Geometric model for fundamental particles

5/22/2018 Geometric model for fundamental particles

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International Journal o f Theoretical Physics V ol . 19 No. 6 1980

G e o m e t r ic M o d e l fo r u n d a m e n t a l P a r t i c l esE . P . B a t t e y P r a t t I a n d T . J . R a c e y

Received October 14 1979A n attem pt is m ade to show that fu ndam ental part ic les are manifes ta tions ofthe geometry of space-time. This is done by demonstra ting the exis tence of apurely geometrical model, which w e have called spherical rotation tha t satisfiesDirac ' s equat ion . The mode l i s deve loped and i l lus t ra ted bo th m athemat ica l lyand mechanically . I t indicates tha t the mass of a p art ic le is entire ly due to thespinning of the space-time continuum. Using the model, we can show thedis tinction between spin-up and spin-down s ta tes and a lso between p artic le andantipartic le s tates. I t sa tisfies E inste in 's cri teria for a m odel that has both waveand partic le properties, and i t does so w ithout introducing a s ingulari ty into thecontinuum.

1 . I N T R O D U C T I O NT h e o l d e s t m e t h o d o f r e c o r d i n g m o t i o n i s q u i t e s i m p l y th e m e t h o d o f

l a y i n g d o w n a t r a i l . T h e t r a i l c o n n e c t s t h e s t a r t to t h e e n d o f t h e j o u r n e y .T a k e t h e e x a m p l e o f a d o g t i e d b y a l o n g r o p e t o a t re e . H e r u n s t w i c ea r o u n d h i s k e n n e l a n d t h e n l i e s d o w n . H i s m a s t e r r e t u r n s , s e es t h e r o p e ,a n d i n s t a n t l y k n o w s t h a t t h e d o g h a s c i r c l e d h i s h o u s e t w i c e .

I n a n y t h e o r y o f t h e c o n t i n u u m t h a t p u r p o r t s t o d e s c r i b e m a t t e r a s ad i s t o r t i o n o f s p a c e i n t h e m a n n e r f i rs t s u g g e s t e d b y W . K . C l i f f o r d , 2 t h em o t i o n s o f t h e m a t t e r m u s t n o t d e s t r o y t h e c o n t i n u i t y o f t h e s p a c e . A n y s e to f c u r v i l i n e a r c o o r d i n a t e s u s e d t o m a p t h e s p a c e i n th e v i c i n i t y o f th ep a r t i c l e m u s t , l i k e t h e r o p e a t t a c h e d t o t h e c ro g, p a r t i c i p a t e i n t h e a l l o w e dm o t i o n s o f t h e s p a ce . T h i s r e q u i r e m e n t , w e h a v e b e e n a b l e to s h o w ( s eeA p p e n d i x ) , i s e q u i v a l e n t t o t h e a s s e r t i o n t h a t t h e o n l y a l l o w a b l e m o t i o n sm u s t b e r e p r e s e n t e d b y a s i m p l y c o n n e c t e d g ro u p n a m e l y , t h e u n i v e r s a lc o v e r i n g g r o u p o f t h e L i e g r o u p u s e d t o d e s c r i b e a l o c a l p a r t o f th e m o t i o n .1Present address: RR 1 , Invera ry , Onta r io K OH 1X0, Canada .2W. K. Clifford (1956). All those who searched for a unified fie ld theory subscribed to a

geom etrical interpretation of ma tter, including Einstein , W eyl, an d Eddington.437

0020-7748/80/06004)437503.00/0 9 198 Plenum Publishing Corporation


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438 Battey Pratt and Racey

This model shows that a persistent particle must be described by a c ompa c tg roup The simplest compact universal covering group is named SU 2).The model to be described has SU 2) as its motion group.

2. THE MECHANICAL MODEL: THE SPHERICALR O T A T O R

The mechanical model that led to the geometric solution of the Diracequation can be constructed roughly by suspending a practice golf ballfrom flexible wires that are, in turn, fixed onto a wooden framework. It issufficient to choose six wires corresponding to the positive and negativeaxes of a three-dimensional coordinate system Figure 1). The ball canalways be rotated indefinitely without the wires becoming entangled. Eachdouble revolution returns the system to its original configuration, providedeach wire is kept in the correct relation to all the others.

The simplest workable relation is one that requires all spheres con-centric with the ball to remain rigid; that is to say, the six points where thewires intersect an imagined spherical shell surrounding the ball must

Fig. 1. Mechanical model for demonstrating spherical rotation.


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Geometric M o d e l f o r u p d a m e n t a l P a r t i c l e s 439

Fig 2 Configuration of the z axis of the model

r e m a i n e q u i d i s t a n t f r o m o n e a n o t h e r , t h o u g h t h e s h e l l a s a w h o l e i s f r e e t ot u r n a b o u t t h e c e n t e r .

W h e r e a s t h e b a l l o f t h i s m o d e l c a n , i n p r i n c i p l e , b e t u r n e d a b o u t a n ya x i s s t a r t i n g f r o m a n y i n i t i a l p o s i t i o n , i n p r a c t i c e , b e c a u s e o f t h e r e s t r a i n t sf l e x i b i l i t y , e x t e n s i b i l i t y , e t c . ) o f t h e m a t e r i a l s u s e d , t h e r e i s a b e s t w a y f o r

d e m o n s t r a t i n g t h e m o t i o n :P o s i t i o n t h e b a l l s o t h a t t h e p o i n t w h e r e e a c h w i r e i s a t t a c h e d t o t h e

b a l l f a c e s th e p o i n t w h e r e i t i s a t t a c h e d t o t h e f r a m e . T h e r e m u s t b ee n o u g h s l a c k i n t h e w i r e s t o a l l o w f o r w h a t f o l lo w s . ) N e x t , t u r n t h e b a l lt h r o u g h 1 80 ~ a b o u t a h o r i z o n t a l a x is , s a y t h e x a x is . T h e v e r t i c a l w i r e s t h a tw e r e i n t h e d i r e c t i o n o f t h e z a x i s n o w t a k e o n t h e c o n f i g u r a t i o n o f F i g u r e2 . N o w r o t a t e t h e b a l l a b o u t i t s v e r t i c a l a x i s . H e r e t h e c o n f i g u r a t i o n o f t h ez a x i s w i r e s r e t a i n s i t s s h a p e a n d r o t a t e s a t h a l f t h e a n g u l a r v e l o c i t y o f t h eba l l .

W e c a n g o o n r o t a t i n g t h e b a l l i n d e f in i t e ly , b u t i t w i ll b e f o u n d t h a ta f t e r e v e r y t w o r o t a t i o n s t h e s y s t e m r e t u r n s t o i t s o r i g i n a l c o n f i g u r a t i o n .F i g u r e 3 i l lu s t ra t e s t h e s e q u e n c e o f c o n f i g u r a t i o n s o f t h e m o d e l d u r i n g o n es u c h c o m p l e t e c y c l e .

T h e w i r e s o f t h i s m o d e l c a n b e t h o u g h t o f a s a s e t o f c u r v i l i n e a rc o o r d i n a t e s u s e d t o d e s c r ib e t h e p o s i t io n s o f p o i n ts i n a m e d i u m t h a ts u r r o u n d s t h e s p i n n i n g c o r e .

F o r a n a l t e r n a t i v e m o d e l , i m a g i n e a l a r g e s p h e r i c a l c a n i s t e r w h i c h i sc o m p l e t e l y f il le d w i t h a g e l a t in o u s m e d i u m . I m a g i n e , a ls o , t h a t t h e r e i s as m a l l m a g n e t i z e d s t e e l b a l l s e t i n t h e c e n t e r . W e w i l l p r e s u m e a r e a s o n a b l ed e g r e e o f a d h e s i o n t o e x i st b e t w e e n t h e j e l l y a n d t h e w a ll s o f t h e c o n t a i n e r ,a n d b e t w e e n t h e j e l ly a n d t h e c e n t r a l b a ll . B y m e a n s o f e x t e r n a l e l ec t r o -m a g n e t s w e n o w i n v e r t t h e s t e e l b a l l b y h a l f a t u r n a b o u t t h e x a x i s o f ac o o r d i n a t e f r a m e c e n t e r e d o n t h e b a l l . B e c a u s e o f t h e r e s i l i e n c e o f t h em e d i u m t hi s c a n b e e f f e c te d w i t h o u t t e a r in g o r l o ss o f a d h e s io n . A g a i n ,u s i n g t h e e x t e r n a l e l e c t r o m a g n e t i c f i e l d , w e s e t t h e b a l l s p i n n i n g a b o u t t h e


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~ c)o )

1 /F i g . 3 . T o p : R o t a t i o n o f t h e c o r e is a n t ic l o c k w i s e v i ew e d f r o m t h e u p p e r p o s i ti v e ) zd i r e c t i o n . a ) I n i t ia l p o s i t i o n . b ) A f t e r a q u a r t e r t u r n o f t h e co r e . c ) A f t e r h a l f a t u r n o f t h ec o r e . d ) T h r e e q u a r t e r s tu r n o f th e c o re . e ) A f t e r o n e fu l l t u r n o f t h e co r e . f ) O n e a n d aq u a r t e r c o r e tu r n s . g ) O n e a n d a h a l f c o r e tu r n s . h ) O n e a n d t h r e e q u a r t e r co r e t u r n s . i )T w o f u ll t u r n s o f t h e c o r e r e t u r n s t h e s y s t e m t o t h e i n it ia l p o s i t io n .


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Geo m etr i c M o de l f o r F unda m enta l P a r t i c l e s 4 4

z a x is . A w a v e o f s t r a in w i ll r o t a t e i n t h e j e l l y a t h a l f th e f r e q u e n c y o f t h es t ee l b a l l. A f t e r e v e r y t w o t u r n s o f t h e s t e e l b a l l, t h e s y s t e m w i ll r e t u r n t oi t s in i t ia l s ta te . T h e je l ly w i ll a c c o m m o d a t e t h e m o t i o n w i t h o u t b e i n g t o r no r d i s r u p t e d i n a n y w a y . 2

3 . T H E M A T H E M A T I C S O F S P H E R I C A L R O T A T I O NI n t he A p p e n d i x w e h a v e s h o w n t h a t e a c h c o n f i g u r a t i o n o f th e

s p h e r ic a l r o t a t io n m o d e l c a n b e r e p r e s e n t e d b y a p o i n t o n a E u c l i d e a nf o u r - d i m e n s i o n a l h y p e r s p h e r e t h e L i e g r o u p sp a c e) . T a k i n g th e h y p e r -s p h e r e t o b e o f u n i t r a d iu s , w e c a n d e s c r ib e t h e t r a n s f o r m a t i o n s f r o m o n ec o n f i g u r a t i o n t o a n o t h e r b y a c l os e d u n i m o d u l a r g r o u p . M a k i n g t hisc h o i c e is , i n th e l a n g u a g e o f q u a n t u m p h y s ic s , e q u i v a l e n t t o t h e p r o c e d u r eo f n o r m a l i z i n g t h e w a v e f u n c t i o n . ) I f w e c e n t e r t h is u n i t h y p e r s p h e r e a t t h eo r ig i n o f a r e c t a n g u l a r C a r t e s i a n c o o r d i n a t e s y s te m , a n d l e t t h e v e c t o r 1 ,0 , 0 , 0 ) f r o m t h e o ri g i n lo c a t e t h e p o i n t c o r r e s p o n d i n g t o s o m e c h o s e ni ni ti al c o n f i g u r a t io n , t h e n a n y o t h e r c o n f i g u r a t i o n w i l l b e g i v e n b y t h ev e c t o r ( a , f l , 6 ) , w h e r e a 2 ~ f l2 A t y 2 - J t ' ~ 2 = 1.3 m r o t a t i o n i n t h e s p h e r i c a lm o d e c a n b e r e p r es e n t e d b y a n y o p e r a t o r t h a t w il l t r a n s f o r m v e c t o r s o ft h is t y p e o r t h e i r e q u i v a l e n t s ) i n t o o n e a n o t h e r . H e r e w e sh a l l d e s c r i b e t w oo f t h e m o s t u s e f u l r e p r e s e n t a t i o n s .

1 ) T h e v e c t o r ( a , f l , y , 3 ) c a n b e w r i t t e n a s t h e q u a t e m i o n~ = a + i f l + j T + k 8

w i t h [~b[ ~- t~*r = 0l 2 + 2 _1. . ./2 _[_ ~ 2 = 1 , whe re ~ i s t h e q u a t e r n i o n i c c o n -j u g a t e .

T r a n s f o r m a t i o n s o f t h is q u a t e r n i o n i n to a n y o t h e r u n i m o d u l a rq u a t e r n i o n c a n b e e f f e c t e d b y m u l t i p l y i n g b y a n o t h e r s u i t a b l e q u a t e r r t i o n .T h u s , u n i m o d u l a r q u a t e r n i o n s d o d u t y f o r b o t h t h e c o n f i g u r a t i o n v e c to ra n d t h e r o t a t i o n o p e r a t o r .

2If we replace the canister b y a surroun ding region of stationary jelly, an d the steel ball by acore o f spinning jelly, then , insofar as there is strain neither in the stationary region no r inthe core, and their potential energies are therefore both zero, we m ay have a n exam ple of asoliton. Fo r a general article on S olitons see Claudio Rebbi 1979), p. 92, with furtherbibliography furnished o n p. 168. This m ight be w orth pursuing though we, ourselves, havenot done so.3For example, referring to Figure 3, let 1, 0, 0, 0) stand for configuration a), 0, 1, 0, 0) fo rconfiguration c), then 0a) wou ld be 2 -1/2, 2 -1/2, 0, 0), d) wou ld be - 2 - I/2 , 2 -1/2, 0, 0),e) wou ld be - 1, 0, 0, 0) etc.


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( 2) T h e q u a t e r n i o n a + i/3 + j ~ , + k 8 c a n b e r e p r e s e n t e d b y t h e c o l u m nv e c t o r

8y/3

T h i s p a r t i c u la r a r r a n g e m e n t o f t h e c o e f f ic i e n ts h a s b e e n c h o s e n t o a c h i e v e,i n w h a t f o l lo w s , c o n f o r m i t y w i t h t h e t r a d i ti o n a l f o r m u l a t i o n s o f q u a n t u mp h y s i c s .

N o w b e c a u s e i ( a + i f l + j y + k r ) = - 13 + i a - j 8 + k 't , l e f t m u l t i p l i c a -t i o n b y i h a s t h e s a m e e f f e c t o n a , f l, y , a n d 6 a s t h e m a t r i x p r o d u c t :

0 0 1 0 =0 - 1 0 01 0 0 0

s i m i l a rl y , j a n d k a r e , re s p e c t i v e l y , r e p r e s e n t e d b y

i 0 - 1 00 0 - 10 0 01 0 0

a n d 1 0 00 0 00 0 - 1

T h u s t h e f u l l o p e r a t o r a + if l + j ~ , + k/~ i s r e p r e s e n t e d b ya - t ~ - y - / 38 a /3 - t

- / 3 ~/3 / - 8 a

T h i s m a t r i x c a n b e p a r t i t io n e d a s s h o w n . E a c h o f t h e q u a d r a n t s isr e c o g n i z a b le as t h e m a t r i x r e p r e s e n t a ti o n o f a c o m p l e x n u m b e r . H e n c e w ec a n w r i t e

[ a + i 8 - - / + i f l ]r y+ i f l a - - i ST h e d e t e r m i n a n t o f ~ , i s a 2 + 8 2 + ,/2 + f l 2 = 1 . T h i s m a t r i x i s u n i t a r y a s w e l l

a s u n i m o d u l a r o r s p e c i a l . T h u s , th is representation o f spherical rotation


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TABLECompsobwehQuenocaSU2Ree

aooS

cRoao

Queno

SU2

Sn

P

cnepeao

oaoa

oa

oao

oa

Opao

p

|

i0~0]

i

i

[

i

0]

[0]

Le

hmo

a

s

Roaethceothmo

tho

1~a

hxas

gh

y

[0

]

E0

~oO.~eooroom~o~

1O

tho

1~a

h

as

[~_

[0

Roaethceothmo

i

tho

1~a

hzas

Inap

o

P

oaanboance

1~a

thxasfom

inap

o

P

oaanboanh

ce1~a

h

asom

inap

o

P

oaanboance

1~a

hzasomh

inap

o


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4 4 4 B a t te y P r a tt a n d R a c e y

consis ts o f the special unitary ma trices o f order 2, f r o m w h ic h co m e s t he n a m eSU(2) .

T h e o p e r a n d f o r m o f ~ isI a + i 8

~ + i f l IThis fo rm is ca l led a spinor a w or d co i ned by P . A . M . D i r ac .T ab l e I s how s a compar i s on be t w een t he qua t e r n i on i c and t he SU ( 2 )r ep r e s en t a ti ons o f s phe r ica l r o t a t ion t oge t he r w i th an i n t e r p r e t a t i on o f t hesymbols .

4 . S P I NRota t ion tha t i s a l inear func t ion of t ime i s r e fe r red to as spin.A m ode l o f s phe r i ca l r o t a t ion w hos e i n i ti a l con f i gu r a t i on is g i ven bythe sp ino r [1] can be ro ta ted in to th e pos i t ion [~ ] by the op era tor [o 0 ] .

This i s a ro ta t ion of the c ore o f the m od el by 180 ~ ab ou t th e z ax is . I t ; sr ep r e s en t ed in t he L i e g roup s pace by a q ua r t e r t u r n a r o un d a g r ea t c i rc l ei n t he f ou r - d i mens i ona l hype r s phe r e . A n i n t e r me d i a t e pos i ti on is g i ven byt he s p i no r [ S 0 o . i O 0 ] [ e ~]w her e 0 i s t he angu l a r d i s p l acemen t a l ong t he g r ea t c i r c l e . N o t e t ha t t h i sr ep r e s en t s a r o t a t i on o f t he co r e o f t he m ode l by 20 ab ou t t he z axis . T her o t a t i on t ha t b r i ngs t he m ode l in t o t h is pos i ti on f r om t he i n it ia l con f i gu r a -t ion i s r epre sen te d by the o pe ra tor [ co8 e~ ] . Thu s , i f we in t ro du ce th e t im epa r am e t e r , t, t hen t he ope r a t o r [e~, _ o ] w i l l gene r a t e t he s p i n [ ~ ~ ]rep rese nt ing co re sp in of an gu lar ~Telocity 2~o. Sim ilar ly ,

cos ~ot i sin tot ]i s in to t co s ta Jgene r a t e s a s p i n i n w h i ch t he co r e o f t he mode l has angu l a r ve l oc i t y 2~ 04The configurationof the z-axis wires of the m odel (Figure 2) r o t a t e s a t ha lf the angularvelocity of the core, tha t is, with velocity~0.


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Geom etr ic Mo d e l for u n d am en ta l Particles 5

a b o u t t h e x a x is , a n d c o s to t - s i n t o t ]s in tot co s tot

d o e s t h e s a m e a b o u t t h e y a x is .I n t h e t r a d i t i o n a l a n a l y s i s o f s p i n , i t i s u s u a l t o i m p l y t h a t t h e p r o c e s s

o f i n v e r t i n g t h e a x i s o f a s p i n n i n g o b j e c t i s i d e n t i c a l t o r e v e r s i n g t h e s p i n .W h e n , h o w e v e r , t h e s p i n n i n g o b j e c t i s c o n t i n u o u s l y c o n n e c t e d t o i t ss t a t i o n a r y e n v i r o n m e n t , t h i s c e a s e s t o b e t r u e ; a n d w e m u s t m a k e a c a r e f u ld i s t in c t i o n b e t w e e n t h e i n v e r s i o n a n d t h e r e v e rs a l o f sp in . T h i s d i s t i n c t io na f f o r d s u s in s ig h t in t o o n e o f t h e m o s t f u n d a m e n t a l p r o p e r ti e s o f e l e m e n -t a r y p a r t i c l e s .

To reverse the z a x i s sp irt [ eo~ o n e c a n r e v e r s e t i m e t - + - t , o r o n e c a nr e v e r s e t h e a n g u l a r v e l o c i t y o f t h e m o t i o n : t o - + - - t o . T h e r e v e r s e d s p i nb e c o m e s [ e ~ ~' ] . T h i s m o t i o n c a n b e g e n e r a t e d b y t h e r e v e r s e s p i n o p e r a t o r[ -o ~ 2 , ] f r o m t h e i n i ti a l c o n f i g u r a t i o n [ o . W e s h a ll s u g g e s t i v e ly c a ll t h isantispin a n d r e f e r t o t h e antispin state a s o p p o s e d t o t h e normal spin state o fs p h e r i c a l r o t a t i o n .

T o invert t h e s p i n a x is o f t h e c o r e o f o u r m o d e l i t is n e c e s s a r y t o t u r nt h e s p i n n i n g c o r e a b o u t o n e o f t h e a x e s p e r p e n d i c u l a r t o t h e s p i n a x i s , f o re x a m p l e , t h e y a x is . T h e o p e r a t o r t h a t w i ll a c h i e v e th i s i s [ 0 01 ] . T h e r e -f o r e , t h e i n v e r t e d s p i n s t a t e i s g i v e n b y[ 0 O 1 ] [ e ~ ~ t] _ ~ [ e iO t lN o w , o l ] E e i o t l = I o l ] E = o e = l [ ]= I ] I e = ~= I e = ] I

e itot 1T h u s , t h e i n v e r t e d s p i n s t a t e i s g e n e r a t e d b y t h e a c t i o n o f t h e r e v e r s e s p i no p e r a t o r o n t h e i n i t i a l c o n f i g u r a t i o n [ o ] . L a s t l y , w e h a v e t h e i n v e r t e da n t i s p i n s t a t e [ e _ ~ ] .

T h e c o n t r a s t b e t w e e n t h e a b o v e f o u r s t a t e s i s i m p o r t a n t f o r t h e l a t e ru n d e r s t a n d i n g o f p a r t i c l e p h y s i c s . H o w e v e r , i t m u s t b e u n d e r s t o o d t h a t


10/39


t h e r e e x is ts a n i n f i n i t y o f i n t e r m e d i a t e s t a te s o f o u r m o d e l . T h e n o r m a ls p i n s t a te a x i s c a n b e o r i e n t e d b e t w e e n t h e e x t r e m e s o f z a x is s p in - u p a n ds p i n - d o w n . T h e i n t e r m e d i a t e i n v e r t e r i s

c o s x - s i n x ]s i n x c o s x ]

w h e r e 2 X i s t h e a x i s a n g l e w i t h r e s p e c t t o t h e s p i n - u p d i r e c t i o n . T h ei n t e r m e d i a t e s p i n o r i s

e i to tC O S Xe i~ o t i n X J

w h i c h i s g e n e r a t e d b ycos to t + i s in to t c os 2 x

i s in tot s in 2 Xi s in tot s in 2 X ]

cos to t - i s in w t cos 2Xf r o m th e i n it ia l p o s i t i o n [ ~~ T h e c o r r e s p o n d i n g a n t i s p i n d a t a a re o b -L ~mx jt a i n e d b y t a k i n g th e c o m p l e x c o n j u g a t e f o r m u l a s . T h e r e i s a l so a c o n -t i n u u m o f s t a t e s i n t e r m e d i a t e b e t w e e n n o r m a l s p i n a n d a n t i s p i n . T h u s ,[ e ] . . . . I r176 ] w ill b e i n t h e n o r m a l_ 0 , o p e r a t i n g o n t h e l m t l a l c o n f i g u r a t i o n sty xsp in s t a t e fo r X = 0 , an d i n t he an t i sp in s t a t e fo r X = o r //2 -

5 . E Q U A T I O N S O F M O T I O N O F T H E S P H E R I C A L S P I NM O D E L

S u p p o s e w e h a v e a m o d e l o f s p h e r i c a l r o t a t i o n w h o s e i n i t i a l c o n f i g -u r a t i o n is g i v en b y t h e s p i n o r

[ a + i ~

W i t h o u t l o s s o f g e n e r a l i t y , w e s h a l l s u p p o s e s p i n t o b e g e n e r a t e d b y t h e[ e , 0 ] T h e c e n t e r o f t h e s p i n n i n g c o r e is s t a t io n a r y , a n d t h eo p e r a t o r o e - ~p h a s e o f t h e r o t a t i o n i s , a t e a c h i n s t a n t , t h e s a m e t h r o u g h o u t t h e s p a c elo c c u p i e d b y t h e m o d e l . I t w i l l b e u s e f u l t o a s s u m e t h a t t h e s t a t i o n a r yf r a m e w o r k i s l a r g e a n d f a r a w a y f r o m t h e c o r e s o t h a t w e c a n a s c r i b e t h ec o n f i g u r a t i o n s p i n o r ( w h i c h d e s c r i b e s t h e p h a s e o f t h e r o t a t i o n ) t o e v e r yp o i n t o f t h e l o c a l r e g i o n .


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Geo m et r ic Mo de l f o r Funda m ent a l Pa r t i c le s 4 4 7

A c c o r d i n g t o s o m e o t h e r o b s e r v e r m o v i n g p a s t o u r s y s t e m w i t hv e l o c i ty - g , t h e s y s t e m is r e p r e s e n t e d b y th e s p i n o r

ito(t -- e '~ /c2)eple 0- i ~ t - e-~/~20 ~2 e B

H e r e w e h a v e u s e d t h e L o r e n t z t r a n s f o r m a t i o n t ' ~ ( t - g 4 / c 2 ) / f l , w h e r ef l = ( 1 - v 2 / c 2 ) 1 / 2 a n d g . f = v x x + V y y + v z z . T h u s , t h e m o v i n g o b s e r v e r se e st h e c e n t e r o f o u r m o d e l m o v i n g p a s t h i m w i t h v e l o c i t y g w h i l e th e p h a s e o ft h e r o t a t i o n v a r ie s n o t o n l y w i t h t i m e b u t a l s o f r o m p l a c e t o p l ac e . I n f a c t ,f r o m t h e e x p o n e n t f a c t o r t - g . f / c 2 , w e c a n d e d u c e t h a t h e s e e s e a c hp a r t i c u l a r p h a s e o f t h e m o t i o n s w e e p i n g f o r w a r d w i t h a v e l o c i ty o f c 2 / [ F [i n th e d i r e c t io n o f g ( F i g u r e 4 ). R e g i o n s o f c o n s t a n t p h a s e f o r m p l a n e sp e r p e n d i c u l a r t o th e m o t i o n o f th e m o d e l .

T h e o b s e r v e r a l so r e c k o n s t h a t t h e c o n f i g u r a t i o n r o t a te s a b o u t t h e c o r ew i t h a n g u l a r v e l o c i t y t o ( 1 - v 2 / c 2 ) 1 /2 . T h i s d e c r e a s e f r o m t h e v a l u e t o is am a n i f e s t a t i o n o f t h e r el a t i v is t ic p r i n c i p l e o f ti m e d i l a t a t i o n . O n t h e o t h e rh a n d , t h is ro t a t i o n c o m b i n e d w i th t h e fi n it e f o r w a r d m o t i o n o f t h e p h a s ep r o d u c e s a c o n f i g u r a t i o n h e l ix w h o s e p i t c h d e c re a s e s w i t h in c r e a s i n g c o r e

p h a s e i n a d v a n c e o f c o r e x x p h a s e l a g g i n g b e h i n d c o r e r e g i o n\ L v ~-

w a v e p h a s e a n d c o r e v e l o c i t y v e c t o r sunit time)

Fig. 4. To p: T he mo tion of the wires of Figure 2. The angular veloci ty of the core is 2~ andtha t of the wire configuration is ,o. Bottom: Th e sam e with its core c e n te r m o v i n g to the right(in the z direction) w ith velocity ,5. Th e seco nd figure shows the situation 8~r/~(l - v 2 / c 2 ) t / 2t i m e units after the f irst . N ote h ow the w ires become helices which ro t a t e co nt ra ry t o acorkscrew w ith respect to the advancing core center . (] he h el ix o f a corkscrew advancesa l o n g a fixed locus.)


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4 4 8 B a t t e y - P r a t t a n d R a c e y

ve loc i ty exem pl i f ied by a s e lec t pa i r o f m od e l w i res in F igure 4 ) .M e a s u r e d a t t h e p o s i t i o n o f t h e o b s e r v e r , th i s he l i ca l conf igu ra t ion ro ta tesw i t h a ng u l a r v el o c it y ~ / 1 - v 2 / c 2 ) 1 / 2 . 5I t s h o u l d b e n o t i c e d t h a t t h e m a t h e m a t i c a l d e s c r i p t i o n o f t h e m o t i o no f o u r m o d e l d e s c r i b e s o n l y t h e c o n f i g u r a t i o n p h a s e : i t d o e s n o t i n d i c a t et h e p o s i t i o n o f t h e c o r e c e n t e r . I t f o l l o w s t h a t a t h e o r y e n t i r e l y b a s e d o nsp inor s and the i r equa t ions w i l l be es sen t i a l ly incomple te .

W i t h i n t h i s i n c o m p l e t e t h e o r y , w e c a n f o r m u l a t e t h e g e n e r a l l a w o fm o t i o n f o r t h e p h a s e o f t h e c o n f i g u r a t i o n s u r r o u n d i n g t h e m o d e l b yder iv ing the d i f f e ren t i a l equa t ion tha t i s independen t o f 15 :

i t ~ t - - ~ . ~ / c 2 )0__~_~= L ga le 13O t O t - i ~ ( t - ~ . ~ /c 2 )ga2e

i c o i ~ ( t - ~ . ~ / c 2 )gal -- ~ e 13( i t o ) - i ' ~ga2 - - ~ e

~ I i t~( t -- ~ '~ /c2)= gal e . 13- - i ( o ( t - - ~ ~ 1 c 2 )-- ga2e lJ

0t 20 i ~0 t 3

i ro( t - -~ .~ /c2)g a t e P- - i ~ ( t - - ~ ~ / c 2 )- - ga2 P

02 2= Sga

O ga = i t O V xax c B

i~ ( t - - ~ '~ /c2)ga l e 13-- i to(t-- 6 ?/C2)-- ga2 /3

h e n c e0 2 g aOX2 = C ~ ga

5 T h e d i f f e re n t b e h a v i o r o f t h e s e t w o a s p e c t s o f t h e a n g u l a r v e l o c i ty i l l u st r a te s t h e d i f f e r e n c ee t w e e n a c o n t r a v a r ia n t a n d a c o v a r i a n t v e c to r in M i n k o w s k i s p a c e, t o o - - v 2 / c 2 ) I /2 i s , a s

a l r e a d y m e n t i o n e d , r e l a te d t o t h e t i m e c o m p o n e n t o f a c o n t r a v a r i a n t s p a c e - t i m e v e c t o r ;w h e r e a s , w e s h a ll , w h e n w e c o m e t o c o m p a r e o u r m o d e l w i t h a n e l e m e n t a r y p a r ti c le , s e e t h a tw / l - v 2 / c 2 ) I /2 i s r e l a te d t o t h e e n e r g y o f a c o v a r i a n t m o m e n t u m v ec t o r.


13/39

G e o m e t r ic M o d e l f o r F u n d a m e n t a l P a r t ic l e s 4 4 9Likewise ,

0 2 ~ 2 23y---- - - -_ C4fl---~b

H e n c e , u s i n g t h e n o t a t i o n

w e h a v e

T H U S ,

_ _ _ ~ ~n d 0 2~b 2 2a z 2 - - c 4 3 2 ,

2 0 2 t~ t 02--'-'-~-~4 - 0 2 ~9 = ~ 3),2 3z----70)2 0)21)2

c B 2 - - c 4 B 20 2r = ~222~ 2 t ~ C 2 O t 2

6. F U N D A M E N T A L P A R T IC L E ST h e r e h a v e b e e n v a r i o u s a t t e m p t s t o d e s c r ib e f u n d a m e n t a l p a r ti c le s a s

f o r m s o f t h e c o n t i n u u m . l B u t , i n c o n v e n t i o n a l t h in k i n g , a p a r a d o x a r is e s i fpa r t i c l e s a r e cons ide red f r ee to sp in . Fo r , i f the re i s indeed a con t inuumtha t a t one po in t i s to be asc r ibed to the pa r t i c l e , and , a t ano the r , to thes u r r o u n d i n g v o i d , t h e n t h e c o o r d i n a t e l i n e s u s e d t o m a p o u t t h e w h o l es p a c e a t a n y g i v e n i n s ta n t w o u l d , w i t h t h e p a s s a g e o f t im e , b e c o m e t w i s t e du p a n d s t r e t c h e d w i t h o u t l i m it . A l t e r n a t iv e l y , t h e y w o u l d r ip , a n d o n e p a r to f t h e c o n t i n u u m w o u l d s l id e p a s t a n o t h e r a l o n g a s u r f a c e o f d i s c o n ti n u i ty .

N o w w e m a i n t a i n t h a t th e c o n c e p t o f r i p p in g th e v a c u u m i s i n tu i-t iv e l y a b s u r d , a n d m u s t b e r e je c t e d , b e c a u s e i t i n t r o d u c e s d i s c o n t i n u i t yi n t o a t h e o r y w h o s e b a s i c a s s u m p t i o n i s t h a t w e s h o u l d b e a b l e t o e x p l a i nthe un ive r se as be ing con t inuous . L ikewise , an in f in i t e deg ree o f tw is t ingu p o f t h e c o n t i n u u m m u s t a l s o b e r e j e c t e d a s u n w o r k a b l e . W e t h e r e f o r ei n t r o d u c e t h e f o l lo w i n g p o s t u l a t e :

Postulate The o n l y a l l o w a b l e p e r s i s t e n t m o t i o n s o f a l o c a l r e g i o n o ft h e c o n t i n u u m w i t h r e s p e c t t o t h e a m b i e n t r e m a i n d e r a r e s u c h t h a t t h e i rm o t i o n g r o u p s a r e s i m p l y c o n n e c t e d a n d c o m p a c t .

U n d e r t h e se c i rc u m s t a n c e s, t h e m o t i o n i n t h e c o n t i n u u m w i ll b ec y c li c , a n d t h e s y s t e m c o n f i g u r a t i o n w i ll r e p e a t e d l y r e t u r n t o e a c h p h a s e o ft h e c y c l e ( s ee A p p e n d i x ) .T h e s i m p l e s t p o s s i b l e s p i n n i n g e l e m e n t i n t h e c o n t i n u u m i s o n e t h a t

r o t a t e s i n t h e s p h e r i c a l m o d e . I m a g i n e , t h e r e f o r e , t h a t t h e w i r e s o f o u rm o d e l a r e G a u s s i a n c o o r d i n a t e l i n e s f o r a r e f e r e n c e f r a m e i n s p a c e . T h e


14/39


o r i g i n i s i n a p a r t o f s p a c e t h a t i s s p i n n i n g ( t h e c o r e ) r e l a t i v e t o t h es u r r o u n d i n g p a r t s . T h e m a t h e m a t i c s t h a t w e h a v e d e v e l o p e d s o f a r i st h e r e f o r e a p p r o p r i a t e . T h e c o n f i g u r a t i o n o f t h e m o d e l b e c o m e s t h e c o n f i g -u r a t i o n o f sp a c e. H e r e w e m u s t s o u n d a c a u t i o n a r y n o t e . T h e m o d e l w a sd e s c r i b e d b y r e f e r en c e t o a r e c t a n g u l a r c o o r d i n a t e s y s t e m i n t h e fl a t s p a c ei n w h i c h t h e m o d e l w a s p r e s u m e d t o e x i s t . N o w w e a r e a s s u m i n g t h a t o u rs p i n n i n g c o n f i g u r a t i o n i s t h e s p a c e i t s e l f . U s e o f t h e m o d e l f o r m u l a st h e r e f o r e m e a n s t h a t w e s h a l l b e d e s c r i b i n g r e a l s p a c e b y r e f e r e n c e t o an o n e x i s t e n t b a c k g r o u n d . T h i s w ill w o r k f o r th e s a m e r e a s o n t h a t e n a b l e dN e w t o n t o d e v e l o p a th e o r y o f g r a v i ta t i o n o n th e a s s u m p t i o n t h a t s p a c ew a s e v e r y w h e r e E u c l i d e a n . B u t , a s w a s t h e c a s e w i t h N e w t o n s t h e o r y , o u rt h e o r y m a y r e q u ir e l a te r r e f i n e m e n t . 6O u r c o n c l u si o n , t h en , is t h a t t h e s p i n n i n g c o n t i n u u m is s u r r o u n d e d b ya n u n d u l a t i n g , w a v e l i k e r e g i o n w h o s e p h a s e , @, s a ti sf i es t h e e q u a t i o n

1 0 2 t ~ 0 2v ' a t 2 =

T h i s l o o k s v e r y l ik e t h e K l e i n - G o r d o n e q u a t i o n f o r a n e l e m e n t a r y p a rt ic l e.W e s h a l l t h e r e f o r e m a k e t h e i d e n t i f i c a t i o n e x a c t . T h e u s u a l f o r m u l a t i o n o ft h e K l e i n - G o r d o n e q u a t i o n i s

1 x~ m cV2x~ X~C Or2 ~2

w h e r e m is t h e m a s s o f t h e p a r t i c l e , h i s P l a n k s c o n s t a n t d i v i d e d b y 2 rr ,a n d a / i s t h e w a v e f u n c t i o n f o r t h e p a r t i c l e . T h e w a v e f u n c t i o n m u s t ,t h e r e f o r e , b e i d e n t i f i e d w i t h t h e s p i n o r t h a t d e s i g n a t e s t h e phase o f t h es p h e r ic a l s p in in s p a c e a n d t im e . W e c a n n o w s ee w h y t h e w a v e f u n c t i o nh a s n o p h y s i c a l ly m e a n i n g f u l m a g n i t u d e , b u t i s u s u a l ly n o r m a l i z e d . T h ei d e n t i f i c a t i o n r e q u i r e s t h a t w e p u t

m c r2 C2

w h e n c e

I m l = l ' 0 1

6For example, a radially dependent gauge transformation can be added to a n exact theory o fparticles (that includes a d escription of the co re location) to bring ab out a contraction in thevolume of the particle s space. This could account fo r the gravitational field.


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G e o ~ M o d e l f o r F u n d a m e n t a l P a r t ic l es 4 5 1

T h u s , w e d e d u c e t h a t w h e n w e h a v e a s p i n n i n g r e g io n o f th e c o n t i n u u m , i ti n t e ra c t s w i th o t h e r s p i n n i n g r e g i o n s in s u c h a w a y t h a t w e f i n d i tn e c e s s a r y t o i n t r o d u c e a m e a s u r e f o r th e i n er t i a o f th e in t e r a c t io n s . W ea r b i tr a r i l y d e f in e t h e u n i t o f m a s s . B u t it w o u l d a p p e a r t h a t t h e i n e r ti a o ft h e s p i n n i n g r e g i o n i s s i m p l y a m a n i f e s t a t i o n o f i ts a n g u l a r v e l o c i t y ; a n d , i ft hi s is m e a s u r e d i n r a d i a n s p e r s e c o n d , t h e n t h e ra t i o o f th e a r b i t r a r y m a s su n i t t o t h e s p i n f r e q u e n c y i s h / c 2 , w h i c h m u s t , th e r e f o re , b e a c o n s t a n t .T h i s i s t h e s i g n i f i c a n c e o f P l a n k ' s c o n s t a n t .

T h e a s c r i p t i o n o f m a s s t o t h e a n g u l a r v e l o c i t y o f th e s p i n i s r e la t iv e t ot h e o b s e r v e r a n d m e a s u r e d a t h i s l o c a t i o n . 7 S o a p a r t ic l e , w h o s e m a s s is m 0w h e n s t a t i o n a r y , h a s a m a s s m o / ( 1 - v 2 / c 2 ) 1 / 2 w h e n i n m o t i o n w i t hv e l o c i t y 6 (s ee p a g e 4 5 8) . A n o t h e r c o n s e q u e n c e o f t h e i d e n t i f ic a t i o nr n c 2 = h t o is t h a t t h e c o n f i g u r a t i o n s p i n , t h a t is , t h e o u t e r m o s t u n d u l a t i o n so f t h e s y s t e m , i s t h e d e B r o g l i e w a v e . T h i s w a v e e m b o d i e s a c o r e r e g io nt h a t , in t h e s t a t i o n a r y p a r t ic l e , s p i n s w i t h f r e q u e n c y t o / ~ . T h i s is t h eZ i t t e r b e w e g u n g t h a t w a s f i r s t s tu d i e d b y E . S c h r 6 d i n g e r ( 1 9 30 ). I n t h em o v i n g p a r t i c l e, t h e Z i t t e r b e w e g u n g f r e q u e n c y s lo w s t o ( t o / ~ r ) ( 1 -V2/C2 1/2 ( s e e p a g e 4 5 8 ) . 8

T o i n c r e a s e o u r in s i g h t i n t o t h e e l e m e n t a r y p a r t i c l e f o r m e d b y t h es p h e r i c a l l y s p i n n i n g m a n i f o l d , w e s h a ll e s t a b l is h a f i r s t -o r d e r d i f f e r e n t i a le q u a t i o n f o r t h e s p i n o r a s is d o n e i n r e la t iv i st ic q u a n t u m m e c h a n i c s . T h es t a n d a r d f i rs t -o r d e r o p e r a t o r i n M i n k o w s k i s p a c e i s t h e q u a d , [ ] , w h o s es e c o n d - o r d e r d e r i v a t i v e is t h e d ' A l e m b e r t i a n [ : f f f i V 2 - O 2 / c 2 0 t 2. T h eK l e i n - G o r d o n e q u a t i o n i s o f t e n w r i t t e n in t h e f o r m

( [ ] 2 - - m 2 c 2 / ~ 2 ) x i ~_ 0

T h e q u a d o p e r a t o r , a s n o r m a l l y u s e d , is a f o u r - d i m e n s i o n a l v e c t o r o p e r a -t o r. W e s h a ll m o d i f y i t s o t h a t i t b e c o m e s a s p i n o r o p e r a t o r . I n t r a d i t io n a lq u a n t u m m e c h a n i c s i t is u s u a l , a t th i s s ta g e , t o i n t r o d u c e a g r o u p o fH e r m i t i a n m a t r i c e s k n o w n a s P a u l i ' s s p in m a t r i c e s . W e s h a l l n o t d o t h a t7We shall use the word observer in the way that is usual when discussing d i f f e r e n t r e f e r e n c eframes. At the level of an elementary particle, the notion of a real observer ceases to h a v emeaning. We shall retain the word, however, as meaning the person or object (in thiscontext, the other interacting particle) t o w h o m the relevant calculations apply.S i n traditional quantum theory, the mathematical formalism that produces the Zitterbewe-gu ng term indicates a frequency th at increases with the particle's mo tion. This is because theformalism i s d e s i g n e d t o predict the results of a c tu a l m e a s u r e m e n t s A m e a s u r e m e n t m a d eo n a moving system is m a d e w i t h r e f e r e n c e t o the stationary f r a m e w o r k o f the measuringdevice. It is the projection of the spin o f the moving core on to the stat ionary measuring f r a m etha t increases in frequency. M ost t e x ts o n relativistic quan tum theory go through the exerciseo f using the established mathem atical form alism to calculate a particle's velocity. W e give, asan example, J. McConneU (1960).


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h e r e a s w e h a v e n o r e a s o n t o u s e P a u l i 's m a t r i c e s a n d n o i n t e r p r e t a t io n f o rt h e m . W e sh a ll t h e r e f o r e i n t r o d u c e t h e o p e r a t o r

0 8 - i - - i ] ~ z[ : ] - - I 0 i 0 1 0 ] -@ Y + [ 0W e h a v e a p h y s i c a l i n t e r p r e t a t i o n f o r e a c h t e r m i n t h i s e x p r e s s i o n . T h em a t r ic e s u s e d f o r m a b a s i s f o r t h e a lg e b r a o f S U ( 2 ) . E a c h o f t h e s p a c et e r m m a t r i c e s t u r n s t h e d e r i v e d s p i n o r i n t h e t e r m t h r o u g h a q u a r t e r t u r n( o r t h e c o r e t h r o u g h a h a lf t u r n ) a b o u t t h e a s s o c i a te d a x is . T h e q u a -t e r n i o n i c f o r m o f t h e o p e r a t o r i s

~ . ~i c ~ t+ i + J ~ y + k -- ~ -w h i c h is s t ro n g l y s u g g e s t iv e o f t h e u s u a l q u a d o p e r a t o r . T h e q u a t e r n i o n i cf o r m r e m i n d s u s t h a t t h e r e is a c o n j u g a t e o p e r a t o r

1 ] i i 1D * = [ O 1 i O ] ~ x - - [ 0 i 0 lW e n o t e t h a t [ - I * D = [ o o ][:]2 , w h e r e E ]2 i s t h e d ' A l e m b e r t i a n o p e r a t o r .L e t u s n o w c a l c u l a t e D e? , w h e r e e? i s t h e s p i n o r f o r s p h e r i c a l s p i n i nt h e m a n i f o l d . T o s a v e sp a c e , s in c e t h e e x p o n e n t i a l p a r t s o f t h e s p i n o r so c c u r s o f r e q u e n t l y , w e s h a l l i n t r o d u c e t h e a b b r e v i a t i o n s

i t o ( t - - 6 . F / c 2 ) - - i t o ( t - - 6 F / c 2 )e + = e # , e - -- -- e #U s i n g t h e d e r i v a t i v e s f r o m p a g e 4 5 7 , w e s e e t h a t

to e?l e + +[ ] e?= - ~ - - e? 2e - c2fl k e? le + o [ 0 e ]

1 + ~ . / c e ?,e~ - _ ~ + i ~ , ) / c-~ e?~e -t ) x ~ i . i . i ; y ) / c e ? l e +- f l z / c e ? 2 e -o r

t oD ~ = c e ? ( A )


17/39

G e o m e t r i c M o d e l f o r F u n d a m e n t a l P a r t i c le s 4 5 3

w h e r e w e h a v e d e f i n e d t h e q u a n t i t y

~ , = f l [ ( l + v J c ) - ( v ~ + i V y ) / C ]( v ~ - i v y ) /c - ( 1 - v J c ) 'pN o w w e h a v e a l r e a d y se e n t h a t [ : ] * [ ~ = [:]2 ~ 0 )2 /C 2)t~ . H e n c e

, t O t ~ 2[] T~ = ~r

~

W e c a n c o m b i n e A ) a n d B ) i n t o o n e e q ua t i on b y w ri ti ng

~

t ~ l et~ 2 e -

( l + v z / C ) ( v ~ + i v y )fl ePle+ c f l( v x iVy ) 1 - v J c )e f t ~ l e + f l

r -

~b2e -

T h e n0 01 00 10 0 o ~o ~ ~ ~1 + i 00 00 00 - i o ] ~i0

+- 1 00 00 00 - 1

yi0+ 00

o o ]i 0 ~

0 - i 0z0 0

0 0= _ ~ 0 0

c 1 00 1

B )

1 ]10 00 0


18/39

454 Battey Pratt and RaceyT h e t o p l e f t q u a r t e r p a r t i t i o n o f t h e l e f t - h a n d s i d e o f t h e e q u a t i o n , b e i n gt h e s t a t e m e n t [[]q~ , i s e q u a t e d t o t h e l o w e r h a l f p a r t i t i o n o f ( t o / c ) ~ b ym e a n s o f t h e p a r t i t i o n - r e v e r s i n g o p e r a t o r1 10 0

10 1 0T h e q u e s t i o n n a t u r a l l y a r is e s, i s th i s equ iva len t to Dira c s eq ua t ion?T o see t ha t i t is , w e wr i t e { i n p l a ce o f 4~, m c / h i n p l a c e o f o~/c ,

m u l t ip l y b y h c , e x t r a c t t h e f a c t o r i f r o m t h e s p a c e t e r m s , a n d r e a r r a n g e t of o r l n

- 100

- 10+ 00

- 1000

o o o0 0 O ~ i0 1 - ~ x + 01

~ 76 760 0 0 {0 1 0 ] - -~z J l+0 0 - 1

- i 00 00 00 - i

0o oi 0y0

0 0 - 1 0 ]0 0 0 - 1 ] m c2{ ,- 1 0 0 00 - 1 0 0

O,I,1 ih= 1 - ~0 0

W e c a n n o w i n t ro d u c e a s i m i la r it y t r a n s f o rm a t i o n . T h e m a t r i x

A = A l =- -2 -1 /2 0 2 -1 /2 0

0 - - 2 - I / z 0 2 1/22 - 1 / 2 0 2 - 1 / 2 0

0 2 - 1 / 2 0 2 - 1 / 2i s i ts o w n i n v e r s e . T h i s g iv e s th e f o l l o w i n g t r a n s f o r m a t i o n s :

A I A - I = I0

A 0- 10 l O 1 [ 1 o0 0 - 1 A - l _ _ 0 1 0 00 0 0 0 0 - 1 0- 1 0 0 0 0 0 - I


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Geom etric M odel for Fundam ental Part ic les 455

( I t w a s t h is r e su l t t h a t d i c t a t e d t h e f o r m o f A . )

A0 - 1 0 O ]

- 1 0 0 0 J A - I =0 0 0 10 0 1 0

0 0 0 1 ]0 0 1 00 1 0 0 '1

0 - i 0 0 ]i 0 0 0 A -A 0 0 0 i0 0 - i 0

1 =0 0 0 i |0 - i 00 i 0 0 '

- i 0 0 0

A- 1 0 0 0 ]0 1 0 0 I A -0 0 1 00 0 0 - 1

0 0 1 00 0 0 - 110 - 1 0 0

H e n c e o u r s p i nn in g m a n i f o ld e q u a t i o n b e c o m e s

+

0 0 10 1 0 OxI '1 0 0 --~-x+0 0 0

0 0 1 00 0 0 - 110 - 1 0 0

0 0 0 i0 0 - i 00 i 0 0- - i 0 0 0

O'Yy

g ; i10 1 0 00 0 - 1 00 0 0 - I

m c 2 , ~

l ~ 7 61 0 ih Ox~0 0 1 at0 0 0

w he r e 'I = A ~ ' .T h i s is D i r a c ' s e q u a t i o n . T h u s , w e h a v e s h o w n t h a t t h e e n t it y f o r m e d

b y a s p h e ri c a ll y r o t a t in g d i s t u r b a n c e o f t h e m a n i f o l d i s a D i r a c p a r t i cl e .


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456 Battey-Pratt and RaceyE v a l u a t i n g x I,, w e h a v e

--2 1 /2 0 2 -1 /2 0I' = 0 2 - 1 / 2 0 2 - 1 / 2

2 - 1 / 2 0 2 - 1 / 2 00 2 - 1 /2 0 2 - 1 /2

~1e +~b2e

l + v , / c )f l r Cfl

( V - - i V y ) (1 -- V / C )cf l P ie+ f l~2e -~2e -

1 l + v ~ / c ) ~ l e + _ ( v x + i v y ~+ 1~ } ~ b l e -- 2 - - ~ + 2 1 / ~ ~bxe-

( ~ l - ~ + l + v z / c ) ~ ? l e + - - ( V x + i V y ) * 2 e -2l /2f l 2 1 / 2 f l Cv x i , , + i 1 1 , .} l e + 2 1 / ~ 2 1 / 2 fl 2 e -

F o r t h e s t a t i o n a r y p a rt ic l e , 6 = 0 , a n d0x it = - 2 1 / 2 @ 2 e - i a2 1 / 2 ~ p l e i ~ O t0

T h e f a c t o r 21/2 i s u n i m p o r t a n t . W e s t a r t e d w i t h t h e n o r m a l i z e d 2 - s p i n o r a n d h a v e d e r i v e d w h a t is c a l le d a 4 - s p in o r , q , w h i c h is n o t

n o r m a l i z e d . T h e n o r m a l i z a t i o n o f t h e 2 - s p i n o r w a s a r b i t r a r y , a n d w a sb a s e d u p o n s e le c t in g a u n i t h y p e r s p h e r e f o r S U ( 2 ). H a d w e c h o s e n ah y p e r s p h e r e o f r a d i u s 2 -1 /2 , t h e n o u r 4 - sp in o r w o u l d h a v e m a g n i t u d ex I * xI -- 17 w h e r e ,I,* i s t h e c o n j u g a t e t r a n s p o s e d r o w m a t r i x . W e c o u l d t h e n


21/39

Geom etr ic M odel fo r Fundam enta l Par t ic les 457

a b s o r b t h e 2 1 2 f a c t o r s b y w r i t i n g f fl = 2 1 2 d ? 1 a n d t~2 = 21/2~b2 to pr o du ce an o r m a l i z e d 4 - s pi n o r,

0_ ~ b 2 e R o t

~ 1 e i~0

i n w h i c h I~1 1~1 + ~ 2 ~ 2 = 1.N o w i f w e h a d b e g u n b y a s s u m i n g o u r m a n i f o l d t o b e in t h e s ta t e[ , ,e - ] a n d r e p e a t e d t h e p r e v i o u s c a l c u l a ti o n s , w e w o u l d h a v eesc r ibe d by [ ,2e + 1f o u n d t h e c o r r e s p o n d i n g 4 - s p i n o r f o r t h e s y s t e m a t r e s t t o b e

_ t ~ l e R o t00

~ 2 e R o t

A l l o t h e r s p i n d i r e c t i o n s a ls o s a t i s f y t h e D i r a c e q u a t i o n . F o r e x a m p l e ,y a x is s p in r e p r e s e n t e d b y t h e s p i n o rs i o l [ 0 ]si n ~0t c o s ~0t ~ 2

( s e e p a g e 4 5 5 ) c o r r e s p o n d s t o t h e r e n o r m a l i z e d D i r a c s p i n o r( - - ~ 1 + i ~ 2 ) e - i t

1 - - i ~ 1 - - ~ 2 ) e - i t2 ( ~ 1 + i ~ 2 ) e i t

( - - i t~ l + ~ 2 ) e i ~ t

f o r a p a r t ic l e a t r e s t. T h e m o s t g e n e r a l D i r a c 4 - s p i n o r f o r a p a r t i c l e a t r e s tis

1{ ( - - 1 4 n ) ~ l + ( i ) ~ 4- ] ) ~ 2 } e - i ~ t

{ ( - / m + 0 ~ , + ( - 1 - , , ) , ~ , } e - ~ '{ ( 1 + n ) t ~ l + ( i r a + l ) ~ 2 } e i ' ~

( ( - i m + l ) t ~ l + ( 1 - n ) t ~ 2 } e i ~ t


22/39

458 Battey Pratt and Raceyb e i n g d e r i v e d f r o m t h e 2 - s p i n o r

c o s o~t + in si n o~tm + i l ) s in o~ t

w h e r e l , m , n , a r e t h e d i r e c t i o n c o s i n e s o f t h e s p i n a x i s .c o s o~t - in si n o~t t ~

7 . P A R T I C L E S T A T E SE a r l i e r ( p a g e 4 5 5 ) w e s e l e c t e d a p a r t i c u l a r c o m b i n a t i o n o f i n i t i a l

c o n f i g u r a t i o n a n d s p i n ax is o f o u r r o t a t i o n a l m o d e a n d l a b e le d i t t h en o r m a l sp i n s t a t e i n o r d e r t o c o n t r a s t i t w i t h t h e a n t i s p i n s t a t e

o b t a i n e d b y r e v e r s i n g t h e s p in . W e al s o d e f i n e d i n v e r s i o n o f t h o s e s t a te sf r o m t h e u p t o t h e d o w n o r i e n ta t io n . W e s h al l n o w c o m p a r e t h o s el a b e l e d s p i n s t a t e s w i t h t h e i r c o r r e s p o n d i n g 4 - s p i n o r s .

B y c o m p a r i n g t h e s e re s ul ts ( T a b l e I I ) w i t h t h o s e o f s t a n d a r d q u a n t u mt h e o r y , w e s ee t h a t w h a t w e h a v e c a ll e d t h e a n t i s p i n s ta t e s p r o d u c es o l u ti o n s th a t a re n o r m a l l y a p p l ie d t o t h e e l e c tr o n . O u r n o r m a l s p ins t a t e s t h e r e f o r e c o r r e s p o n d t o t h e p o s i t r o n . T h e t h e o ry is s y m m e t r i c a l . W en o l o n g e r n e e d t o p o s t u l a t e t h e e x i s t e n c e o f h o l e s i n a s e a o f n e g a t i v ee n e r g y s t a t e s f o r t h e p o s i t r o n a s i s d o n e i n s t a n d a r d t h e o r y . W e a s s e r t t h a ta ll m a s s a n d e n e r g y a re c y c l ic a l d i s t u r b a n c e s o f t h e c o n t i n u u m , a n d t h a tt h e i r m e a s u r e s a r e p r o p o r t i o n a l t o t h e i r fr e q u e n c i e s . T h e r e i s, t h e r e fo r e , n os u c h t h i n g a s n e g a t i v e e n e r g y a n y m o r e t h a n t h e r e i s n e g a t i v e t e m p e r a t u r e ,i n t h e a b s o l u t e s e n s e .

T h e r e i s n o i m m e d i a t e l y o b v i o u s r e a s o n w h y t h e s p i n n i n g c o n t i n u u mm u s t , i n p r a c t i c e , b e e i t h e r a p a r t i c l e o r a n a n t i p a r t i c l e . P r e s u m a b l y , s o m eb a s i c r e s t r i c t i o n f o r b i d s t h e c o n t i n u u m o f i n t e r m e d i a t e s t a t e s i n f r e ep a r t i c l e s s u g g e s t e d b y t h e f o r m u l a a t t h e e n d o f s e c t i o n 4 ( p a g e 4 5 6 ) . I tw o u l d a p p e a r t h a t t h e r o t a t i o n a x is is l o c k e d i n t o t h e c o n f i g u r a t i o n a t a n yg i v e n i n s t a n t . O f s i g n i f i c a n c e is th e f a c t t h a t t h e a n t i s p i n s t a t e s a r e m i r r o ri m a g e s o f t h e c o r r e s p o n d i n g n o r m a l s p i n s t a t e s ( t h i s c a n b e s e e n b ye x a m i n i n g t h e m o d e l o f s p h e r i c a l r o t a t i o n ) .

W e h a v e p r o d u c e d a t h e o r y o f p a r ti c le s a n d t h e ir s ta te s t h a t, c o n t r a r yt o tr a d i ti o n a l q u a n t u m t h e o r y , is i n d e p e n d e n t o f m e a s u r e m e n t s m a d e o nt h e p a r t i c l e s . A p a r t i c l e , a s s u c h , i s u n o b s e r v a b l e ( s e e A p p e n d i x , p a g e 4 5 6 ) .M e a s u r e m e n t s c a n o n l y b e m a d e b y i n t e r a ct i n g w i t h t h e p a r ti c le , a n d t h eo n l y t o o l s fo r i n v e s t i g a t i o n a r e o t h e r r e g i o n s o f d i s t o r t e d c o n t i n u u m ( w eh a v e h e r e m a d e t h e p r e s u m p t i v e l e a p o f a s s u m i n g p h o t o n s t o b e u n d u l a -t i o n s o f t h e s p a c e - t i m e s t r u c t u re ) . I n t e r a c t i o n s w i ll b e g o v e r n e d b y r u l es o fw a v e i n t e r f e re n c e a n d b y w h a t e v e r g e o m e t r i c c o n s e r v a t i o n la w s t u r n o u t t ob e f u n d a m e n t a l l aw s o f th e u n i v e rs e .


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Geom etr ic Mo d e l for Fu n d am en ta l Par t i c le s 59

TA BL E II. Com parison of 2-Spinor and 4-Spinor Spin StatesNam e of state 2-Spinor 4-Spinor[ ~o a m . p u p I e o l 0e i~toI ~

e i ot

Pmtispin up o0

~The negative coefficients of the antispin 4-spinors are of no

significance and can be eliminated by a tim e shift of ~r/to.

I t i s c l e a r t h a t , i n o r d e r t o f i n d t h e p o s i t i o n o f t h e p a r t i c l e , i t i sn e c e s s a r y t o l o c a t e t h e c e n t e r o f t h e s p i n n i n g c o r e . T h i s m e a n s t h a t t h ew h o l e d i s t u r b a n c e t h a t c o n s t i t u t e s t h e p a r t i c l e h a s t o b e c o n f i n e d . T h i sc o n f i n i n g p r o c e s s , w h i c h i s i m p l e m e n t e d b y o t h e r m a n i f o l d d i s t u r b a n c e s ,c o n s i s t s i n s o e n r i c h i n g t h e h a r m o n i c c o n t e n t o f t h e p a r t i c l e ' s d e B r o g l i ew a v e t h a t i ts m o m e n t u m b e c o m e s u n p r e d i c t a b l y a l t er ed . Thus the mea sur-i n g p r o c e ss i s l i m i t e d b y t h e i n d e t er m i n a n cy p r i n c i p l e - - n o t w i t h s t a n d i n gw h i c h , t h e p a r t i c l e d o e s a t a l l t i m e s h a v e a n e x a c t l o c a t i o n , a n d , w h e nc o m p l e t e l y f re e o f in t e r a c t i o n s w i t h o t h e r p a r t ic l e s , h a s a v e r y d e f i n i t em o m e n t u m .

I m p l i e d b y t h e p r i n c i p l e s t h a t h a v e b e e n f o r m u l a t e d i n t h i s p a p e r i st h e id e a t h a t t h e re c a n b e m o r e c o m p l e x d is t u r b a n c e s t h a t d o n o t d i s r u p tt h e c o n t i n u i t y o f s pa c e. A n a t o m i n a g i v e n e n e r g y s t a te i s p r o b a b l y a ne x a m p l e o f t h is h i g h e r - o r d e r c o m p l e x i t y . T h i s b e i n g s o , th e s a m e p r i n c i p let h a t p r e v e n t s s p h e r ic a l s p i n f r o m e x c h a n g i n g r o t a t i o n a l e n e r g y w i t h i tss u r r o u n d i n g s w o u l d a l s o i n h i b i t t h e a t o m f r o m r a d i a t i n g e n e r g y . T h i sa l l o w s f o r t h e e x i s t e n c e o f s t e a d y s t a t e s o f a t o m s w i t h i n t h e s t r u c t u r e o f ac l a s s i c a l t h e o r y o f t h e c o n t i n u u m , a n d t h u s r e n d e r s N e i l s B o h r ' s a d h o ch y p o t h e s i s u n n e c e s s a ry .


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A P P E N D I X

A .1 L I E G R O U P S A N D T H R E E - D I M E N S I O N A L R O T A T I O N 9I n o r d e r t o a n a l y z e t h e n a t u r e o f t h r e e - d i m e n s i o n a l r o t a ti o n , w e s h al l

c o n s t r u c t w h a t i s c a l l e d t h e Lie group s p a c e f o r t h e r o t a t io n .L e t S b e a b a l l f re e t o r o t a t e a b o u t i t s c e n t e r C . L e t t h e i n it ia l p o s i t i o n

o f S b e r e p r e s e n t e d i n t h e L i e g r o u p s p a c e b y t h e p o i n t O . I f S i s r o t a t e dt h r o u g h a n g l e 0 a b o u t a n a x i s / , i ts n e w p o s i t i o n w i ll b e d e s i g n a t e d b y t h ep o i n t i n t h e L i e g r o u p s p a c e t h a t l i e s 0 u n i t s f r o m O o n t h e l i n e t h r o u g h Op a r a l l e l t o t h e a x i s l , a n d i n t h e d i r e c t i o n t h a t a r i g h t - h a n d e d c o r k s c r e ww o u l d m o v e if i t w e r e p l a c e d a l o n g l a n d t u r n e d t h r o u g h t h e a n g l e 0. Ar o t a t i o n 0 i n t h e r e v e r s e d i r e c t i o n a b o u t l i s r e p r e s e n t e d b y a p o i n t 0 u n i t sf r o m O b u t i n th e o p p o s i t e d i r e c ti o n f r o m t h e p r e v io u s p o i n t . B e c a u s e ar o t a t i o n o f ~r u n i t s a b o u t l w i ll t a k e t h e b a l l S i n t o e x a c t l y t h e s a m ep o s i t i o n a s t h e c o u n t e r r o t a t i o n - q r , i t f o l lo w s t h a t i n t h e L i e g r o u p s p a c e ,w e n e e d o n l y g o a s f a r a s ~r u n i t s f r o m O i n e a c h o f t h e d i r e c t io n s p a r a l l e lt o 1 i n o r d e r t o e n d u p w i t h a l in e a r a r r a y o f p o i n t s r e p r e s e n t i n g e v e r yp o s s i b l e p o s i t i o n o f S t h a t l e a v e s 1 f i x ed . W e c a n , t h e r e f o r e , i d e n t i f y t h et w o e n d p o i n t s er a n d - ~ r a s o n e a n d t h e s a m e p o i n t . T h i s c o n s t r u c t io n isr e p e a t e d f o r a ll th e p o s s i b l e o r i e n t a t i o n s o f t h e a x i s l.

T h e L i e g r o u p s p a c e , th e n , c o n s i s ts o f a s p h e r e o f r a d i u s ~r u n i t sc e n t e r e d o n O w i t h d i a m e t r i c a l ly o p p o s i t e p o i n t s id e n ti fi ed . T h e n a m e o ft h i s g r o u p i s t h e proper orthogonal group in three dimensions a n d i s d e n o t e db y t h e sy m b o l 0 ( 3 ) + . I t is a l so s o m e t i m e s c a l le d th e th r e e - d i m e n s i o n a lr o t a t i o n g r o u p , b u t t h i s is a m i s n o m e r a s w e s h a ll s ee i n d u e c o u r s e .

A L i e g r o u p i s, f r o m a m a t h e m a t i c a l s t a n d p o i n t , a f a i rl y c o m p l i c a t e ds tr u ct u re . I t c a n b e t h o u g h t o f a s b e i n g a c o m p a t i b l e c o m b i n a t i o n o f m o r eb a s i c s t r u c t u r e s . T h u s i t i s , a m o n g s t o t h e r t h i n g s , a topological s p a c e , a n df o r o u r p u r p o s e s i t w i l l b e s u f f i c i e n t t o d e s c r i b e a t o p o l o g i c a l s p a c e a s t h ev e h i c l e f o r t h o s e e l e m e n t s o f g e o m e t r y t h a t a r e u n a f f e c t e d b y p l a s t i cd e f o r m a t i o n s - - s u c h a s t h e c o n t i n u i t y o r connectedness o f t h e s p a c e : ap r o p e r t y t h a t i s r e l a t e d t o i t s homotopy type

I n a t o p o l o g i c a l s p a c e , th e p a t h s b e t w e e n t w o p o i n t s a r e s a i d to b ehomotopic if t h e o n e p a t h c a n b e c o n t i n u o u s l y d e f o r m e d i n t o t h e o th e r . I tis u n d e r s t o o d t h a t a l l p o i n t s o n t h e p a t h r e m a i n p o i n t s o f th e t o p o l o g i c a ls p a c e d u r i n g t h e t r a n s i t io n . I f t h e in i ti a l a n d t e r m i n a l p o i n t s o f a p a t hh a p p e n t o co i n c id e , i t b e c o m e s a closed path B y e x a m i n i n g t h e c l o s e dp a t h s t h a t b e g i n a n d e n d o n a p a r t ic u l a r p o i n t , w e c a n g a i n s o m e i d e aa b o u t t h e t o t a l o r global s t r u c t u r e o f t h e s p a c e .9D Speiser (1964). This article was the precurso r of ou r th eory.


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Geo m et r ic Mo de l f o r Fund a m ent a l Pa r t i c le s 4 6 1

hese two

c o n s i d e r e do n e a n d t h e8~e

Fig. 5. The rotating object underconsideration.

0 / / - ~ ;

Fig 6. The Lie group space of the rotations.

A s a n i l lu s t r a ti o n , l e t u s t a k e t h e t w o - d i m e n s i o n a l s p a c e o f t h e s u r f a c eo f a t o r u s ( a f i g u r e s h a p e d l i k e a n a n c h o r r i n g o r d o u g h n u t ) . F i g u r e 9 i s ap i c t u r e o f it. E m a n a t i n g f r o m t h e p o i n t P w e s e e t w o p a t h s, p~ a n d p 2 , t h a tc a n c l e a r ly b e d e f o r m e d i n t o o n e a n o t h e r ; t h a t is to s a y, th e y a r e h o m o -t o p ic . T h e c o l l e c t io n o f a ll cl o s e d p a t h s t h r o u g h P t h a t a r e h o m o t o p i c t oo n e a n o t h e r f o r m s a homotopic c lass , p~ a n d P 2 ar e t h u s i n t h e s a m eh o m o t o p i c c la ss . T h e p a t h s P 3 a n d P4, o n t h e o t h e r h a n d , c a n n o t b ed e f o r m e d i n t o o n e a n o t h e r o r i n t o p l o r p 2 ; th e y a r e m e m b e r s o f t w o o t h e rh o m o t o p i c c la ss e s. T h e f i r s t c l as s m e n t i o n e d , t h a t o f p ~ a n d P 2 , h a s t h eu n i q u e f e a t u r e o f i n c l u d i n g t h e d e g e n e r a t e p a t h , n a m e l y , t h e p o i n t P i ts e lf .P a t h s l ik e P l a n d P 2 c a n b e c o n t r a c t e d c o n t i n u o u s l y u n t il o n l y t h e p o i n t Pr e m a i n s . T h e p o i n t P , t h o u g h t o f a s a d e g e n e r a t e p a t h , i s c a l l e d a n u l l p a t h .T h e s e t o f a ll h o m o t o p i c c l a s se s t o g e t h e r w i t h t h e i r in t e r r e l a t i o n s h i p s ( t h e yf o r m a m a t h e m a t i c a l g r o u p c h a r a c t e r i z e t h e s p a c e a n d d e s i g n a t e i t sh o m o t o p y t y p e .

O f s p e c i a l i m p o r t a n c e t o L i e g r o u p s p a c e s i s t h e t y p e i n w h i c h t h e r e i so n l y o n e h o m o t o p i c c l a s s a s s o c i a t e d w i t h e a c h p o i n t . T h i s c l a s s n e c e s s a r i l y

Fig. 7. Two paths which canbe continuouslydeformed ntoo n e a n o t h e r a r e h o m o t o p i c

F ig 8 A c lo sed pa t h


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4 6 2

F i g . 9 . P a t h s o n t h e s u r f a c e o f a to r u s .

B a O . e y P r a tt a n d R a c e y

F i g . 10 . T h e L i e g r o u p s p a c eof 0(3) + .

i n c l u d e s t h e n u l l p a t h : i n o t h e r w o r d s , e v e r y c l o s ed p a t h c a n b e c o n t r a c t e di n t o a p o i n t . A t o p o l o g i c a l s p a c e o f t h i s t y p e i s d e s c r i b e d a s b e i n g s imp lyconnec ted . I f t h e t o p o l o g i c a l s p a c e i s a l s o a L i e g r o u p , t h e n i t i s c a l l e d aun iv e r sa l c ov e r ing group b e c a u s e t h e r e i s o n l y o n e s i m p l y c o n n e c t e d L i eg r o u p w i t h a g i v e n l o c a l s t r u c t u r e , a n d b e c a u s e a ll o t h e r L i e g r o u p s h a v i n gt h e s a m e l o c a l s t r u c t u r e , b u t w h i c h a r e n o t s i m p l y c o n n e c t e d , a r e h o m o -m o r p h i s m s of i t .

T h e s e p r o p e r ti e s a n d d e f i n it i o n s w ill n o w b e e x e m p l i f i ed b y r e t u r n i n gt o t h e d i s c u s s i o n o f t h e o r t h o g o n a l g r o u p i n t h r e e d i m e n s i o n s . W e r e c a l lt h a t t h e g r o u p s p a c e o f 0 ( 3 ) + w a s a s p h e r e o f r a d i u s ~r w i t h d i a m e t r i c a l l yo p p o s i t e p o i n t s i d e n t i fi e d .

C o n s i d e r t h e c l o s e d p a t h s t h a t b e g i n a n d e n d a t O i n F i g u r e 1 0. Ap a t h P l t h a t r e m a i n s e v e r y w h e r e w i t h i n t h e s p h e r e c a n b e c o n t r a c t e d i n t ot h e n u l l p a t h a t O . B u t w h a t o f a p a t h t h a t c r o s s e s t h e b o u n d a r y o f t h es p h e r e ? L e t P 2 b e t h e l o c u s o f a p o i n t t h a t s t a rt s f r o m O a n d p a s s e s o u tt h r o u g h t h e s u r f a c e o f t h e s p h e r e a t p o i n t P . S i n c e P i s t h e s a m e a s t h ed i a m e t r i c a l ly o p p o s i t e p o i n t P ' , t h e c o n t i n u a t i o n o f th i s p a t h i s t r a c e d b y ap o i n t m o v i n g i n w a r d s f r o m t h e p o i n t P ' . I f th is p a t h f i n al l y r e t u rn s t o O ,w e h a v e a c l o s e d p a t h . C a n t hi s p a t h b e c o n t i n u o u s l y c o n t r a c t e d i n t o t h en u l l p a th ? T h e a n s w e r i s no . A n y a t t e m p t t o s o c o n t r a c t P 2 m u s t, a t s o m es ta g e , r e d u c e t h e p a t h t o o n e t h a t is e v e r y w h e r e w i t h i n t h e b o u n d a r y o f t h es p h e r e, l ik e p x . B u t i n o r d e r t o d o t h a t s m o o t h l y a n d c o n t i n u o u s l y , t h ep o i n t P m u s t s o o n e r o r la t e r m o v e c l o s e r t o P ' u n t il t h e y m e e t . H o w e v e r ,t h is is im p o s s i b le , b e c a u s e P ' is a l w a y s d i a m e t r i c a l l y o p p o s i t e P . T h u s , a llt h e p a t h s t h a t c u t t h e b o u n d a r y onc e a n d t h e n r e t u r n t o O f o r m ah o m o t o p i c c l a s s d i s t i n c t f r o m t h e c l a s s t h a t i n c l u d e s t h e n u l l p a t h . H e n c e ,w e h av e s h o w n t h a t 0 ( 3 ) + is n o t s i m p ly c o n n e c t ed .

T h e r e d o e s , h o w e v e r , e x i st a L i e g r o u p s p a c e t h a t is l o c a l l y t h e s a m ea s O ( 3 ) + , b u t w h i c h is s i m p l y c o n n e c t e d . F o r , c o n s i d e r t w o s p h e r e s o f


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Geom etr ic Mod e l for u n d am en ta l Par t i c le s 46

Fig. 11. The Lie group space of SU (2), which is simply connected, but locally the same as0(3) .

r a d iu s ~r c e n t r e d o n O a n d O , b o t h r e p l ic a s o f 0 ( 3 ) + e x c e p t th a t a p o i n to n t h e b o u n d a r y o f t h e f i r s t s p h e r e b e c o m e s i d e n t i f i e d w i t h t h e p o i n t o nt h e s e c o n d s p h e r e t h a t c o r r e s p o n d s t o i t s d i a m e t r i c a l l y o p p o s i t e p o s i t i o n( F i g u r e 1 1). I n t h is s p a c e, w e c a n t r a c e a p a t h p f r o m O t h a t l e a v e s t h e f ir s ts p h e r e a t a b o u n d a r y p o i n t P a n d r e a p p e a r s e n t e r i n g t h e s e c o n d s p h e r ef r o m P . A f t e r c r o s s i n g t h e s e c o n d s p h e r e , t h e p a t h l e av e s a t a p o i n t Q ,a n d r e a p p e a r s i n t h e f i r s t s p h e r e a t Q , w h e n c e i t r e t u r n s t o O . I t i s n o wp o s s ib l e to c o n t r a c t th e p a t h p t o w a r d s O , si n ce w e a r e f r ee t o a l lo w P t oa p p r o a c h Q ( a nd , s im u l t a n e o u s l y , P m u s t m o v e t o w a r d s Q ) . A n y c l o s edp a t h t h r o u g h O c a n , t h e r e fo r e , b e c o n t r a c t e d i n t o t h e n u ll p a th . H e n c e , t h es p a c e i s s i m p l y c o n n e c t e d . T h i s L i e g r o u p s p a c e i s t h e u n i v e r s a l c o v e r i n gg r o u p o f 0 ( 3 ) + , a n d is c a ll e d S U ( 2 ) f o r r e a s o n s t h a t w e re g i ve n o n p a g e454 .

A .2 P H Y S I C A L I N T E R P R E T A T I O N (B olk er , 1 97 3)W e r ec a ll t h a t th e g r o u p s p a c e 0 ( 3 ) + w a s a g e o m e t r ic a l d e s c r i p t i o n

o f th e r o t a t i o n s o f a b a ll , S , i n w h i c h e v e r y p o s i t i o n o f S w a s r e p r e s e n t e du n i q u e l y b y a p o i n t i n t h a t s p a ce . T h e i n i ti al p o s i t io n o f S w a s r e p r e s e n t e db y t h e p o i n t O .

L e t u s f i n d t h e p h y s i c a l m e a n i n g o f t ra v e r s in g a s m a l l c l o se d p a t ht h r o u g h O . A s w e l e a v e O i n a g i v e n d i re c t i o n , t h e b a ll S t u r n s a b o u t t h ea x i s p a r a l l e l t o t h a t d i r e c t i o n . I f w e n o w t r a v e l a c r o s s t h e L i e g r o u p s p a c ek e e p i n g r o u g h l y t h e s a m e d i s t an c e f r o m O , th e b a l l S w i ll r o t a t e a b o u tv a r i o u s a x e s i n s u c h a w a y t h a t i t r e m a i n s a t r o u g h l y t h e s a m e a n g u l a rd e v i a t i o n f r o m t h e i n i t i a l p o s i t i o n . F i n a l l y , a s w e r e t u r n t o O , t h e b a l ld e c r e a s e s i t s a n g u l a r d e v i a t i o n u n t i l i t i s r i g h t b a c k i n t h e p o s i t i o n f r o mw h i c h i t s t a r t e d .


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m o v e m e n t o fp o i n t o n t h e

s p h e r e

Fig. 12. A small wob ble of the sphere S p icks out a small path through O in the group space0 ( 3 ) + .

H o w w o u l d o n e d e s c ri b e th e m o t i o n e x e c u t e d b y S ? C l e ar ly , th ea n s w e r is th a t S w o b b l e d ; t h a t i s t o s a y , i t e x e c u t e d th e s o r t o f m o t i o n t h a tc o u l d h a v e b e e n i m p a r t e d t o t h e b a l l if i t w e r e h e l d f i rm l y i n t h e h a n d , t h ew r i s t a n d e l b o w g i v e n a f e w f l ex e s a n d t w is ts , a n d t h e n r e l a x e d b a c k i n t ot h e o r i g i n a l p o s i t i o n .

W e c a n n o w g iv e a p h y s ic a l in t e r p r e t a t i o n o f h o m o t o p y . A c l o se dp a t h o n O t h a t is j u s t a s h a d e d i s p l ac e d f r o m t h e p r e v i o u s p a t h r e p r e s e n tsa w o b b l e o f S i n w h i c h t h e m o t i o n s o c c u r a r o u n d a x e s t h a t a r e s li g h tl yd i s p l a c e d f r o m t h e p r e v i o u s a x e s, a n d t o a n e x t e n t t h a t is a s h a d e d i f f e r e n t .T h e t r a n s i t i o n f r o m o n e p a t h t o t h e o t h e r , th e n , c o n s i s ts o f e i t h e r ex -a g g e r a t i n g o r d i m i n i s h i n g t h e m o t i o n . T h u s , t h e s e q u e n c e o f p a t h s i n th eL i e g r o u p s p a c e c o n s t it u t in g t h e s m o o t h t r an s i t io n b e t w e e n t w o h o m o t o p i cp a t h s r e q u i r es t h a t t h e m o t i o n s o f t h e b al l S u n d e r l y i n g t h e o n e p a t h b ec a p a b l e o f s m o o t h a n d g r a d u a l e x a g g e r a t io n u n t il t h e y c o in c i d e w i t h t h em o t i o n s u n d e r l y i n g t h e fi n a l p at h . A l l t h e p a t h s o f t h e h o m o t o p i c c la sst h a t i n c l u d e s t h e n u l l p a t h a r e d e s c r i b a b l e a s w o b b l e s o f a g r e a t e r o r l e ss e re x t e n t ; t h a t is, t h e y c a n b e d e m o n s t r a t e d w i t h S h e l d f i r m l y i n th e h a n d .

I n 0 ( 3 ) + , a c lo s e d p a t h t h a t le a ve s O , t h a t m o v e s st r ai g h t a lo n g ar a d i u s u n t i l i t l e a v e s t h e s u r f a c e o f t h e s p a c e , t h a t r e a p p e a r s a t t h ed i a m e t r i c a ll y o p p o s i te p o in t , a n d t h e n m o v e s s t r ai g h t b a c k t o O c o r r e -s p o n d s t o o n e c o m p l e t e r e v o l u t i o n o f S a r o u n d a n a x is p a ra l le l to t h e p a t h .T h i s p a t h i s n o t i n t h e s a m e h o m o t o p i c c l a s s a s t h e n u l l p a t h . I n S U ( 2 ) ,h o w e v e r , t h e L i e g r o u p s p a c e o f 0 ( 3 ) + is r e p r e s e n t e d tw i ce . A c l o s eds t r a i g h t p a t h t h a t l e a v e s O a l o n g a r a d i u s c r o s s e s t h e s e c o n d s p a c e v i a t h ed i a m e t e r th r o u g h O b e f o r e r e a p p e a r i n g i n th e f ir st sp a c e a n d r e t u r n i n g t oO . T h e b a l l S , i n t h is c as e , m a k e s t w o c o m p l e t e t u r n s a r o u n d a n a x i sp a r a l l e l t o t h e p a t h . B u t S U ( 2 ) i s s i m p l y c o n n e c t e d ; t h e r e f o r e , t h e p a t h


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Geom etr ic M od e l for u n d am en ta l Particles 6 5

Fig. 13. A closed path in SU(2) consisting of two parallel diameters through O and Orepresents a double rotation of S.

d e s c r i b e d i s h o m o t o p i c t o , a n d , s o , c o n t r a c t i b l e i n t o t h e n u l l p a t h . T h i sm e a n s t h a t t h e m o t i o n o f S i.e ., a c o m p l e t e d o u b l e r o t a t i o n a r o u n d a na x i s ) c a n b e a r r i v e d a t b y a s m o o t h e x a g g e r a t i o n o f a w o b b l e , w h i c h , i nt u r n , m e a n s t h a t i t is p o s s i b le t o g r a s p a b a l l i n t h e h a n d , a n d , b y f l e x in go f w r i s t a n d e l b o w j o i n t s , e t c ., t o e x e c u t e a c o m p l e t e d o u b l e b u t n o ts in g le ) r e v o l u t i o n o f t h e b a l l a b o u t a f i x e d a x is , a n d t h e r e b y e n d u p i n t h es a m e s t a n c e a s o n e b e g a n .

T h i s is , i n f a c t , t h e b a s is o f a n o l d p a r t y t r ic k i n w h i c h t h e j o k e r t w i rl sa b o w l o f s o u p a r o u n d i t s v e r t i c a l a x i s s o a s n o t t o s p i l l a d r o p . T h e t r i c kr e q u i r e s r a i s i n g a n d l o w e r i n g t h e b o w l s o m e w h a t d u r i n g t h e g y r a t i o n t oa c c o m m o d a t e t h e a w k w a r d j o i n t i n g o f t h e a r m ; b u t i t is, ne v e rt h el e ss , a ni l l u s t r a t i o n o f t h e a b o v e t h e o r y . T h e b o w l m a k e s t w o f u l l t u r n s i n e x e c u t -i n g t h e c o m p l e t e g y r a t i o n .

W e c a n r e p la c e th e h u m a n a r m a n d t h e s o u p b o w l b y a l o n g c o i le ds p r i n g a n d t h e b a l l, S . T h e t o p o f t h e s p r in g c a n b e c l a m p e d t o a s t a t i o n a r yf r a m e w o r k in im i t a t i o n o f t h e s h o u l d e r c o n n e c t i o n , a n d t h e b o t t o m e n d

S

Fig. 14. Coiled spring simulating a human arm.


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466 B attey Pr at t and Racey

i I 1 1 ~ - ~ .I < v . L ; I J

a

h X /

~ / // I/_.Fig. 15. Eac h half turn o f S forms a tw is t in the two tracers (dotte d l ines), which can only beundo ne by the displacem ent shown (solid l ines).

c a n b e b e n t a r o u n d a n d b r o u g h t u p u n d e r n e a t h t h e b a l l i n i m i t a t i o n o f t h eh a n d ( F i g u r e 1 4). T h e u n i f o r m f l e x i b i l i t y o f t h e s p r i n g a l l o w s th e b a l l t os p i n w i t h o u t t h e a w k w a r d v e r t i c a l m o v e m e n t t h a t o c c u r s i n th e s o u p b o w lt r i c k .

T h e m o t i o n o f t h e b a l l a n d s p r i n g c a n b e u n d e r s t o o d i n d e ta i l b yc o n s i d e r i n g tw o l in e s d r a w n o p p o s i t e o n e a n o t h e r d o w n t h e l en g t h o f t h eu n s t r a i n e d s p r i n g . T h e s p r i n g i s f o r c e d t o b e n d i n a s s u m i n g t h e a b o v ec o n f i g u r a t i o n , b u t i t w i l l s w i n g i n t o t h e p l a n e t h a t m i n i m i z e s i ts to r s i o n .E x a m i n a t i o n o f th e d i a g r a m s o f F i g u r e 15 s h o w s t h a t th e c o n f i g u r a t i o n o ft h e s p r i n g w i l l r o t a t e a t h a l f t h e a n g u l a r v e l o c i t y o f t h e b a l l . 1~

N o w , b e c a u s e o f th e s h a p e o f th i s s y s t e m , w e c a n a d d a n o t h e r s p r i n gs y m m e t r i c a l l y o p p o s i t e to t h e o n e a b o v e . I t , t o o , m u s t a d v a n c e a t h a l f t h ea n g u l a r v e l o c i t y o f t h e b a l l a n d w i ll t h e r e b y m a i n t a i n i ts r e l a t i o n s h i p w i t ht h e f i r s t s p r i n g ( F i g u r e 16 ). T h e q u e s t i o n n a t u r a l l y a r is e s : h o w c o m p l i c a t e da s y s t e m o f sp r i n g s c a n p a r t i c i p a t e in t h e m o t i o n ? T o a n s w e r t h is , w e g ob a c k t o f ir s t p r i n c i p l e s .

C o n s i d e r a b a l l t o w h i c h i s a t t a c h e d a l a r g e i n d e f i n i t e n u m b e r o f l o n gn a r r o w s p r in g s r u n n i n g r a d i a l l y o u t w a r d s a n d t i e d a t t h e i r o u t e r e n d s t o af i x e d f r a m e w o r k ( F i g u r e 1 7). S i n c e t h e s p r i n g s a r e f l e x ib l e , i t is p o s s i b l e f o rt h e b a l l t o e x e c u t e s m a l l w o b b l e s a b o u t i ts c e n t e r w i t h o u t c a u s i n g a n y

10Mentioned in Bolker (1973) a re bo th Dirac s sp an ne r-- a demon s tra tion o f spher ica lrota tion using s trings a ttached to a n asymmetric w ren ch- -an d, a lso, D. A. A dam s spatented device for transmitt ing e lectric i ty to a rota ting turntable without the use of s l iprings . The Adams device is identical in i ts symmetry to our incipient model. A full andwell-i llustra ted report on i t appe ared in Ad am s (1975). Just fo r the reco rd, we would l ike topo in t ou t tha t our mode l dep ic ted in F igure 1 was cons t ruc ted and demons tra ted in theM athem atics De partm ent a t Q ueen s Univers ity , O ntario, Ca nad a in the fa ll of 1966. E. P.Battey -Pratt wou ld l ike to acknowledge his indebted ness to Prof. A. J . Colem an forinspira tion and to Ha ns Ku mm er who, in conversation, helped c larify the role of universalcovering groups in the theory.


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G e o m e t ri c M o d e l f o r F u n d n m e n t a l P a r t i c l e s 4 6 7

F i g . 1 6 . A s y m m e t r ic p a i r o f s p r i n g s . F i g . 1 7 . R a y s o f s p r i n g s e m a n a t i n gf r o m a b a l l .e n t a n g l e m e n t o f t h e s p ri ng s , B u t w e h a v e s e e n t h a t a d o u b l e r o t a t i o n o f th eb a ll c a n b e d e v e l o p e d b y t h e c o n t i n u o u s e x a g g e r a t i o n o f a s m a l l w o b b l e .N o w , t h e t r an s i ti o n f r o m a m o t i o n t h a t d o e s n o t c a u s e t h e s p ri n gs t o t a n g l ea n d k n o t u p t o o n e t h a t d o e s is n e ce s s a ri l y a b r u p t a n d d i s c o n t in u o u s , a n d ,t h er e fo r e ; c a n n o t o c c u r d u r i n g o u r c o n t i n u o u s d e v e l o p m e n t o f th e w o b b l ei n t o a d o u b l e r o t a t i o n .

T h u s , w e s ee t h a t t h e s p r i n g s i n F i g u r e 1 6 a r e b u t t w o o f a n i n f i n i ten u m b e r o f r a y s t h a t c a n p a r t i c i p a t e i n t h e m o t i o n o f t h e b a l l w i t h o u tk n o t t i n g u p . T o f i n d t h e d i s p o s i t i o n o f t h e o t h e r r a y s , w e s i m p l y n o t e t h a t ,s t a r ti n g f r o m t h e m o d e l o f F i g u r e 1 7 w e m u s t g i v e t h e b a l l a h a l f t u r na b o u t a h o r i z o n t a l a x i s i n o r d e r t o p u t t h e v e r t i c a l s p r i n g s i n t o t h ec o n f i g u r a t i o n o f F i g u r e 16. T h e b a l l m a y t h e n b e r o t a t e d a b o u t i ts v e r ti c a la x is . T h i s is th e e x p l a n a t i o n f o r t h e v a l i d i t y o f t h e m o d e l s o f S e c t i o n 2 o ft h e p a p e r ( p a g e 4 4 8 ).

2:

F i g . 1 8 . S p r i n g s i n t h e p l a n e p e r p e n d i c u l a r F i g . 1 9 . A s t h e b a l l i s r o t a t e d a b o u t t h e zt o t h e i n i ti a l h a l f t u r n a b o u t t h e x a x is . a x is t h e a x i s o f t h e i n i t i a l h a l f t u r n a ls o

r o t a t e s


32/39


N o w l et u s g o b a c k a n d f o l lo w t h r o u g h t h e s a m e a r g u m e n t s f o r th ec a s e o f t w o - d i m e n s i o n a l r o t a t i o n . W e s h a l l t h e n p r o d u c e a g e o m e t r i c a la n a l o g y t h a t w i l l i l l u m i n a t e t h e r e l a t i o n s h i p b e t w e e n t h e a b o v e a n a l y s i sa n d t h e c o n v e n t i o n a l v i e w o f th r e e - d i m e n s i o n a l r o t a t i o n .

A .3 T W O - D I M E N S I O N A L R O T A T I O NT h e L i e g r o u p s p a c e f o r t h e t w o - d i m e n s i o n a l r o t a t i o n o f a f la t o b j e c t

a b o u t s o m e p o i n t i n t h e p l a n e i s v e r y s im p l e. F o r c o n v e n i e n c e , le t th e f l a to b j e c t b e a c i r c u l a r d i s k D r o t a t i n g a b o u t i t s c e n t e r . L e t t h e i n it ia l p o s i t i o no f th e d i s k b e r e p r e s e n t e d b y t h e p o i n t O i n t h e L i e g r o u p s p a c e . Ac o u n t e r c l o c k w i s e r o t a t i o n o f D t h r o u g h a n a n g l e 0 is r e p r e s e n t e d b y ap o i n t 0 u n i t s t o t h e f i g h t o f O i n t h e g r o u p s p a c e . S i m i l a r l y , a c l o c k w i s er o t a t i o n , - 0 , o f D i s r e p r e s e n t e d b y a p o i n t 0 u n i t s t o t h e le f t o f O . S i n c e ar o t a t i o n ~r b r in g s D t o t h e s am e p o s i t i o n a s th e c o u n t e r r o t a t i o n - , r , w ec a n i d e n t i f y t h e se t w o p o i n t s . T h u s , t h e L i e g r o u p s p a c e is a li n e o f l e n g t h2~r u n i t s w i t h e n d p o i n t s i d e n t i fi e d . T h i s m a y b e t h o u g h t o f a s b e i n g t h ec i r c u m f e r e n c e o f a c i rc l e w i t h O a n d t h e d u a l p o i n t ~r, - ~ r d i a m e t r i c a l l yo p p o s i t e . T h e n a m e o f t h i s L i e g r o u p i s the proper orthogon l group in two

Fig. 20. The ro tating disk.

T h e s e t w o p o i n t st o b e i d e n t i f i e d

F ig 21. The Lie group space 0 2)+ .


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Geom etr ic M od e l for Fu n d am en ta l Par t i c le s 69

Fig. 22. Th e group space R derived fro m 0 2)+.

dimensions or 0 2)+. It is often referred to, erroneously, as the two-dimensional rotation group.Since the Lie group space is one-dimensional, the only type of paththat is homotopic to the null path through O is one that leaves O, goes tosome other point, and then returns to O again by tracking back along itsoutward path. 0 2) + is, therefore, not simply connected since it is possi-ble, by completing one or more circuits of the space, to return to O withoutback tracking.

In order to construct the universal covering group of 0 2) + thelocally isomorphic but simply connected group space), we duplicate thespace as we did for 0 3)+, and make the end point of the original spaceconterminal with the beginning point of the repeated segment rather thanterminating with the other end of itself. We see, however, that we stillcannot rejoin the other end point of the repeated space to the open end ofthe original because to do so would produce a circle, which is not simplyconnected. We have, therefore, to repeat the segment once more. Again,we come to an end that cannot be joined back to the first segment withoutcreating a circular, noncontractible path. Thus, the repetitions of thesegment never end, and the Lie group space of 0 2 )+ must be extendedindefinitely in repetitions of itself in either direction. The universal cover-ing group of 0 2) + is, therefore, an infinite line. Because every point onan infinite line can be represented by a real number, the symbol for thisgroup is R. It is the same as the Euclidean one-dimensional continuum.

A closed path in R is effectively jus t a line segment that is traversedboth ways, both out and back again to the starting point. Corresponding tosuch a path, the disk D turns by the specified amount, and then turns backagain by the same amount to its starting position. That is to say, if one endof a string were affixed to the rim of D and the other end to a stationaryframework, then a motion that corresponds to a closed path in R would beone in which, at the completion of the motion, the string would be back inits original configuration. For, if D makes several full turns in onedirection, then the string which, remember, is confined to the plane in thiscase) becomes wrapped around D that many times. But D then has to turnback again by the same number of turns in order to complete the motionwhile the string unwinds itself again and so returns to its starting config-uration.


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4 7 0 B a t te y Pr a tt a n d R a c e y

P h y s i c a l l y s p e a k i n g , j u s t a s t h e m o t i o n s o f t h e b a ll S c o r r e s p o n d i n g t oe v e r y c l o se d p a t h i n S U ( 2 ) c a n b e d e v e l o p e d b y t h e c o n t i n u o u s e x a g g e r a -t i o n s o f a s li gh t w o b b l e , s o c a n t h e m o t i o n s o f th e d i s k D c o r r e s p o n d i n g t oc l o se d p a t h s i n R b e d e v e l o p e d b y t h e c o n t i n u o u s e x a g g e r a ti o n s o f a s m a l lo s c i l l a t i o n o f D .

A .4 C Y L I N D R I C A L A N D S P H E R I C A L R O T A T I O NH i t h er to , m a t h e m a t i c ia n s h a v e r e g a rd e d 0 ( 2 ) + a n d 0 ( 3 ) + a s r o ta -

t i o n g r o u p s. T h e c o n c e p t o f r o t a t i o n i m p l ie s m o t i o n s o t h a t w e a re asm u c h c o n c e r n e d w i t h t h e d y n a m i c p r o c e s s o f a b o d y g e t ti n g f r o m i ts i n it ia lt o i t s f i n a l p o s i t i o n a s w e a r e w i t h t h e p o s i t i o n s t h e m s e l v e s . A t t h eb e g i n n i n g o f t hi s p a p e r , w e d e s c r ib e d h o w t h e m o v e m e n t s m a d e b y a d o gt ie d b y a l o n g l e a d c o u l d b e t r a c e d b y t h e fa c t th a t t h e r o p e w a s w r a p p e da r o u n d h i s k e n n e l . T h i s i s a v e r y o l d i d e a t h a t d e s e r v e s t o b e r a i s e d t o ap r i n c i p l e : the final configuration of a line connecting a moving point to anobserver s stationary framework represents the motion of that point.

B e c a u s e , s u p e r f i c i a l l y , w e d i s c e r n t h e u n i v e r s e a s b e i n g d i v i d e d i n t om a t t e r a n d e m p t y s p a ce , a n d b e c a u s e w e h a b i t u a l ly t

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