QM Graphic Dynamics

7/28/2019 QM Graphic Dynamics

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Fou nda t ions o f Physics , VoL 18, No. 7, 1988

Quantum Graphic DynamicsS t a n l e y P . G u d d e r 1Rece i ved June 12 , 1986A d i sc re t e quan t um mechan i c s i s deve l oped an d used t o cons t ruc t mode l s f o r d is -c re t e space - ti me an d f o r t he i n t e rnal dynam i cs o f e lemen t ary par t i cl e s . T h i sdynam i cs i s g i ven i n t e rms o f par t i c l e s pe r f orm i ng a q uant um random wal k on amult igraph.

1 . I N T R O D U C T I O ND a v i d B o h m p u b l i s h e d a f a s c i n a t i n g a r t i c l e 2 5 y e a r s a g o i n w h i c h h ep r o p o s e d a t o p o l o g i c a l f o r m u l a t i o n o f q u a n t u m t h e o r y / ~ ) H e c a l le d th i sf o r m u l a t i o n a t o p o - c h r o n o l o g y t h e o r y , b y w h i c h h e m e a n t t h e s t u d y o fo r d e r a n d r e l a t i o n s h i p i n p r o c e ss . T h e b a s is o f h i s t h e o r y is t h a t e v e r ya c t i o n i s a r e s u l t o f d i s c r e te p r i m a r y a c t i o n s w h i c h t a k e p l a c e a t t h e s p e e dof l i gh t c . A ge nera l ac t i o n a t a speed l e s s t han c cons i s t s o f a z ig -zag se r ie so f p r i m a r y c o n t a c t s a t t h e s p e e d e , a n d a ll a p p a r e n t l y c o n t i n u o u sm o v e m e n t s a t s p e e d s le ss t h a n c a r e t o b e e x p l a i n e d a s t h e r e s u l t o f s u c h at r e m b l i n g m o v e m e n t . A c c o r d i n g t o B o h m : " T h i s f i t s i n w i t h m o d e r nr e la t iv i st ic q u a n t u m m e c h a n i c a l i d e as c o n c e r n i n g t h e m o v e m e n t o f th e e le c-t r o n . D i r a c ' s e q u a t i o n i m p l i e s , f o r e x a m p l e , t h a t t h e e l e c t r o n m o v e s a t t h es p e e d o f li g h t in t r e m b l i n g m o v e m e n t s ( c al le d Zitterbewegungen). ( T h ea v e r a g e v e l o c i ty , w h i c h is l es s t h a n t h a t o f l ig h t , t h e n c o r r e s p o n d s t o a k i n do f c i r c u l a t i o n i n a sp i r a l p a t h t h a t i s r e s p o n s i b l e f o r t h e p h e n o m e n aa s s o c i a t e d w i t h t h e e l e c t r o n s p i n . ) " T h i s e x p l a i n s w h y c i s a b s o l u t e . I n f a c t ,c is the only p o s s i b l e s p e e d a n d a ll o t h e r s a r e o n l y a p p a r e n t . A s a s im p l i fi e de x a m p l e , s u p p o s e a p a r t i c l e m o v i n g i n o n e d i m e n s i o n t r a v e l s t w o s t e p s

1Department of Mathematics and Computer Science, University of Denver, Denver,Colorado 80208.751

0015-9018/88/0700-0751506.00/0 1988 Plenum PublishingCorporation


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forward and one step backward in a repeated motion. Although its actualspeed is c, it has an apparent speed of c / 3 .

Bohm then proposes a mathematical model for his topo-chronologytheory. He represents every actual movement by a point (or vertex) andevery actual process of immediate contact between such moments by a line(or edge). This corresponds mathematically to a graph. Relationshipswhich define each moment in terms of past moments are described by cer-tain kinds of matrices. He indicates that these matrices have a close connec-tion with fermions and bosons and they might provide an explanation forthe structure of elementary particles. In the present paper we intend toexpand upon Bohm's ideas by constructing various concrete mathematicalmodels. However, instead of calling this a topochronology theory we shalluse the term quantum graphic dynamics.

As its name implies, quantum graphic dynamics (QGD) consists oftwo main ingredients, a graph and a quantum dynamics. The graph (ormultigraph) is interpreted as a generalized discrete phase space in whichthe vertices represent discrete positions which a particle can occupy and theedges represent discrete directions that a particle can propagate. The quan-tum dynamics is induced by a transition amplitude which generates aquantum random walk on the graph.Our developmment of QGD has two purposes.. First, QGD can beused to describe the motion of a particle in discrete space-time. In this case,the graph represents an actual discrete phase space and the discrete time isgiven by the time steps of the random walk. Second, QGD can be used todescribe the internal dynamics of "elementary particles." In this case, thevertices represent quarklike constituents of a particle, and edges representinteraction paths for gluons which are emitted and absorbed by the ver-tices. A vertex can emit or absorb a gluon at each time step of the randomwalk.In Section 2 we begin the development of QGD for an abstract mul-tigraph. Section 3 illustrates this theory by considering two models of dis-crete space-time. The first model is a cubic lattice, and the second is a treemodel. Section 4 develops a multigraph model for elementary particles, andSection 5 derives the gluon dynamics for these multigraphs. For otherapproaches to discrete quantum mechanics, the reader is referred to Refs.1-4 and 6-9.

2. ABSTRACT MULTIG RAPHSIn a previous paper, ~5) we began to develop a theory of discrete quan-tum mechanics. To make the present work self-contained, we shall review


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Quantum Graphic Dynamics 753s o m e o f t h a t m a t e r i a l h e r e. M o r e o v e r , i n R ef. 5 w e o n l y c o n s i d e r e d g r a p h sw h i l e i n t h i s w o r k w e s h a l l r e q u i r e a t h e o r y o n m u l t i g r a p h s .A multigraph i s a p a i r G = ( V , E ) , w h e r e V is a n o n e m p t y s e t a n d E i sa c o l l e c t i o n o f o n e - a n d t w o - e l e m e n t s u b s e t s o f V . T h e e l e m e n t s o f V a r ec a l l e d vertices, t h e t w o - e l e m e n t s e t s i n E a r e c a l l e d edges, a n d t h e o n e -e l e me n t s e t s i n E a r e c a l l e d loops. T h e s e t s i n E m a y b e r e p e a t e d , a n d ar e p e a t e d p a i r o f e d g e s is ca l l e d a biedge. T h e degree o f a v e r t e x is t h e n u m -b e r o f e d g e s c o n t a i n i n g ( o r i n c i d e n t t o ) i t. A s f a r a s t h e d e g r e e is c o n c e r -n e d , a l o o p i s c o n s i d e r e d t o b e t w o e d g e s i n c i d e n t t o a v e r t e x . T h u s , e a c hl o o p i n c i d e n t t o a v e r t e x a d d s t w o t o i t s d e g r e e .

S u p p o s e t h a t V = { v j : j ~ J } a n d t h a t t h e e d g es c o n t a i n i n g v j a r ed e n o t e d b y e j k , k ~ K ( j ) . I f {vr , vs} ~ E , we w r i te vr 2_ vs an d say th a t v r , v~a r e adjacent (r m a y e q u a l s ) . T h e phase space on G i s t he s e t

S = { ( v j , ej~): j ~ J , k ~ K ( j ) } ~ V x EI f q = (vj , e jk) , q ' = (vj, , e j,k ,) e S , an d vj 2_ v / w e w ri te q 2_ q ' . F o r n E ~ , ann-path i s a s e q ue n c e o f no t ne c e s sa r i l y d i s t i nc t e l e m e n t s q0 ,..., q , ~ S w i thqj 2_ qj+ 1, J = 9 . .. .. n + 1. W e ca ll q0, q , th e initial and fina l e l e m e n t s o f t h en - p a t h , r e s p e c t iv e l y . D e n o t e t h e s e t o f n - p a t h s w i t h i n i t i al e l e m e n t q o a n df i n a l e l e m e n t q b y ~ , ( q o , q ) .W e i n t e r p r e t V a s a se t o f d i s c r e t e p o s i t i o n c o o r d i n a t e s f o r a p a r t i c l ea n d a d j a c e n t v e r t ic e s c o r r e s p o n d t o " n e a r e s t n e i g h b o r " p o s it i o n s . A n ed g ec o r r e s p o n d s t o a d i r e c t i o n t h a t a p a r t i c l e c a n m o v e , s o i t r e p r e s e n t s a d i r e c -t io n o f m o m e n t u m . I f e = { v l, v 2 } ~ E , t h e n a p a r t i c l e l o c a t e d a t V l c a nm o v e a l o n g e t o v2 i n o n e t i m e s t ep . A n n - p a t h p ~ ~ n ( q 0, q ) is a p o s s i b l et r a j e c t o r y f o r a p a r t i c l e m o v i n g i n a d i s c r e te p h a s e s p a c e f r o m q o t o q in nt ime s t e ps .

A f u n c t i o n A : S S ~ C is a transition amplitude i f A ( q , q ' ) = O i fq J_ q ' a nd f o r e ve r y q l , q2 ~ S we ha ve

A(q l , q ) A( q2 , q ) = ~ a(q, qv) A(q, q2) = ( ~ q l q 2 (1 )q q

w h e r e . ~ d e n o t e s t h e c o m p l e x c o n j u g a t e o f A . I f p = { qo ,..., q , } ~ ~ , ( q o , q ),t h e amplitude of p i s

A (p )= A(qo, q~) A(q~, q2) . . . A (q , 1, q,)F o r qo , q ~ S , t he n-ste p transition amp litude fro m qo to q is

A,(qo, q ) = Z { A ( p ) : P ~ , ( q o , q )}a n d b y c o n v e n t i o n A o (q o , q ) = ( ~ q o q " T h e n-step transition probability fr o mqo to q is P,,(qo, q ) = [ A n (q o , q )j 2.

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W e c a n m o t i v a t e o u r d e f i n i t io n o f a t r a n s i t i o n a m p l i t u d e p h y s i ca l ly .F i r s t , w e i n t e r p r e t A ( q , q ' ) a s t h e p r o b a b i l i t y a m p l i t u d e t h a t a p a r t i c l em o v e s f r o m q t o q ' i n o n e t i m e s t ep a n d w e i n t e r p r e t A ( q , q ' ) a s t h ep r o b a b i l it y a m p l i t u d e t h a t a p a r t ic l e m o v e s f r o m q ' t o q i n m i n u s o n e t i m es teps . Al te rna t ive ly , i f a p ar t i c le i s a t q ' a t t im e t , the n .,/ (q , q ' ) i s thep r o b a b i li ty a m p l i t u d e th a t i t w a s a t q a t ti m e t - 1 . T h e n A ( q, q ' ) = 0 ifq l _ q ' m e a n s t h a t a p a r t i c l e c a n o n l y m o v e t o a d j a c e n t v e r t i c e s i n o n e t i m es t e p . M o r e o v e r , w e t h e n i n t e r p r e t A ( q l , q ) A ( q 2 , q ) a s t h e p r o b a b i l it ya m p l i t u d e t h a t a p a r t i c l e m o v e s f r o m q l t o q 2 v i a q i n z e r o t i m e s t e p s . I tf o l l o w s t h a t Z q A ( q l , q ) A ( q 2 , q ) i s t h e t o t a l p r o b a b i l i t y a m p l i t u d e t h a t apa r t i c l e moves f rom q~ t o q2 i n ze ro t ime s t eps . I t i s c l ea r t ha t t h i s shou lde q u a l 6q~q2. S i m i l a r r e a s o n i n g a p p l i e s t o t h e o t h e r e q u a l i t y i n E q . ( 1 ) .

T h e f o l l o w i n g r e s u l t i s a s l ig h t g e n e r a l i z a t i o n o f a t h e o r e m p r o v e d i nR e f. 5 . I t s t a t e s t h a t p r o b a b i l i t y is c o n s e r v e d a n d t h a t a n a m p l i t u d eC h a p m a n - K o l m o g o r o v e q u a l i t y h o l d s .

T h e o r e m 1 . ( a ) ~q IA , (qo , q )l 2 = Z q [A~(q , qo )12= 1 , n ~ ~ . (b ) I fm , n ~ ~ , m ~< n, th en

A n ( q o , q ) : ~ Am(qo,q ' ) An m(q ' , q )q 'Le t 12 (S ) be t h e H i lbe r t space o f fun c t i on s { f : S ~ C : 52q I f (q ) ] 2 < oo }

w i th i nn e r p r o d u c t ( f , g ) = ~ f ( q ) g ( q ) . A n o r t h o n o r m a l b a sis f o r / 2 ( S ) i s{ 6 q : q s S } , w h e r e 6 q ( q ' ) = 6 q q , . I t f ol lo w s f ro m T h e o r e m l ( a ) t h a tA , ( q o , . ) , A , ( . , q o ) a r e u n i t v e c t o r s i n / 2 ( 8 ) . L e t T , U b e l i n e a r o p e r a t o r s o n/2(8) d e f i e d b y( T f ) ( q ) = ~ A ( q , q ' ) f ( q ' ) ,q ' ( U f ) ( q ) = ~ A ( q ', q ) f ( q ' )q '

W e c a l l T a n d U p r o p a g a t o rs fo r A . The nex t r e su l t s l i gh t l y genera l i zes at h e o r e m i n R e f . 5 .

T h e o r e m 2 . ( a ) T a n d U a r e u n i t a r y o p e r a t o r s a n d U = T * . ( b)A n ( q o , q ) = ( T"3q, ( ~ q o ) = ( ( ~ q , U n b q o ) .F i x q o ~ S a n d l e t S n ( q ) = A n( qo , q ) b e t h e w a v e f u n c t i o n . W e h a v e

a l r e a d y n o t e d t h a t ~ n i s a u n i t v e c t o r i n I2(S). A p p l y i n g T h e o r e m l ( b ) , w eo b t a i n

~P~+ ( q ) = ~ A n ( q o , q ' ) A ( q ' , q ) = ~ A ( q ' , q ) t~ , ( q ' ) (2 )q' q 'W e s h a l l l a t e r s h o w t h a t t h e d i f fe r e n c e e q u a t i o n ( 2 ) is a d is c r e te a n a l o g o ft h e D i r a c e q u a t i o n . W e c a l l E q . ( 2 ) t h e d i s cr e t e wa ve eq u a t io n .


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o f M w i t h m u l t i p li c i ty m a n d t h e f o ll o w i n g a r e c o r r e s p o n d i n g o r t h o n o r m a te i g e n v e c t o r s :

( 0 2 , 0 . . . . . 0 ) / , , / L . . , ( 0 , 0 , . . . , 0 , 0 2 ) / , / 2M o r e o v e r , a + b e ~ + 2(m - 1 ) c e ~ is a n o n d e g e n e r a t e e ig e n v a l u e o f M w i t hc o r r e s p o n d i n g n o r m a l i z e d e i g e n v e c t o r

( 0 1 , 0 1 .. .. .F i n a l l y , a + b e ~ - 2 c e ~ is a n e i g e n v a l u e w i t h m u l t i p l i c i t y m - 1, a n d t h ef o l lo w i n g a r e c o r r e s p o n d i n g o r t h o n o r m a l e i g e n v e ct o rs :

( 0 1 , - 0 1 , 0 ..... 0 ) / 2( 0 1 , 0 1 , - 2 0 1 , 0 ..... 0 ) / x / J 2

( 0 1 , 0 1 ..... 0 1 , - ( r n - 1 ) 0 1 ) / , J 2 ( m - 1 )mTheorem 4 . T h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t :( a ) A is a t r a n s i t i o n a m p l i t u d e .(b ) l a - bei l = la + b e ~ - 2ceeOl -= la + be i + 2(m - 1) ce~O = 1.( c ) M is a u n i t a r y m a t r i x .P r o o f T h a t ( a ) im p l i es ( b ) f o l lo w s f r o m e x p a n d i n g t h e s q u a r e s o f t h e

e x p r e s s i o n s i n ( b ) a n d a p p l y i n g T h e o r e m 3. I f (b ) h o l d s , t h e n M is an o r m a l m a t r i x w i t h e i g e n v a lu e s o f m o d u l u s o n e . I t f o ll o w s f r o m t h es p e c t r a l t h e o r e m t h a t M i s u n i t a r y a n d h e n c e ( c ) h o l d s .

N o w s u p p o s e M is u n i t a r y , a n d d e n o t e i ts e n t ri e s b y M i k , j ,k = 1, .. ., 2m . T h e n ~.k M~k-Myk= ~jj. , j , j ' = 1 ,..., 2m. I f j = j ' , we ob ta in

1 = ~ I M j k l 2 = a 2 + b 2 + 2 ( m - 1 ) c 2k

a n d h e n c e C o n d i t i o n ( a ) o f T h e o r e m 3 h o l d s . I f j C j ' , t h e r e a r e t w o c a se s .In t he f i r s t case , I J - J ' l = 1 a n d m i n ( j , f ) is o d d . F o r t h i s c a se , w e h a v e

0 = E M )e~ / i J ' k = 2ab cos 0 + 2 (m - 1 )c 2k

a n d h e n c e ( b ) o f T h e o r e m 3 h o l ds . O t h e r w i s e , w e h a v e t h e s e c o n d c a s ew h i c h g i v e s

0 = ~ m j k M j ,k = 2 a c cos + 2 b c cos (0 - - ~b) + 2(m - 2) c 2k


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Quantum Graphic Dynamics 757I f c 0 a n d m ~> 2 , w e o b t a i n ( c ) o f T h e o r e m 3 . O t h e r w i s e , c = 0 a n d m = l ,s o t h i s c o n d i t i o n is v a c u o u s . |

W e n o w c o n s i d e r t h e d is c r e te w a v e e q u a t i o n i n t h i s c o n te x t . A s s u m e Ai s a t r a n s i t i o n a m p l i t u d e a n d w r i t e E q . ( 2 ) a s f o l l o w s :

2m~n+ l(J for)= 2 A((J--fC s'~s)'(J ' fCr))~ tn(J--ks' fCs) ( 3 )S=lI f we use th e n o ta t io n ~S(n , j ) = ~ , , ( j , /~ , ) , s = 1 ,.. ., 2m, w e ca n w ri te (3) inm a t r i x f o r m :

" = M " ( 4 ),,j)j [/2m(n,j-k2m)where M is the unitary matrix considered above. In a rough sense, Eq. (4)is a discrete analog of Dirac's equation since the left-hand side has theposition variable j fixed and the time variable n is incremented, while theright-hand side has the time variable n fixed and the position variable j isi n c r e m e n t e d .

F o r s i m p l i c i ty , e t u s c o n s i d e r t h e o n e - d i m e n s i o n a l c a s e, m = 1. I n t h i sc a s e , t h e p a r a m e t e r s c a n d ~ d o n o t a p p e a r a n d t h e c o n d i t i o n s f o r u n i t a r i t yi n T h e o r e m 3 be c o m e : a 2 + b 2 = 1 , a b c o s 0 = 0 . I t f o ll o w s t h a t b = ( 1 - a Z ) 1/2a n d 0 = ~ / 2 o r 3 r r/2 . F o r c o n c r e t e n e s s , s u p p o s e 0 = r e/2 . W e t h u s h a v e o n l yo n e f i'e e p a r a m e t e r 0 < a < 1. I n t h i s c as e , E q . ( 4 ) i s e q u i v a l e n t t o t h ef o l l o w i n g t w o e q u a t i o n s , w h e r e j s Z :

O ~ (n + 1 , j ) = a O l ( n , j - 1 ) + i b O 2 (n , j + 1) (5)02 (n + 1 , j ) = i b O ' ( n , j - 1 ) + a~ ,2 ( n, j + 1) (6)

F r o m E q . ( 5 ) w e o b t a i ni ia~ tZ (n, j ) = - ~ ~ l ( n + l , j - 1 ) + ~ - ~ l ( n , j - 2 ) ( 7)

S u b s t i t u t i n g ( 7 ) in t o ( 6 ) a n d s i m p l i f y in g g i ve s0 1 ( n + l , j ) + ~ k l ( n - l , j ) = a [ ~ t ( n , j + l ) + ~ l ( n , j - 1 ) ] (8 )

A s i m i la r p r o c e d u r e s h o w s t h a t E q . ( 8 ) a ls o h o l d s f o r ~ 2 ( n , j ) . N o t i c e t h a tE q . ( 8) is a d i s c re t e a n a l o g o f th e K l e i n - G o r d o n e q u a t i o n .

T h e s o l u t i o n o f (4 ) i n c l o s e d f o r m a p p e a r s t o b e q u i t e d i ff ic u l t. T h i sh a s b e e n s o l v e d f o r m = 1 i n R e f . 5. A l t h o u g h t h e m e t h o d e m p l o y e d t h e r e


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c a n b e e x t e n d e d t o h i g h e r d i m e n s i o n s , t h e w o r k i n v o l v e d i s v e r y t e d i o u s .H o w e v e r , a r e l a t e d p r o b l e m w h i c h i s p h y s i c a l l y r e l e v a n t c a n b e e a s i l ys o l v e d . W e c a l l t h i s t h e m o m e n t u m p r o b l e m . L e t

O n ( s ) = ~ t pS ( n, j ) = ~ A n ( q o , ( j , ~ s ) )J /

W e c a l l ~ b . ( s ) t h e m o m e n t u m w a v e f u n c t i o n a n d n o t e t h a t i t g i v e s t h ep r o b a b i l i t y a m p l i t u d e t h a t a p a r t ic l e m o v e s in t h e /~ . d i r e c t i o n a t t i m e ng i v e n t h a t i t b e g a n a t q o a t t i m e 0 .

I f w e s u m E q . (4 ) o v e r j , w e o b t a i n

~ + 1 ( 2 m ) L ~ b n ( 2 m ) ]= M ~ + l (9 )l + o ( i )

S i nc e ~ bo(S i s c l e a r l y a u n i t ve c t o r i n C 2m a n d M i s un i t a r y , i t f o l l ow s t ha t~bn(s) i s a un i t ve c t or in C 2m.

S i n c e w e k n o w t h e e i g e n p a i r s f o r M ( t h e s e w e r e l i s t e d r i g h t b e f o r eT h e o r e m 4 ), w e c a n d i a g o n a li z e M a n d s ol ve th e m o m e n t u m p r o b le m . T of a c i l it a t e t h i s, d e f i n e t h e 2 x 2 m a t r i c e s - - I ' 1 ' d ,T h e n t h e p r o j e c t i o n s P1, P2, P3 o n t o t h e e i g e n s p a c e s f o r t h e e i g e n v a l u e s21 = a - be i, 22 = a + b d + 2 ( m - 1 ) e g , 23 = a + b e ~ - 2 c d ~ , r e s p e c t i v e l y ,b e c o m e

i i o . . . l 10 E . - . 1 D D . . . De l = ~ , P 2 = ~m m0 . . . D D D"1( m - 1 )D - D . . . . /

1 [ _ ~ ( m - - 1 ) D . . . . _ D

o 13 =~mm " - -

- - O . . - ( m 1)a n d f r o m t h e s p e c t r a l t h e o r e m w e h a v e

M = 2 1 P ~ + 2 2 P 2 + )~3P3 (10)


9/26

Quantum Graphic Dynam ics 759

S u p p o s e t h e i n i t ia l m o m e n t u m is i n t h e / ~ r d i r e c t i o n . T h e n ~ bo (S )= 6 ,r a n d ,by Eq . (9 ) , q~ , ( s )= ( M ~ o ) ( s ) . A p p l y i n g E q . ( 1 0 ) g i v e s t h e f o l l o w i n g e x p l i c its o l u t i o n t o t h e m o m e n t u m p r o b l e m :

~ ( s ) = ( a - - b e i ) ~ ( P l ). ,. r + [ a + b e i + 2(m - 1) c e i ~ ] " ( P 2 ) s r+ ( a + b e i - 2 c e i ) n ( P 3 ) , r

T h e c u b i c la t t ic e s j u s t c o n s i d e r e d d o n o t g i v e a v e r y r e a l is t ic m o d e ls in c e t h e d i r e c t i o n s o f m o t i o n a r e q u i t e l i m i te d . F o r e x a m p l e , i n t h e t h r e e -d i m e n s i o n a l c a s e , a p a r t i c l e c a n o n l y m o v e i n s i x d i r e c t i o n s . H o w e v e r , w ec a n u s e s o m e o f o u r p r e v i o u s r e s u l ts i n th e n e x t m o d e l . I n t h is m o d e l a p a r -t ic l e c a n m o v e i n a n a r b i t r a r y f i n it e n u m b e r o f d i r e c t io n s . W e s h a l l,h o w e v e r , r e s t r i c t o u r a t t e n t i o n t o t h e t w o - a n d t h r e e - d i m e n s i o n a l c a s e s .

W e b e g i n i n t w o d i m e n s i o n s . L e t /~ 1 b e a u n i t v e c t o r i n R 2 w h i c h w et a k e f o r c o n c r e t e n e s s t o b e i n t h e h o r i z o n t a l d i r e c t i o n . L e t n e N , a n d l e tc~rr/n b e a n a n g l e . F o r m t h e u n i t v e c t o r s ~ 1 , -. ., /~ 2 ~ , w h e r e / ~ j - / ~ l =c o s ( j - 1 )~ , j = 2 , . . . , 2 n. T h u s , t h e /~s a r e u n i t v e c t o rs a n d e a c h f o r m s a na n g l e ~ w i t h i t s p r e d e c e s s o r . W e c a l l a p o i n t x ~ ~ 2 a c c e s s i b l e i f i t has t hef o r m x = ~ k s , k j~ {/~1 .... /~2~}. L et V be the se t o f access ib le po in t s an d ,for x , y ~ V, def ine x _1_ y i f x - y e { /c~,...,/~2 ,} . Le t G = (V, E ) b e the g ra phi n w h i c h t h e e d g e s a r e r e p r e s e n t e d b y t h e s t r a i g h t l i n e s b e t w e e n a d j a c e n tv e r t i c e s . T h e p h a s e s p a c e S c a n b e r e p r e s e n t e d a s f o l l o w s :

s = { ( x , k ) : v ,W e n o w d e f in e a f u n c t i o n A : S x S ~ C a s f o l lo w s :

A ( ( x , k ) , ( x + k , k , ) ) = { a i f k ' = kb e ~ i f k ' kw h e r e a , b > 0 , 0 ~ [ 0, 2 re ) a n d A ( ( x , k ) , ( x ' , k ' ) ) = 0 o th e rw i s e. P h y s ic a l ly ,t h is m a y b e i n t e r p r e t e d a s f o l lo w s . I f a p a r t i c l e i s m o v i n g i n d i r e c t i o n k ,t h e n a t t h e n e x t t i m e s t e p i t c o n t i n u e s i n t h i s d i r e c t i o n w i t h a m p l i t u d e aa n d c h a n g e s t o o n e o f t h e o t h e r p o s s ib l e d i re c t io n s w i t h a m p l i t u d e b e ~. I tis e a s y t o c h e c k t h a t A is a t r a n s i t i o n a m p l i t u d e i f a n d o n l y i f i t s a ti sf ie s

a 2 + ( 2 n - - 1 ) b 2 = 1 ( 1 1)a c o s 0 + ( n - 1 ) b = 0 ( t2 )

H e n c e , t h e r e i s o n l y o n e f r e e p a r a m e t e r w h i c h w e t a k e t o b e a . S o l v i n g ( 1 1 )an d (12 ) in t e rm s o f a g ives

{ 1 - a 2 ~ 1 / ' 2b = ( 1 3 )c o s 0 = (1 - n ) ( 1 - - a 2 ) "] i/2a \ 2 - ~ ~ - - ] ( 1 4 )


10/26

760 Gudder

B e c a u s e o f E q . ( 1 4 ) , a c a n n o t h a v e a r b i t r a r y v a l u e s in t h e i n t e r v a l ( 0, 1 )b u t m u s t s a t i s f y

( n - - 1 ) / n < . a < 1I n th e l im i t ing c a se a = ( n - 1 ) /n , we ob ta i n

(15)

b = 1 / n , 0 = g ( 1 6 )I n t h i s m o d e l , t h e d i s c r e t e w a v e e q u a t i o n is s i m i l a r t o E q . ( 4 ) e x c e p t

t h a t t h e m a t r i x M is re p l a c e d b y t h e 2 n x 2 n m a t r i x M ' w i t h a ' s o n t h em a i n d i a g o n a l a n d be i e l s e w h e r e . T h e s o l u t i o n t o t h e m o m e n t u m p r o b l e mi s s i m i l a r t o o u r p r e v i o u s w o r k . I n f a c t , a l l w e h a v e t o d o i s r e p l a c e ce ~ b ybe ~ w h e n e v e r i t a p p e a r s .

T h e e x t e n s i o n o f t h e a b o v e t o t h r e e d i m e n s i o n s is s t r a i g h t f o r w a r d . F o rn ~ N we a ga in l e t ~ = rein. F o r m a s p h e r i c a l c o o r d i n a t e s y s t e m a n d l e t / ~ 1b e a u n i t v e c t o r i n t h e v e r t i c al d i r e c ti o n . W e n o w h a v e n p o l a r a n g l es ,~ , 2~,.. ., n~ , an d for ea ch of the n - 1 p o l a r ang les ~ , 2~,.. ., (n - t )~ w e ha ve2 n a z i m u t h a l a n g l e s 0, c~ .... ( 2 n - 1 )~ . C o n s t r u c t a u n i t v e c t o r ~ i n e a c h o ft h e s e 2 n ( n - 1 ) d i r e ct i o n s . I n c l u d i n g t h e u n i t v e c t o rs ~ i , ~ r = - k l , w e n o wh a v e r = 2n(n - 1 ) + 2 un i t ve c to r s /~ t ,..-, ~ r . W e no w de f ine G = ( V , E ) a n dA : S x S - ~ C i n a s i m i l a r w a y a s w e d i d i n t w o d i m e n s i o n s . T h e u n i t a r i t yc o n d i t i o n s n o w b e c o m e

a 2 q - [ n 2 + ( n - 1 ) 2 ] b 2 = 1a c o s O + n ( n - - 1 ) b = O

( 1 7 )( t 8 )

S o l v i n g ( 1 7 ) a n d ( 1 8 ) i n t e r m s o f a g iv e sb = I 1 - a 2 ] 1/2n2+ -l?J

c o s = a n 2 + ~ - 1 ) zA g a i n b e c a u s e o f E q . ( 2 0 ) w e m u s t h a v e

[ ( n - t ) 2 + n - 2 + l ] - 1 ~ < a < l

( 19 )( 20 )

I n t h e l i m i t i n g c a s e w e o b t a i nb = I n 2 + ( n - 1)2 + n 2 ( n - 1 ) 2 ] - 1 , 0 = r e


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Quantum Graphic Dyna mics 761

4. P A R T I C L E M O D E L SI n t h is s e c ti o n w e a p p l y t h e t h e o r y o f Q G D t o c o n s t r u c t m o d e l s f o r

t h e i n t e r n a l d y n a m i c s o f p a r ti c le s . B y a " p a r t i c l e " w e m e a n a m a s s i v e l ep -t o n o r a h a d r o n . W e b e g i n b y r e p r e se n t i n g e a c h p a r t i c le b y a m u l t ig r a p h .T h e v e rt ic e s o f t h e m u l t i g r a p h r e p r e s e n t q u a r k t i k e c o n s t i tu t e n t s o f th e p a r -t i c l e , a n d e d g e s r e p r e s e n t i n t e r a c t i o n p a t h s f o r g l u o n s w h i c h a r e e m i t t e d o ra b s o r b e d b y t h e v e r t i c e s . W e a s s u m e t h a t t h e v e r t i c e s a r e r e l a t i v e l ys t a t io n a r y a n d t h a t t h e d y n a m i c s is d e s cr ib e d b y t h e Q G D t h e o r y o f t h eg l u o n s .

T h e e l e c t r o n is r e p r e s e n t e d b y a m u l t i g r a p h c o n s i s t i n g o f a s i n g lev e r t e x a n d a l o o p . T h e m o r e m a s s i v e l e p t o n s a r e f o r m e d b y a d j o i n i n g a no d d n u m b e r o f l o o p s to t h e e l e c tr o n m u l t i g r a p h . T h u s , th e m u o n h a s t w oloops , t he t au has fou r l oops , e t c . These a re i l l u s t r a t ed i n F ig . 1 .

I n F i g . 1, w e h a v e d e n o t e d t h e s in g l e v e r t e x b y a 1 a n d h a v e n u m b e r e dt h e e d g e s . U s i n g t h i s n u m b e r i n g s c h e m e w e c a n r e p r e s e n t t h e i r p h a s espaces as fo l l ows :

S~= {(1, 1) , (1, 2)}S~ = {(1, 1), (1, 2), (1, 3), (1, 4) }St = {(1, t), (1, 2) . . . . (1, 8)}

T h e p i o n is r e p r e s e n t e d b y a m u l t i g r a p h c o n s i s ti n g o f t w o v e rt ic e s a n da b ie d ge . M o r e m a s si ve m e s o n s a re f o r m e d b y a d j o i n i n g a n o d d n u m b e r o fl o o p s t o o n e o r b o t h o f th e v e r t ic e s . A f e w m e s o n s a r e i l l u s t r a t e d i n F i g . 2 .

A l t h o u g h w e s h a ll n o t h a v e m u c h n e e d f o r i t h e re , t h e m e m b e r s o f am u l t i p l e t c a n b e d i s t i n g u i s h e d b y l a b e l i n g t h e v e r t i c e s a c c o r d i n g t o t h e i re le c tr ic c h a rg e . W e c a n f o l l o w th e u s u a l q u a r k c o n v e n t i o n o f a s s ig n i n gc h a r g e s o f + 1 /3 a n d + 2 / 3 t o t h e v er ti c es . M o r e o v e r , t h e v a r i o u s s p i ns t a te s c a n b e d i s t i n g u i s h e d b y l a b e l i n g e a c h v e r t e x w i t h a s p i n u p (1 ") o r a

1 4

2 3

Ix

8

3 '

'r

7

1 4

Figure 1


12/26

76 2 Gadder2 1 2 1 4

1 2 ] 2 3zr K

3 2 1 4

4 1 2 3T I ,

Figure 2

s p i n d o w n ( $ ). T h e p h a s e s p a c e s f or t h e m u l t i g r a p h s i n F i g . 2 c a n b er e pr e se nt e d a s f o l l o ws :

S . = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) }S ~ = { ( 1 , 1 ) , ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 2 , 4 ) }

S , , = { ( 1 , t ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 2 , 4 ) }F i na l l y , t he nuc l e o n i s r e pr e se nt e d by a m ul t i g r a ph wi t h t hr e e v e r t i c e s

a n d t h r e e b i e d g e s . M o r e m a s s i v e b a r y o n s a r e f o r m e d b y a d j o i n i n g a n o d dn u m b e r o f l o o p s t o o n e o r m o r e o f th e v er tic es . A f e w b a r y o n s a r ei l l u s t r a t e d i n F i g . 3 . Ag a i n , t he m e m be r s o f a m ul t i p l e t c a n be d i s t i ng u i she dby labe l ing the ver t i ces wi th an e l ec tr i c charge and a spin .

T h e ph a se sp a c e s f o r t he m ul t i g r a ph s i n F i g . 3 c a n be r e pr e se nte d a sf o l l o ws :

3

4 1

1 4

N

31 4 1 ~4 1 4 1

61 ~ 2 1 _ .t 4

Figure 3


13/26

Quantum Graphic Dynam ics 763SN= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1),

(3, 2), (3, 3), (3, 4)}S z = {(1, 1) , (1 , 2) , (1 , 3) , (1 , 4) , (2 , 1) , (2 , 2) , (2 , 3) , (2 , 4) , (3 , 1 ) . .. . (3 , 6)}S z = {(1, 1) , (1 , 2) , (1 , 3) , (1 , 4) , (2 , 1) ,. .. , (2 , 6) , (3 , t ) , . .. , (3 , 6) }

I t w i l t b e c o n v e n i e n t t o h a v e a n o t a t i o n d e s c r i b i n g t h e m u l t i g r a p h su n d e r c o n s i d e r a t i o n . W e d o t h is b y g iv i n g t h e n u m b e r o f l o o p s a t e a c hv e r t e x i n t h e o r d e r o f t h e l a b e ls o n t h e v e r t ic e s a n d c a ll t h is t h e type o fm u l t i g r a p h . T h e m u l t i g r a p h s i n F ig s . 1 , 2 , a n d 3 h a v e t h e fo l l o w i n g ty p e s :

e = (1 ), / t = (2 ), ~ = ( 4 )re = (0 , 0 ) , ~c= (0 , 1 ) , r / '= (1 , 1 )

N = ( 0, 0 , 0 ), Z = ( 0, 0 , 1 ), ~ = ( 0 , 1, l )N o t i c e t h a t i n t h i s s c h e m e t h e r e i s n o n e e d t o p o s t u l a t e v a r i o u s

g e n e r a t i o n s o f q u a r k s . A l l w e n e e d i s t h e f ir st g e n e r a t i o n c o n s i s t in g o f av e r te x w i th c h a r g e _ + 1 / 3 a n d a v e r te x w i th c h a r g e + 2 / 3 . T h e l a te rg e n e r a t i o n s a r e a u t o m a t i c a l l y d e s c r ib e d b y t h e n u m b e r o f l o o ps . A s w es h a ll se e, o n e l o o p c o r r e s p o n d s t o s t ra n g e , t h r e e t o c h a r m , fi v e t o b o t t o m ,s e v e n t o t o p , e t c . T h e r e m a y b e p h y s i c a l r e a s o n s f o r t h i s s e q u e n c e t ot e r m i n a t e , b u t t h e r e i s n o m a t h e m a t i c a l r e a s o n .

A l t h o u g h w e s h a l l n o t c o n s i d e r e x c i t e d s t a t e s h e r e , w e p r o p o s e , f o rl a t e r in v e s t i g a t io n , t h a t t h e s e a r e f o r m e d b y a d j o i n i n g p a i r s o f v e r t ic e s t ob i e d g e s . T w o e x c i t e d s ta t e s a r e i l l u s t r a t e d i n F i g . 4 .

W e n o w d e s c r i b e t h e g l u o n d y n a m i c s . O n c e t h e m u l t i g r a p h i s g i v e n ,t h e d y n a m i c s i s d e t e r m i n e d b y t h e t r a n s i t i o n a m p l i t u d e s o r e q u i v a l e n t l y b yt h e p r o p a g a t o r o r t r a n s i t i o n m a t r i x T . W e s h a l l a s s u m e t h a t t h e t r a n s i t i o na m p l i t u d e s h a v e t h e s i m p l e s t n o n t r i v i a l v a l u e s . F i r s t n o t i c e t h a t t h e v e r t i c e so f t h e m u l t i g r a p h s u n d e r c o n s i d e r a t i o n a l l h a v e e v e n d e g r e e . I n f a ct , t h ep o s s ib l e d e g r e e s f o r a l e p t o n o r m e s o n a r e 2 , 4 ,..., 4 n , a n d f o r a b a r y o n t h e yar e 4 , 6 . .. .. (4n - 2 ) . W e ca l l a ve r t e x basic i f i t h a s d e g r e e 2 in a l e p t o n o r

< 2 >Figure 4


14/26

7 6 4 Gudder

m e s o n o r d e g r e e 4 i n a b a r y o n . L e t v 1, v2 b e a d j a c e n t v e r ti c es , l e t e l b ei n c i d e n t t o v l a n d v 2, a n d l e t e 2 b e i n c i d e n t t o v 2. F o r b r e v i t y , le t A ( I , 2 ) =A ( ( ~ ) I , e l ) , ( v 2 , e 2 ) ) . I f v2 i s ba s i c , w e de f ine A ( 1 , 2 ) -- - 612 f o r a l e p ton a n df o r a h a d r o n A ( 1 , 2 ) - - 1 i f e l , e 2 a r e d i f f e r e n t e d g e s o f t h e s a m e b i e d g e a n dA ( 1 , 2 ) = 0 o t h e r w i s e . N o w s u p p o s e v 2 i s n o t b a s i c a n d h a s d e g r e e n . F o r al e p t o n , d e f in e A ( 1 , 2 ) = ( 2 - n ) / n i f e l , e 2 a r e " d i f f e r e n t " e d g e s o f t h e s a m el o o p a n d A ( 1 , 2 ) = 2 /n o t h e rw i s e . F o r a h a d r o n , d e f in e A ( 1, 2 ) = ( 2 - n) / n i fe l = e2 a n d A ( t , 2 ) = 2 in o t h e r w i s e . I n a l l o t h e r c a s e s , w e d e f i n e A ( ( v l , e l ) ,( v 2 , e 2 ) ) = 0 . N o t i c e t h a t t h e s e v a l u e s c o r r e s p o n d t o t h e l i m i t i n g c a s e i nE q . ( 1 6 ). I t i s e a s y t o s h o w t h a t A i s i n d e e d a t r a n s i t i o n a m p l i t u d e .M o r e o v e r , i t i s t h e u n i q u e t r a n s i t i o n a m p l i t u d e i n w h i c h t h e " f o r w a r d "a m p l i t u d e s a r e e q u a l a n d h a v e m a x i m a l v a lu e .

I n F i g . 5 w e i l l u s t r a t e t h e t r a n s i t i o n a m p l i t u d e s f o r v a r i o u s c a se s . I nt h i s f i g u r e w e l a b e l t h e i n i t i a l e d g e w i t h a d o u b l e a r r o w a n d l a b e l a d j a c e n te l e m e n t s w i t h s i n g l e a r r o w s t o g e t h e r w i t h t h e i r t r a n s i t i o n a m p l i t u d e s .

I f t h e e l e m e n t s o f t h e p h a s e s p a c e a r e e n u m e r a t e d 1 , 2 ..... m , t h e n t h em a t r i x T w i t h e n t r i e s T o = A (j , i ) , i , j = 1, 2, . .. , m , i s c a l e d the t rans i t ionm a t r i x . T h i s i s a u n i t a r y m a t r i x w h i c h d e s cr i b e s th e e v o l u t i o n o f a g l u o n . I nf a c t , i f a g luo n is in i t i a l l y i n a s t a t e ~ 0 , t he n a f t e r n t ime s t e ps i t w i l l be i nt h e s t a t e T " O o . W e n o w c o n s t r u c t t h e t r a n s i t i o n m ~ itric es f o r t h e m u l -t i g r a p h s r e p r e s e n t i n g l e p t o n s a n d h a d r o n s . I n d o i n g t h i s w e a s s u m e t h a tt h e e l e m e n t s o f t h e p h a s e s p a c e a r e o r d e r e d a s w e h a v e d o n e p r e v i o u s ly . Itw i ll f ir s t b e c o n v e n i e n t t o d e f i n e t h e 2 x 2 m a t r i c e sE j nA n = ( 1 - n ) / n , n = l , 2 ....

~

1

o

1

F i g u r e 5


15/26

Quantum G r a p h i c D y n a m i c s 7 6 5

Also, we let I = A 1 and B.= (1 In ) I , n = 2 , 3 .... It turns o ut that T can berepresented in terms of these matrices. Wh en a mult igraph has type (i,...),we write the corresponding transition matrix as T(i,...). The number ofrows (and columns) in a matrix is placed at its right-hand corner and whenan A , (and hence Bn's) appear, the numbe r of such matrices in that columnor row is always n.

The leptons have the following transition matrices: T (1 )= I

T ( n ) B . A . B .= . , n = 2, 4, 6,,..n B , , A n 2n

The mesons have the following transition matrices:

T ( 0, 0 ) = 0 4

T(O, n) =

0 I 0 .. . 0A , + I 0 B~+1 ... B, +IB,+ 1 0 A,+ I . . . B, +IB , + I 0 B ~ + 1 " '" B , + I

B . + I 0 B . + 1 . . . A . + 2(n + 2)

n=1,3,5 . . . .

T ( m , n ) =

0 B m + l " " B m + l A , . + l 00 A m + l " " B , . + l B , . + l 00 B,~+I " " B m + I B m + I 00 B i n + t . . . A m + t B i n + 1 0

A . + I 0 . . . 0 0 B . + IB~+I 0 ... 0 0 A.+IB,+I 0 ... 0 0 B,+IB,+I 0 ... 0 0 B,+i

. . 000

0B n + 1O n + tB n + 1

A n + l ~ ( r n+ n + 2)m ,n = 1,3,5 ....

825/18/7-6


16/26

I

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+ + C~

+ 4 P + +

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q, 0 0


17/26

Quan tum Graphic Dynam ics 767

~ l , m , n ) =0 0 B t + 2 " ' B I + 2 0 Al+2 0 .." 0 B t + 2 0 0 ' . - 00 0 B I+ z. . . BI + 2 0 BI+ 2 0 -." 0 A + 2 0 0 , - - 00 0 AI +2 .' .B I+ 2 0 Bl+2 0 . . , 0 B l + 2 0 0 . - - 0

0 0 B~ +2 "" At +~ 0 BI+2 0 .. . 0 B l + 2 0 0 - . . 0Bm+2 0 0 . . . 0 0 0 B~ +2 .. .B m+ 2 0 A , , + 2 0 - - . 0A m + 2 0 0 . . . 0 0 0 B m + 2 " " B , , + 2 0 Bin+ 2 0 ,. . 0Bin+2 0 0 . ., 0 0 0 A m +2 . .. B m + 2 0 Bin+ 2 0 .. . 0

Bin+ 2 0 0 .. . 0 0 0 B m+ 2 . .. A m+ 2 0 Bin+ 2 0 ... 00 A n + 2 0 . - , 0 B n + 2 0 0 - - - 0 0 0 B ~ + z ' " B , , + 20 B ~ + 2 0 " " 0 A ~ + 2 0 0 - . . 0 0 0 B # + 2 . . . B , + 20 Bn+ 2 0 "" 0 Bn+2 0 0 "" 0 0 0 A, ~+ z' ,. Bn +2

0 Bn+ 2 0 ," 0 B,+2 0 0 . . , 0 0 0 Bn +2 "- An +2 2(rn+,~+[+61l , m , n = l , 3 , 5 , . . .

5 . G L U O N D Y N A M I C SI n S e c t i o n 4 w e c o n s t r u c t e d t h e t r a n s i t i o n m a t r ix T fo r a m a s s i v e l e p -

t o n o r h a d r o n . T o d e s c r ib e t h e g l u o n d y n a m i c s w e n e e d t o f in d T ", n e N .E s s e n t i a ll y t h e o n l y w a y t h is c a n b e d o n e i s b y d i a g o n a l i z i n g T or ,e q u i v a l e n t ly , f in d i n g t h e e ig e n p a i r s o f T . W e d e v o t e t h e p r e s e n t s e c t i o n t oa c c o m p l i s h i n g t h i s .

N o t i c e t h a t w e w i s h t o d i a g o n a l i z e a 2 m x 2 m m a t r i x T w h o s e e n t r ie sm a y b e t h o u g h t o f a s 2 x 2 m a t r i c e s w h i c h h a v e t h e s p e c ia l f o r m

O u r n e x t r e s u lt s s h o w t h a t b e c a u s e o f t h is s p e ci a l fo r m w e c a n r e d u c e o u rw o r k t o d i a g o n a l i z i n g t w o m x m m a t r ic e s . F i rs t d e f in e t h e m a t r i c e s

L e m m a 5 . L e t A = ( A ij ), i , j = 1 , 2 , b e a 2 x 2 c o m p l e x m a t r i x . T h e n


18/26

768 Gudder

t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t . ( a ) A c o m m u t e s w i t h K . ( b ) Ac o m m u t e s w i t h L . ( c ) A H = A 2 2, A 12 = A z l .

P r o o f S t r a i g h t f o r w a r d . iD e f i n e t h e v e c t o r s ~ = (1 , 1 ), ~ 2 = ( 1 , - 1 ) a n d d e f in e t h e f o l lo w i n gv e c t o r s i n C 2 " :

= 0 . . . . . 0 ) , . . . , = ( 0 , 0 , . . . , 0 ,~ 1 : ( 0 2 , 0 , . . . , 0 ) , . . . , ( ~ r n ( 0 , 0 , . . . , 0 , 0 2 )

L e m m a 6 . L e t M b e a 2 m x 2 m m a t r ix o f t h e f o r mM = [ A ( t , 1 ) .: . .. A ( 1 . , m ) ]

k A (m , 1) A(m , rn) Jw h e r e t h e 2 x 2 m a t r i c e s A(i , j ) h a v e t h e f o r m

[ a i j b i j ]A ( i , j ) = b ~ a ~ j A

T h e n M is n o r m a l a n d i ts e i g e n v e c t o r s h a v e t h e fo r m Z e i ~ o r Z cq q~ ,cci~C, i= l ,. .. ,m.P r o o f L e t J / a n d ~ b e t h e s u b s p a c e s o f C 2m g e n e r a t e d b y ~ 1 ..... ~

a n d q ~ ..... ~ m , r e s p e c ti v e l y . N o t i c e t h a t J / a n d Y a r e m u t u a l l y o r t h o g o n a ls u b sp a c es a n d c 2 m = j / @ J V ". L e t K m, L ~ b e t h e f o ll ow i n g 2 m x 2 mm a t r i c e s :

K m = d i a g ( K , . . . , K ) , L , , = d i a g ( L , . .. , L )I t is e a s y t o c h e c k t h a t t h e p r o j e c t i o n s o n t o J g a n d ~ /" a r e Kin~2 a n d L m / 2 ,r e s p e c t iv e l y . M o r e o v e r , i t i s n o t h a r d t o v e r if y t h a t M i s n o r m a l a n d t h u sh a s a c o m p l e t e s e t o f e i g e n v e ct o r s. B y L e m m a 5 , M c o m m u t e s w i t h K m a n dLm, a n d h e n c e t h e e i g e n v e c t o r s o f M li e e i th e r in J / t o ~ . T h e r e s u l t n o wf o l l o w s . I

T h e o r e m 7 . L e t M b e a 2 m x 2 m m a t r i x w i t h th e p r o p e r t i e s g i v e n i nL e m m a 6 . L e t h ~ , N~ b e t h e m x m m a t r i c e s w i t h e n t r i e s ~ 0 = a o + b ~ j, .~T . =a ~ - b i ~ . T h e n t h e e i g e n v a lu e s o f M a r e p re c i se l y t h e u n i o n o f t h e ei g e n-v a l u e s o f . ~ a n d N . M o r e o v e r , i f 2 ( 2 ' ) i s a n e i g e n v a l u e o f 2 14 (N ) w i t hc o r r e s p o n d i n g e i g e n v e c t o r ( e l , . . . , e ,~ ) ( (/ ~l ... .. / ~m ) ), t h e n 2 ( 2 ' ) i s a n e i g e n -v a l u e o f M w i t h c o r r e s p o n d i n g e i g e n v e c t o r ~2 c q ~ ( Z Piq~i).


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Quantum Graphic Dynamics 769P r o o f A p p l y i n g L e m m a 6 , t h e e i g e n v e c to r s o f M h a v e t h e f o r m

5 2 e ~ i o r ~ ]e iq ~i- S u p p o s e 5 2 ~ i ~ is a n e i g e n v e c t o r o f M w i t hc o r r e s p o n d i n g e i g e n v a l u e 2 . I f f o l lo w s f r o m t h e e q u a t i o n M ( 5 2 c ~ i t ~ ) =) ~ ( Z O~i~l ) t h a t

~ . A ( i , j ) c ~ j ( k j = 2 c q O , , i , j = 1 , .. ., mJ

H e n c ec~j(a~ + 1 0) = 2c q, i, j = 1 .... m

J

I t fo l l ows t ha t M (a l ..... c~ m )= 2( ~ ..... am). T he con ver se i s eas i l y ve r i f ied .S im i l a r r ea son ing app l i e s t o t he e igenv ec to r s o f t he fo rm 52 a i~i. |

W e a r e n o w r e a d y t o e v a l u a t e t h e e i g e n p a i r s f o r t h e t r a n s i t i o nm a t r i c e s c o n s t r u c t e d i n S e c t i o n 4 . W e b e g i n w i t h t h e m s s iv e l e p to n s . T h e2 x 2 m a t r i x T ( 1 ) = I h a s d e g e n e r a t e e i g e n a l u e 1 a n d w e t a k e , b y c o n v e n -t i o n , i ts e i g e n v e c t o r s to b e 0 ~ , 0 2 . N e x t c o n s i d e r t h e 2 n x 2 n m a t r i c e s T ( n ) ,n = 2 , 4 , 6 .... A p p l y i n g T h e o r e m 7, t h e p r o b l e m r e d u c e s t o f i n d i n g t h ee i g e n p a i r s f o r t h e n x n m a t r i c e s M , N , w h e r eF 2 2 2 ] I 1 017 /= 1 2 - n 2 ~ = 0 1 . .

t/2 - .. 2 - n 0 0 . .

01

N o w N h a s t h e n - f o l d d e g e n e r a t e e i g e n v a l u e 1 . W e t a k e t h eco rre sp on di ng e ige nv ecto rs to be (1 , 0 ,..., 0 ) . .. .. (0, 0 .... . 0 , i ) . N ex t M ha st h e n o n d e g e n e r a t e e i g e n v a l u e 1 w i t h c o r r e s p o n d i n g e i g e n v e c t o r ( 1, 1,... , 1 ).F i n a l ly , f / h a s t h e ( n - 1 )- fo ld d e g e n e r a t e e i g e n v a lu e - 1 a n d w e t a k e t h eco rr es po nd in g eig en ve cto rs to be (1, - 1 , 0 , 0 . .. . 0) , (1, 0 , - 1 , 0 ,. .. , 0) . .. ..(1 , 0 , 0 , . . . , 0 , - 1 ) . N o t e t h a t t h e se l a t t e r e i g e n v e c t o rs a r e n o t m u t u a l l yo r t h o g o n a l . H o w e v e r , t h e y a r e l i n e a r ly i n d e p e n d e n t , t h e y h a v e t h e s i m p l e s tf o r m , a n d a s s h a l l l a t e r s ee , th e y h a v e p h y s i c a l si g n if ic a n c e . M o r e o v e r , w eh a v e n o t b o t h e r e d t o n o r m a l i z e t h e se e i g e n v e c t o r s s in c e th i s w il l n o t b en e c e s s a r y f o r o u r p u r p o s e s . A g a i n , a p p l y i n g T h e o r e m 7, t h e e i g e n p a ir s f o rT ( n ) a r e g i v e n i n T a b l e I .

W e n e x t c o n s i d e r t h e m e s o n s . T h e e i g e n p a i rs f o r t h e 4 x 4 m a t r i xT ( 0, 0 ) a r e e a s i ly c o m p u t e d a n d a r e s h o w n i n T a b l e II .


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770 Gudder

Ta ble I . Eigenpairs for T(n)Eigenvectors Eigenvector(s)

W e n o w c o m e t o t h e 2 ( n + 2 ) x 2 ( n + 2 ) m a t r i c e s T ( 0 , n ) , n = 1 , 3 , 5 .. ...T h e ( n + 2 ) x ( n + 2 ) m a t r i c e s ~ / , )V a r e g i v e n b y

1M = n + l

0 n + l 0 . . . 01 - n 0 2 . . . 2

2 0 1 - n . . . 22 0 2 . . . 2

2 0 2 . . . 1 - n

, N =

- 0 1 0 . . . 0 -1 0 0 . - - 00 0 1 - - . 00 0 0 - - . 0." ~

_ 0 0 0 . . . 1F o r N , th e ( n + l ) - f o l d d e g e e r a t e e i g e n v a l u e 1 h a s e i g e n v e c t o r s( 1 , 1 , 0 , .. ., 0 ) , ( 0 , 0 , 1 , 0 , .. ., 0 ), .. ., ( 0 , 0 ..... 0 , 1 ) w h i l e t h e n o n d e g e n e r a t e e i g e n -v a l u e - 1 h a s e i g e n v e c t o r ( 1 , - 1 , 0 ,, .. , 0 ) . T o f i n d t h e e i g e n p a i r s f o r 3 7 /, l e t( e l ..... ~ , + 2 ) b e a n e i g e n v e c t o r s w i t h c o r r e s p o n d i n g e i g e n v a l u e 2 , W e t h e no b t a i n t h e e q u a t i o n s

~2 = )~0~t( 1 - - n ) c~1 + 2c% + 2 0~ 4 - t - " ' " q - 2 % + 2 = ( n + 1 ) 20~ 22 ~ 1 + ( t - - n ) 0 % + 2 ~ 4 + . - . + 2 0 t , + 2 = ( n + l ) 2 ~ 32 cq + 2 c % + 2 c ~ 4 + - - - + ( 1 --n)o:,,+2=(n+ 1 ) 2 c ~ , + 2

Tab le I I . Eigenpairs fo r T(O, 0)Eigenvalue Eigenvector

- 1 , 6 t - 6 2, i


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Qua ntum Gra phic D y na mics 7 7 !

S up pos e e~ = 1 an d henc e c~2 = 2. F r o m t h e s e c o n d a n d t h i r d e q u a t i o n s , w eo b ta in ~ 3 = ( 1 + 2 2 ) / ( 1 + 2 ) (n ot ic e t h a t 2 - 1 ) . I n a s im i la r w a y w eob t a in e j= (1 + 22 ) /(1 + 2 ), j = 4 ,..., n + 2 . S u bs t i t u t i ng t hese va lues i n to t hes e c o n d e q u a t i o n g i v e s

( 1 - n ) + 2 n (1 + 2 2 ) = ( n + 1)221 + 2H e n c e ,

( n + 1 )2 3+ ( 1 - n ) 2 2 - ( 1 - n ) 2 - ( n + 1 ) = 0T h e r o o t s o f t h is e q u a t i o n a r e 1 a n d

- 1 +_ i [ n ( n + 2)] ~/2n + l

N o w s u p p o s e n > 1, cq = 7 2 = 0 , a n d l et ~ 3 = 1. I t f o ll o w s f r o m t h e s e c o n dtwo eq ua t i on s t ha t 2 = -1 . I f we le t ~4 = -1 , an d ~s = 0 , j = 5 .... n + 2 , t he na l l t he e qu a t i o ns a re s a t is f ied . In a s im i l a r wa y w e can l e t c~5 = -1 , an de s = 0 , j = 4 , 6 , 7 ,..., n + 2 , e tc . T h u s 2 = - 1 is a n ( n - 1 ) - f o l d d e g e n e r a t ee i g e n v a l u e . L e t t i n g

1 + I , t y ) =/L+= 1 + 2 ~we l i s t t he e igenpa i r s fo r T (0 , n ) i n Tab l e I I I .

F in al l y , w e h av e th e 2 ( m + n + 2 ) 2 ( m + n + 2 ) m a t r ix T ( m , n ) , m , n =1 , 3 , 5 .... W e sha l l no t d i s p l a y M an d N s ince t he i r co ns t ru c t i o n i s s imi l a rt o t hose do ne p rev ious ly . F o r ~" t he (m + n + 1 ) - fo ld degene ra t e e i genva lue 1h as ei ge nv ec to rs (1, 0,. .. , 0, I , 0, .. ., 0), (0, 1, 0, .. ., 0) . .. .. (0, 0 . .. .. 0, 1) w he re th es e c o n d 1 i n t h e fi rs t v e c t o r is a t t h e ( m + 2 ) - e n t r y . T h e n o n d e g e n e r a t ee i g e n v a l u e - 1 h a s e i g e n v e c t o r ( 1, 0 ,..., 0 , - 1 , 0 ..... 0 ) w h e r e t h e - 1 is a tt h e ( m + 2 ) - e n t r y .

Table III. Eigenpairs or T(0, n)Eigenvalue Eigenvector(s)

1- 11

- I ( if n> 1)i i i i i i i i i i i i i i i

(' 1+(' 2+ "'" +('~+2

'3- ('4, ~' 3- ~ ..... ~3-( ,+~


22/26

772 GudderT o f i n d t h e e i g e n p a i r s f o r 3 ~ r, l e t (c q . . . . . 0 ~ m + n + 2 ) b e a n e i g e n v e c t o r

w i t h c o r r e s p o n d i n g e i g e n v a l u e L W e t h e n o b t a i n t h e e q u a t i o n s2 ~ 2 + . . . + 2 ~ m + l + ( 1 - - m ) C ~ m + 2 = ( m + l ) 2 ~ q( 1 - - m ) C ~ z q - . . . + 2 0 ~ m + l + 2 0 ~ m + 2 = ( m + l ) ~ , ~ 22C~2 + " ' " + (1 - - m ) e,~ + 1 + 2 ~ m + Z = ( m --1 - 1 ) ,~ 0~m + 1( 1 - - n ) c q + 20~m+ 3 q - " " nt" 2 0 ~ m + n + 2 = ( n + 1 ) J . 0 ~ m + 22~1 + ( 1 - n ) 0~m+ 3 -t" " " q - 2 ~ m + n + 2 = (/7 q - 1 ) ~ ' ~m+ 3

2Cq + 2~ m + 3 q- '- 1- (1 --n)O~m+n+ 2 = (n-t- 1 ) 2 ~ m + n + 2S u p p o s e cq = 1. F r o m t h e fi rs t t w o e q u a t i o n s w e h a v e ( a s s u m i n g 2 ~ - 1 )

2 1

I n a s i m i l a r w a y , a j , j = 3 .... m + 1 , h a v e t h i s s a m e v a l u e . ' S u b s t i t u t i n g t h e s ev a l u e s i n t o t h e f i rs t e q u a t i o n g i v e s

W e t h e n o b t a i n

2 [ ( m + 1 ) 2 + (1 - m ) ]a m + 2 = ( l - m ) 2 + ( m + 1 )

2 2c ~ J - ( 1 - m ) 2 + ( m + 1 ) ' j = 2 , 3 ..... m + l

F r o m t h e f o u r t h a n d f i f t h e q u a t i o n s w e h a v e( m + 1 ) 2 2 - 2 m 2 + (m + t )

e r a + 3 = ( 1 - m ) 2 + ( m + l )I n a s i m i l a r w a y , e j , j = m + 4 ,. . . , m + m + 2 , h a v e t h i s s a m e v a l u e . S u b -s t i t u t in g t h e s e v a l u e s in t o t h e f o u r t h e q u a t i o n g i v e s

( n + 1 ) ( m + 1 ) 2 3 - (3m n + m + n - 1 ) 2 2 + ( 3 m n + m + n - 1 ) 2- ( n + 1 ) ( m + 1 ) = 0

T h e r o o t s o f t h is e q u a t i o n a r e 1 a n d( m n - 1 ) _ + i [ ( n + 1 ) 2 ( m + 1 ) z - ( I - r a n ) 2 ] m+2 t i n ' " ) - - ( n + 1 ) ( m + 1 )


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Quantum Graphic Dyn amics 773

Table IV. Eigenpairs for T(m, n)Eigenvalue Eigenvector(s)

1- 1

1

- t

+

+ 7 ( m , n ) m + 3 + " ' " " ~ " 7 (r n , n ) ~ . . . . + 2

N o w l e t 2 = - 1 a n d ~ 1 = ~ , . + 2 = 1. T h e n a ll t h e e q u a t i o n s a r e s a t i s fi e d i fw e l e t ~ 2 = ~ + 3 = - 1 a n d t h e o t h e r ccs 's e q u a l to 0 . I n a s i m i l a r w a y w ec a n ~ et ~ 2 = ~ m + 4 = - -1 , .. ., ~ 2 = ~ m + . + 2 = - - 1 , a n d t h e o t h e r ~ j 's e q u a l t o 0 .F i n a l l y , w e c a n l e t ~ 3 = ~ m + 3 = - - 1 ..... ~ m + l = ~ m + 3 = - -1 a n d t h e o t h e r ~ j 'se q u a l t o 0. T h u s , - 1 is a n ( m + n - 1 ) - f o ld d e g e n e r a t e e i g e n v a l u e . I f w ed e f i n e

+2 2 ~ , n )c o L ' " ) = ( 1 - m ) 2 i ~ , , . ) + ( m + I )

+ +f l (~ , . ) = 2 6 . , . ) [ ( m + 1 ) 2 6 ~ , . ) + ( 1 - - m ) ](1 - m ) 2 ( . ~ ,~ ) + ( m + 1 )

+ 2 +( m + 1 ) [ 2 & . . ) ] - 2 m 2 ~ , . ) + ( m + i )- m ) 2 ~ , m + ( m + 1 )

t h e n t h e e i g e n p a i r s f o r T(rn, n) a r e l i s te d i n T a b l e I V .W e f in a l l y c o n s i d e r t h e b a r y o n s . T h e e i g e n p a i r s f o r th e 1 2 1 2 m a t r i x

T ( 0 , 43, 0 ) a r e s h o w n i n T a b l e V .Table V. Eigenpairs for T (0, 0, 0)

Eigenvalue Eigenvector(s)

-1 ,-,, ~-~, ~-o


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7 7 4 G u d d e r

T a b l e VI. Eige npairs for T (0, 0, n)E i g e n v a l u e E i g e n v e c t o r ( s )

- 1 ~ - &, ~ - ~ , , & - &

. - @ 2 - { - @ 3 " } - ] . / ; @ S ~ - t ~ - ~ 6 - ' } - / ~ ; @ 7 - t - - - - - l - /~ Z @ n + 6i ( I 2 - - ~ , + i ( t , - - i ( t 6,i i

W e ne x t ha v e the 2 ( n + 6 ) x 2 ( n + 6 ) m a t r i x T ( 0 , 0 , n ) , n = 1 , 3 , 5 ....T h e c o m p u t a t i o n o f t h e e i g e n p a i rs i s s i m i la r t o t h a t f o r 7 ( 0 , n ). L e t t i n g

2 ++_ [ n ( n -t- 4 )] 1/2 1 + (2 +) 2/~ '~ = n + 2 ' f l+ = 1 + 2 , {

w e l is t t h e e i g e n p a i r s f o r T ( 0 , 0 , n ) i n T a b l e V I .T h e e i g e n p a i r p r o b l e m f o r t h e n e x t t r a n s i t i o n m a t r i x T ( O , m , n ) ,

m, n = 1 , 3 , 5 ,..., i s m u c h m o r e d i f fi c u l t. E xc e p t f o r s om e spe c ia l c a se s , i ta p p e a r s t h a t i t c a n o n l y b e s o l v e d n u m e r i c al l y . A s u s u al , d i a g o n a t i z i n g ? ~ iss t r a i g h t f o r w a r d . H o w e v e r , s i x o f t h e e i g e n v a l u e s o f h~r t u r n o u t t o t h e r o o t so f t h e f o l l o w i n g e q u a t i o n :

( m + 2 ) ( n + 2 ) ) ; 6 - ( 2 r a n - 8) 25 + ( 3 r a n + 2 m + 2 n + 8 ) ) . 4 - ( 4 m n - 8 ) 2 3+ ( 3 m n + 2 m + 2 n + 8 ) 2 2 - ( 2 m n - 8 ) 2 + ( m + 2 ) ( n + 2 ) = O ( 2 1 )

W e l i st t h e e i g e n p a i r s f o r T ( 0 , m , n ) i n T a b l e V I I . W e c a l l th e r o o t s o f ( 2 1 )( w h i c h w e h a v e n o t e v a l u a t e d ) 2~ ..... 2 6. O f c o u r s e , t h e c o r r e s p o n d i n g

Eigenvalue1

- 112 1 ,..., ),6--1

Table V II. Eigenpairs for T(O,m, n)I I I I I I I I I I I I I I I I I I I I I I I

Eigenvector(s)

~ 1+ " ' " + ~ . . . . 6~ 1 3 " } - ~ l m + 6 - - ~ l S - - ~ l m + 7 , . . , , 3 3 - ~ ~ m + 6 - - ~ l S - - ~ l m + n + 6 ,

.,. ,u,,,,,,,, .,,., m......,,,


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Q u a n t u m G r a p h i c D y n a m i c s 7 7 5

e i g e n v e c t o r s c a n b e c o n s t r u c t e d o n c e t h e se e i g e n v a lu e s a r e e v a lu a t e d . F o r2 ~ { 2 ~ , . . . , 2 6 } , t h e c o r r e s p o n d i n g e i g e n v e c t o r is ( e l ,- - . , e m + ~ + 6 ) , w h e r e~ 1 = 1 , g 4 = 2 , a n d 1 + 2 2

~ 5 = e 6 . . . . = 0 ~ m + 4 = 1 + 2( m + 2)(J~ 4 + 1 ) - - 2m 2( 22 - ,~ + 1 )

~ 2 = 2 2 ( 1 + 2 2 )m 2 3 + m ~ . 2 - - ( m - - 2 ) 2 + ( m + 2 )

~ 3 = - 22 (1 + 2)~ m + 5 = " ~ 2

( m + 2 ) i t 3 + (2 - m ) 2 2 + m ) . - mO~m+6= 2 ( 1 + R )

( I + )})~20 ~ m + 7 = 0 ( m + 8 . . . . -~" 0 [r n + n + 6 l + J .W e c a n s o l v e ( 2 1 ) i n t h e s p e c i a l c a s e m = n = 1 . I n t h i s c a s e ( 2 t ) h a s

t h e f a c t o r i z a t i o n( 3 2 4 + 2 )~ : + 3 ) ( 3 2 2 + 2 2 + 3 ) = 0

T a b l e V I I I . E i g e n p a i r s f o r T ( 1 , 1 , 1 )

E i g e n v a l u e E i g e n v e c t o r ( s )1

- - 11

- - 1

- - i

2 ~1 + ~3 + 2 ~4 + t~6 + 2 ~7 + t~9- ~ , + ~ - ~ + ~ - ~ + ~ ,- ~ + ~ + ~ . - ~ - ~ + ~ ,~ 1 - - ~ 2 - - 3 ~ 4 + ~ 5 -}- 2 ~ 6 - - ~ 7 - - 3 ~ 8 + 4 ~ 9

- ,e ( 1 + i ) r ( i + 3 ) ~ .~ ( 1 + 3 i ) ~(I - , ~v ,, - - 7 - - w + ~ , - - - - ~ ~ , - ,v , e + - - - 5 - - q , ~(1 + i )- 5 - - ~ 8 + ( i - 1 )~9

i ~ 1 - - i ~ 2 + (1 - -i) q~ 4 + ~ 5 + (1 + i ) ~ 6 - - ~ 7 + ( i + 1 ) ~ 8 - (1 + i ) ~ 9

+ ~ 8 - - ( 1 + i ) ~ 9it ~l -}- i ~ 2 + ( 1 + i ) ~ 4 + ~ 5 + ( 1 - - i ) ~ 6 - - ~ 7 q - ( 1 - - i ) ~ 8 + ( i - 1 ) ~ 9


26/26

776 Gudder

We then obtain the six eigenvalues given by2 = - ~ i2 ,,f23 2 2 = - ~ + i2x/~3

As the reader would expect , the eigenvatue problem for T ( l , m , n ) ism uch m ore invo lved then i t was for T(0 , m, n) . Fo r th i s reason, we sha llon ly cons ide r the spcial case T(1 , 1, 1) an d this is given in T able V III .

R E F E R E N C E S1. D. B ohm, "A p roposed topo log ica l fo rmu la t ion o f quan tum theory ," in T h e S c i e n t i s t

S p e c u l a t e s , I . J . Go od , ed. (Basic B ooks, N ew Y ork, 1962), pp. 302-314.2. A. Das, J . M a t h . P h y s . 7, 52 (1966).3 . D. Finkleste in , P h y s . R e v . D 9, 2219 (1974).4 . R. Friedberg and T. D. Lee, N u e l . P h y s . B 225, 1 (1983).5 . S. Gud der , " Disc re te quan tum mechan ics ," J . M a t h . P h y s . 27, 178 2 (1986).6 . S . Gudder and V. Narod i t sky , I n t . J . T h e o r . P h y s . 20, 619 (1981).7 . R. Jagan na tha n an d F . San thanam , I n t . J . T h e o r . P h y s . 20, 755 (1981).8 . M. Lorente , I n t . J . T h e o r . P h y s . 11, 213 (1974).9. D. Shale, F o u n d . P h y s . t2, 661 (1982).

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