa n e d o u b t e r s , who s u g g e s te d t h a t some o f

3 t h e d i f f i c u l t i e s a r o s e from t h e m a th e m a tic s u se d t o s o lv e 12 1^ t h e e q u a tio n s * A le n g th y p a p e r a p p e a re d show ing t h a t t h e e q u a tio n s o f p e r t u r b a t i o n th e o r y n eed n o t p o s s e s s s o l u t i o n s w hich a r e a n a l y t i c t o t h e I n t e r a c t i o n c o n s t a n t s , w hich I s th e fu n d a m e n ta l a s s u n p t i o n o f p e r t u r b a t i o n th e o ry * 14 The n o w - c la s s lc e x p e rim e n t on t h e d e v i a t i o n o f t h e sp e c tru m o f h y d ro g en from t h a t p r e d i c t e d by t h e D ira c th e o r y c a u se d t h e r e c o n s i d e r a t i o n o f a l l t h e s e e f f e c t s *

F irs t

I t was shown t h a t t h e Lamb s h i f t can be e x p la in e d by t h e I n t e r ­ a c t i o n o f t h e e l e c t r o n w ith t h e z e r o p o i n t f l u c t u a t i o n s o f t h e e le c tr o m a g n e tic f i e l d , p ro v id e d t h a t t h e e n e rg y o f t h e e l e c t r o n I n a f r e e s t a t e i s s u b t r a c t e d from t h a t o f t h e e l e c t r o n 1*5 i n t h e bound s t a t e * J T h is was r e c o g n iz e d a s b e in g e q u iv a le n t t o a r e n o r m a li z a t io n o f t h e m ass o f t h e e l e c t r o n . 1**

The

r e l a t l v l s t l c c o rre c tio n s to t h i s b a s ic a lly n o n - r e la tlv ls tlc 17 t h e o r y w ere d e v e lo p e d . The s p e c i f i c e f f e c t s o f t h e z e r o 18 p o i n t f l u c t u a t i o n s w ere c o n s id e re d * The th e o r y was th e n p u t I n t o a c o m p le te ly r e l a t l v l s t l c 19 20 s e ttin g * P a p e rs d e v e lo p in g t h e m ethods o f c a l c u l a t i o n and 21 a l s o a p p ly in g them t o m eson f i e l d s th e n a p p e a re d I n p ro fu s io n * A g e n e r a l m ethod o f " r e g u l a t o r s 1' 22 was in tr o d u c e d w hich i s p r e s e n t l y c o n s id e r e d c a p a b le o f e l i m i n a t i n g a l l d i f f i c u l t i e s * A l l t h e s e r e c e n t t h e o r i e s hav e t h e f o llo w in g p r o p e r ­ ty * th e y t a k e t h e d iv e r g e n c e a n d e x h i b i t i t , th e n show i t may be e li m in a te d by s u i t a b l e r e n o r m a li z a t io n s o f m ass and c h a rg e , a n d , i n t h e c a s e o f meson f i e l d s , t h e m a g n e tic movements o f t h e n u c le o n s .2 ^

B ut t h i s p r o c e d u r e does n o t e l i m i n a t e t h e

4

d iv e r g e n c e , i t m e re ly c o n c e a ls i t i n " u n o b s e rv a b le f a c t o r s * " T h ere have b een some a tte m p ts t o d e v e lo p t h e o r i e s w hich a r e c o m p le te ly f r e e from d iv e rg e n c e s *

None o f t h e s e a p p e a rs

s a t i s f a c t o r y t o i t s a u th o r* I n v iew o f a l l t h e d i f f i c u l t i e s t h a t have b e s e t quantum e le c tr o d y n a m ic s , i t a p p e a re d w o rth w h ile t o c o n s id e r a q u an tu m -m e c h an ica l sy ste m w hich w ould p o s s e s s t h e same d iv e r g e n c e s , b u t whose s o l u t i o n s c o u ld be fo u n d e x p l i c i t l y * F o r t h i s p u rp o s e t h e f o llo w in g sy ste m was c h o se n : a v i b r a t i n g s t r i n g e l a s t i c a l l y c o u p le d t o a harm onic o s c i l l a t o r * W hile t h e c l a s s i c a l th e o r y o f t h e v i b r a t i n g s t r i n g i s u n i v e r s a l l y known, v e ry l i t t l e h a s b e en done w ith t h e q u a n tiz e d s t r i n g *

T h e re e x i s t two s h o r t n o t e s , 2-* i n th e

f i r s t o f w hich t h e th e o r y o f t h e f r e e s t r i n g i s I n d i c a t e d , and i n t h e se co n d o f w hich a p o i n t p a r t i c l e i s r i g i d l y a t t a c h e d t o t h e s t r i n g and t h e t h e o r y o f t h a t sy ste m p r e s e n te d *

In

t h e l a t t e r n o te i t i s i n d i c a t e d t h a t The mean z e r o p o i n t k i n e t i c e n e rg y o f t h e p a r t i c l e t u r n s o u t t o be i n f i n i t e . H ow ever, i t i s i n t e r e s t i n g t o n o te t h a t , w h ile a f i r s t o r d e r p e r t u r b a t i o n c a l c u l a ­ t i o n le a d s t o a q u a d r a t ic d iv e rg e n c e o f t h e z e r o - p o i n t e n e r g y , t h e e x a c t c a l c u l a t i o n g iv e s o n ly a lo g a r ith m ic d iv e rg e n c e * I n a d d i t i o n t o t h e s e n o t e s , i t h a s b een b ro u g h t t o t h e a u t h o r 's a t t e n t i o n t h a t t h e s t r i n g h a s b een c o n s id e r e d i n 26 u n p u b lis h e d work* The p ro b le m o f t h e o n e -d im e n s io n a l v i b r a t i n g s t r i n g e l a s t i c a l l y c o u p le d t o a h arm o n ic o s c i l l a t o r w i l l occupy o u r a t t e n t i o n th ro u g h o u t t h e body o f t h i s p a p e r*

The t h e o r y w i l l

b e d e v e lo p e d f o r t h e p u rp o s e o f s h e d d in g a s much l i g h t a s p o s s i b l e o n to t h e d iv e rg e n c e s *

What I s p r e s e n t e d h e r e I s

o n ly a b e g in n in g , and t h e r a m i f i c a t i o n s a r e b e in g e x p lo re d * We a s s e r t , i f one a s k s a s e n s i b l e q u e s t i o n a b o u t a s e n s i b l e s y s te m , one m ust o b t a i n a s e n s i b l e answ er*

I f e ith e r

t h e q u e s t io n o r t h e sy ste m i s i n a d m i s s i b l e , t h e a n sw e r c a n n o t b e e x p e c te d t o be re a s o n a b le *

C o n s e q u e n tly , i f one o b ta in s

d iv e r g e n c e s , t h e s e m ust be due t o l ) a s k in g a p h y s i c a l q u e s ­ t i o n w hich In v o lv e s m aking n o n -p e rfo rm a b le e x p e r im e n ts , o r 2) a s k in g a r e a s o n a b le q u e s t io n an d t h e n u s in g I n c o r r e c t m a th e ­ m a t i c a l t e c h n iq u e , o r 3) t r e a t i n g a p h y s i c a l sy ste m w hich i s n o n s e n s i c a l , su c h a s t h e one t r e a t e d i n A ppendix A* a p p ly t h i s c r i t e r i o n th ro u g h o u t t h e work*

We s h a l l

In th e s t r i n g p ro b ­

lem h e r e i n , t h e c l a s s i c a l d e s c r i p t i o n o f t h e sy ste m i s su c h a s t o e n a b le t h e c o n s t r u c t i o n o f a s e n s i b l e p h y s i c a l s i t u a t i o n th u s e l i m i n a t i n g one o f t h e t h r e e m e n tio n e d s o u r c e s o f d i v e r ­ g e n c ie s * The t h e o r y b e g in s w ith t h e L a g ra n g la n o f t h e sy s te m , u s in g a s v a r i a b l e s t h e s t r i n g d is p la c e m e n t and t h e o s c i l l a t o r d is p la c e m e n t*

We f i r s t d e v e lo p t h e c l a s s i c a l t h e o r y f o r a

s t r i n g o f f i n i t e le n g th fix e d a t i t s e n d s.

The e q u a tio n s o f

m o tio n a r e o b ta in e d and t h e H a m ilto n ia n i s s e t up*

We th e n

o b t a i n t h e n orm al modes o f b o th t h e u n c o u p le d an d t h e c o u p le d s y s te m s , and a n a ly z e t h e i r p r o p e r t i e s *

I t i s i n d i c a t e d how

t h e n orm al modes o f t h e c o u p le d sy ste m may be o b ta in e d by a t r a n s f o r m a t i o n w hich d l a g o n a ll z e s t h e H a m ilto n ia n * i c a l s o l u t i o n o f t h e i n i t i a l v a lu e p ro b lem i s giv en *

The c l a s s ­

6

The sy ste m i s th e n q u a n tiz e d I n a c c o rd a n c e w ith t h e 27 s t a n d a r d m ethods o f th e quantum t h e o r y o f boson f i e l d s * 1 The H a m ilto n ia n I s d e s c r ib e d I n te rm s o f t h e tim e - v a r y in g a m p li­ tu d e s o f t h e norm al m odes, w ith a l l v a r i a b l e s e x p re s s e d I n th e H e is e n b e rg r e p r e s e n t a t i o n * I n c a r r y i n g o u t t h e t r a n s i t i o n t o t h e quantum t h e o r y , we may o r d e r t h e v a r i a b l e s I n an y d e s i r e d m anner w ith o u t d i s t u r b i n g t h e c o rre s p o n d e n c e w ith t h e c l a s s i c a l th e o ry * s o o r d e r t h e o p e r a t o r s t h a t I n th e lo w e s t e n e rg y s t a t e

We

(in

w hich no modes o f t h e c o u p le d sy ste m a r e e x c i t e d ) t h e e n e rg y I s z e ro *

T h is d e f i n e s t h e vacuum a s t h a t s t a t e I n w hich th e

e n e rg y I s a n a b s o l u te minimum*

28

T h is vacuum s t a t e I s n o t t h e

same a s t h e s t a t e I n w hich no modes o f t h e uncotqpled sy ste m a r e e x c i t e d , an d w h ile th e y a r e t h e same when th e c o u p lin g re d u c e s t o z e r o , we w i l l show l a t e r t h a t t h e a p p ro a c h t o e q u a l i t y I s n o t a n a l y t i c , so I t I s n o t p o s s i b l e t o o b t a i n one s t a t e by a pow er s e r i e s I n t h e c o u p lin g w ith o p e r a t o r c o e f f i ­ c i e n t s a p p li e d t o t h e o t h e r s t a t e . We t h e n I n v e s t i g a t e t h e u n c o u p le d sy s te m vacuum s t a t e v e c to r*

We show t h a t I t h a s a f i n i t e e n e rg y , composed

o f t h r e e te r m s , a s e l f - e n e r g y o f t h e s t r i n g due t o t h e a t t a c h ­ in g o f t h e co tq p lin g and p r o p o r t i o n a l t o t h e l e n g t h o f t h e s t r i n g , a s e l f - e n e r g y o f t h e o s c i l l a t o r due t o th e change i n i t s fu n d a m e n ta l f r e q u e n c y , an d a c o u p lin g e n e rg y p r o p o r t i o n a l t o t h e lo g a r ith m o f t h e s t r i n g le n g th *

T h is r e s u l t I s e s t a b ­

l i s h e d u n d e r t h e a ss u m p tio n t h a t t h e c o u p lin g fre q u e n c y I s s m a ll com pared t o t h e s m a l l e s t f re q u e n c y t h a t can b e t r a n s ­

7 m it t e d b y t h e s t r i n g .

I f t h e c o u p lin g I s s t r o n g e r , t h e s t r i n g

s e lf-* e n e rg y i s p r o p o r t i o n a l t o t h e l o g a r ith m o f t h e s t r i n g le n g th *

The c o u p lin g e n e rg y te n d s t o a c o n s t a n t v a lu e a s f a r

a s t h e s t r i n g l e n g t h i s c o n c e rn e d f o r l a r g e c o u p lin g , b u t t h e v a lu e d ep en d s on th e lo g a r ith m o f t h e c o u p lin g c o n s ta n t*

Con­

s e q u e n t l y , I f we had c a l c u l a t e d I t by p e r t u r b a t i o n t h e o r y , we would h av e o b ta in e d a n I n f i n i t e r e s u l t *

T h is i n f i n i t y i s due

t o t h e a tte m p t t o expand a n o n - a n a l y ti c f u n c t i o n by p e r t u r b a ­ t i o n t h e o r y , and I s a co n seq u e n ce o f t h e m a th e m a tic s u s e d , r a t h e r th a n o f t h e p h y s ic s o f t h e system * The p o s s i b i l i t y o f t h e a p p e a ra n c e o f t h e lo g a r ith m 29 o f t h e c o u p lin g c o n s ta n t h a s b een I n d i c a t e d b e f o r e , but not i n c o n n e c tio n w ith a p h y s i c a l problem *

E f f e c t s in v o lv in g t h e

lo g a r ith m o f t h e c o u p lin g have a p p e a re d i n t h e th e o r y o f h y p e r 30 fin e s tr u c tu r e , b u t t h e r e th e y a r e e s t im a t e s o f t h e e f f e c t o f c u t t i n g o f f t h e Coulomb p o t e n t i a l a t s h o r t d i s t a n c e s from a n u c le u s .

T h is n o n - a n a l y t l c l t y o f t h e e n e rg y r a i s e s t h e

p o s s i b i l i t y t h a t a s i m i l a r e f f e c t o c c u rs i n quantum e l e c t r o ­ d y n a m ic s, b u t t h a t i t i s m asked by t h e i m p o s s i b i l i t y o f p e r ­ fo rm in g c a l c u l a t i o n s by means o t h e r th a n p e r t u r b a t i o n th e o ry * T h is vacuum e n e rg y i s f i n i t e b e c a u s e o f t h e r e o r d e r ­ in g o f t h e c o m p le te H a m ilto n ia n *

I f o n ly t h e u n c o u p le d p a r t o f

th e H a m ilto n ia n i s r e o r d e r e d , a s i s c u sto m a ry i n quantum e le c tr o d y n a m ic s , t h e e n e rg y o f t h i s s t a t e i s l o g a r i t h m i c a l l y d iv e r g e n t a t h ig h f r e q u e n c i e s .

T h is s u g g e s ts t h a t t h e i n f i n i t e

e n e rg y c o n te n t o f t h e vacuum i n quantum e le c tr o d y n a m ic s i s p a r t l y a s p u r io u s i n f i n i t y , o f m a th e m a tic a l r a t h e r th a n p h y s i c a l c h a ra c te r.

8 A lth o u g h t h e e n e rg y o f t h i s s t a t e i s f i n i t e , we n e x t show t h a t t h e e x p e c t a t i o n v a lu e o f t h e s q u a re o f t h e e n e rg y I s I n f i n i t e , d iv e r g i n g a t h ig h f r e q u e n c ie s l i k e t h e s q u a r e o f a lo g a r ith m .

T h is means t h a t I n S c h rB d in g e r r e p r e ­

s e n t a t i o n t h e r a t e o f change o f t h e s t a t e v e c t o r w i l l be In fin ite .

C o n s e q u e n tly , t o p r e p a r e t h i s s t a t e v e c t o r r e q u i r e s

an I n f i n i t e am ount o f e n e rg y t o be I n s t a n ta n e o u s ly a v a i l a b l e , o r , a l t e r n a t e l y , t h a t t h e m e a su rin g a p p a r a tu s hav e I n f i n i t e b an d w id th .

T h is m eans t h a t we a r e now a s k in g a n I n c o r r e c t

p h y s i c a l q u e s t i o n , f o r we a r e r e q u i r i n g o u r m e a su rin g a p p a r a tu s t o have n o n - r e a l l z a b l e p r o p e r t i e s . W hile t h e vacuum sy ste m may be h a n d le d f o r t h e f i n i t e s t r i n g , t o t r e a t th e s t a t e I n w hich t h e sy ste m I s e x c i t e d i t i s much s im p le r t o p e rfo rm t h e c a l c u l a t i o n s f o r a s t r i n g o f I n ­ f i n i t e le n g th .

H ere t h e r e I s no sim p le d e s c r i p t i o n i n te rm s

o f modes o f th e c o u p le d s y s te m , b u t we work w ith t h e t r a n s ­ f o r m a tio n w hich d l a g o n a ll z e s t h e H a m ilto n ia n , t h u s o b t a i n i n g th e n e c e s s a r y t o o l s . We r e c o n s i d e r t h e vacuum s t a t e , and show t h a t t h i s t im e , e v en when t h e r e a r e no u n c o i l e d modes e x c i t e d , t h e e n e rg y I s I n f i n i t e , h a v in g a lo g a r ith m ic d iv e rg e n c e a t low f r e ­ q u e n c ie s .

T h is i s i n d i c a t e d by t h e work on t h e f i n i t e s t r i n g .

T h is d iv e rg e n c e has t h e p h y s i c a l i n t e r p r e t a t i o n t h a t we a r e a tt e m p t in g t o p e rfo rm m easurem ents o v e r a n i n f i n i t e r e g io n o f s p a c e , an d c o n s e q u e n tly low f r e q u e n c i e s , w hich c o rre s p o n d t o b o d i ly d i s p l a c i n g t h e s t r i n g , w i l l g iv e an i n f i n i t e c o n t r i ­ b u t io n when t h e s t r i n g I s p a r t l y bound.

9 We n e x t c o n s i d e r t h e s t a t e s I n w hich th e o s c i l l a t o r I s e x c i t e d t o I t s f i r s t quantum s t a t e *

We I n v e s t i g a t e how

t h i s s t a t e w i l l b eh av e I n t im e , and show I t w i l l d e c a y e x p o n e n t i a l l y , w ith a d e c a y c o n s t a n t w hich f o r s m a ll c o u p lin g c o n s ta n t i s I d e n t i c a l w ith t h a t g iv e n by s ta n d a r d p e r t u r b a t i o n th e o ry *

We a l s o f i n d t h e e m is s io n s p e c tru m , t h a t i s , th e

p r o b a b i l i t y t h a t some mode o f t h e s t r i n g w i l l be e x c i t e d , and show t h a t t h e e x a c t fo rm u la g iv e s t h e s t a n d a r d l i n e s h a p e -' when t h e c o u p lin g c o n s ta n t I s sm all* We n e x t c o n s i d e r t h e e n e rg y o f t h e e x c i t e d s t a t e * We c a l c u l a t e t h i s by s ta n d a r d se c o n d o r d e r p e r t u r b a t i o n t h e o r y , an d show t h a t t h e r e s u l t , w h ile f i n i t e , i s t h e d i f f e r e n c e b etw een tw o I n f i n i t e q u a n t i t i e s *

C o n s e q u e n tly , we m ust be

c a r e f u l a b o u t th e I n t e r p r e t a t i o n o f t h e f i n i t e v a lu e and c a n n o t t r u s t th e re s u lt* N e x t, we c a l c u l a t e t h e e f f e c t o f c u t t i n g o f f t h e sy ste m a t h ig h and low f r e q u e n c i e s .

The h ig h - f re q u e n c y c u t o f f

I s a c h ie v e d by p l a c i n g r e s t r i c t i o n s on t h e m e a su rin g a p p a r a tu s s o t h a t I n s t e a d o f i t s m e a su rin g t h e s t a t e o f t h e e y ste m a t a g iv e n t im e , t h e m easurem ent, t a k e s a f i n i t e tim e , and c o n s e ­ q u e n tly I t d e te rm in e s t h e a v e ra g e o f t h e s t a t e o v e r a tim e i n ­ t e r v a l w ith a s u i t a b l e w e ig h tin g fu n c tio n *

We o b ta in a lo w -

f r e q u e n c y c u t o f f by m e a su rin g o n ly q u a n t i t i e s c o n ta in e d i n a f i n i t e r e g io n o f s p a c e , w hich i s a l s o r e q u i r e d by c o n s i d e r a t i o n s of re la tiv ity .

The r e l a t l v l s t l c in v a r i a n c e o f t h e s e c u t o f f s

an d t h e i r e f f e c t s i n e l i m i n a t i n g d iv e r g e n c e s a r e c o n s id e r e d . We f i n a l l y p r e s e n t a summary o f r e s u l t s and c o n c lu ­ s io n s .

The v a r i o u s f i n i t e and I n f i n i t e r e s u l t s a r e g iv e n , e a c h

10

w ith I t s p h y s i c a l i n t e r p r e t a t i o n . The b a s i c i d e a s h e r e i n a r e t h a t c e r t a i n I n f i n i t i e s a r e o f p h y s i c a l o r i g i n , and t h a t th e y may be t r e a t e d by c o n ­ s i d e r i n g t h e b e h a v io r o f th e m e a su rin g a p p a r a t u s , and show ing t h a t th e l a t t e r i s r e q u i r e d t o have n o n - r e a l i z a b l e p r o p e r t i e s i n o r d e r t o a s k t h e q u e s t io n w hich l e a d s t o an i n f i n i t e a n sw e r; w h ile o t h e r i n f i n i t i e s a r e o f m a th e m a tic a l o r i g i n , and may be t r e a t e d by im provem ents i n t h e m a th e m a tic a l te c h n iq u e .

We do

n o t s e e k t o change e q u a tio n s o f m o tio n , a s i s som etim es done i n t r e a t i n g quantum e le c tr o d y n a m ic s , b u t I n s t e a d we r e s t r i c t t h e num ber o f q u e s tio n s we c an a s k t h e sy ste m t o a n sw e r.

The

m a th e m a tic a l m ethods a r e d e s ig n e d t o t r e a t th e a d m is s ib le q u e s t io n s w ith o u t assu m in g t h a t t h e c u sto m a ry p e r t u r b a t i o n t h e o r y m ethods a r e v a l i d , and h en ce g iv e s f i n i t e a n sw e rs where i n f i n i t i e s would a p p e a r a s a co n seq u en ce o f t h e p e r t u r b a t i o n th e o ry . The work p r e s e n t e d h e re i s r e g a r d e d a s m e re ly a s ta rt.

H ow ever, we do r e g a r d i t a s s u g g e s tin g a p o s s i b l e

m ethod o f t r e a t i n g d iv e rg e n c e s a r i s i n g a s a c o n seq u en ce o f a s k in g u n a s k a b le q u e s t i o n s .

F u tu r e work w i l l a tte m p t t o d e v e l ­

op t h e s e c o n s i d e r a t i o n s f u r t h e r .

The a u th o r i s d e e p ly in d e b te d t o P r o f . G e ra ld G o e r tz e l, who s u g g e s te d th e p ro b le m and p ro v id e d I n e s tim a b le h e lp i n th e c o u rs e o f i t s d e v e lo p m e n t.

I a l s o m ust b e l a t e d l y i n d i c a t e my

g r a t i t u d e t o t h e l a t e P r o f . I . S. Lowen, who in tr o d u c e d me t o re s e a rc h i n th e o r e tic a l p h y s ic s .

CHAPTER I I THE FINITE STRING - CLASSICAL THEORY The n o t a t i o n u se d i n t h i s s e c t i o n i s g iv e n i n T a b le I (P . 1 2 ) . The sy stem i n q u e s t io n c o n s i s t s o f a s t r i n g o f l e n g t h 2L, f i x e d a t i t s e n d s , w ith a h arm o n ic o s c i l l a t o r e l a s t i c a l l y c o u p le d t o i t a t th e c e n t e r .

P ic to ria lly :

The L a g ra n g ia n o f t h i s sy ste m i s g iv e n b y : 2 .1

^o

*L

I*

- i '- v * -

The e q u a tio n s o f m o tio n a r e g iv e n by r e q u i r i n g t h a t I- be s ta tio n a r y f o r a r b itr a r y v a ria tio n s o f y

and y , s a t i s f y i n g

t h e c o n d it i o n t h a t t h e v a r i a t i o n s v a n is h a t t h e e n d p o in ts o f th e in te r v a l of in te g r a tio n .^ 2

The momenta (m ore a c c u r a t e l y ,

xf i s a momentum d e n s i t y ) a r e fo u n d by fo rm in g p a r t i a l d e r i v a ­ t i v e s o f

t h e d e n s i t y o f t h e s t r i n g - assum ed uniform *

x

t h e t e n s i o n i n t h e s t r i n g - assum ed uniform *

2L

th e le n g th o f th e s trin g *

m

t h e m ass o f t h e o s c i l l a t o r *

k

t h e b in d in g c o n s t a n t o f t h e o s c i l l a t o r *

X °

t h e fu n d a m e n ta l fre q u e n c y o f t h e o s c i l l a t o r when i t i s n o t c o u p le d t o t h e s t r i n g *

e

t h e c o u p lin g c o n s t a n t o f t h e system *

x'

t h e fu n d a m e n ta l f re q u e n c y o f t h e o s c i l l a t o r when i t I s c o u p le d t o t h e s t r i n g *

t,

t h e L a g ra n g ia n o f t h e system *

H

t h e H a m ilto n ia n o f t h e system *

^n

th e fre q u e n c y o f t h e n ' t h mode o f t h e u n c o u p le d system *

j/ V

. 5 'rX~

t h e n o rm a liz e d e ig e n f u n c t i o n s b e lo n g in g fr e q u e n c y •

to

th e

a A , bA

t h e a m p litu d e o f t h e n ' t h u n c o u p le d mode com ponent o f f , 7r , r e s p e c t iv e ly *

a0 , bc

t h e a m p litu d e o f t h e fu n d a m e n ta l u n c o u p le d mode o f q , p r e s p e c t iv e ly *

^n 'V U) -

I

•

We c o n tin u e t o c a l l t h e l e n g t h o f t h e s t r i n g 2L

when I t I s m easu red I n t h e s e u n i t s . 2 .3

1H =

The H a m ilto n ia n I s nows e (.-Krt-v)1

We may now s e p a r a t e t h e v i b r a t i o n s o f th e s t r i n g I n t o tw o t y p e s , th o s e I n w hich f I s a n e v en f u n c t i o n o f x and th o s e I n w hich I t I s odd*

F o r t h e l a t t e r , y (0 ) I s z e r o ,

s o th e y a r e n o t c o u p le d t o t h e o s c i l l a t o r * I n i t i a l c o n d it i o n s so t h a t V and y

I f we choose th e

a r e I n i t i a l l y e v en f u n c tio n s

o f x , no odd modes w i l l e v e r be e x c ite d *

A c c o r d in g ly , we

s h a l l n o t c o n s i d e r t h e odd modes f u r t h e r , and s h a l l r e s t r i c t o u r s e lv e s t o y

b e in g a n ev en f u n c t i o n o f x*

The e q u a tio n s o f m o tio n may now b e fo u n d , e i t h e r I n H a m ilto n ia n o r L a g ra n g la n form*

The H a m ilto n ia n form I s fo u n d

by t h e p r i n c i p l e o f l e a s t a c t i o n , a n d t h e L a g ra n g la n form by f in d in g th e v a r ia tio n o f X

w ith r e s p e c t t o f

an d q*

In

H a m ilto n ia n form : S' ~ Tr

2*4

i *• p/'**-

tr * ai y - e^x)Ct(o)-v) p -

V +«

The L a g ra n g la n form I s e q u iv a le n t t o e l i m i n a t i n g y a n d p from t h e s e , and 1 s t 2*5 a )

y t efl>0 0K o)-v )

b) We s h a l l f i r s t d i s c u s s t h e c a s e e = 0 , c o rr e s p o n d in g t o t h e u n c o u p le d system *

I n t h a t c a s e 2*5a and 2*5b a r e In d e p e n d e n t*

The s o l u t i o n o f 2 . 5b I s e v i d e n t l y : tX .t

- i \ at

14 We may f i n d t h e elgenm odes o f 2*5a f o r e ■ 0 by assu m in g a s o l u t i o n o f t h e form t x ( x ) e ^ t * 2 .7 a )

S u b s t i t u t i n g g iv e s *

y ^ x (x ) - 0

b)

txCO

o

A s o l u t i o n o f 2 . 7a w hich I s e v en I n x I s c l e a r l y s a t i s f y t h e b o u n d a ry

c o n d it i o n s 2 . 7b g iv e s

e lg e n f r e q u e n c ie s a r e

g iv e n by A^=

cosAx*

To

cosAL = 0 , so th e '*-s 1* 2 , 3* • • •

The n e g a tiv e s o f t h e s e f r e q u e n c ie s a r e a l s o e lg e n f r e q u e n c ie s * b u t we w i l l show l a t e r t h a t t h e l a t t e r s e t may be e lim in a te d * We s h a l l n o rm a liz e th e s o l u t i o n s

In such a

way t h a t 2 .8 H ere

J Lt x C x ) t : w i x = 4 * ^ -L

&

V

A*v

I s t h e K ro n e c k e r sym bol;

(.x - *£•)

I- - - - - - - - - -

x+y)

F irs t,

32

e n a b le s t h e se co n d te rm I n t h e sum m ation t o be w r i t t e n a s : 3 ,2 9

~

The f i r s t sum h e re I s one t h a t we h av e e v a lu a te d p r e v i o u s l y . We have from 2 .4 8 t h e v a lu e : 5 “3Q 0 0

IT XqL 3L X . L / r

~ v-x.^

_ _

Xoi-

T h is c a n c e ls th e l a s t te rm o f 3*27, l e a v in g 3*3i

0,

+

^5 } + n>>.

The f i r s t te rm I n t h e sum I s c l e a r l y o f o r d e r L .

We

s h a l l show t h a t t h e se c o n d s e r i e s I s o f o r d e r lo g L f o r L la rg e .

The f i r s t s e r i e s may be Im m e d ia te ly e v a l u a t e d I n te rm s

o f t h e Rlemann Z e ta f u n c t i o n . ^5 3 .3 2

in d e e d we have d i r e c t l y

L cJ& 'T W '* '"* To e v a l u a t e t h e se c o n d s e r i e s , we u se t h e d i r e c t

re la tio n : I

°°

^

I

**

J

3 .3 3 -^+ * * 0 ZuJZ) which may be p ro v e n by s e p a r a t i n g t h e even and odd te rm s on t h e r i g h t , and th e n we use t h e fo rm u la : 3*34

t

H ere t ( t )

I s th e l o g a r i th m i c d e r i v a t i v e o f t h e f a c t o r i a l

f u n c t i o n ^ and C I s E u l e r 's c o n s t a n t .

U sin g t h i s g iv e s f o r

t h e s e r i e s 3»33* i

.

c + x + ( x . O - t(x)

Z

3 .3 5

'NH ence, we f i n a l l y h av e forx (A)

+

-i ^XlJ-X

x »]

We s h a l l in tr o d u c e t h e eigenm odes o f t h e sy ste m by means o f t h e i n t e g r a l e q u iv a le n t o f t h e t r a n s f o r m a ti o n 2 .3 8 . T h is i s : A ( a ) T j°°s(A; X) ol( x)^X + ff'CAV 0 o

We n o te t h a t now we a r e w r i t i n g th e i n d i c e s o f t h e tr a n s f o r m a t i o n a s a rg u m e n ts r a t h e r th a n s u b s c r i p t s .

The c o n d it i o n s on

42 th e tra n s fo rm a tio n fu n c tio n a re t h a t i t r e p r e s e n ts an o r th o ­ g o n a l t r a n s f o r m a t i o n and r e n d e r s t h e H a m ilto n ia n d ia g o n a l*

The

o r t h o g o n a l it y c o n d it i o n s s t a t e : 4.5

J“ 5(/|;X)Sto,X'UA -o

rf-/\ - 0 O 5M;)0 ^ a^(A~) A-A ~ I o H ere we a r e u s in g t h e c o n tin u o u s

^ -fu n c tio n .

T h ese g iv e f o r

t h e i n v e r s e t r a n s f o r m a ti o n t o 4 . 4 : 4.6

olCA)

S(AjX) chs A*

frp

I n s e r t i n g t h e fo rm u la f o r S (a ,X ) and p e rfo rm in g t h e i n t e g r a t i o n o v e r t h e p a r t i n v o lv i n g t h e 4 .4 7

& -fu n c tio n s g iv e s :

n * , x ) ‘ fpflfe5 [(^-X o^c^/xx ^

To e v a l u a t e t h e l a s t I n t e g r a l , we ru n i t from m inus I n f i n i t y t o i n f i n i t y , and b r e a k up i n t o p a r t i a l f r a c t i o n s . 4 .4 8

Jo *A c_

xx^

k’

' M 0O‘U t~ * x

wX

We p e rfo rm t h e t r a n s l a t i o n X*x'~A X s X'+ a

in th e second.

Thus

~ i n t h e f i r s t i n t e g r a l , and

Then

4 .4 9 - xK ^ '

Ax J

XX

A

1 a ] s

t [U (A ),U

] ' I r

The r e l a t i o n s betw een t h e o p e r a t o r s o f t h e c o u p le d sy ste m and th o s e o f t h e u n c o u p le d a r e g iv e n by t h e a n a lo g u e o f 3*14, w ith I n t e g r a l s r e p l a c i n g sums an d t h e

X0 te rm s e p a r a t e d .

We r e o r d e r

t h e H a m ilto n ia n a s b e f o r e , so we u s e a s t h e q u a n tiz e d H am il­ to n ia n : 4 .5 5

H5

AA o We now a g a in c o n s i d e r t h e vacuum s t a t e s

£„

and

T h ese a r e d e fin e d a s b e f o r e by t h e c o n d itio n s

$T0 • U(a)£b-o.

A g ain £, I s t h e s t a t e o f minimum e n e r g y , w ith I n f a c t z e ro e n e rg y .

The e n e rg y o f t h e s t a t e $0' i s g iv e n i n c o m p le te a n a lo g y

w ith 3*19 b y : 4 .5 6

[f

#. : i

a

(a-x)VA +^

('''■Xo)1/A0]

S u b s t i t u t i n g f o r S and ^ g iv e s t h e r e s u l t : 4.57

]

63 I f we make a n a p p ro x im a te c a l c u l a t i o n f o r e s m a ll, t h e o n ly te rm s w hich s u r v iv e a r e t h e f i r s t and fo u r th *

T hese a r e

a p p ro x im a te ly : 4*84

\J ~

Now I f we n e g l e c t t h e e f f e c t o f t h e dam ping, t h i s I s approx** lm a te ly :

-L (X ^')t ^ £ iL! X- X'

4 - 85

The f a c t o r I n b r a c k e t s g iv e s t h e c o n s e r v a tio n o f e n e r g y , w h ile t h e f a c t o r i n f r o n t j u s t g iv e s t h e m a tr ix e le m e n t f o r e m is s io n o f t h e s t a n d a r d p e r t u r b a t i o n th e o ry *

I f we I n ­

c lu d e t h e dam p in g , e q u a tio n 4 .8 4 g iv e s j u s t t h e w ell-know n fo rm u la s f o r t h e l i n e sh ap e and fre q u e n c y d isp la c e m e n t* ■We have t h u s shown t h a t t h e e x a c t r e s u l t s re d u c e t o th o s e o f s ta n d a r d p e r t u r b a t i o n th e o r y when e te n d s t o zero * B u t, a s b e f o r e , we. may c o n s id e r l a r g e v a lu e s of e i n t r e a t i n g 4*83*

Now f o r e l a r g e , a l l th e te rm s I n t h e b r a k c e t a r e o f

o r d e r e - 1 , and h ence t h e t o t a l e x p r e s s io n W becom es in d e p e n d ­ en t of e fo r e s u ffic ie n tly la rg e . tw o c a s e s \ =

1 and

1*

We a g a in m ust s e p a r a t e t h e

In case

\> 1 > we hav e f o r

e s u f f ic ie n tly la rg e ,

- i Xt

t

4 .8 6 i!r/>OiX0

1

.

X.

.

i. / , a.

SIK

\

I ____

e I L.

V

I'TvT rl+^X }

[\Iv"i - i.

T h is r e s u l t may b e i n t e r p r e t e d a s a f o r c e d o s c i l l a t i o n

64 a t t h e o b s e r v a tio n fre q u e n c y p lu s a t r a n s i e n t . o r d e r exp - e t h a s b e e n o m itte d . b e h a v io r I s a s f o llo w s .

A gain a te r m o f

C o n s e q u e n tly , t h e g e n e r a l

D u rin g tim e I n t e r v a l s o f th e o r d e r o f

m ag n itu d e o f e ”1 , r a p i d s u r g e s o f e n e r g y ta k e p l a c e .

T hen

a t r a n s i e n t o s c i l l a t i o n a p p e a r s , w ith a fre q u e n c y n e a r t h e o s c i l l a t o r fre q u e n c y and a dam ping c o n s t a n t d e te rm in e d by t h e o s c i l l a t o r m a ss, p l u s a f o r c e d o s c i l l a t i o n .

I f we w a it a n

i n t e r v a l o f tim e lo n g enough f o r t h e t r a n s i e n t t o d i s a p p e a r , we w i l l b e l e f t w ith J u s t t h e f i r s t te r m of 4 .8 6 .

B e ca u se o f

t h e d e n o m in a to r, t h e a m p litu d e o f t h i s te rm h as a re s o n a n c e n e a r X =A0

(a c tu a lly a t

\ = ^

^

)•

I f th e p a r t i c l e i s

f a i r l y h e a v y , t h i s te rm i s a p p ro x im a te ly 4x0/ [ ( \ - X0)a‘+rt^. ] ^ e x p t ( - \ t + 0 ) , where © i s a n a n g le c l o s e t o T r / 2 .

H ence, t h e

h a l f - w i d t h o f t h e re s o n a n c e c u rv e i s g iv e n by th e dam ping c o n s t a n t o f t h e t r a n s i e n t te rm , e x a c t l y a s I n th e s t a n d a r d p e r tu r b a tio n th e o ry .

VJe s h a l l n o t d i s c u s s th e c a s e \ < l ,

w hich I s s i m i l a r and o f l e s s I n t e r e s t . We now o b s e rv e t h a t 4 .8 3 , an d a l l a p p ro x im a tio n s d e ­ r i v e d from i t , d i s p l a y s a s i n g u l a r b e h a v io r a t low f r e q u e n c i e s . The t r a n s i t i o n p r o b a b i l i t y \w !2 w i l l b e p r o p o r t io n a l t o l / A f o r v e r y low f r e q u e n c i e s .

C o n s e q u e n tly , i f t h e r e a r e no low

fre q u e n c y q u a n ta a t t ■ 0 , th e p r o b a b i l i t y o f t h e sy s te m u n d e r ­ g o in g a t r a n s i t i o n i n w hich a lo w -fre q u e n c y quantum i s e m i t te d i s v e ry l a r g e , and t h e r e f o r e we c a n n o t e x p e c t th e sy ste m t o re m a in i n t h e p u re e x c i t e d s t a t e , b u t i t

w i l l im m e d ia te ly

i n c lu d e a n a sse m b la g e o f lo w -fre q u e n c y q u a n ta . T h is b e h a v io r i s c h a r a c t e r i s t i c o f th e e x c i t e d s t a t e .

65 I f we had c o n s id e r e d t h e vacuum s t a t e

■r- /

$ 0 , th e p r o b a b ility o f

s i n g l e quantum e m is s io n v a n is h e s , and o n ly p r o c e s s e s i n w hich two q u a n ta , one o f w hich may be a s t r i n g quantum and one an o s c i l l a t o r quantum , a r e p ro d u c e d g iv e r i s e t o n o n -v a n is h in g m a tr ix e le m e n ts t o t h e lo w e s t o r d e r o f p e r t u r b a t i o n th e o r y . T h ese p r o c e s s e s a l s o a r e s i n g u l a r a t lo w - f r e q u e n c ie s , w ith a l / \ fre q u e n c y d e p e n d e n c e . T hese c o n s i d e r a t i o n s show v e ry n i c e l y t h e n a tu r e o f t h e s i n g u l a r i t y o f t h e s t a t e s o f th e u n c o u p le d sy s te m .

If

we ta k e any s t a t i o n a r y s t a t e o f t h e u n c o u p le d sy s te m , we w i l l f i n d a l a r g e p r o b a b i l i t y f o r i t t o c h a n g e , b e c a u se o f th e c o u p lin g , w ith e m is s io n o r a b s o r p t io n o f lo w -fre q u e n c y q u a n ta . T h is e f f e c t a r i s e s from im p o sin g im p o s s ib le r e q u i r e ­ m ents upon t h e m e a su rin g a p p a r a t u s .

We a r e a tt e m p t in g t o

d e te rm in e t h e s t a t e o f t h e sy ste m o v e r t h e e n t i r e l e n g t h o f th e s t r i n g .

But no m e a su rin g a p p a r a tu s c a n b e c o n s t r u c t e d t o

m easure a l l f r e q u e n c i e s , e i t h e r h ig h o r lo w .

H ence, t h e lo w -

fre q u e n c y q u a n ta w i l l be l n d e t e c t a b l e , an d we s h o u ld n o t im pose i n i t i a l c o n d it i o n s on t h e i r num bers.

G iv in g a f i n i t e b an d w id th

t o th e m e a su rin g a p p a r a t u s , o r e q u i v a l e n t l y , o n ly p e r m i t t i n g i t t o make m easurem ents o v e r f i n i t e r e g io n s o f sp a c e f o r f i n i t e t im e , w i l l e li m in a te t h e d iv e rg e n c e by p r o v i d in g a c u t o f f . We n e x t c o n s i d e r what w i l l be t h e e n e rg y o f th e s ta te

d /.

To t h e f i r s t a p p ro x im a tio n t h i s i s g iv e n by th e

e x p e c ta ti o n v a lu e o f H i n t h e s t a t e s i o n a n a lo g o u s t o 3«12, f o r

3?i •

We I n s e r t t h e e x p a n ­

U ( a ) , e t c . , I n te rm s o f-w (X ),

e t c . , in to th e e x p e c ta tio n v a lu e , o b ta in in g :

66

^ ■ •8 7

E,

SM,A)/|jX [f/VX)''*- (Mf ^'A)/w-(A)j f o'(/>yvTo

+ (/\'A 0)/w-0 j j

l^\S(A}X)/(y [M^'Wv)^M'X'KlAV]^/

^

^(

i a a '-Aa- xa'

J ^

A ^ ' x 4 * ~ y ' x * ( 3- A- XA ~ A a ;)

We s h a l l b r e a k t h i s up i n t o t h r e e p a r t s .

co«

\

x c (x }A x

We have f o r

t h e f i r s t te rm I n t h e f i r s t p a r e n t h e s i s :

SAx'^A-A" A

5«11

^ y x^ \ " x

A / ', X - » X',A"=I

T h is may be summed im m e d ia te ly o v e r 5 .1 2

xL H £ X xs» x',x";/

a

and

a *»

o b ta in in g :

a^-X'x ^ -A " x 4/O-A ( L-|x/jJ

OO ~

- X A/A/a

Hence we have f o r t h e e n e rg y d e n s i t y : 5.19

< £ t K > 0 ) o, : £ V l - £ X 'l

a / a/*

A-»

We se e im m e d ia te ly t h a t t h e e n e rg y d e n s i t y i s u n ifo rm . T h is i s c l e a r l y a c o n seq u en ce o f th e u n c e r t a i n t y p r i n c i p l e . S in c e t h e num bers o f q u a n ta and t h e i r p h a s e s a r e c a n o n i c a ll y c o n ju g a te v a r i a b l e s , i f we s p e c i f y th e num bers p r e c i s e l y we l o s e a l l know ledge a b o u t th e p h a s e s .

T h e r e f o r e , we have no

know ledge o f t h e l o c a t i o n o f t h e q u a n ta , and th e e n e rg y d e n s i ty m ust b e u n ifo rm , e x c lu d in g t h e o r i g i n , w here t h e c o u p lin g i s a p p l i e d and t h e o p e r a t o r H(x) doeB n o t r e p r e s e n t t h e c o m p lete e n e rg y d e n s i t y .

T h is c o n c lu s io n i s t r u e f o r any s t a t e w ith a

f i x e d num ber o f q u a n ta .

79 We now d i s c u s s t h e h ig h - f re q u e n c y and lo w -fre q u e n c y b e h a v io r o f th e e x p r e s s io n .

F o r X>>£c , we hav e t h e a p p r o x i ­

m a tio n s v a l i d t o se c o n d o r d e r : 5 .2 0 5 .2 1

N a ~ L + ^^ ~ XV,L — * a/a/a ^

At Ao

f + - - 1^ 1- ^ VLJ

—

When t h i s I s s u b s t i t u t e d I n t o ( h ( x ) ) 0/ , t h e X/L te rm s c a n c e l , and we a r e l e f t w ith a s e r i e s w hich a t h ig h f r e q u e n c ie s i s o f o r d e r >T3 and h en ce c o n v e rg e s .

H ence, th e e n e rg y d e n s i t y i s w e l l -

beh av ed a t h ig h f r e q u e n c i e s .

The a d d i t i o n a l te rm

A °/n a

rem oves t h e c o n t r i b u t i o n t o t h e s e r i e s made by t h e te rm s i n th e n e ig h b o rh o o d o f \ = X„ , i n d i c a t e d by t h e te rm i n ( A /n ^ .

^ in

T h is may be shown by d e t a i l e d a n a l y s i s w hich i s o f th e

same n a tu r e a s t h a t em ployed t o s i m p li f y < H)0, , and i s n o t n e c e s ­ sa ry to p re s e n t h e re . A t low f r e q u e n c i e s , t h e n o r m a liz a tio n c o n s t a n t r e ­ d u c es t o L( l +

)» when t h e fre q u e n c y

an d 2 /e L i s s m a ll.

H ence, t h e lo w -fre q u e n c y c o n t r i b u t i o n t o HU ) i s a bounded q u a n t i t y , o f o r d e r e /L . bounded f o r L l a r g e .

H ence, th e e n t i r e e n e rg y d e n s i t y i s

T h is i s a v e ry r e a s o n a b le r e s u l t , f o r

s in c e t h e e n e rg y o f t h e whole s t r i n g i s o n ly o f o r d e r e lo g L, t h e e n e rg y d e n s i t y m ust be ro u g h ly o f o r d e r e / L , t o make i t s i n t e g r a l o v e r t h e w hole s t r i n g o f o r d e r e lo g L . T h is th ro w s more l i g h t on t h e n a tu r e of t h e i n f r a - r e d c a ta s tro p h e .

The e n e rg y d e n s i t y o f a s t r i n g o f l e n g t h 2L

s h o u ld be a b o u t t h e same a s h a l f th e d e r i v a t i v e o f t h e t o t a l e n e rg y w ith r e s p e c t t o L.

The lo w -fre q u e n c y c o n t r i b u t i o n t o

80

t h e e n e rg y d e n s i t y I s th e n j u s t su c h a s t o g iv e a te rm o f o r d e r e /L I n t h e e n e rg y d e n s i t y , w hich th e n c o rre s p o n d s t o a te rm o f o r d e r e l o g L I n t h e t o t a l energy#

A lth o u g h t h e e n e rg y

d e n s i t y goes t o z e r o a s t h e l e n g t h goes t o I n f i n i t y , t h e a p p ro a c h I s n o t s u f f i c i e n t l y r a p i d t o p ro d u c e a f i n i t e t o t a l e n e rg y .

H ow ever, t h e m e a su rin g a p p a r a tu s c a n n o t d e t e c t what

h ap p en s o u ts id e a f i n i t e l e n g t h of s t r i n g , so t h e e n e rg y d e n ­ s i t y i n th e vacuum s t a t e s h o u ld n o t be u n ifo rm o u t s i d e t h i s l e n g t h , b u t s h o u ld d ro p t o z e r o .

Then t h e te rm e lo g L i n

t h e t o t a l e n e rg y i s c u t o f f a t some v a lu e e lo g Lf , s i n c e no v i b r a t i o n o f fre q u e n c y lo w e r th a n fT /2 L , can b e d e t e c t e d . We may r e g a r d t h e e f f e c t o f a lo w -fre q u e n c y c u t o f f a l s o a s m o d ify in g t h e num ber o f lo w -fre q u e n c y q u a n ta p r e s e n t . I f t h e a p p a r a tu s c a n n o t d e t e c t t h e p r e s e n c e o f su c h q u a n ta , we s h o u ld l i f t t h e c o n d it i o n a».(\ ) £• = 0 when X i s l e s s th a n some low f r e q u e n c y X,

.

The sy ste m may th e n c o n ta in a n a r b i t r a r y

num ber o f q u a n ta o f fr e q u e n c y l e s s th a n X,

•

The p r e s e n c e of

t h e s e q u a n ta m o d ifie s t h e t o t a l e n e rg y , and may make I t f i n i t e . H ow ever, t h e p ro b lem o f d e te r m in in g how many lo w -fre q u e n c y q u a n ta , and o f what f r e q u e n c i e s , a r e n eeded t o com pensate th e d iv e r g e n c e o f ( h ) 0'

i s e q u iv a le n t t o f i n d i n g t h e lo w -fre q u e n c y

b e h a v io r o f t h e f u n c t i o n A (a ,a * ) in tr o d u c e d i n th e d i s c u s s i o n o f h ig h - f r e q u e n c y c u t o f f s , and t h a t t o t h i s d a te i s a n i n ­ tra c ta b le ta s k . W ith t h i s , we c lo s e t h e d e t a i l e d c o n s i d e r a t i o n s o f th e s t r l n g - o s e i l l a t o r s y s te m , and s h a l l p ro c e e d t o sum m arize o u r r e s u l t s an d p r e s e n t t h e c o n c lu s io n s t h a t c a n be draw n from them .

CHAPTER VI RESULTS AND CONCLUSIONS

We s h a l l now p r e s e n t a summary o f r e s u l t s and th e v a r i o u s c o n c lu s io n s w hich c an he draw n from th em .

We have

f i r s t fo rm u la te d a sy ste m whose c l a s s i c a l b e h a v io r c an be fo u n d d i r e c t l y and w hich h a s no d iv e r g e n c e s c l a s s i c a l l y .

We

hav e worked o u t th e c l a s s i c a l s o l u t i o n f o r th o f i n i t e s t r i n g in d e ta il. I n th e c l a s s i c a l c o n s id e ra tio n s o f th e f i n i t e s t r i n g , we showed t h a t f o r s m a ll c o u p lin g b etw een t h e s t r i n g and th e o s c i l l a t o r , th e e ig e n v a lu e s an d e ig e n f u n c t i o n s o f t h e c o u p le d sy ste m d i f f e r b u t s l i g h t l y from t h o s e o f t h e u n c o u p le d s y s te m , a s lo n g a s t h e fre q u e n c y o f t h e mode in v o lv e d i s l a r g e r th a n th e c o u p lin g fre q u e n c y c o n s id e r a b ly b elow

is *

When t h e mode h a s a fre q u e n c y

•£. e, t h e c o u p le d mode i s q u i t e d i f f e r e n t

from th e u n c o u p le d m ode.

I n te rra s o f a fre q u e n c y a n a l y s i s , t h e

fre q u e n c y o f t h e nt h c o u p le d mode d i f f e r s from t h a t o f t h e nt h u n c o u p le d by a q u a n t i t y o f o r d e r e / 2 a^L when ^ e l / ( XJ*)2 i s s m a ll , w h ile f o r

t . e l / ( X^L)2 l a r g e th e fre q u e n c y o f t h e nt h i> u

c o u p le d mode i s ir/2 L g r e a t e r th a n t h a t o f t h e n mode.

u n c o u p le d

S i m i l a r l y , t h e p h a s e s h i f t o f t h e s t r i n g c o u p le d e i g e n ­

f u n c t i o n s from t h e c o rr e s p o n d in g u n c o u p le d f u n c t i o n s i s s m a ll fo r

i_e/X s m a ll , p a s s e s th ro u g h t t/ 4 n e a r

r / 2 f o r \ s m a ll com pared t o

t.©*

X = £ .e , and te n d s t o

8S

S in c e t h e sy s te m i s w e ll-b e h a v e d c l a s s i c a l l y , we hope t h a t when i t i s t r e a t e d q u a n tu m -m e c h a n ic a lly i t w i l l a l s o b e w e ll-b e h a v e d *

T h u s, we q u a n tiz e d t h e sy s te m i n a c c o rd a n c e

w ith t h e s ta n d a r d p r e s c r i p t i o n s o f bo so n f i e l d t h e o r y , t r e a t i n g t h e c l a s s i c a l f i e l d v a r i a b l e s a s o p e ra to r s *

We showed t h a t th e

e q u a tio n s o f m o tio n o f t h e c l a s s i c a l th e o r y a r e a l s o g iv e n by q u a n t Tim t h e o r y , u s i n g t h e q u a n tu m -m e c h an ica l e q u a tio n o f m o tio n i t ?

* [F ? H ], t h e com m utation r u l e s b etw een t h e v a r io u s

o p e r a t o r s a s s e t up by s t a n d a r d m eans, a n d t h e d i r e c t t r a n ­ s c r i p t i o n o f t h e H a m ilto n ia n from t h e c l a s s i c a l th e o r y i n t o t h e H e is e n b e rg r e p r e s e n t a t i o n *

T h ese r e s u l t s , o f c o u r s e , a r e

n e c e s s a r y f o r f u r t h e r w ork, and m e re ly show t h a t o u r sy ste m i s am enable t o a q u a n tu m -m e c h a n ic a l d e s c r i p t io n * We th e n c o n s id e r e d t h e vacuum o f th e system *

We d e ­

f i n e d two vacuum s t a t e s , i n one o f w hich t h e e ig e n v a lu e s o f th e o c c u p a tio n num bers o f a l l t h e c o u p le d modes a r e z e r o , and th e o t h e r c o r r e s p o n d in g ly f o r u n c o u p le d modes*

N e x t, we a v a i le d

o u r s e lv e s o f t h e p r i v i l e g e o f r e o r d e r i n g th e o p e r a t o r s i n th e H a m ilto n ia n i n an y m anner w ith o u t d i s t u r b i n g th e c o rre s p o n d e n c e w ith t h e c l a s s i c a l th e o ry *

The c h o ic e was so made t h a t i n th e

c o u p le d sy ste m vacuum s t a t e t h e e x p e c t a t i o n v a lu e o f t h e e n e rg y i s zero *

The H a m ilto n ia n i s th e n p o s i t i v e d e f i n i t e , w ith t h e

lo w e s t e ig e n v a lu e zero* A d e t a i l e d s tu d y was th e n made o f t h e vacuum s t a t e o f th e u n c o u p le d system * v a lu e o f i t s e n erg y *

We f i r s t e v a lu a te d th e e x p e c ta ti o n

T h is p ro v e d t o be f i n i t e , an d we e v a l u ­

a t e d i t i n th e tw o c a s e s

eL s m a ll and ^ e L l a r g e .

When i e L

83 I s s m a ll , we o b ta in e d t h e r e s u l t 3*38, t h a t i s : 6 .1

t e ^ .134 l + T h is e n e rg y was I n t e r p r e t e d a s f o l lo w s , t h e f i r s t

te rm r e p r e s e n t s t h e e n e rg y o f t h e s t r i n g due t o I t s p a r t i a l b in d in g a t t h e c e n t e r . £ e L re m a in s s m a ll.

I t I n c r e a s e s l i n e a r l y w ith L a s lo n g a s

The se co n d te rm r e p r e s e n t s t h e c o u p lin g

e n e rg y b etw een t h e s t r i n g an d th e o s c i l l a t o r , v a r y in g v e ry s lo w ly w ith L , b e c a u se o f t h e l o g a r ith m ic te r m .

The t h i r d I s

t h e e n e rg y o f t h e o s c i l l a t o r due t o t h e change i n i t s fu n d a ­ m e n ta l fre q u e n c y o f v i b r a t i o n . The f o n a u la 3*38, o r 6 . 1 , h a s a v e r y l i m i t e d sc o p e o f a p p lic a tio n .

The re q u ire m e n t t h a t i eL I s sm a ll means t h a t

t h e c o u p lin g fre q u e n c y i s s m a ll com pared t o t h e lo w e s t f r e ­ q u en cy w hich c a n b e t r a n s m i t t e d by th e s t r i n g .

T h is c o u p lin g

I s th e n so weak t h a t i t I s u n l i k e l y t h a t I t c an p ro d u c e an y s ig n if ic a n t p h y s ic a l e f f e c ts .

I t I s much more I n t e r e s t i n g and

I m p o rta n t t o o b t a i n t h e e x p e c t a t i o n v a lu e o f t h e e n e rg y when t h e c o u p lin g i s a p p r e c i a b l e , so

i.e L I s a l a r g e num ber, th o u g h

we s t i l l assum e t h a t e i s s m a ll com pared t o mxj*-, so t h e change i n o s c i l l a t o r fre q u e n c y i s s m a ll com pared t o X„ i t s e l f .

U nder

t h e s e c o n d i t i o n s , th e e x p e c ta ti o n v a lu e o f H f o r t h e uncotqpled vacuum i s g iv e n b y 3.49# w hich we g iv e h e re a s :

6 .2 The f i r s t te rm h e re i s t h e e n e rg y o f b in d in g o f t h e s trin g .

I n c o n tr a s t t o 6 .1 , i t i s p ro p o rtio n a l to e in s te a d

o f e 2 , and i n c r e a s e s o n ly l o g a r i t h m i c a l l y w ith L .

T h is i s

84

p l a u s i b l e , b e c a u se f o r a v e ry lo n g s t r i n g what h ap p en s a t one p o i n t s h o u ld a f f e c t o n ly s l i g h t l y th e d i s t a n t p o r t i o n s , an d h en ce th e e n e rg y s h o u ld I n c r e a s e v e ry s lo w ly w ith L*

The m a jo r

c o n t r i b u t i o n t o t h i s te rm i s from lo w -fre q u e n c y te r m s , s i n c e f o r t h e h ig h - f re q u e n c y te rm s t h e c o u p lin g i s r e l a t i v e l y s m a ll and we have e f f e c t i v e l y a v e ry s m a ll p e r t u r b a ti o n *

In d e e d , t h i s

te rm d iv e r g e s when t h e l e n g t h o f t h e s t r i n g te n d s t o i n f i n i t y , b u t we w i l l d i s c u s s t h a t e f f e c t i n i t s p la c e * The se c o n d te rm o f 6 * 2 , t h e c o u p lin g e n e rg y betw een th e s t r i n g and o s c i l l a t o r , i s f i n i t e and c o n s ta n t f o r L v e ry la r g e *

H ow ever, b e c a u se o f t h e l o g a r i th m i c f a c t o r , t h i s te rm

i s n o t an a n a l y t i c f u n c t i o n o f e f o r e sm all*

I f we had c a l ­

c u l a t e d i t by p e r t u r b a t i o n t h e o r y t o seco n d o r d e r , we w ould g e t an i n f i n i t e answ er*

T h is i n f i n i t y i s c o m p le te ly s p u r i o u s ,

and a r i s e s from t h e breakdow n o f t h e p e r t u r b a t i o n th e o r y r a t h e r th a n from any d e e p - s e a te d l i m i t a t i o n s o f th e system *

T h is r e ­

s u l t s u g g e s ts t h a t i n o t h e r sy ste m s t h e same ty p e o f phenomenon may a r i s e , i n w hich a n i n f i n i t y a p p e a rs i n p e r t u r b a t i o n t h e o r y b e c a u se o f an a tte m p t t o expand a n o n - a n a l y ti c f u n c t i o n i n pow er s e rie s .

How ever, t h e d i f f i c u l t y o f p e rfo rm in g t h e c a l c u l a t i o n s

f o r o t h e r sy ste m s h a s so f a r p r e c lu d e d f i n d i n g t h i s phenom enon, i f i t e x is ts * We n e x t showed t h a t t h e e x p e c ta ti o n v a lu e o f th e e n e rg y i n th e s t a t e

i s o n ly f i n i t e b e c a u se o f t h e r e o r d e r i n g

o f th e c o m p lete H a m ilto n ia n .

When o n ly th e u n c o u p le d p a r t o f

t h e H a m ilto n ia n i s r e o r d e r e d , t h e e x p e c ta ti o n v a lu e o f H d iv e r g e s l o g a r i t h m i c a l l y a t h ig h f r e q u e n c i e s .

T h is i n f i n i t y

85 a r i s e s from t h e i n c o n s i s t e n t t r e a tm e n t o f H, in t h a t o n ly p a r t o f i t i s re o rd e re d *

J u s t su c h a p r o c e s s i s a p p lie d i n quantum

e le c tr o d y n a m ic s , w ith a s a r e s u l t a n am biguous a n sw e r, w hich s u g g e s ts t h a t t h e a m b ig u ity m ig h t b e rem oved by p r o p e r r e o r d e r ­ in g .

H ow ever, t h e r e i t i s im p o s s ib le t o f i n d t h e t r u e e ig e n ­

f u n c t i o n s , so t h e p r o c e s s can o n ly be c a r r i e d o u t i n th o u g h t* W hile

i s f i n i t e f o r t h e f i n i t e s t r i n g when th e

p r o p e r r e o r d e r i n g i s c a r r i e d o u t , we showed t h a t

is

d iv e r g e n t a t h ig h f r e q u e n c i e s , b e h a v in g l i k e th e s q u a re o f a lo g a rith m *

T h is we I n t e r p r e t a s show ing t h a t th e s t a t e f u n c tio n 0

c a n n o t be p r e p a r e d .

S in c e H ^ 0* i n p r o p o r t i o n a l t o

in

t h e S c h rU d in g e r r e p r e s e n t a t i o n , t h e d iv e r g e n c e o f