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Numerical analysis of a unitary particle model. Kennedy, Edith Mary 1955

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' NUMERICAL ANALYSIS OP A UNITARY PARTICLE MODEL EDITH MARY KENNEDY B . S c , M c G i l l U n i v e r s i t y , 1955 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Physios We accept t h i s t h e s i s as conforming t o the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1955 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree th a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree th a t permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e . I t i s under-stood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia, Vancouver Canada. - i i -ABSTRACT Thi s t h e s i s i n v e s t i g a t e s a c l a s s i c a l u n i t a r y f i e l d model c o n s i s t i n g of a complex s c a l a r source f i e l d coupled t o a r e a l s c a l a r massless f i e l d , i n an attempt t o describe the f a m i l y of heavy p a r t i c l e s c o n s i s t i n g of nucleons and hyperone. The d i s c u s s i o n i s l i m i t e d t o s p h e r i c a l l y symmetric f i e l d s and approximate methods are used t o solve the n o n - l i n e a r f i e l d equations* The s o l u t i o n s may be c l a s s i f i e d by the number of nodes i n the source f i e l d . Por each type of s o l u t i o n , the energy i n the f i e l d i s c a l c u l a t e d as a f u n c t i o n of the s p a t i a l extension of the p a r t i c l e d e s c r i b e d by that s o l u t i o n . A s t r i k i n g f e a t u r e of the theory i s t h a t f o r each type of s o l u t i o n the energy may be minimized w i t h respect t o the extension of the p a r t i c l e . The p a r t i c u l a r s o l u t i o n which y i e l d s the minimum energy i s i n t e r p r e t e d as representing a p a r t i c l e i n i t s normal s t a t e . I n t h i s way, a d i s c r e t e mass spectrum i s obtained. The mass r a t i o s y i e l d e d by the theory compare reasonably w e l l w i t h the experimentally observed mass r a t i o s f o r nucleons, A p a r t i c l e s and H p a r t i c l e s . A comparison of t h e o r e t i c a l and experimental values determines the "bare" mass of the source f i e l d t o correspond t o 1185 e l e c t r o n masses, and y i e l d s a value f o r the r a t i o of the n o r m a l i z a t i o n constant of the source f i e l d t o the coup l i n g constant of the two f i e l d s , but does not s p e c i f y these l a t t e r constants s e p a r a t e l y . i l i -L i m i t a t i o n s of t h i s c l a s s i c a l model are st a t e d * A p o s s i b l e improvement of the model i s i n v e s t i g a t e d i n an attempt t o improve the t h e o r e t i c a l mass r a t i o s ; however, the s i g n i f i c a n t features of the model remain unchanged, and the change i n the mass r a t i o s , while i n the r i g h t d i r e c t i o n , i s very s l i g h t . v i -ACKNOWLEDGEMENTS I am g r a t e f u l t o Dr. P. A. Kaempffer f o r suggesting t h i s problem and f o r h i s continued i n t e r e s t and valuable advice throughout the performance of the research. I am a l s o Indebted t o the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l a s s i s t a n c e i n the form of a Bursary. - i v -TABLE OF CONTENTS ACKNOWLEDGEMENTS v i I . IRTRODUCTION 1* Program 1 2. C l a s s i f i c a t i o n o f U n i t a r y Theories 2 3.. Formalism f o r Theories o f Type (1) 3 I I . IMPORTANCE OF NON-LINEAR TERMS IN THE FIELD EQUATIONS 1. S t a b i l i t y 5 2. Q u a l i t a t i v e D i s c u s s i o n of a C l a s s i c a l Model of Type ( l a ) 6 a) Equation to be solved b) S o l u t i o n s of the equation c) P a r t i c l e s produced I I I . NUMERICAL ANALYSIS OF A CLASSICAL MODEL OF TYPE ( l b ) 1. The S p e c i f i c Model and F i e l d Equations 9 2. Approximate S o l u t i o n of the Equations 11 3. « N o r m a l i z a t i o n Procedure 13 4« The Mass Equation 13 5. Numerical R e s u l t s 14 a) Energy of the system b) Mass and c l a s s i c a l r a d i u s c) Parameters c h a r a c t e r i z i n g the model d) "Quasicharge" e) Summary of r e s u l t s 6. L i m i t a t i o n s o f the Model 1$ -V-TABLE OF CONTENTS IV. POSSIBLE IMPROVEMENT OF THE MODEL 1. S p e c i f i c Model and F i e l d Equations 21 2 . Approximate S o l u t i o n o f the Equations 22 3.. Numerical R e s u l t s 23 GRAPHS Graphs 1 and 2 f a c i n g page 7 Graphs 3 and 4 f a c i n g page 11 Graphs 5, 6, 7, and # f a c i n g page 12 Graphs 9, 10, 11, and 12 f a c i n g page 13 Graph 13 f a c i n g page 15 Graphs 14 and 15 f a c i n g page 22 Graphs 16 and 17 f a c i n g page 23 TABLES Table I f a c i n g page 14 Table I I f a c i n g page 24 APPENDICES Dimensional A n a l y s i s 25 Sample C a l c u l a t i o n 26 BIBLIOGRAPHY 27-I . rUTRODUCTION 1. Program Ever s i n c e Mie ( 1 9 1 2 ) ^ p u b l i s h e d h i s paper on the theory of matter, u n i t a r y t h e o r i e s of the s o - c a l l e d elementary p a r t i c l e s have remained i n what might be c a l l e d a s t a t e of animated suspension. New attempts i n the s p i r i t of Mie keep appearing i n the l i t e r a t u r e , but although they r a i s e h i g h hopes by e s t a b l i s h i n g the f e a s i b i l i t y of a u n i t a r y theory, none of them has yet succeeded i n p r e d i c t i n g q u a n t i t a t i v e l y the outcome of a s i n g l e experiment. The aim of the present i n v e s t i g a t i o n i s not t o enlarge on the general f e a s i b i l i t y of u n i t a r y p a r t i c l e theory, but r a t h e r t o study the numerical f e a s i b i l i t y of a known model which appears a t t r a c t i v e from a general p o i n t of view, and, i f p o s s i b l e , t o develop a f e e l i n g f o r the order of magnitude of the parameters c h a r a c t e r i z i n g t h i s model. I n p a r t i c u l a r , the f a m i l y of heavy p a r t i c l e s , c o n s i s t i n g of nucleons and hyperone, w i l l be taken as the r e a l i t y t o which the model under c o n s i d e r a t i o n should correspond. The reason f o r t h i s p a r t i c u l a r choice i s the f a c t t h a t the model t o be discussed below describes a " p a r t i c l e " i n terms of s e l f - m a i n t a i n e d s o l u t i o n s of c e r t a i n n o n - l i n e a r d i f f e r e n t i a l equations, which a l l o w f o r the existence of a "ground s t a t e " together w i t h " e x c i t e d s t a t e s " , a s i t u a t i o n resembling most c l o s e l y the r e l a t i o n between the various members of the heavy p a r t i c l e f a m i l y . 2-I t w i l l be found t h a t one can obta i n reasonable values f o r the mass spectrum. However, the d e s c r i p t i o n of the decays of the various p a r t i c l e s and the p r e d i c t i o n of t h e i r l i f e t i m e s i s beyond the reach of present u n i t a r y t h e o r i e s . 2. C l a s s i f i c a t i o n of U n i t a r y Theories E x i s t i n g u n i t a r y t h e o r i e s may be separated n a t u r a l l y i n t o three groups: (1) P u r e l y c l a s s i c a l f i e l d t h e o r i e s , i n which the cohesion of p a r t i c l e s i s provided by e i t h e r ( l a ) c o upling of the f i e l d t o i t s e l f , or (lb ) coupling of two f i e l d s , each a c t i n g as the glue which holds the other together. Models r e s u l t i n g from t h i s approach have been st u d i e d by Rosen ( 1 9 3 9 ) ( 2 ) , P i n k e l s t e i n (1949)< 3), and others ( * ) - ( 7 ) # problems i n v o l v e d i n the q u a n t i z a t i o n of such t h e o r i e s have been s t u d i e d by P i n k e l s t e i n ^ 3 ^ . (2) Quantum f i e l d t h e o r i e s , w i t h coupling of the f i e l d t o i t s e l f . T h i s approach has been fo l l o w e d by Heisenberg (1954)( Q) and others. (3) F i e l d t h e o r i e s i n which a c l a s s i c a l "source" i s coupled t o a quantized "glue". The f e a s i b i l i t y of t h i s intermediate type of model was f i r s t p o i n t e d out by Heber ( 1 9 5 5 ) ( 9 ) . -3-The i n t e r p r e t a t i o n of models r e s u l t i n g from the approach (2) meets w i t h great d i f f i c u l t i e s , as has been shown by K I t a ( 1 9 5 6 ) ( 1 0 ) , and so f a r no model of type (3) i s known which w i l l correspond t o a p a r t i c l e w i t h p o s i t i v e mass. Por these reasons, .the present work w i l l t r e a t only models of type ( 1 ) . 3. Formalism f o r Theories of Type (1) We summarize the formalism used t o d e s c r i b e t h e o r i e s of types ( l a ) and ( l b ) . The c o n d i t i o n s t h a t the theory be gauge i n v a r i a n t and have no s i n g u l a r i t i e s are imposed. A Lagrangian d e n s i t y oC - cC , - J i s d e f i n e d , and the f i e l d equations are de r i v e d from i t by r e q u i r i n g t h a t the a c t i o n i n t e g r a l 1.1) T = f<£. d 4x be an extremum. The energy-momentum d e n s i t y t e n s o r , T^ ^ , and the charge-current d e n s i t y v e c t o r , s^, are introduced: 1.2) T , = . tt; + X. i.3) = ie[Mu vL\ - J a £ - v ^ * J The conservation laws f o r the momentum, energy, and e l e c t r i c charge f o l l o w from the i n v a r i a n c e p r o p e r t i e s of the Lagran-g i a n , and take the form -4-1.5) | ^ = o -F i n a l l y , any s o l u t i o n of the f i e l d equations which i s f r e e from s i n g u l a r i t i e s i s i n t e r p r e t e d as a p a r t i c l e of mass I t i s evident from the d e f i n i t i o n of the charge-current d e n s i t y v e c t o r t h a t a theory w i t h r e a l f i e l d s can only describe an e l e c t r i c a l l y uncharged f i e l d and t h e r e f o r e n e u t r a l p a r t i c l e s , and t h a t charged p a r t i c l e s may be obtained only i f at l e a s t one of the f i e l d v a r i a b l e s i s complex and time-dependent• 1.6) and charge 1.7) I I . IMPORTANCE OP NON-LINEAR TERMS IN THE FIELD EQUATIONS 1. S t a b i l i t y The common feature of a l l three types of u n i t a r y t h e o r i e s i s that the s t a b i l i t y of the p a r t i c l e i s provided by the n o n - l i n e a r i t y of the f i e l d equations. I t i s w e l l known t h a t i n any p h y s i c a l l y reasonable l i n e a r f i e l d theory which y i e l d s plane wave s o l u t i o n s , e , a wave packet spreads i n the absence of e x t e r n a l f o r c e s : the wave packet at time t i s gi v e n i n terms of the i n i t i a l wave packet as I I . l ) where the form of co depends on the p a r t i c u l a r f i e l d theory under d i s c u s s i o n , and determines the e f f e c t e" d 3 ^ on Ivp^t)! • Using any conven-i e n t l y d e f i n e d width of | * ^ ( x / t ) | versus X as a measure of the spread of the wave packet, i t can be shown t h a t the p h y s i c a l l y reasonable r e l a t i o n s II.2a) u) oc fe ( n o n - r e l a t i v i s t i c Schrodinger equation) II.2b) co J^ u.*+ ( r e l a t i v i s t i c Klein-Gordon and Dir a c equations) both l e a d t o spreading of any i n i t i a l wave packet. Thus we see that a f r e e wave packet c h a r a c t e r i z e d by \ ip(Jt, ~t)( * and i n t e r p r e t e d as a f r e e , s p a t i a l l y - c o n f i n e d p i e c e of matter flows apart or i s unstable. -6-We expect t o o b t a i n a s t a b l e l o c a l i z e d s o l u t i o n i f there i s a source which leads t o n o n - l i n e a r f i e l d equations* We i l l u s t r a t e t h i s general f e a t u r e by d i s c u s s i n g i n some d e t a i l a model of type ( l a ) * , 2. Q u a l i t a t i v e D i s c u s s i o n of a C l a s s i c a l Model of Type ( l a ) a) Equation t o be solved: As a simple example of a theory of type ( l a ) we consider the r e a l f i e l d c h a r a c t e r i z e d by the Lagrangian d e n s i t y where ( , and ^ determines the " s t r e n g t h " of the source term. The corresponding f i e l d equation i s I I . 4 ) a*cj/ = - yip"*" • I f we i n v e s t i g a t e s t a t i c , s p h e r i c a l l y symmetric s o l u t i o n s vf/=if/(*.) »• the f i e l d equation becomes n . 5 ) K > i f O ~ ' * The s u b s t i t u t i o n s I I . 6 ) reduce the f i e l d equation t o II.7) d% = „ft_ £ K * * j £ V We r e q u i r e that i\> be f i n i t e everywhere and t h a t a l l observable i n t e g r a l s be f i n i t e ; t h i s imposes two boundary c o n d i t i o n s on IJ: 11*8) <^  = o at x = o o as x —> aa . -7-The second c o n d i t i o n i m p l i e s that <^>° 9 L e t II.9) C^(<j-K~~^yn so t h a t 1^ now s a t i s f i e s the n o n - l i n e a r equation H . I O , Y * l ( ' - ( f f ) -b) S o l u t i o n s of the equation: To di s c u s s equation 11.10) we d i s t i n g u i s h between the cases t\ odd and v i even. I f n i s odd, d i v i d e the h a l f - p l a n e X&o i n t o three regions by the l i n e s tj=x and I f *> i s even, d i v i d e the h a l f - p l a n e i n t o f o u r regions by the l i n e s /jj=x , y=o , and xp-x. The s i g n o f -y" i n each region i s obtained by i n s p e c t i o n of the d i f f e r e n t i a l equation, and i s marked i n graphs 1 and 2, which show q u a l i t a t i v e sketches of the d i f f e r e n t types of s o l u t i o n s of equation 11.10) which s a t i s f y the f i r s t boundary c o n d i t i o n ; the s o l u t i o n s which s a t i s f y i n a d d i t i o n the second boundary c o n d i t i o n are drawn more h e a v i l y . I t i s seen that f o r n odd there i s one. such s o l u t i o n , while f o r n even there are two. c) P a r t i c l e s produced: The energy-momentum d e n s i t y tensor i s i i . u ) X „ = - ^ + X <5_ . "(It) >** Since ^ = ^ f'O * 11.12) •f-f and the f i e l d equation i s 11.13) O + * dl£ _ K*d/ = - QK\JN+' 8-I n t e g r a t i o n by p a r t s and a p p l i c a t i o n of the boundary co n d i t i o n s and the f i e l d equation y i e l d n.14) ~ -LJTU - % f c o J * ~ S * . Thus i f Y\ i s odd, the theory y i e l d s one p a r t i c l e of p o s i t i v e mass and no charge, while i f Y\ IS even, i t y i e l d s two p a r t i c l e s of d i f f e r e n t , p o s i t i v e , mass and no charge. I I I . NUMERICAL ANALYSIS OF A CLASSICAL MODEL OF TYPE ( l b ) 1. The S p e c i f i c Model and F i e l d Equations As an example of a model of type ( l b ) , consider a complex s c a l a r source f i e l d , vL> , coupled t o a r e a l s c a l a r massless f i e l d , (Q , w i t h Lagrangian d e n s i t y 111.1) ol = - dA t^*- t c > v|>~ - *P djp +a^^~<q . The corresponding f i e l d equations are 111.2) • * > - K * > = - ^ By analogy w i t h orthodox quantum f i e l d theory one may t h i n k of these as d e s c r i b i n g a "bare" nucleon f i e l d <f» of "bare" mass TM= x k coupled t o a massless meson f i e l d i£ • The main c. reason f o r the choice of t h i s model i s i t s s i m p l i c i t y and the f a c t t h a t p a r t i c l e s of p o s i t i v e mass r e s u l t from i t . I t w i l l be shown l a t e r that assignment of a "bare" mass ytt t o the meson f i e l d does not a l t e r a p p r e c i a b l y the features of t h i s model, provided /±<& I • fife i n v e s t i g a t e s t a t i c , s p h e r i c a l l y symmetric s o l u t i o n s of the type I I I . 3 ) where i s r e a l . The f i e l d equations become in.4) , dY^ ce) = - «V<t . St-- 1 0 The s u b s t i t u t i o n s I I I . 5 ) ^ reduce the f i e l d equations t o u. _ «*x* III.6) The boundary c o n d i t i o n s on u. and v f o r p a r t i c l e - l i k e s o l u t i o n s a r e : UL. = o ? \J = o Oct x = o I I I . 7 ) — V o OLS X — * < » • * X I n the r e g i o n x » i the d i f f e r e n t i a l equations become I I I . 8 ) u " = u_ w i t h s o l u t i o n s I I I . 9 ) V = o u = £ e " x + J ) e x v = A + S x The boundary c o n d i t i o n s r e q u i r e B=D=0. Thus f o r x » i u. «= Ce"" I I I . 1 0 ) v = A . By i n s p e c t i n g the d i f f e r e n t i a l equations, we see that U"< o i f 5 > « I I I . 1 1 ) u " > o i f * < I v " < o f o r a l l x > o The equations w i t h the boundary con d i t i o n s Imposed w i l l i n general a l l o w s o l u t i o n s w i t h u. having no nodes, one node, two nodes, e t c . I f each such s o l u t i o n i s i n t e r p r e t e d as a a Graph 3. Nodeless solutions of equations HI.6) u versus x. > x Graph 4.- Nodeless solutions of equations HI.6) v versus x. 1 1 -p a r t i c l e , we have a model which a l l o w s , i n p r i n c i p l e , the c a l c u l a t i o n of a mass spectrum of va r i o u s members of a f a m i l y of e l e c t r i c a l l y n e u t r a l p a r t i c l e s described by these equations. 2 . Approximate S o l u t i o n of the Equations Consider f i r s t the nodeless s o l u t i o n , of which a q u a l i t a t i v e sketch i s given i n graphs 3 and 4 . The exact s o l u t i o n of the equations appears q u i t e hopeless, so we adopt the f o l l o w i n g approximation procedure: l e t '= <xx f o r x £ x , I I I . 1 2 ) a x . f o r x £ x, Then the f i r s t of equations I I I . 6 ) becomes . ( i - a j O f o r x. I I I . 13) U. fc] f o r x > x „ . The CL and x„ which have been Introduced are constant parameters which w i l l be determined l a t e r . We may r e s t r i c t a now, however, s i n c e f o r o.< I , we have from I I I . 13) I I I . 14) u . " > o f o r a l l x , while f o r Q.-1 , we have u." = o f o r x $ x„ I I I . 1 5 ) UL" y o f o r *> . In e i t h e r case, ^ does not v a n i s h as x-*•«>, thus v i o l a t i n g the boundary c o n d i t i o n . Therefore we r e q u i r e I I I . 1 6 ) a > l Equation I I I . 1 3 ) now y i e l d s x} f o r x $ x„ C e " x I I I . 1 7 ) a s y m p t o t i c a l l y f o r x » I . V G r a p h 5 : G r a p h 6 : A p p r o x i m a t e s o l u t i o n s o f e q u a t i o n s I L L 6)-. n o d e l e s s u . xfj i f > A. G r a p h 7: G r a p h 8: A p p r o x i m a t e s o l u t i o n s o f e q u a t i o n s I E . 4): n o d e l e s s -12 As a f u r t h e r rough approximation, we now l e t 111.18) U s V ' l u 4 = C e f o r x> , and r e q u i r e t h a t u and u ' be continuous at x = x-0 . These co n d i t i o n s y i e l d 111.19) f< = C e~*° 111.20) X o = - t o n - ' ^ V o ^ i l , where we must have 111.21) S < Ja^T \, < 7Z f o r the s o l u t i o n w i t h no nodes I n it • The approximate u. and v and the corresponding uV and ^ , 4/z = C e ~ f o r X,, III.22) r<*,= ^ f o r ^ < ^ <P„ » f o r ^ are shown i n graphs 5, 6, 7, and 8. I t should be p o i n t e d out at t h i s stage that the pre-ceding d i s c u s s i o n a l s o a p p l i e s t o the case of u. having one node, or two nodes, e t c . , i f we agree t o keep the same approximation f o r v and i f we understand t h a t r e l a t i o n I I I . 2 0 ) between CL and x„ becomes subject t o 3£ <: J<f- i Xo < SCK f o r the one-node s o l u t i o n I I I . 2 3 ) srn</ Xa < 37t f o r the two-node s o l u t i o n e t c . i n s t e a d of t o I I I . 2 1 ) . G r a p h I h G r a p h 12: A p p r o x i m a t e u a n d f o r t h e t w o - n o d e c a s e . -13-The approximate u and ifs f o r these next two cases are shown i n graphs 9, 10, 11, and 12. 3. Nor m a l i z a t i o n Procedure The model i s now c h a r a c t e r i z e d by the f o u r parameters C, *• , ,"and ^  • I t i s convenient, however, t o express C. i n terms of a new constant, n , by a n o r m a l i z a t i o n procedure: now the dimensions of q u a n t i t i e s appearing i n are s p e c i f i e d , s i n c e JL must be an energy d e n s i t y ; a dimensional a n a l y s i s reveals t h a t f c^ /"*"d3x has the dimensions of ~- , and so we l e t 111.24) / ^ d 3 * = ^ where n i s a pure number (see Appendix 1 ) . When we set 111.25) 4KJ + A-Kj^l^ = , we f i n d 111.26) C = Jl j p - 5 — - • The model i s now c h a r a c t e r i z e d by v* , X«, , 1< , and • 4. The Mass Equation The energy-momentum d e n s i t y tensor i s 111.27) ~ C „ = p! &*+**S p . . bx^ 3xy <*x*> Por V|/ = V|>(»L) we have 111.28) = - (vv^r+Cv^j • TABLE I VALUES OF Ha AND CORRESPONDING VALUES OF x e nx FOR THE NODELESS, ONE-NODE, AND TWO-NODE SOLUTIONS NODELESS SOLUTION ONE-NODE SOLUTION TWO-NODE SOLUTION X. K at 0>70 3>377 1.50 4.249 2.50 4.409 1.20 2.336 2.75 2.546 4.00 2.7O8 2.00 1«950 4.5O 2.033 6.00 2.153 2.40 1.9186 5..00 1.9942 8.00 2.001 2.45 I .914S 5..50 1.9777 8150 1.9911 2.50 1.9165 5*75 1.9759 9.00 I .9884 2.60 1.921 6.00 1.9778 9.50 1.9921 3.00 1.950 6.50 1.9906 11.00 2.O3O 4.50 2.211 7..00 2.013 18 .00 2.479 7.50 2.987 10.00 2.270 10.00 3.713 -14-I n t e g r a t i o n by p a r t s and a p p l i c a t i o n of the boundary co n d i t i o n s and the f i e l d equations y i e l d H = — - = - / T ^ d3* I I I . 2 9 ) where III . 3 0 ) «o*W«ST xa) ' *<zx, u. ^3 - I -+- ' £/o?w x„ ~cas(Ja^ijC)s:r,QaF 5. Numerical R e s u l t s a) Energy of the system: For each of the three cases I n v e s t i g a t e d , namely no nodes, one node, and two nodes i n U or i f , values of the dimensionless q u a n t i t y H%- were c a l c u l a t e d f o r a range of values of x c . A t y p i c a l c a l -c u l a t i o n i s shown i n Appendix 2, and the r e s u l t s are t a b u l a t e d i n Table I and p l o t t e d i n graph 13 (opposite page 15 ). Thus f o r each case, we have a f a m i l y of approximate s o l u t i o n s of the d i f f e r e n t i a l equations, each y i e l d i n g a c h a r a c t e r i s t i c value of Hc^  , corresponding t o d i f f e r e n t values of X,,. However, the ^ versus x„ curves show that there i s one value of X© I n each case f o r which tt|^> and hence the energy, H , i s a minimum. Graph 13: The variation of Ha^ with x 0 . At the beginning of each curve is shown the number of nodes in the - f ie ld. -15 b) Mass and c l a s s i c a l r a d i u s : The exis t e n c e of a minimum f o r the energy i s a s t r i k i n g f eature of the theory, and enables us t o obtain a d i s c r e t e mass spectrum* We i n t e r p r e t the p a r t i c u l a r s o l u t i o n s which y i e l d the minimum energy i n the three cases as r e p r e s e n t i n g three n e u t r a l p a r t i c l e s i n t h e i r normal s t a t e s , and we use the corresponding values of •To= as a measure of the s i z e of these p a r t i c l e s * 1C The mass r a t i o s are independent of n , K , and w i t h i n the l i m i t s of the approximations used and are found t o be 111*31) m ; yv\M : tv\ = /.03S : 1.032. ' I The r a t i o s of the r a d i i are a l s o independent of n , K , and ^ i n the same sense, and are 111.32) /t z : A , : Fl* - 3.^7 • 2*4- I The values of X„ used are those found by approximating Hc^xL) by a parabola i n the v i c i n i t y of the minimum. When we compare the t h e o r e t i c a l mass spectrum w i t h the mass spectrum of the known n e u t r a l members of the heavy p a r t i c l e f a m i l y , 111.33) m E : ™ A : »^ N= «. ifc : I - I f : / , the t h e o r e t i c a l values appear reasonable i n view of the s i m p l i c i t y of the model and the crudeness of the approximations used. c) Parameters c h a r a c t e r i z i n g the model: We f u r t h e r r e l a t e t h i s model t o r e a l i t y by s p e c i f y i n g the c l a s s i c a l radius and mass of the l i g h t e s t p a r t i c l e produced by the theory t o be those of a nucleon: -13 fza =r o.S x IO cm. I I I . 34) _ a 4 yri0 - j.fc7 x »o < j r » > -- 1 6 In t h i s way we ob t a i n III.35) 13 ~K_ - 3 . Ofe x to O W N . -17 3 - a . H 2..STS x - ' O d m . C r v \ . s e c . This value of K. , which c h a r a c t e r i z e s the ^  - f i e l d , cor-responds t o a "bare" nucleon f i e l d r e s t mass of 111.36) = I I SST -m , , Thi s comparison of the theory w i t h experimental values does not y i e l d n and ^ s e p a r a t e l y . d) "Quasicharge": Since we are at present d i s c u s s i n g s t a t i c s o l u t i o n s of the f i e l d equations, the p a r t i c l e s described are n e c e s s a r i l y e l e c t r i c a l l y n e u t r a l . However, the concept of a "quasicharge" of the *p - f i e l d p a r t i c l e s , a property of the p a r t i c l e s which i s "seen" by the - f i e l d , may be introduced. The massless Cf - f i e l d must be regarded as a hidden v a r i a b l e of the theory which produces s t a b i l i t y of the (J/-field p a r t i c l e s . I t has been introduced i n analogy w i t h an e l e c t r o -s t a t i c f i e l d , but cannot be i d e n t i f i e d w i t h the l a t t e r . However, an i n t e r e s t i n g f e a t u r e of the - f i e l d i s brought to l i g h t when the analogy i s continued and - ^ 7 i < p d 3 x i s c a l c u l a t e d f o r the s o l u t i o n s r e p r e s e n t i n g p a r t i c l e s I n t h e i r normal s t a t e s . With Gauss 1 theorem i n mind, we set 111.37) -yV^e«/*x = "«C<£ , and c a l l the quasicharge of the p a r t i c l e . From one of the f i e l d equations I I I . 4 ) w i t h vp*if^)we have --/Vc?d3X = <?Tfo/ Ik'V, At'dk. +«fro c ^2iQ^<U, 111.38) © •%• -17 where Y, and V3 are de f i n e d I n 1 1 1 , 3 0 ) , and I t i s found that the quasicharges of the three p a r t i c l e s represented by the nodeless, one-node, and two-node s o l u t i o n s are roughly equal: <^a = l . 4 o l v i ^ (^^»w. t»vi . sec. J I I I . 4 0 ) <£r = r » g ^ ( g m . c m . S e c } = l . 4 o | (gm- cm* See"*) i 2. e) Summary of r e s u l t s : T h i s i n v e s t i g a t i o n of time-independent s o l u t i o n s has produced three p a r t i c l e s w i t h the f o l l o w i n g d e s c r i p t i o n : NODES IN 4 » o f 2. REST MASS 2.. 130 2-J98 »0 njk. am. 9 , RADIUS 2 - 4 5 6-. 7^ ELECTRIC CHARGE o o e.s. u_. QUASICHARGE Z.4-ol 1.^ -32. «-^ o 1 ^"(grw.cm. Sec. However, there i s no apparent l i m i t t o the number of p a r t i c l e s which can be obtained by i n v e s t i g a t i n g s t a t i c s o l u t i o n s and in c r e a s i n g the number of nodes i n vj; , u n t i l the approximations used are no longer v a l i d . The tre n d observed i n the f i r s t three s o l u t i o n s i n d i c a t e s that the mass spectrum might tend t o a l i m i t . •18-6. L i m i t a t i o n s of the Model The preceding i n v e s t i g a t i o n was r e s t r i c t e d t o s t a t i c s p h e r i c a l l y symmetric s o l u t i o n s . The methods used are e a s i l y extended t o s o l u t i o n s of the type I I I . 4 1 ) ^ ; (S. = <Q M where ipo and *o are r e a l . The f i e l d equations are now The s u b s t i t u t i o n s at1'- vC- fsi" > o w. = V5. <k> III.43) c * ' reduce the f i e l d equations t o the form I I I . 6 ) again; the same approximate s o l u t i o n s are accepted, g i v i n g us <//,and c% : _ C e ~ A ° s<Vi fee? o^^) f o r ^ $ I I I . 4 4 ) ^ A = C . §L f o r ^J. ^  ><* c? = f o r A i Xo (0^ = ax^ f 0 r A where n o r m a l i z a t i o n 111.45) Jv~ty d 3x = 2L y i e l d s 111.46) C " = i i e ^ j -as before. We now use these s o l u t i o n s t o determine the mass and e l e c t r o s t a t i c charge of the p a r t i c l e s d e s c r i b e d . -19 III.48) With ^ = «li(*0 c"*** we have 111.47) = " f + ^ " ^ ^ ^ - < ? + - f • I n t e g r a t i o n by pa r t s and a p p l i c a t i o n of the f i e l d equations and boundary c o n d i t i o n s l e a d t o wi t h V, ^ , and ^  as defined i n I I I . 3 0 ) . We see that f o r s o l u t i o n s w i t h a gi v e n number of nodes and with a f i x e d value of OJ , the curve t*$- versus x c has the same shape as the curve versus xe f o r the corresponding s t a t i c case, and the minimum occurs f o r the same value of x 0 as before. However, the whole curve i s r a i s e d by the amount z-at^* The mass YV\' of a p a r t i c l e corresponding t o an o s c i l l a t o r y s o l u t i o n w i t h frequency cu i s III.49) m ' = - 2 ^ n . + m o < C* oC^ XC where n\ i s the mass of the corresponding p a r t i c l e w i t h a s t a t i c s o l u t i o n . The c l a s s i c a l r adius i s increased by the f a c t o r * • X The charge d e n s i t y f o r an o s c i l l a t o r y s o l u t i o n i s III.50) S. = i £ <±> t h * • ^ c T h i s y i e l d s an e l e c t r o s t a t i c charge of ni.5i) = £ g>J>; d 3 x -20-where £ la a. s o a l i n g f a c t o r which cannot be determined by the theory. C l e a r l y , s i n c e to i s not r e s t r i c t e d t o a d i s c r e t e spectrum but t o a continuous range of v a l u e s , <*><<c, and s i n c e £ i s u n s p e c i f i e d , i t i s not p o s s i b l e to grasp w i t h t h i s c l a s s i c a l model the q u a n t i z a t i o n of charge. I t i s a l s o obvious that the i n s t a b i l i t y and l i f e t i m e s of the heavy p a r t i c l e s are beyond the c a p a c i t y of t h i s model• 21 IV. POSSIBLE IMPROVEMENT OF THE MODEL 1. S p e c i f i c Model and F i e l d Equations I n an attempt t o improve the t h e o r e t i c a l mass spectrum we a s s i g n a "hare" mass t o the ($ - f i e l d as w e l l as t o the U / - f i e l d . Thus we consider two coupled f i e l d s w i t h Lagrangian d e n s i t y i v . l ) px^Zxr ax^ax^ 0 and we re q u i r e t h a t IV.2) £ \ « I • The f i e l d equations i v . 3 ) ay-<y.-^ become ^ IV. 4) , i - , v - , _ f o r s t a t i c , s p h e r i c a l l y symmetric s o l u t i o n s IV. 5) The s u b s t i t u t i o n s IV. 6) l e a d t o IV.7) G r a p h 14: N o d e l e s s solutions o f e q u a t i o n s 1 2 . 7 ) u v e r s u s x . G r a p h 15: N o d e l e s s s o l u t i o n s o f e q u a t i o n s v v e r s u s x . 22 The "boundary c o n d i t i o n s on u. and v are OL-OJ V = O cut X = o X ' X IV. 8) I n the region x » f we have IV. 9) w i t h acceptable s o l u t i o n s -x IV. 10) _ Q u a l i t a t i v e s o l u t i o n s of equations IV.7) w i t h boundary cond i t i o n s IV.8) are drawn i n graphs 14 and 15.. 2. Approximate S o l u t i o n of the Equations We must again adopt approximate methods of s o l u t i o n because of the n o n - l i n e a r i t y of the equations. We put v = ax IV.11) /• \ u = A s.Vi (/a*-1 x) i n the region of s m a l l x , and j o i n these t o the asymp-t o t i c expressions IV.10) at x , • The approximate s o l u t i o n s are shown i n graphs 16 and 17. C o n t i n u i t y of u. , v , and u / at x 0 r e q u i r e IV.12) sin(7ser x,) i v . i 3 ) x ; s W f ^ i j , J a % 7 which l e a d t o : G r a p h 16: A p p r o x i m a t e s o l u t i o n o f e q u a t i o n s 3 2 " . 7 ) f o r t h e n o d e l e s s c a s e : u v e r s u s x . v » x G r a p h 1 7 : A p p r o x i m a t e s o l u t i o n o f e q u a t i o n s I S . 7 ) f o r t h e n o d e l e s s c a s e : v v e r s u s x . -23-IV.14) a. = £ej for ^„^£, /*- * T V n = x „ f = i < for IV.15) q _ J ^ K l(Qx = ax. e* ^ L . for A„ £ ^ A . K Since the v|/ - f i e l d i s the same as I I I . 2 1 ) , n o r m a l i z a t i o n gives expression I I I . 2 6 ) again f o r C*. To c a l c u l a t e the energy of the system we form IV. 16) ~CU = " L(<*- 1<0 V + ^ V * ( ( v <f] and reduce IV.17) H = TWC* = - f%+ c/\ by I n t e g r a t i o n by p a r t s and use of the boundary c o n d i t i o n s and f i e l d equations, t o f i n d : i v . i e ) - ^(Vofc) where Y, and Y3 are de f i n e d i n I I I . 3 0 ) and IV.19> * - 3 a \ - ^ S ( ^ ) t ^ f £ < ^ • 3. Numerical R e s u l t s We now t i e t h i s model down t o r e a l i t y by c o n s i d e r i n g the *\* - f i e l d as the d e s c r i p t i o n of a bare nucleon, and the < ? - f i e l d as the d e s c r i p t i o n of a bare meson f i e l d a c t i n g as glue . I n p a r t i c u l a r , we assume t h a t the bare mass r a t i o r e f l e c t s approximately the observable mass r a t i o of the corresponding dressed p a r t i c l e s , and we s u b s t i t u t e iv.20) \ - / £ N = i • TABLE I I VALUES OF Hj| AND CORRESPONDING VALUES OF Xa FOR THE NODELESS AND ONE-NODE SOLUTIONS NODELESS SOLUTION ONE-NODE SOLUTION h K % 0.4285 5.092 • 0.3573 2.813 1.800 3>479 1.2860 2.206 3..001 2.368 1.7140 1.9825 4.028 2.072 2.1000 1.9085 4..7I6 1.993 2.3580 1.8950 5.143 1.9695 2.4000 1.8948 5..400 I..963 2.4S56 1.8957 5.571 I.96O 2.7000 I .9050 5..700 I.96O 2,.87l6 1.9188 6.000 1.963 3.4286 1-991 6.427 I .976 5.356O 2.404 -24-G a l o u l a t i o n s of H^. were c a r r i e d out f o r d i f f e r e n t x„ and f o r the case of having no nodes and one node* The r e s u l t s are tabulated i n Table I I * We see th a t Hg-(x„) again has a minimum i n each case; the minimum values and the corresponding values of x„ are Xo= - 2 - i - (nodeless case) x ^ s y.fe (one-node case) x„ = 2.Hi (nodeless case) x„ = sns (one-node case) i n the massless <? - f i e l d model* The mass r a t i o of the two l i g h t e s t p a r t i c l e s has Increased only very s l i g h t l y , from 1.032 : 1 t o 1.034 : 1* Since no s i g n i f i c a n t changes I n the features of the model have been produced, t h i s model was not i n v e s t i g a t e d any f u r t h e r * The assumption t h a t bare mass r a t i o s r e f l e c t dressed mass r a t i o s i s very dubious i n any case* IV.21) as compared IV.22) «9- = ft-w i t h ».??s f o r l.«?6o f o r /. f o r f o r -25-APPENDIX I Dimensional A n a l y s i s of Q u a n t i t i e s Occurring i n the Type ( l b ) Model Discussed I I I . l ) ^ = _ dj}>Zjf- t c > i f ~ - W ^ +c?W*cf A l s o , since must be an energy d e n s i t y , we have From t h i s we conclude t h a t = L ( M L 3 T ~ l ) - i - 2 6 -APPENDIX I I A Sample o f the C a l c u l a t i o n s Made f o r the Model i n which $ i s a Massless F i e l d , C a l c u l a t i o n o f HJ| corresponding to x„ - 5.75" f o r the case of «f having one node. Wf-/^7") 5.5183 cos(7^Tx.") 0.72149 J^T 0.9597 sin(^Tx.)cos(J5%7x^ O..4996 K = * » & m 5 > 7 5 o o s i n T i > T 2 - ° * 5 7 2.x„ 11.500 >-SL 2.0017 /"-ST^u. 8.1498 x 1 0 " 7 22.0915 98716. 4(tf-x.)a 254.05 e2t4V^ 0.080452 ^ ) V ^ ^ 20.439 0..9210 ^ X ^ - ^ ^ T ^ n ^ x . ) 6.0179 1.9210 X 25.124 Jc^Tx. 5.5183 K I . 6 5 3 S^(Ja?T>C) - O..69242 V 3 13.552 t±S- = X-^Vx = 1.9759 • n K >^  Tables which were found u s e f u l were: 1 ) B r i t i s h A s s o c i a t i o n f o r the Advancement o f Science Mathematical Tables: Volume I ( f o r the exp o n e n t i a l i n t e g r a l ) ; 2) U. S. Department o f Commerce, N a t i o n a l Bureau of Standards, A p p l i e d Mathematics S e r i e s , No. 26: Table of A r c t a n x. -27 BIBLIOGRAPHY PAPERS (1) Mie, G. Annalen der Physik, 37, 511, 1912 39, 1, 1912. (2) Rosen, N. Phys. Rev. 55, 94, 1939. (3) F i n k e l s t e i n , R. J . Phys. Rev. 75, 1079, 1949. (4) Menius, A. C. and Rosen, N. Phys. Rev. 62, 436, 1942. (5) D r e l l , S. D. Phys. Rev. 79, 220, 1950. (6) F i n k e l s t e i n , L e L e v i e r , and Rudennan Phys. Rev. 83, 326, 1951. (7) Rosen, N. and Rosenstock, H. B. Phys. Rev. 85, 257, 1952. (8) Heisenberg, W. Z e i t s , f . Naturforschg. 9a, 292, 1954 10a, 425, 1955. (9) Heber, G. Annalen der Physik, 16, 43, 1955. (10) K l t a , H. Progr. Theor. Phys. (Japan), 15, 83, 1956. BOOKS FOR GENERAL REFERENCE Wentzel, G. "Quantum Theory of F i e l d s " Schweber, Bethe, and de Hoffman "Mesons and F i e l d s " Volume 1: F i e l d s . 

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