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UBC Theses and Dissertations

Non-existence of geometrodynamical analog to electric charge Davenport, Michael Richard 1982

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NON-EXISTENCE OF GEOMETRODYNAMICAL ANALOG TO ELECTRIC CHARGE by MICHAEL RICHARD DAVENPORT B . S c , The U n i v e r s i t y of C a l g a r y , 1978 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r 1982 © M i c h a e l R i c h a r d D a v e n p o r t , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P S t c  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 0 1% I ( U l A b s t r a c t A " G e o m e t r o d y n a m i c a l A n a l o g t o E l e c t r i c C h a r g e " ( o r "p-c h a r g e " ) i s d e f i n e d (as i n t h e e a r l i e r p a p e r by Unruh,[ Gen.  R e l . and G r a v . , 2, ( 1 9 7 1 ) , pp 27-33 ]) t o be t h e p e r i o d on a p-c y c l e (p = 1, 2, o r 3) o f a p - f o r m w h i c h i s c o n s t r u c t e d out of o n l y t h e Riemann t e n s o r or i t s d e r i v a t i v e s . A p r e v i o u s l y - u n p u b l i s h e d p r o o f by U n r uh i s b r i e f l y summarized w h i c h p r o v e s t h a t no n o n - z e r o p - c h a r g e s can e x i s t on a c o m p l e t e l y u n r e s t r i c t e d m e t r i c f i e l d . The m e t r i c f i e l d i s t h e n c o n s t r a i n e d t o obey E i n s t e i n ' s e q u a t i o n s f o r empty s p a c e , and s e t s of 1 i n e a r l y - i n d e p e n d e n t , p u r e l y - g r a v i t a t i o n a l p - f o r m s a r e a n a l y z e d t o d e t e r m i n e i f p-c h a r g e s c a n be d e f i n e d under t h e s e c o n d i t i o n s . A scheme i s d e v e l o p e d , b a s e d on t h e s p i n - t e n s o r r e p r e s e n t a t i o n of t h e g r a v i t a t i o n a l f i e l d , t o g e n e r a t e c o m p l e t e s e t s o f s u c h p - f o r m s , arid c a l c u l a t e t h e i r d e r i v a t i v e s , w i t h a s y m b o l i c - m a n i p u l a t i o n computer p r o g r a m . I t i s shown t h a t no g r a v i t a t i o n a l p - f o r m s t h a t a r e l i n e a r c o m b i n a t i o n s o f l e s s t h a n f i v e Riemann t e n s o r s and l e s s t h a n n i n e d e r i v a t i v e s w i l l r e s u l t i n p - c h a r g e s . T a b l e o f C o n t e n t s A b s t r a c t i i T a b l e Of C o n t e n t s . . i i i L i s t Of T a b l e s v i L i s t Of F i g u r e s v i i L i s t Of APL F u n c t i o n - C o d e s In A p p e n d i x B v i i i A cknowledgement i x I n t r o d u c t i o n 1 1. G e n e r a l i z a t i o n Of The C o n c e p t Of " C h a r g e " 5 1.1 I n t r o d u c t i o n 5 1.2 Review Of Homologous P - C y c l e s And DeRham's Theorems 8 1.3 D e f i n i t i o n Of P - c h a r g e 13 1.4 I n t e r p r e t a t i o n Of P - c h a r g e In A 4 - M a n i f o l d 15 2. N o n - E x i s t e n c e Of C h a r g e s F o r U n r e s t r i c t e d M e t r i c S p a c e t i m e s 21 2.1 The S t r u c t u r e Of The S p a c e t i m e And M e t r i c F i e l d 23 2.2 R e p r e s e n t a t i o n Of M e t r i c F i e l d s In The Base Space 'A' 26 3. E x i s t e n c e Of H o m o t o p i e s In The Space "A" Of U n i t a r y , L o r e n t z - S i g n a t u r e M a t r i c e s 29 3.1 E x i s t e n c e Of H o m o t o p i e s In 'A' F o r 1 - c h a r g e s ... 30 3.2 E x i s t e n c e Of H o m o t o p i e s In 'A' F o r 2 - c h a r g e s ... 35 3.3 E x i s t e n c e Of H o m o t o p i e s In 'A' F o r 3 - c h a r g e s ... 37 4. R e s t r i c t i n g The M e t r i c To Obey A F i e l d E q u a t i o n 40 5. S p i n - T e n s o r R e p r e s e n t a t i o n Of T e n s o r s In F l a t S p a c e -Time 43 5.1 Example Of A Second-Rank S p i n - T e n s o r 44 5.2 L o r e n t z T r a n s f o r m a t i o n s Of S p i n o r s 46 5.3 E x t e n s i o n To H i g h e r O r d e r S p i n - T e n s o r s 49 5.4 The S p i n - T e n s o r " M e t r i c " 50 5.5 The Complex C o n j u g a t e Of A S p i n - T e n s o r 52 5.6 Symmetry And A n t i s y m m e t r y In S p i n - T e n s o r s 53 5.7 D e r i v a t i v e s Of S p i n - T e n s o r s 55 6. S p i n - T e n s o r R e p r e s e n t a t i o n Of N o n - F l a t S p a c e - T i m e s .. 56 6.1 C o v a r i a n t D e r i v a t i v e s 57 6.2 The C u r v a t u r e S p i n - T e n s o r s 59 6.3 The B i a n c h i And R i c c i I d e n t i t i e s 62 6.4 E i n s t e i n ' s E q u a t i o n s 64 7. E x p a n s i o n Of P u r e l y G e o m e t r i c S p i n - T e n s o r s 65 7.1 ' C l a s s i f i c a t i o n Of G e o m e t r i c S p i n - T e n s o r s 65 7.2 A Complete S e t Of G e o m e t r i c S p i n - T e n s o r s 68 7.3 R e p r e s e n t i n g T e n s o r s As P r o d u c t s Of B a s i s S p i n -T e n s o r s 71 8. Computer S e a r c h F o r C l o s e d , N o n - E x a c t P-Forms 75 8.1 O v e r v i e w Of The P r o j e c t 75 8.2 C o m p u t e r - C o m p a t i b l e N o t a t i o n And S u p p o r t F u n c t i o n s 78 8.3 C a n o n i c a l Form 83 8.4 C a l c u l a t i o n Of The D e r i v a t i v e s ' 91 9. R e s u l t s Of Computer S e a r c h F o r P - Charges 94 9.1 C o m p l e t e L i s t s Of I n d e p e n d e n t S p i n - T e n s o r s 94 9.2 T a b l e s Of D e r i v a t i v e s Of S p i n - T e n s o r Terms 97 9.3 R e s u l t s Of The S e a r c h F o r 1-Charges 104 9.4 R e s u l t s Of The S e a r c h F o r 2-Charges .....105 9.5 R e s u l t s Of The S e a r c h F o r 3-Charges 106 Summary 108 R e f e r e n c e s 110 B i b l i o g r a p h y 112 A p p e n d i x A. Computer A l g o r i t h m s G e n e r a t i n g S p i n t e n s o r s And D e r i v a t i v e s 114 A . 1 The D r i v i n g Program 116 A. 2 The G e n e r a t i n g F u n c t i o n VCREATEV 1.16 A.3 C a l c u l a t i n g The D e r i v a t i v e s 118 A.4 R e p l a c i n g Terms By S t a n d a r d - F o r m Terms 126 A p p e n d i x B. L i s t i n g Of The APL S o u r c e Codes 129 A p p e n d i x C. C o m p l e t e L i s t Of S p i n t e n s o r s To D e g ree 8 ...148 v i L i s t of T a b l e s 9.1 Rank And Degree Of S p i n - T e n s o r s W i t h No N o n - z e r o Terms 95 9.2 D e r i v a t i v e s Of Rank-0 Degree-2 S p i n - T e n s o r s 100 9.3 D e r i v a t i v e s Of Rank-0 Degree-4 S p i n - T e n s o r s 100 9.4 D e r i v a t i v e s Of Rank-0 D e g r e e - 6 S p i n - T e n s o r s 101 9.5 D e r i v a t i v e s Of Rank-1 D e g r e e - 5 S p i n - T e n s o r s 102 9.6 D e r i v a t i v e s Of Rank-2 D e g r e e - 6 S p i n - T e n s o r s 102 9.7 D e r i v a t i v e s Of Rank-3 Deg r e e - 3 S p i n t e n s o r s 103 9.8 D e r i v a t i v e s Of Rank-3 D e g r e e - 5 S p i n - T e n s o r s 103 C.1 Rank 0 Degree 2 S p i n - T e n s o r Terms 150 C.2 Rank 1 Degree 3 S p i n - T e n s o r Terms 150 C.3 Rank 0 Degree 4 S p i n - T e n s o r Terms 150 C.4 Rank 1 D e g r e e 5 S p i n - T e n s o r Terms 151 C.5 Rank 0 D e g r e e 6 S p i n - T e n s o r Terms 151 C.6 Rank 2 D e g r e e 6 S p i n - T e n s o r Terms 152 C.7 Rank 1 D e g r e e 7 S p i n - T e n s o r Terms 153 C.8 Rank 0 D e g r e e 8 S p i n - T e n s o r Terms 154 C.9 Rank 2 D e g r e e 8 S p i n - T e n s o r -Terms 155 C.10 Rank 1 Degree 9 S p i n - T e n s o r Terms 158 v i i L i s t o f F i g u r e s 1.1 E l e c t r i c " c h a r g e " G e n e r a t e d By M u l t i p l e -c o n n e c t e d n e s s 6 1.2 Two Non-homologous 1 - c y c l e s 9 1.3 A 2 - M a n i f o l d I l l u s t r a t i n g L o c a l E x a c t n e s s W i t h o u t G l o b a l E x a c t n e s s 12 1.4 A Loop Of S u p e r c o n d u c t i n g W i r e As An Example Of 2-c h a r g e 18 2.1 Example Of A D i s c o n t i n u o u s M e t r i c F i e l d . .. 22 2.2 Example Of A T i m e - O r i e n t e d M a n i f o l d 24 2.3 Example Of A N o n - T i m e - O r i e n t a b l e M a n i f o l d 25 2.4 A s s o c i a t i o n Of E a c h P o i n t In M W i t h A P o i n t In 'A' . 27 3.1 The 1 - c y c l e G e (Cj ) On The Space 'A' 32 3.2 D e f o r m a t i o n Of A P e r i o d - 1 M e t r i c To A P e r i o d - 2 M e t r i c 34 3.3 The Two 2 - D i m e n s i o n a l Homology C l a s s e s In 'A' 36 3.4 T o p o l o g i c a l S t r u c t u r e Of The F r i e d m a n n And M i n k o w s k i M a n i f o l d s 38 3.5 C a r t e s i a n T e t r a d F i e l d A p p l i e d To The M a n i f o l d s Of F i g u r e 3.4 38 3.6 R a d i a l T e t r a d F i e l d A p p l i e d To The M a n i f o l d s Of F i g u r e 3.4 39 9.1 G r a p h Of The Number Of S p i n - T e n s o r s Of E a c h D e gree . 95 9.2 Example Computer P r i n t o u t Of Terms In A D e r i v a t i v e . 97 v i i i L i s t of APL Function-Codes i n Appendix B B. 1 7UNRAVV . 1 30 B.2 VRERAV7 1 30 B.3 vWRITE V 130 B.4 VWRITEL00P7 131 B.5 7AAA V 131 B.6 7CREATE7 1 32 B.7 vCOMBINV 132 B.8 7ALLV 133 B.9 VNAMESV 1 33 B. 1 0 VNMSFNV 133 B. 1 1 VPMU7 134 B. 12 7PRTTYPEV 134 B.1 3 VCROSS2 V 1 34 B. 1 4 "vGENRTV ,135 B. 1 5 7SPEC0RD7 135 B. 16 7CR0SS3 7 136 B. 1 7 VPERM 7 136 B. 1 8 7EXADV7 136 B. 1 9 VDERIVSV 137 B.20 VDVSOFSUMV 138 B.21 7DELOPV 1 38 B.22 7REPLACE V' 1 38 B.23 VINDEL7 . . . 1 39 B.24 7UP0RIENTV 1 40 B.25 VCOMMUTE V 140 B.26 VCOMMLP7 141 B.27 VPAIRORDV 141 B.28 7HI0RD17 142 B. 29 70RDALL7 143 B. 30 7TERMORD7 143 B.31 7HIER V ' 144 B.32 7SPECREPV 145 B.33 7COMPLIST7 145 B.34 VCNSDT7 146 B.35 7SYMV 146 B.36 7ZTERMSV 147 Acknowledgement T h e , a u t h o r w i s h e s t o thank D r . W.G. Unruh f o r s u g g e s t i n g t h e s u b j e c t of t h i s t h e s i s and f o r t h e e n l i g h t e n i n g d i s c u s s i o n s we have had a t many p o i n t s d u r i n g t h e r e s e a r c h . T hanks a l s o t o D r . P. R a s t a l l f o r h i s i n p u t and i n t e r e s t d u r i n g t h e c o u r s e of t h e work. Thanks a l w a y s t o J . and C. f o r t h e i r c o n s t a n t s u p p o r t and i n s p i r a t i o n . T h i s work was made p o s s i b l e by f u n d i n g i n t h e form of a P o s t g r a d u a t e S c h o l a r s h i p from t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l . 1 I n t r o d u c t i o n " I f [ a s y m p t o t i c a l l y M i n k o w s k i a n ] c o o r d i n a t e s do not e x i s t . . . one must c o m p l e t e l y abandon t h e q u a n t i t i e s t h a t r e l y on them f o r d e f i n i t i o n : t h e t o t a l mass, momentum, and a n g u l a r momentum of t h e g r a v i t a t i n g s o u r c e . " ( M i s n e r T h o r n e and W h e e l e r ( 1 } ) T h i s t h e s i s i s about c o n s e r v e d q u a n t i t i e s i n c u r v e d s p a c e -t i m e s . More c o r r e c t l y , i t i s an i n v e s t i g a t i o n of a s p e c i f i c t y p e of c o n s e r v e d q u a n t i t y w h i c h i s i n s t r u c t u r e s i m i l a r t o e l e c t r i c c h a r g e , b u t c o n s t r u c t e d e x c l u s i v e l y o u t of t h e e l e m e n t s of t h e geometry ( i e : t h e Riemann ' t e n s o r and i t s d e r i v a t i v e s ) . The l a t t e r r e q u i r e m e n t ( t h a t t h e c o n s e r v e d q u a n t i t y be c o n s t r u c t e d p u r e l y f r o m components of t h e ge o m e t r y ) i s v e r y much i n t h e s p i r i t of W h e e l e r ' s " g e o m e t r o d y n a r n i c s " ( 2 ) , wh i c h l e a d s t o t h e r a t h e r c l u m s y l a b e l w h i c h i s t o be u s e d f o r t h e c o n s e r v e d q u a n t i t i e s : " G e o m e t r o d y n a m i c a l A n a l o g s t o C h a r g e " . B e f o r e p r o c e e d i n g , i t i s p r o f i t a b l e t o r e v i e w b r i e f l y what form t h i s t y p e of c o n s e r v e d q u a n t i t y t a k e s , i n what way s u c h a q u a n t i t y i s " c o n s e r v e d " , and why t h e c l a s s i c a l c o n s e r v e d q u a n t i t i e s a r e not c o n s e r v e d i n c u r v e d s p a c e - t i m e s . Assume t h a t t h e r e i s an u n s p e c i f i e d g r a v i t a t i n g body a r o u n d w h i c h measurements of t h e g r a v i t a t i o n a l f i e l d c a n be made. F a r 2 enough away f r o m t h e body, where s p a c e t i m e i s a s y m p t o t i c a l l y f l a t , c o n s e r v e d q u a n t i t i e s (mass, a n g u l a r momentum, 4-momentum) c a n be d e f i n e d w h i c h s e r v e p h y s i c s v e r y w e l l i n d e s c r i b i n g t h a t body. One c a n c a l c u l a t - e t h e s e q u a n t i t i e s by i n t e g r a t i n g c e r t a i n t e n s o r components o f t h e l i n e a r i z e d g r a v i t a t i o n a l f i e l d o v e r a c l o s e d s u r f a c e e n c l o s i n g t h e body < 2 5 ' , and t h e i r v a l u e s w i l l be i n d e p e n d e n t o f t h e p a r t i c u l a r s u r f a c e c h o s e n . Any c h a n g e s t o t h e s e q u a n t i t i e s i n t i m e c a n o n l y r e s u l t f r o m some f l u x t h r o u g h t h e s u r f a c e o f i n t e g r a t i o n , and t h i s c a n be t a k e n i n t o a c c o u n t a s a s o u r c e - t e r m i n t h e c a l c u l a t i o n of t h e c o n s e r v e d q u a n t i t y . The q u a n t i t i e s a r e t h u s c o n s e r v e d i n time and c o n s e r v e d i n t h e s e n s e t h a t t h e y a r e i n d e p e n d e n t o f t h e s p e c i f i c c h a r a c t e r i s t i c s of t h e s u r f a c e of i n t e g r a t i o n . In s t r o n g g r a v i t a t i o n a l f i e l d s , however, t h e s e c o n s e r v a t i o n laws b r e a k down b e c a u s e : 1) t h e y a r e g l o b a l s t a t e m e n t s about an e x t e n d e d r e g i o n a r o u n d t h e body. 2 ) t h e y a r e d e f i n e d i n t e r m s of t e n s o r s . 3) t h e g l o b a l p r o p e r t i e s of t e n s o r s c a n n o t c o n s i s t e n t l y be d e f i n e d u n l e s s t h e s p a c e t i m e i s a s y m p t o t i c a l l y f l a t . The a p p r o a c h t a k e n i n t h i s t h e s i s o vercomes t h i s p r o b l e m by d e f i n i n g g e o m e t r o d y n a m i c a l a n a l o g s t o c h a r g e n o t i n t e r m s o f t e n s o r s , b u t r a t h e r i n t e r m s of d i f f e r e n t i a l f o r m s , w h i c h do behave w e l l g l o b a l l y i n c u r v e d s p a c e - t i m e s . The work i s b a s e d on two p a p e r s by W.G.Unruh < 3 ' " > , one o f w h i c h i s u n p u b l i s h e d , , i n w h i c h he d e f i n e s t h e s e " c h a r g e s " and 3 p r o v e s : 1) t h a t none e x i s t when t h e m e t r i c i s n o t c o n s t r a i n e d t o obey any f i e l d e q u a t i o n s . and 2) t h a t i n a s p a c e t i m e c o n s t r a i n e d t o obey E i n s t e i n ' s e q u a t i o n s , none e x i s t t h a t c an be c o n s t r u c t e d o ut o f a s e t o f r e l a t i v e l y s i m p l e c o m b i n a t i o n s of t h e Riemann t e n s o r and i t s f i r s t t h r e e d e r i v a t i v e s . C h a p t e r s 2 and 3 v e r y b r i e f l y r e v i e w s U n r u h ' s p r o o f t h a t no c h a r g e s e x i s t when t h e m e t r i c i s u n r e s t r i c t e d . The p u r p o s e of t h e d i s c u s s i o n i s t o i l l u m i n a t e t h e p h y s i c a l p r i n c i p a l s i n v o l v e d w i t h o u t g e t t i n g m i r e d i n a l o n g d i s c u s s i o n o f t h e m a t h e m a t i c a l p r o o f s . Some t i m e i s s p e n t l o o k i n g a t t h e s t r u c t u r e imposed on t h e m a t r i x r e p r e s e n t a t i o n of t h e m e t r i c by d i f f e r e n t c h o i c e s o f t e t r a d f i e l d s on t h e m a n i f o l d . The b u l k of t h e t h e s i s ( c h a p t e r s 4 t h r o u g h 9 and t h e a p p e n d i c e s ) a p p r o a c h e s t h e p r o b l e m of t h e e x i s t e n c e o f t h e s e c h a r g e s w i t h " b r u t e f o r c e " . E a c h p o s s i b l e 1-, 2-, and 3-form i s g e n e r a t e d and i t s d e r i v a t i v e c a l c u l a t e d , i n t h e hope t h a t when e v e r y p o s s i b l e t e r m i s t e s t e d , some c o m b i n a t i o n may be f o u n d o u t of w h i c h a c o n s e r v e d c h a r g e c a n be b u i l t . T h i s s e a r c h was c a r r i e d o ut w i t h t h e use of a s y m b o l i c -m a n i p u l a t i o n computer p r o g r a m w h i c h was d e v e l o p e d as p a r t o f t h i s r e s e a r c h . To make t h e r e p r e s e n t a t i o n o f geometrodynamic q u a n t i t i e s a s s i m p l e as p o s s i b l e f o r t h e computer, i t was d e c i d e d : 1) t o use t h e s p i n - t e n s o r r e p r e s e n t a t i o n o f t h e 4 c u r v a t u r e , as d e v e l o p e d by P e n r o s e , among o t h e r s , and 2) t o i n v e n t a new v e r s i o n o f t h i s s p i n - t e n s o r n o t a t i o n w h i c h was i n form c o m p a t i b l e w i t h t h e t y p e s o f c a l c u l a t i o n s t h a t t h e computer was p e r f o r m i n g . C h a p t e r s f i v e t h r o u g h s e v e n g i v e an o v e r v i e w of t h i s s p i n - t e n s o r a p p r o a c h t o g e n e r a l r e l a t i v i t y . C h a p t e r e i g h t and A p p e n d i x A d e s c r i b e i n some d e t a i l t h e work t h a t was done m o d i f y i n g t h e n o t a t i o n i n t o t h e r e q u i r e d c o m p u t e r - c o m p a t i b l e form, and t h e a l g o r i t h m t h a t was d e v e l o p e d t o e n a b l e t h e computer t o m a n i p u l a t e t h e s p i n - t e n s o r s as r e q u i r e d . C h a p t e r n i n e p r e s e n t s t h e r e s u l t s of t h i s s y s t e m a t i c s e a r c h f o r a g e o m e t r o d y n a m i c a l a n a l o g t o c h a r g e . 5 C h a p t e r j_ G e n e r a l i z a t i o n o f t h e C o n c e p t o f " C h a r g e " In s t r o n g g r a v i t a t i o n a l f i e l d s , t h e c o n v e n t i o n a l c o n s e r v a t i o n laws b r e a k down so t h a t s u c h q u a n t i t i e s a s t o t a l 4 -momentum o r t o t a l a n g u l a r momentum of a s y s t e m c a n n o t be u n a m b i g u o u s l y d e f i n e d . T h i s p a p e r i s a s e a r c h f o r a s p e c i f i c t y p e o f c o n s e r v e d q u a n t i t y , i n s t r u c t u r e s i m i l a r t o e l e c t r i c c h a r g e b ut c o n s t r u c t e d o f p u r e l y g e o m e t r i c a l components. T h i s c h a p t e r f i r s t a t t e m p t s t o e s t a b l i s h an i n t u i t i v e c o n c e p t of t h e s e g e o m e t r o d y n a m i c a n a l o g s t o e l e c t r i c c h a r g e , t h e n r e v i e w s deRham's theorems i n s e c t i o n 1 .2 as a b a s i s f o r a more c a r e f u l d e f i n i t i o n o f t h e c h a r g e s i n t h e f i n a l s e c t i o n . 1.1 I n t r o d u c t i ion We b e g i n by d i s c a r d i n g t h e c o n c e p t o f c h a r g e as a q u a n t i t y w h i c h e x i s t s a t a c e r t a i n p o i n t . T h i s can b e s t be u n d e r s t o o d by f i r s t d e m o n s t r a t i n g t h a t e l e c t r i c c h a r g e c a n be d e s c r i b e d w i t h o u t s o u r c e t e r m s i n M a x w e l l ' s e q u a t i o n s . W h e e l e r ( 5 ) , among o t h e r s , has shown t h a t e l e c t r i c c h a r g e can be e x a c t l y d e s c r i b e d by c l o s e d f r e e - f i e l d l i n e s t r a p p e d i n a m u l t i p l y - c o n n e c t e d ("wormhole") s p a c e t i m e . T h i s model can be summarized a s f o l l o w s : 1) Assume i n t h e s p a c e two d i s j o i n t 2 - s p h e r e s A and B. 6 2) Remove from t h e s p a c e t h e i n t e r i o r s of A and B. 3) E q u a t e e v e r y p o i n t on A w i t h a p o i n t on B, i n a c o n t i n u o u s manner. 4) Impose on t h i s m u l t i p l y - c o n n e c t e d m a n i f o l d f r e e e l e c t r i c f i e l d l i n e s , some o f w h i c h p e n e t r a t e t h e s p h e r e s , as shown i n f i g u r e 1 . 1 . fl 6>J F i g u r e 1 . 1 : E l e c t r i c " c h a r g e " g e n e r a t e d by m u l t i p l e -c o n n e c t e d n e s s To an o b s e r v e r o u t s i d e s p h e r e A, who s e e s t h e e l e c t r i c f i e l d l i n e s e m a n a t i n g i n a l l d i r e c t i o n s , t h e r e i s no c h a r a c t e r i s t i c of t h e f i e l d w h i c h would t e l l him t h a t t h e r e i s no c h a r g e d body i n s i d e t h e s p h e r e . He would i n t e g r a t e t h e r a d i a l component of t h e E l e c t r i c f i e l d , E, o v e r a G a u s s i a n s u r f a c e S and use Q = k \ E - i s ( i.D s t o d e t e r m i n e t h a t t h e r e i s a t o t a l c h a r g e Q e n c l o s e d by t h e s p h e r e . A c t u a l o b s e r v a t i o n s o f c h a r g e a r e , of c o u r s e , made i n 7 j u s t t h i s manner. T h i s example i l l u s t r a t e s two p o i n t s : a) t h a t " c h a r g e " need not be a p r o p e r t y o f a s p e c i f i c o b j e c t o r p o i n t i n s p a c e , but r a t h e r a g l o b a l p r o p e r t y o f t h e f i e l d o v e r a G a u s s i a n s u r f a c e e n c l o s i n g t h a t o b j e c t o r p o i n t , and b) t h a t t h e t o p o l o g y of t h e s p a c e t i m e p l a y s an i m p o r t a n t r o l e i n making s u c h c h a r g e s p o s s i b l e . D e f i n e t h e g e o m e t r i c a n a l o g t o e l e c t r i c c h a r g e i n a s i m i l a r way. In t h e s p i r i t of G e o m e t r o d y n a m i c s , ( " a l l i s g e o m e t r y " ) , and f o l l o w i n g Unruh ( 3 ) , we c o n s t r u c t t h e s e c h a r g e s p u r e l y i n terms of t h e m e t r i c t e n s o r , ( g ^ ) , t h e Riemann t e n s o r R^vps- and i t s c o v a r i a n t d e r i v a t i v e s (R^v^e--,*. > R c*f t* j * r e t c ) . The g o a l i s t o f i n d some c o n s t r u c t i o n w h i c h , when i n t e g r a t e d o v e r any s u r f a c e e n c l o s i n g a s p e c i f i e d r e g i o n , w i l l a l w a y s y i e l d t h e same i n t e g r a l - t h e " c h a r g e " o f t h a t r e g i o n . C o n s i d e r , as t h e most d i r e c t example, t h e c h a r g e s w h i c h most r e s e m b l e e l e c t r i c c h a r g e , w h i c h a r e t o be c a l l e d "2-c h a r g e s " . They a r e t h e i n t e g e r v a l u e s f o u n d by i n t e g r a t i n g " g e o m e t r i c a l 2 - f o r m s " ( i e : a n t i s y m m e t r i c 2 - t e n s o r s c o n s t r u c t e d of Rpvfr and i t s d e r i v a t i v e s ) o v e r c l o s e d 2 - s u r f a c e s . A c e r t a i n r e g i o n J i s d e f i n e d t o have a 2 - c h a r g e Q i f t h e i n t e g r a l o v e r any s u r f a c e w h i c h e n c l o s e s J (and no o t h e r s i m i l a r l y " c h a r g e d " r e g i o n s ) w i l l e q u a l Q. In o r d e r t o d e f i n e t h e s e c h a r g e s more c a r e f u l l y and more g e n e r a l l y , i t i s n e c e s s a r y t o r e v i e w b a s i c homology t h e o r y f o r a 8 s p a c e o f n - d i m e n s i o n s . A f t e r t h i s d i g r e s s i o n , we w i l l r e t u r n t o a c a r e f u l d e f i n i t i o n o f t h e g e o m e t r i c a l a n a l o g t o c h a r g e . 1.2 Review of Homologous p - C y c l e s and deRham's Theorems In o r d e r t o d e f i n e t h e above " 2 - c h a r g e s " and more g e n e r a l " p - c h a r g e s " more c a r e f u l l y , i t w i l l be n e c e s s a r y t o d e f i n e them u s i n g some c o n c e p t s of homology and homotopy t h e o r y . I t i s assumed t h a t t h e r e a d e r i s f a m i l i a r w i t h t o p o l o g i c a l m a n i f o l d s and w i t h e l e m e n t s of homology t h e o r y s u c h as c h a i n s , c y c l e s . , and t h e l i k e . T h e s e a r e d i s c u s s e d w e l l i n many t e x t b o o k s on t h e s u b j e c t 1 6 J . The c o n c e p t s of homologous and h o m o t o p i c p - c y c l e s a r e c e n t r a l t o t h e d i s c u s s i o n which f o l l o w s i n s e c t i o n 1.3, as a r e deRham's t h e o r e m s , so we d i g r e s s d u r i n g t h i s s e c t i o n t o b r i e f l y r e v i e w t h e s e and r e l a t e d c o n c e p t s . No a t t e m p t i s made a t m a t h e m a t i c a l r i g o u r i n t h i s d i s c u s s i o n but i n s t e a d t h e r e a d e r i s r e f e r r e d t o t h e r e f e r e n c e s w h i c h w i l l be n o t e d as t h e c o n c e p t s a r e m e n t i o n e d . We t a k e t h e a r e n a of p h y s i c s t o be a 4-d i m e n s i o n a l d i f f e r e n t i a l m a n i f o l d , o r " f o u r - m a n i f o l d " . C o n s i d e r two p - c y c l e s c J, and c£ w h i c h a r e p a r a m e t r i z e d w i t h t h e p a r a m e t e r " A " . c^(X) and c £ ( X ) a r e s a i d t o be " h o m o t o p i c " i f t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n H ( A , p ) ( c a l l e d an "homotopy") w h i c h c o n t i n u o u s l y d e f o r m s c M A ) i n t o c £ ( X ) . T h a t 9 H(A,l)=C*(X) «s a p-cvfcle a\\ valuer of (1.2) F o r a more c a r e f u l d e f i n i t i o n of t h i s c o n c e p t , t h e r e a d e r i s r e f e r r e d , f o r example, t o Croom ( 7 ) . A s e t w h i c h c o n s i s t s of a l l p - c y c l e s w h i c h a r e h o m o t o p i c t o one a n o t h e r i s an "homolopy c l a s s of o r d e r p " . Two p - c y c l e s a r e h o m o t o p i c t o one a n o t h e r i f and o n l y i f t h e y b e l o n g t o t h e same homotopy c l a s s . Two p - c y c l e s c p and c^ a r e s a i d t o be "homologous" i f t h e r e e x i s t s a ( p + l ) - c h a i n V p + 1 s u c h t h a t : Cp - Cp = WP+1 where i s t h e b o u n d a r y o f Vp + 1. I f , f o r example, c l p e n c l o s e s a s i n g u l a r i t y o r wormhole, w h i l e c J does n o t , t h e y a r e not homologous (as i n f i g u r e 1.2). A p - c y c l e i s "homologous t o z e r o " i f i t i s t h e boundary of a (p+1) c h a i n , t h a t i s , i f i t c o n t a i n s no s i n g u l a r i t i e s , "wormholes", or o t h e r t o p o l o g i c a l s u r p r i s e s . In f i g u r e 1.2, c* i s an example o f a 1 - c y c l e homologous t o z e r o . Any s e t c o n s i s t i n g o f a l l p - c y c l e s t h a t a r e homologous t o e a c h o t h e r i s a "homology c l a s s o f o r d e r p " . Two p - c y c l e s a r e , t h e r e f o r e , homologous t o e a c h o t h e r i f and o n l y i f t h e y b e l o n g (1.3) , 10 or F i g u r e 1 . 2 : Two non-homologous 1 - c y c l e s t o t h e same homology c l a s s . Any two p - c y c l e s c a n be added t o g e t h e r u s i n g p o i n t w i s e a d d i t i o n ('°> and t h e i r sum w i l l be a n o t h e r p - c y c l e . I t i s beyond t h e s c o p e of t h i s t h e s i s t o d i s c u s s t h i s a d d i t i v i t y i n d e t a i l ; f o r s u c h a d i s c u s s i o n t h e r e a d e r i s r e f e r r e d t o t h e l i t e r a t u r e ( 8 ) . The s e t o f a l l p - c y c l e s forms a g r o u p " Z F " w i t h t h i s o p e r a t i o n o f p o i n t w i s e a d d i t i o n i n d u c e d by t h e i n t e g e r s . One s u b g r o u p o f Zv i e t h e g r o u p Bp o f a l l p - c y c l e s w h i c h a r e b o u n d a r i e s o f ( p + 1 ) - c h a i n s ( t h e members of B p a r e a l l "homologous t o z e r o " ) . The p - d i m e n s i o n a l homology g r o u p H p i s d e f i n e d t o be t h e q u o t i e n t g r o u p < 8 ' 9 ) : C o n c e p t u a l l y , t h i s means t h a t e v e r y member of H p i s an homology c l a s s of p - c y c l e s . The i d e n t i t y e l e m e n t of H p i s , f o r example, t h e homology c l a s s B p o f a l l p - c y c l e s w h i c h a r e homologous t o z e r o . F o r a t h o r o u g h d i s c u s s i o n of homology g r o u p s , t h e r e a d e r i s r e f e r r e d t o one of t h e t e x t b o o k s on t h e s u b j e c t ( , 0 ) . We now q u o t e ( w i t h o u t p r o o f ( 1 1 > ) deRham's Theorems, w h i c h w i l l f o r m t h e t h e o r e t i c a l b a s i s f o r t h e d e f i n i t i o n o f ( 1 . 4 ) 11 g e n e r a l i z e d c h a r g e t o be p r e s e n t e d i n s e c t i o n 1 . 3 . The " s e c o n d " t h e o r e m o f deRham i s p r e s e n t e d f i r s t , f o l l o w e d by t h e " f i r s t " . T h e s e s t a t e m e n t s o f t h e theorems a r e w r i t t e n w i t h t h e a s s u m p t i o n t h a t t h e r e a d e r i s f a m i l i a r w i t h t h e c o n c e p t s of d i f f e r e n t i a l f o r m s , w h i c h a r e w e l l p r e s e n t e d i n many t e x t s on t h e s u b j e c t The s e c o n d t h e o r e m of DeRham e s t a b l i s h e s a g u a r a n t e e d l i n k between c y c l e s ( w h i c h c o r r e s p o n d t o s u r f a c e s i n p h y s i c a l s p a c e ) and d i f f e r e n t i a l f orms ( w h i c h c o r r e s p o n d t o a n t i s y m m e t r i c t e n s o r f i e l d s ) . I t p r o v e s t h a t one can a l w a y s d e f i n e a f i e l d of p - f o r m s s u c h t h a t t h e i n t e g r a l o f t h e s e p - f o r m s w i l l g i v e a s p e c i f i c " c h a r g e " when t h e i n t e g r a t i o n i s o v e r a c l o s e d s u r f a c e . S p e c i f i c a l l y , i t s a y s t h a t i f (c' p , C p , . . . c p ) a r e j non-homologous p - c y c l e s o f a m a n i f o l d M, t h e n f o r any j r e a l numbers Q- ( c a l l e d t h e " p e r i o d s " of c l P ) t h e r e e x i s t s a p - f o r m a p w h i c h i s c l o s e d ( d a p = 0) and d i f f e r e n t i a b l e t h r o u g h o u t M and w h i c h s a t i s f i e s a l w a y s have a p e r i o d e q u a l t o z e r o < 2 6 ) from w h i c h i t f o l l o w s t h a t a l l c y c l e s homologous t o e a c h o t h e r have t h e same ( 1 2 ) f o r e a c h c p . A p - c y c l e c* w h i c h i s homologous t o z e r o w i l l 12 p e r i o d [ $ ] • The i n t e g r a l 'of a p i s t h u s d e p e n d e n t s o l e l y on t h e homology c l a s s of t h e s u r f a c e of i n t e g r a t i o n . The f i r s t theorem o f deRham p r o v e s t h a t a c l o s e d p - f o r m a p ( i e : d a p = 0) i s e x a c t i f and o n l y i f a l l i t s p e r i o d s v a n i s h . The term " e x a c t " means t h a t a p can be w r i t t e n a s t h e e x t e r i o r d e r i v a t i v e of a ( p - l ) - f o r m (a p=dw p~'). T h i s t h e o r e m i s a g l o b a l s t a t e m e n t t h a t u n l e s s a p = 0 f o r a l l p - c y c l e s i n t h e m a n i f o l d , t h e r e d o e s n ot e x i s t a wp_1 s u c h t h a t a p=dw p~' e v e r y w h e r e . Note, however, from t h e same t h e o r e m t h a t t h e r e a l w a y s e x i s t s some wp-' s a t i s f y i n g a p =dwp~' l o c a l l y . T h i s f o l l o w s from _ t h e p a r a c o m p a c t n e s s of t h e m a n i f o l d w h i c h a s s u r e s t h a t t h e r e w i l l a l w a y s be a n e i g h b o u r h o o d w h e r e i n a l l p e r i o d s o f a p v a n i s h . As an i l l u s t r a t i o n o f a 1-form w h i c h i s l o c a l l y b u t not g l o b a l l y e x a c t , c o n s i d e r t h e f o l l o w i n g . Assume a m a n i f o l d a s shown i n f i g u r e ( 1 . 3 ) , i n w h i c h a l l p o i n t s where (P = 0 a r e i d e n t i f i e d w i t h p o i n t s where (f = 2TT. The f i e l d E ( w h i c h i s c o n s t a n t i n t h e ^ - d i r e c t i o n ) c a n be w r i t t e n a t most p o i n t s a s : E = ( V ^ ^ and so i s l o c a l l y e x a c t . G l o b a l l y , however, i t i s not e x a c t b e c a u s e o f t h e d i s c o n t i n u i t y of if a t cf = 0 . No c o o r d i n a t e s y s t e m [$] T h i s f o l l o w s b e c a u s e any two c y c l e s c'p and c* w h i c h a r e homologous t o ea c h o t h e r t o g e t h e r f o r m a b o u n d a r y : But so t h e p e r i o d s must be t h e same. 13 F i g u r e 1.3: A 2 - M a n i f o l d I l l u s t r a t i n g L o c a l E x a c t n e s s W i t h o u t G l o b a l E x a c t n e s s c a n be f o u n d where t h i s f i e l d i s g l o b a l l y e x a c t , b e c a u s e t h e of t h e 1-form ( v e c t o r ) E ^ a r o u n d t h e c l o s e d c y c l e ( r = c o n s t . , 0 < (f <2TT) i s not z e r o . 1.3 D e f i n i t i o n of p - c h a r g e We can g e n e r a l i z e t h e i d e a o f e l e c t r i c c h a r g e s (and what we c a l l e d " 2 - c h a r g e s " i n s e c t i o n 1.1) t o d e f i n e o t h e r t y p e s of c h a r g e s , w i t h a n a l o g o u s s t r u c t u r e , w h i c h w i l l be c a l l e d "1-c h a r g e s " and " 3 - c h a r g e s " . The e x i s t e n c e of e l e c t r i c c h a r g e , Q, r e q u i r e s t h e e x i s t e n c e of a 2-form ( t h e e l e c t r o m a g n e t i c 2-form F) whose v a l u e upon i n t e g r a t i o n o v e r any 2 - c y c l e c^ , d epends o n l y on t h e homology c l a s s o f c* . I f we remove a l l s o u r c e t e r m s from M a x w e l l ' s e q u a t i o n s , t h i s c h a r g e must r e s u l t f r o m f i e l d l i n e s b e i n g t r a p p e d i n t h e m u l t i p l e - c o n n e c t e d n e s s of t h e m a n i f o l d . The v a l u e o f Q c o r r e s p o n d s t o t h e c h a r g e a s s o c i a t e d w i t h t h e i n t e g r a l ( 1.6) o 14 e n c l o s e d wormhole, and i s t h u s a l i n e a r f u n c t i o n a l o v e r t h e s e c o n d - o r d e r homology g r o u p o f t h e s p a c e [ f ] . F o l l o w i n g Unruh < 1 3> , we d e f i n e t h e g e n e r a l i z e d c h a r g e , l i k e t h e e l e c t r i c c h a r g e , t o be a l i n e a r f u n c t i o n a l o v e r one o f t h e homology g r o u p s of t h e m a n i f o l d . L e t us r e f e r t o t h e c h a r g e as a " p - c h a r g e " i f i t i s a f u n c t i o n a l o v e r t h e homology g r o u p of o r d e r p. From deRham's s e c o n d t h e o r e m i t i m m e d i a t e l y f o l l o w s t h a t i f a p - c h a r g e e x i s t s , t h e r e w i l l a l w a y s be a c l o s e d p - f o r m d e f i n e d and d i f f e r e n t i a b l e a t a l l p o i n t s i n t h e m a n i f o l d whose p e r i o d o v e r a g i v e n p - c y c l e i s t h e p - c h a r g e " e n c l o s e d " by t h a t p - c y c l e . B e c a u s e t h e s e a r e geometrodynamic a n a l o g s t o c h a r g e , t h e y must be d e f i n e d i n terms o f t h e geometry o f t h e s p a c e t i m e . S p e c i f i c a l l y , we r e s t r i c t o ur a t t e n t i o n t o p - c h a r g e s w h i c h a r e t h e p e r i o d s of p - f o r m s w h i c h a r e c o n s t r u c t e d e n t i r e l y o ut of t h e Riemann t e n s o r and i t s d e r i v a t i v e s . T h e s e p - c h a r g e s , i f t h e y a r e t o be c h a r a c t e r i s t i c of t h e g e o m e t r y , c a n n o t be i d e n t i c a l l y z e r o . From deRham's f i r s t t h e o r e m , t h e r e f o r e , i t f o l l o w s t h a t t h e p - f o r m s w h i c h d e f i n e t h e c h a r g e must n o t be g l o b a l l y e x a c t . In t h e same v e i n , any p-[ f ] I t i s l i n e a r b e c a u s e g i v e n S, ,SX,...S% i n g e n e r a l non-homologous s u r f a c e s and a 2 - f o r m F, i t f o l l o w s t h a t : where k-t a r e i n t e g e r s and k^S;, r e p r e s e n t s k> s u r f a c e s a l l homologous t o S^ 15 c h a r g e w h i c h has t h e same v a l u e f o r a l l p o s s i b l e g e o m e t r i e s i s u n s a t i s f a c t o r y , b e c a u s e i t g i v e s no i n f o r m a t i o n a b o u t t h e g e o m e t r y . Our s e a r c h has now been n a r r o w e d t o a s e a r c h f o r a p - f o r m h* w i t h t h e f o l l o w i n g p r o p e r t i e s : 1) I t i s c l o s e d . T h a t i s , ( d h p = 0) 2) I t i s not e x a c t . T h a t i s , t h e r e d o e s n o t e x i s t any ( p - l ) - f o r m G s u c h t h a t g l o b a l l y ( h p = d G ) . 3) I t i s c o n s t r u c t e d e n t i r e l y from t h e Riemann t e n s o r R * ^ and i t s d e r i v a t i v e s . 1.4 I n t e r p r e t a t i o n of p - c h a r g e i n a 4 - M a n i f o l d In a 4 - d i m e n s i o n a l m a n i f o l d , f i v e t y p e s of c h a r g e s can be c o n s t r u c t e d : 0-, 1-, 2-, 3-, and 4 - c h a r g e s . The f i r s t of t h e s e , 0 - c h a r g e s , a r e u n i n t e r e s t i n g b e c a u s e " i n t e g r a t i o n o v e r a 0-c y c l e " l e a v e s t h e s c a l a r a t t h a t p o i n t unchanged, so t h a t 0-c h a r g e s w o u l d have no f e a t u r e s t h a t s c a l a r s a l o n e d i d n o t h a v e . S i m i l a r l y 4 - c h a r g e s a r e n o t v e r y i n t e r e s t i n g . The 4-form r e q u i r e d must be a t e a c h p o i n t some m u l t i p l e o f t h e L e v i - C i v i t a t e n s o r , a s t h a t i s t h e o n l y f u l l y a n t i s y m m e t r i c 4 - t e n s o r . Such a 4 - c h a r g e w o u l d be t h e i n t e g r a l of t h e 4-form o v e r a c l o s e d 4-d i m e n s i o n a l s u r f a c e ( i e , t h e whole m a n i f o l d ) , and c o u l d not be c o n s i d e r e d t o be " c o n s e r v e d " i n any i n t u i t i v e s e n s e b e c a u s e t h e r e i s no r e m a i n i n g d i r e c t i o n i n w h i c h t o be c o n s e r v e d . The o t h e r t h r e e c h a r g e s a r e more i n t e r e s t i n g , as w i l l be d i s c u s s e d b e low. 16 F o r 3 - c h a r g e s t o e x i s t , we r e q u i r e a c l o s e d 3-form |_ (1.7) whs r s • w h i c h means: o r , e q u i v a l e n t l y [ t 1 (1.10) where: . . To e n s u r e t h a t f_ i s not e x a c t we r e q u i r e t h a t t h e r e e x i s t no a n t i s y m m e t r i c t e n s o r h ^ s u c h t h a t g l o b a l l y : ^L^1»^1 (1.12) o r , i f e q u a t i o n ( 1 . 1 1 ) . i s u s e d , t h a t t h e r e e x i s t no h'W^ s u c h t h a t g l o b a l l y : K ' , ( 1 = ^ 1 (1.13) [ f ] T h i s i s e q u i v a l e n t t o 1.8 b e c a u s e : i m p l i e s t h a t : -f = 0 17 E q u a t i o n s (1.12) or (1.13) e n s u r e t h a t t h e 3-form fr_^ v^1 i s n o t e x a c t . C o n s i d e r what s u c h a c h a r g e would " l o o k l i k e " . I f an o b s e r v e r i n t e g r a t e d f_ o v e r some c l o s e d 3-volume ( s u c h as t h e u n i v e r s e , a s s u m i n g i t i s c l o s e d ) a t a s p e c i f i c t i m e - s l i c e i n t h e m a n i f o l d , he would f i n d t h a t t h e i n t e g r a l w ould be i n d e p e n d e n t o f h i s c h o i c e of t i m e - s l i c e . The 3 - c h a r g e might be r e l a t e d t o t h e t o t a l number of b l a c k h o l e s i n t h e u n i v e r s e , f o r example, o r t h e t o t a l e l e c t r i c c h a r g e of t h e u n i v e r s e , v a l u e s w h i c h would be e x p e c t e d t o r e m a i n c o n s t a n t . The i n t e g r a l c o u l d e q u a l l y w e l l be o v e r one t i m e d i m e n s i o n and two s p a c e d i m e n s i o n s , a s l o n g as t h e h y p e r s u r f a c e of i n t e g r a t i o n i s c l o s e d . As an example, s u r r o u n d a p o i n t w i t h a 2 - s p h e r e o f r a d i u s r , and i n t e g r a t e some c l o s e d 3-f o r m o v e r t h a t s p h e r e f o r a l l t i m e ( a g a i n we assume a c l o s e d u n i v e r s e ) . The i n t e g r a l w i l l be i n d e p e n d e n t of r . The e x i s t e n c e of a 2 - c h a r g e r e q u i r e s a c l o s e d 2-form f_: w h i c h i m p l i e s a f u l l y a n t i s y m m e t r i c 2 - t e n s o r f ^ v s u c h t h a t : W . P = 0 " * f" S \ s -0 ( L I S ) To e n s u r e t h a t f ^ v i s n o t e x a c t , we f u r t h e r r e q u i r e t h a t t h e r e be no 1 - t e n s o r h „ w h i c h g l o b a l l y s a t i s f i e s : 18 V\r , ="f „ (1.16) or e q u i v a l e n t l y , t h a t t h e r e be no 3 - t e n s o r h'^-g w h i c h g l o b a l l y s a t i s f i e s : = (1.17) A d e s c r i p t i o n of what a 2 - c h a r g e would " l o o k l i k e " has a l r e a d y been p r e s e n t e d i n s e c t i o n 1.1. C h a r a c t e r i s t i c of 2-c h a r g e s i s t h a t t h e y a r e c o n s e r v e d i n two d i m e n s i o n s w h i l e b e i n g i n t e g r a t e d o v e r t h e o t h e r two, but i t i s n o t n e c e s s a r y t h a t b o t h t h e l a t t e r be s p a c e l i k e d i m e n s i o n s . As an example of what a 2-c h a r g e r e s e m b l e s when i n t e g r a t e d o v e r one t i m e and one s p a c e d i r e c t i o n , c o n s i d e r a l o o p of s u p e r c o n d u c t i n g w i r e as shown i n f i g u r e ( 1 . 4 ) . Ampere's law e n s u r e s t h a t t h e i n t e g r a l a r o u n d t h e F i g u r e 1.4: A l o o p o f s u p e r c o n d u c t i n g w i r e as an example of 2 - c h a r g e 2 - c y c l e c ^ , w i l l be i n d e p e n d e n t o f t h e r a d i u s o r p o s i t i o n ( s o l o n g as i t s t i l l e n c l o s e s t h e w i r e e x a c t l y o nce) o f t h e 2 - c y c l e , 19 a s s u m i n g a c o n s t a n t c u r r e n t f l o w s t h r o u g h t h e w i r e . I f we now assume a n o n - d i s p e r s i v e p u l s e of c u r r e n t i s f l o w i n g a r o u n d t h e l o o p , t h i s i s no longer- t r u e . I t i s n e c e s s a r y t o i n t e g r a t e b o t h a r o u n d t h e l o o p and t h r o u g h a l l t i m e i n o r d e r t o e n s u r e t h a t t h e i n t e g r a l i s i n d e p e n d e n t of t h e p o s i t i o n and r a d i u s of t h e 2-c y c l e , and we must assume, t o o , t h a t s p a c e t i m e i s c l o s e d , o r t h e i n t e g r a t i o n i s o v e r a n o n - c l o s e d s u r f a c e . The c o n s e r v a t i o n e f f e c t h e r e i s o v e r two s p a t i a l d i m e n s i o n s r a t h e r t h a n i n t h e t i m e d i m e n s i o n . In o r d e r t o have 1 - c h a r g e s , one r e q u i r e s a c l o s e d 1-form f_: w h i c h i m p l i e s a v e c t o r f ^ s u c h t h a t : f r = 0 (1 .19) or e q u i v a l e n t l y a 3 - t e n s o r * f c » ^ w h i c h s a t i s f i e s : *WS* "•2°> (1.18) To e n s u r e t h a t i_ i s not e x a c t , t h e r e must be no s c a l a r f i e l d h s u c h t h a t g l o b a l l y : o r , e q u i v a l e n t l y , no 4 - t e n s o r h ' k * ? * ^ ( w h i c h w i l l be a t e a c h 20 p o i n t a m u l t i p l e of £ e t ^ f ^ ) s u c h t h a t g l o b a l l y : (1.22) A 1-charge would c o r r e s p o n d t o t h e l i n e i n t e g r a l of t h e v e c t o r f ^ o v e r a c l o s e d l o o p . B e c a u s e 1 - c y c l e s homologous t o z e r o c a r r y z e r o c h a r g e , t h e t o p o l o g y of t h e s p a c e w o u l d have t o i n c l u d e an e x t e n d e d s i n g u l a r i t y , s u c h as an i n f i n i t e l i n e of s i n g u l a r i t y o r a c l o s e d c i r c l e of s i n g u l a r i t y . The c h a r g e e n c l o s e d by t h e l o o p ' would be i n v a r i a n t w i t h r e s p e c t t o a) m o t i o n of t h e l o o p a l o n g t h e l i n e o f s i n g u l a r i t y , b) i n c r e a s e s i n t h e s i z e of t h e l o o p , and c) t h e t i m e of o b s e r v a t i o n . 21 C h a p t e r 2 N o n - E x i s t e n c e of C h a r g e s f o r U n r e s t r i c t e d M e t r i c S p a c e t i m e s Unruh ( 4 ) has shown t h a t when t h e m e t r i c f i e l d i s not r e s t r i c t e d by any f i e l d e q u a t i o n s , no p - c h a r g e s can e x i s t . He shows t h i s by e x a m i n i n g how f r e e l y t h e m e t r i c c a n change between two homologous p - c y c l e s c p and c 2,. The r e a s o n i n g b e h i n d t h i s a p p r o a c h i s summarized below. I f a p - f o r m f_ can be u s e d t o d e f i n e p - c h a r g e s , t h e n i t s p e r i o d "Q 1" on c p must be e q u a l t o i t s p e r i o d "Q 2" on c 2 , a c c o r d i n g t o t h e d e f i n i t i o n o f p - c h a r g e s i n s e c t i o n 1.3. So: 1) Q 1 must e q u a l Q 2 f o r e v e r y m e t r i c f i e l d w h i c h can be d e f i n e d o v e r Cp and c p Assume now t h a t t h e m e t r i c c a n v a r y c o m p l e t e l y a r b i t r a r i l y ( a t t h i s p o i n t we do not even r e q u i r e t h a t t h e m e t r i c be c o n t i n u o u s ) between c p 1 and c 2 . T h i s i m p l i e s t h a t : 2) G i v e n a s p e c i f i c m e t r i c a t c p , t h e m e t r i c f i e l d on Cp c o u l d be a n y t h i n g . T h e r e f o r e , f r o m (1) and ( 2 ) : 3) The v a l u e of Q 2 i s t h e same f o r e v e r y p o s s i b l e m e t r i c f i e l d on c 2,. w h i c h i m p l i e s t h a t : 4) Q 2 i s not a v a l i d p - c h a r g e b e c a u s e i t i s n o t i n any way d e t e r m i n e d by t h e m e t r i c f i e l d . I f , on t h e o t h e r hand, i t c a n be shown t h a t t h e m e t r i c does 22 n o t v a r y c o m p l e t e l y a r b i t r a r i l y , and t h a t t h e r e i s some p r o p e r t y of t h e m e t r i c o v e r c p w h i c h i s c h a r a c t e r i s t i c o f t h e homology c l a s s o f c p , t h e n i t may be p o s s i b l e t o d e f i n e a p - c h a r g e w h i c h i s d e p e n d e n t p r e c i s e l y on t h a t p r o p e r t y . The q u e s t i o n i m m e d i a t e l y a r i s e s " i f t h e m e t r i c i s not c o n s t r a i n e d t o obey f i e l d e q u a t i o n s , how c a n i t be t h a t c e r t a i n c o n f i g u r a t i o n s of t h e m e t r i c f i e l d a r e ' d i s a l l o w e d ' on p - c y c l e s o f a c e r t a i n homology c l a s s ? " . The answer i s t h a t a l t h o u g h t h e m e t r i c has no f i e l d e q u a t i o n , i t may be c o n s t r a i n e d by c e r t a i n o t h e r r e s t r i c t i o n s , s u c h as t h e r e s t r i c t i o n t h a t i t be c o n t i n u o u s . F i g u r e 2 . 1 : Example of a D i s c o n t i n u o u s M e t r i c F i e l d F i g u r e 2 .1 i s an i l l u s t r a t i o n of a 2 - m a n i f o l d w h i c h i s a 2 - p i a n e t h a t has no s i n g u l a r i t i e s o r removed r e g i o n s . The m e t r i c f i e l d shown on i t i s n o t c o n t i n u o u s b e c a u s e somewhere i n t h e r e g i o n marked "Z" i t must change d i r e c t i o n d i s c o n t i n u o u s l y . We c a n s t a t e w i t h c o n f i d e n c e , t h e r e f o r e , t h a t on a 2 - m a n i f o l d w h i c h has t h e t o p o l o g y of a 2 - p l a n e w i t h no s i n g u l a r i t i e s o r m i s s i n g r e g i o n s , and w i t h a m e t r i c t h a t i s c o n t i n u o u s , t h e r e c a n be no 23 2 - c y c l e w h i c h w i l l have a m e t r i c f i e l d s u c h as t h a t w h i c h c^ i s shown a s h a v i n g i n f i g u r e 2.1. I f , on t h e o t h e r hand, t h e m a n i f o l d i s a 2 - p l a n e w i t h t h e r e g i o n Z removed, a m e t r i c o f t h e t y p e shown i n f i g u r e 2.1 i s c o n t i n u o u s . In t h i s c a s e , t h e p e r i o d of a p - f o r m f_ on any p - c y c l e w h i c h e n c l o s e s Z may be d i f f e r e n t f r o m t h e p e r i o d o f f_ on a p - c y c l e which d o e s n o t e n c l o s e Z. A c c o r d i n g t o t h e d i s c u s s i o n a b ove, t h i s g i v e s some hope t h a t 2 - c h a r g e s may be d e f i n e d on t h i s m a n i f o l d . The above c a s e i s , i n f a c t , a d i r e c t a n a l o g y f o r a 2 - m a n i f o l d o f t h e " f r a m e - d e p e n d e n t c h a r g e l i k e q u a n t i t i e s " w h i c h a r e d i s c u s s e d i n s e c t i o n s 3.3 and 3.4. B e f o r e t h e s e a r e d i s c u s s e d , however, we r e v i e w v e r y q u i c k l y t h e p h y s i c a l arguments w h i c h U n ruh u s e s t o s i m p l i f y t h e p r o c e s s of c o m p a r i n g t h e m e t r i c f i e l d s of two d i s j o i n t p - c y c l e s . C h a p t e r 3 w i l l t h e n l o o k a t p a r t i c u l a r examples o f m e t r i c s w h i c h show some p o t e n t i a l o f h a v i n g t h e p r o p e r t i e s r e q u i r e d f o r t h e e x i s t e n c e of p - c h a r g e s . 2.1 The S t r u c t u r e of t h e S p a c e t i m e and M e t r i c F i e l d A l t h o u g h Unruh's p r o o f was f o r m e t r i c f i e l d s u n c o n s t r a i n e d by f i e l d e q u a t i o n s , t h i s d o e s n o t i m p l y t h a t t h e m e t r i c f i e l d i s c o m p l e t e l y d e v o i d of s t r u c t u r e . Some s t r u c t u r e i s r e q u i r e d b e f o r e any d i s c u s s i o n o f t h e p r o p e r t i e s o f t h e s p a c e t i m e and i t s m e t r i c c a n t a k e p l a c e . U n r u h made t h e f o l l o w i n g a s s u m p t i o n s ( 1 *' a b o u t t h e s t r u c t u r e of t h e m e t r i c and t h e s p a c e t i m e : 1) The s p a c e t i m e i s a C 0 0 f o u r - m a n i f o l d "M". 24 2) T h e r e i s a L o r e n t z - s i g n a t u r e m e t r i c a s s o c i a t e d w i t h e v e r y p o i n t i n M w h i c h i s c o n t i n u o u s o v e r M. 3) The m a n i f o l d p o s s e s s e s an e v e r y w h e r e - c o n t i n u o u s f i e l d o f t e t r a d s ( 1 5 ) . T h i s a s s u r e s t h a t t h e m e t r i c c a n be r e p r e s e n t e d by a L o r e n t z s i g n a t u r e m a t r i x g p v . The e l e m e n t s o f g K V a r e d e f i n e d by: where r ( T ( l i ,T(p ) i s t h e m e t r i c d o t p r o d u c t of two arms of t h e t e t r a d . The L o r e n t z - s i g n a t u r e m e t r i c f i e l d d e f i n e s t h e l i g h t c o n e s a t e a c h p o i n t i n t h e m a n i f o l d . C e r t a i n m e t r i c f i e l d s w i l l be t i m e - o r i e n t a b l e , w h i c h means t h a t one c a n d e f i n e u n a m b i g u o u s l y w h i c h i s t h e f u t u r e l i g h t cone e v e r y w h e r e i n t h e m a n i f o l d . F i g u r e 2.1 i s a s k e t c h of a 2 - m a n i f o l d w i t h a m e t r i c f i e l d w h i c h i s t i m e - o r i e n t a b l e . Note t h a t t h e r i g h t and l e f t e d ges i n t h e d i a g r a m a r e i d e n t i f i e d w i t h e a c h o t h e r a s shown by t h e a r r o w s . A = V l ? Li) >T t\) ) F i g u r e 2.2: Example o f a T i m e - O r i e n t e d M a n i f o l d f i e l d o f " b i a d s " i s s k e t c h e d i n t h e f i g u r e w h i c h p l a y s t h e same r o l e a s a f i e l d o f t e t r a d s would i n a 4 - m a n i f o l d . 25 O t h e r m e t r i c f i e l d s a r e n o t t i m e - o r i e n t a b l e b e c a u s e any d e s i g n a t i o n of t h e f u t u r e n u l l - c o n e must be d i s c o n t i n u o u s . F i g u r e 2.2 i s a s k e t c h of a 3 - m a n i f o l d w i t h a " t r i a d " f i e l d and w i t h a f i e l d of n u l l c o n e s as i n d i c a t e d . N ote t h a t t h e r i g h t and l e f t f a c e s of t h e d i a g r a m a r e i d e n t i f i e d w i t h e a c h o t h e r i n t h e way shown by t h e a r r o w s . Any a t t e m p t t o c o n s i s t e n t l y s p e c i f y F i g u r e 2.3: Example Of A N o n - T i m e - O r i e n t a b l e Mani f o l d w h i c h i s t h e f u t u r e n u l l cone w i t h t h i s m e t r i c w i l l be u n s u c c e s s f u l b e c a u s e o f i t s i n h e r e n t " t w i s t " . U n r u h ' s p a p e r c o n s i d e r s two c a s e s : 1) t h e c a s e where the m e t r i c i s c o n s t r a i n e d t o be t i m e - o r i e n t a b l e and 2) t h e c a s e where t h e m e t r i c i s n o t c o n s t r a i n e d i n t h i s way. 26 2.2 R e p r e s e n t a t i o n of M e t r i c F i e l d s i n The Base Space 'A' L e t G be t h e s p a c e of a l l L o r e n t z - s i g n a t u r e m a t r i c e s and l e t g be some e l e m e n t of G. U n r u h has shown ( 1 6 ) t h a t any m a t r i x g c a n be w r i t t e n as t h e p r o d u c t o f two o t h e r m a t r i c e s <* and t : where ot i s u n i t a r y and L o r e n t z - s i g n a t u r e and t i s p o s i t i v e d e f i n i t e H e r m i t i a n . E v e r y m a t r i x g i s a s s o c i a t e d u n i q u e l y w i t h one m a t r i x c< ( t h i s i s i m p l i e d by U n r u h ' s c l a i m t h a t G has f i b e r - b u n d l e n a t u r e < 1 6 ) ) T h i s i m p l i e s t h a t a t e v e r y p o i n t i n t h e m a n i f o l d M t h e r e i s e x a c t l y one u n i t a r y L o r e n t z - s i g n a t u r e m a t r i x <* w h i c h s a t i s f i e s ( 2 . 1 ) . L e t "A" d e n o t e the s p a c e of a l l u n i t a r y L o r e n t z - s i g n a t u r e m a t r i c e s <* . Unruh { 1 7 ) shows t h a t t h e t o p o l o g y of A w i l l be: a) t h a t of a 3 - s p h e r e i f t h e m a t r i x i s t i m e -o r i e n t a b l e o r b) t h a t of a 3 - s p h e r e w i t h o p p o s i t e p o i n t s i d e n t i f i e d ( i e : t h e p r o j e c t i v e p l a n e P3) i f t h e m e t r i c i s not t i m e - o r i e n t a b l e . We now ask t h e f o l l o w i n g q u e s t i o n : " G i v e n a p - c y c l e c p ( p a r a m e t r i z e d by a s e t o f p a r a m e t e r s A ) i n M, and a s s o c i a t e d w i t h e a c h p o i n t c p ( X ) i n c p a m a t r i x c x . ( c p ( X ) ) / t h e s e t o f a l l t h e <x's w h i c h a r e on c p w i l l f o r m what s o r t o f a p a t t e r n on t h e 3 - s p h e r e 'A'?" The answer i s t h a t t h e s e t o f a l l t h e s e &'s w i l l 27 form a p - c y c l e "E " i n A [ f l . F i g u r e 2.4 i l l u s t r a t e s t h i s c o n c e p t , ( w i t h some of t h e d i m e n s i o n s s u p p r e s s e d , of c o u r s e ) . F i g u r e 2.4: A s s o c i a t i o n o f E a c h P o i n t i n M W i t h a P o i n t i n 'A' E v e r y p o i n t i n C± " c a s t s a shadow" o n t o t h e 3 - s p h e r e A. I f , f o r example, a l l t h e m a t r i c e s " oc" a t e a c h p o i n t i n c, c o r r e s p o n d t o t h e same m a t r i x oc. =^o_,""l°iJ ' T H E N E P be a s i n g l e p o i n t on t h e 3 - s p h e r e A. In f u t u r e s e c t i o n s , we w i l l use t h e n o t a t i o n o c ( c p ) t o r e p r e s e n t t h e p - c y c l e E p w h i c h i s t h e image i n A o f t h e m e t r i c a t e a c h p o i n t on c p . A s p e c i f i c e l e m e n t of E p i s ' w r i t t e n as o i ( c p , X ) and c o r r e s p o n d s t o t h e m a t r i x oc w h i c h i s a s s o c i a t e d w i t h t h e p o i n t c p ( \ ) . We c a n now s i m p l i f y t h e q u e s t i o n of whether t h e r e i s some p r o p e r t y o f t h e m e t r i c f i e l d o v e r a p - c y c l e C p w h i c h i s c h a r a c t e r i s t i c of t h e homology c l a s s of C p . f t ] In what f o l l o w s , a l l p - c y c l e s i n A a r e d e n o t e d by t h e l e t t e r "E" whereas p - c y c l e s i n M a r e d e n o t e d by t h e l e t t e r "C". 28 C o n s i d e r a p - f o r m "f_" ( c o n s t r u c t e d out o f t h e m e t r i c and i t s d e r i v a t i v e s ) , and two d i s t i n c t p - c y c l e s c p and c p w i t h t h e i r images i n A, " « * ( c p ) " and " c c f c * ) " . Unruh shows ( 1 8 ) t h a t i f o t ( c p ) i s h o m o t o p i c t o <x(c p) on t h e m a n i f o l d A, t h e n t h e p e r i o d o f f_ o v e r c{, i s e q u a l t o t h e p e r i o d o f t_ o v e r c p . T h i s i m p l i e s t h a t t h e p e r i o d o f f_ o v e r any p - c y c l e ( i n M) c p w i l l be d e t e r m i n e d c o m p l e t e l y by t h e homotopy c l a s s o f t h e p - c y c l e * ( C p ) . The q u e s t i o n of whether o r not p - c h a r g e s can be d e f i n e d h a s , t h e r e f o r e , been s i m p l i f i e d t o a q u e s t i o n of how many p-d i m e n s i o n a l homotopy c l a s s e s t h e r e a r e i n t h e s p a c e A. I f t h e r e i s o n l y one, t h e n t h e p e r i o d of any p - f o r m f_ w i l l be t h e same on a l l p - c y c l e s , and so no u s e f u l " p - c h a r g e " can be d e f i n e d . I f t h e r e a r e two or more homotopy c l a s s e s o f p - c y c l e s on A, t h e n i t may be p o s s i b l e t o d e f i n e a p - c h a r g e s u c h t h a t e a c h d i f f e r e n t homotopy c l a s s i n A c o r r e s p o n d s t o a d i f f e r e n t c h a r g e . 29 C h a p t e r 3_ E x i s t e n c e of H o m o t o p i e s i n t h e Space "A" o f U n i t a r y , L o r e n t z -S i q n a t u r e M a t r i c e s T h i s c h a p t e r r e v i e w s U n r u h ' s i n v e s t i g a t i o n o f t h e 1 - , 2-and 3 - d i m e n s i o n a l homotopy c l a s s e s o f t h e s p a c e " A " ( a s d e f i n e d i n c h a p t e r 2, i n wh i c h he c o n s i d e r s : 1) how many homotopy c l a s s e s t h e r e a r e o f e a c h d i m e n s i o n , and what a r e t h e i r c h a r a c t e r i s t i c s . 2) whether p - c h a r g e s c a n be d e f i n e d on t h e b a s i s of the homotopy c l a s s of e a c h < * ( c p ) . The i n t e n t i o n of t h i s d i s c u s s i o n i s t o g i v e a c o n c e p t u a l p i c t u r e of t h e s e homotopy c l a s s e s i n t h e s p a c e A. To t h a t end, some t i m e i s s p e n t d i s c u s s i n g h o m o t o p i e s between 1 - c y c l e s , b e c a u s e t h e y b e s t l e n d t h e m s e l v e s t o b e i n g drawn i n d i a g r a m s , and b e c a u s e example h o m o t o p i e s c a n be g i v e n w h i c h a r e r e l a t i v e l y s i m p l e . As w i l l be s e e n , however, t h e most i n t e r e s t i n g s e t of homotopy c l a s s e s ( i n t h i s c o n t e x t ) a r e t h o s e f o r 3 - c y c l e s , w h i c h a r e d i s c u s s e d i n s e c t i o n s 3.3 and 3.4. The t o p o l o g y of t h e 3 - s p h e r e A i s s u c h t h a t any p - c y c l e s on A w h i c h a r e homologous a r e a l s o h o m o t o p i c . T h i s f o l l o w s f r o m t h e H u r e w i c z I s o m o r p h i s m Theorem, f o r w h i c h t h e r e a d e r i s r e f e r r e d t o Croom ( 1 9 ) f o r example. T h i s w i l l a i d t h e d i s c u s s i o n i n t h e f o l l o w i n g s e c t i o n s , b e c a u s e we w i l l have t h e c h o i c e o f f i n d i n g 30 e i t h e r t h e number o f homotopy c l a s s e s o r , e q u i v a l e n t l y t h e number of homology c l a s s e s , on A. 3.1 E x i s t e n c e o f H o m o t o p i e s i n 'A' f o r 1 - c h a r g e s I f we assume t h e m e t r i c i s t i m e - o r i e n t a b l e , t h e n t h e t o p o l o g y o f A i s e q u i v a l e n t t o t h a t o f a 3 - s p h e r e . The image " o< ( c , ) " of a 1 - c y c l e c, must i t s e l f be a 1 - c y c l e i n A. A l l 1-c y c l e s on a 3 - s p h e r e a r e homologous t o z e r o , so a l l 1 - c y c l e s a r e h o m o t o p i c t o e a c h o t h e r and no 1 - c h a r g e s can be d e f i n e d . I f t h e m e t r i c i s not t i m e - o r i e n t a b l e , t h e t o p o l o g y of A i s t h a t o f t h e p r o j e c t i v e p l a n e P 3 , and e x a c t l y two homology c l a s s e s f o r <x(C,) e x i s t . " C l a s s 0" i s homologous t o z e r o , and f o r t h e s e t h e r e a l w a y s e x i s t s an homotopy t o r e d u c e them t o a p o i n t . " C l a s s 1" c o n s i s t s o f 1 - c h a i n s c o n n e c t i n g p o i n t s t o t h e i r a n t i p o d a l p o i n t s . C o n s i d e r t h e f o l l o w i n g example of a c l a s s - 0 m e t r i c on C^ (a c l o s e d 1 - c y c l e ) w i t h p a r a m e t r i z a t i o n G such t h a t B+ 2 t t = 9 [ t ] : cos 19 0 o Qje) = s i * 16 0 0 (3.1 ) o o -I o 31 Note t h a t G e ( C j ) s a t i s f i e s t h e r e q u i r e m e n t s t o be an e l e m e n t of A, i n t h a t i t i s u n i t a r y , s y m m e t r i c , and has L o r e n t z s i g n a t u r e f o r a l l v a l u e s of 9 . To d e m o n s t r a t e t h e L o r e n t z - s i g n a t u r e we must show t h a t f o r a l l Q t h e r e e x i s t s a c o o r d i n a t e 6 t r a n s f o r m a t i o n w h i c h makes t h e m e t r i c d i a g o n a l and s i g n a t u r e -two. The l i n e - e l e m e n t c a l c u l a t e d u s i n g (3.1) i s : ds*« cos ze at1 +z*\* zo<m*-copied*} -df-d-i <3-2) We l o o k f o r some c o o r d i n a t e t r a n s f o r m a t i o n i n e a c h domain o f w h i c h makes t h i s l i n e e l e m e n t e x p l i c i t l y L o r e n t z - s i g n a t u r e . In t h e r e g i o n s where cos(2S) i s p o s i t i v e , d e f i n e t h e c o o r d i n a t e t r a n s f o r m a t i o n (3.3) c o s 2 e ' & X and t h e l i n e - e l e m e n t becomes: (3.4) [ f ] The m e t r i c G shown has been c h o s e n t o be an e l e m e n t of A a t a l l p o i n t s . T h i s i s n o t n e c e s s a r y , b e c a u s e a c c o r d i n g t o t h e d i s c u s s i o n i s c h a p t e r 1, any m e t r i c G ( c P ) c a n be u n i q u e l y r e p r e s e n t e d i n A by some « * ( c p ) . C h o o s i n g G t o be on A w i l l s i m p l i f y t h e d i s c u s s i o n b e c a u s e we c a n use t h e word " m e t r i c " where we would o t h e r w i s e have to. use t h e words "element of A c o r r e s p o n d i n g t o t h e m e t r i c " . F u r t h e r m o r e , t h e r e e x i s t s a t r i v i a l homotopy w h i c h maps any g t o i t s o c ( * ' . 32 w h i c h i s s i g n a t u r e - t w o as r e q u i r e d . In t h e r e g i o n where c o s ( 2 0 ) i s n e g a t i v e , d e f i n e : d x 2 t , 1 . . . a t (3.5) dt~ ± (/-cosZe'dx + p ^ = r at) and t h e l i n e e l e m e n t becomes: d s ^ d ^ - J ^ - c K f - d * * (3.6) w h i c h i s as r e q u i r e d . When c o s ( 2 © ) i s z e r o , t and x a r e n u l l c o o r d i n a t e s , so i f we l e t x = ( t - x ) and t s ( t + x) t h e n th e l i n e -e l e m e n t i s s i g n a t u r e - t w o as r e q u i r e d . F i g u r e 3.1: The 1 - c y c l e G e ( C i ) on t h e s p a c e 'A' (one d i m e n s i o n s u p p r e s s e d ) We a r e a s s u r e d t h a t G E i s c l a s s 0 by t h e f a c t t h a t G 6 ( TT ) and G„(0) a r e a n t i p o d a l p o i n t s , w h i c h i m p l i e s t h a t G 0(2TT ) i s 33 t h e same p o i n t on A a s G „ ( 0 ) . T h i s i s c o n f i r m e d by t h e e i g e n -v e c t o r of G e ( w / 2 ) b e i n g o r t h o g o n a l t o b o t h G o ( 0 ) and G 0 ( TT ). See f i g u r e 3.1 f o r a s k e t c h o f G 0 ( C i ) . As e x p e c t e d , t h e 1 - c y c l e G 0 ( C 1 ) can be c o n t i n u o u s l y d e f o r m e d w i t h i n A i n t o a p o i n t . The d o t t e d l i n e s i n f i g u r e 3.1 i n d i c a t e p r o g r e s s i v e s t a g e s i n t h i s d e f o r m a t i o n . To d e m o n s t r a t e t h i s , d e f i n e t h e h o m o t o p y t t ] : K (©,<%„) = l " jS iK20( l4cos*/i ) c o s © s'«* <P/Z 0 Sin 9 S m i y j 0 - ce* 0 o -1 (3.7) J s o t h a t (3.8) -I -I w (3.9) A c a r e f u l l o o k a t K( 9 ,^/2-ir) con f i r m s t h a t i t i s u n i t a r y , L o r e n t z - s i g n a t u r e and sym m e t r i c f o r a l l 8 and cf . [ t l Note t h a t t h e example homotopy g i v e n by Unruh ( 2 0 > i s not c o r r e c t b e c a u s e i t doe s not e q u a l G 0 a t <f = 0 nor i s i t L o r e n t z s i g n a t u r e a t a l l p o i n t s . 34 One s i g n i f i c a n t i m p l i c a t i o n of our a b i l i t y t o c o n t r a c t G(C^) t o a p o i n t i s t h a t we.can c o n t i n u o u s l y d e f o r m m e t r i c s w i t h d i f f e r e n t p e r i o d i c i t y on i n t o e a c h o t h e r . F i g u r e 3.6 s k e t c h e s s u c h a t r a n s f o r m a t i o n . The s k e t c h shows how c o n t r a c t i b i l i t y i s F i g u r e 3.2: D e f o r m a t i o n of a p e r i o d - 1 m e t r i c t o a p e r i o d - 2 m e t r i c n e c e s s a r y f o r s u c h a d e f o r m a t i o n . An example of a c l a s s - 1 m e t r i c i s : cos G S\n 9 o o Sin & -COS© 0 0 0 0 -I 0 0 0 o -1 (3.10) As 0 goes from 0 t o 2tt, G , ( 0 ) r e t u r n s t o i t s i n i t i a l v a l u e as r e q u i r e d , b u t G, (2TT) i s now t h e a n t i p o d a l p o i n t t o G , ( 0 ) . Note t h a t , a s e x p e c t e d , t h e e i g e n v e c t o r of G, (TT) i s o r t h o g o n a l t o t h e 35 e i g e n v e c t o r of G t ( 0 ) and G , ( 2 r r ) . Any homotopy o f G, (IT) must l e a v e t h e e n d p o i n t s o f G,(C,) on a n t i p o d a l p o i n t s o r t h e r e w i l l be a d i s c o n t i n u i t y i n t h e m e t r i c a t 0 = 2TT. I t has been e s t a b l i s h e d , t h e r e f o r e , t h a t f o r n o n - t i m e -o r i e n t e d m e t r i c s , t h e r e c a n be two homotopy c l a s s e s of 1 - c h a i n s i n A. From t h e d i s c u s s i o n i n c h a p t e r 2, t h i s c r e a t e s some hope t h a t we c o u l d have a s y s t e m w i t h two 1-charge s t a t e s , w i t h t h e c h a r g e d e t e r m i n e d by whether t h e m e t r i c i s c l a s s 1 or c l a s s 0. Imagine s u c h a s y s t e m , and l e t c l a s s - 0 m e t r i c s have c h a r g e r and c l a s s - 1 m e t r i c s have c h a r g e s. C h a r g e r would have t o be z e r o b e c a u s e i t s p a t h i n A i s h o m o l o p i c a l l y e q u a l t o z e r o . I n t e g r a t i n g any c l a s s - 1 m e t r i c t w i c e a r o u n d t h e 1 - c y c l e f o l l o w s t h e same p a t h i n A as i n t e g r a t i n g some c l a s s - 0 m e t r i c once a r o u n d , and t h e r e f o r e must g i v e z e r o . Our c o n c e p t o f 1 - c h a r g e s s p e c i f i e s t h a t t h e y be l i n e a r f u n c t i o n a l s o v e r t h e f i r s t o r d e r homology g r o u p of t h e s p a c e ( s e e s e c t i o n 1.3), t h a t i s , t h a t 2s=0 o n l y i f s=0. I t f o l l o w s t h a t c h a r g e s must a l s o be z e r o . From t h i s we c o n c l u d e , as does U nruh < 2 0 ) , t h a t no u s e f u l 1-c h a r g e s c a n be d e f i n e d u s i n g t h i s scheme. 3.2 E x i s t e n c e of H o m o t o p i e s i n 'A' f o r 2 - c h a r g e s I f we assume t h e m e t r i c i s t i m e - o r i e n t e d , so t h e s p a c e A has t h e t o p o l o g y o f a 3 - s p h e r e , no 2 - c h a r g e s c a n be d e f i n e d . To show t h i s we n o t e t h a t f o r e v e r y 2 - c y c l e CL t h e r e must e x i s t some p o i n t i n A w h i c h i s e x t e r i o r t o ,<*(C 2). Remove t h i s p o i n t from t h e m a n i f o l d A, and A becomes e q u i v a l e n t t o a t h r e e 36 dimensional plane, which i s c o n t r a c t i b l e . There always exi s t s , therefore, some homotopy K(C1,-V) which maps any a^(Cz) into any other, so a l l metrics give the same 2-charge. Assume now that the metric i s non-time-orientable ( i e : antipodal points are i d e n t i f i e d on A). There are two 2-dimensional homology classes in A: those which are homologous to zero and those which run from point to antipoint. An example element of each of these homology classes i s sketched i n , respectively, figure 3.2(a) and figure 3.2(b), where one dimension of "A" has been suppressed. Note that some of the points in ot(c^) are antipodal, but none of the points in ot(c°) are. Figure 3.3: The Two 2-Dimensional Homology Classes in 'A' By arguments exactly analogous to those presented in section 3.1, however, i t follows that a l l charges defined on the basis of these homology classes must be zero. We conclude, as does Unruh ( 2 1 ) , that no 2-charges can be constructed out of 37 s u c h u n r e s t r i c t e d m e t r i c s . 3.3 E x i s t e n c e o f H o m o t o p i e s i n 'A' f o r 3 - c h a r q e s When l o o k i n g f o r 3 - c h a r g e s , we a r e l o o k i n g a t 3 - c y c l e s i n M w i t h t h e i r images i n t h e 3 - s p h e r e "A", and t h i n g s become more i n t e r e s t i n g . We f o l l o w Unruh i n a s s u m i n g t h e m e t r i c i s t i m e -o r i e n t a b l e . We c a n c l a s s i f y a l l <*(C 3) i n t o homotopy c l a s s e s (and t h e r e a r e an i n f i n i t e number o f them) and d e a l o n l y w i t h r e p r e s e n t a t i v e s of e a c h c l a s s . The s i m p l e s t s u c h c l a s s i s %i0, t h e c l a s s of a l l ociC^) homologous t o z e r o i n A. Any m e t r i c f i e l d w h i c h does not c o v e r e v e r y p o i n t i n A b e l o n g s t o t h i s c l a s s b e c a u s e i f one p o i n t can be removed from A , t h e s p a c e i s c o n t r a c t i b l e i n w h i c h c a s e a l l 3 - c h a i n s a r e homologous t o a p o i n t . I f we r e s t r i c t our view t o o n l y m e t r i c s i n JbD , t h e n t h e r e a l w a y s e x i s t s an homotopy K(C 3,|4) and no 3 - c h a r g e s c a n be d e f i n e d i n t e r m s of t h e m e t r i c . C e r t a i n maps o<(C 3) c o m p l e t e l y c o v e r t h e 3 - s p h e r e A i n s u c h a way t h a t t h e s p a c e i s n o t c o n t r a c t i b l e and h o m o t o p i e s from one p e r i o d i c i t y t o a n o t h e r a r e no l o n g e r p o s s i b l e ( f i g u r e 3.2 s k e t c h e s t h e r o l e t h a t c o n t r a c t i b i l i t y p l a y s i n s u c h h o m o t o p i e s ) . F o r example, i f a 3 - s p h e r e C*3 ( i n M) i s p a r a m e t r i s e d by (<P,9,y) and A by (^ ',e',f') , (where cf +2n= (f and c/)'+2Tr=^) t h e n {<*(</>,*,Y) = h{<P,B,f)}, { *(^,6> rf) = A(2</>,e,<f)}... { *(tf,e f«f) = A(n(/>,G>,y), (n an i n t e g e r ) } a l l r e p r e s e n t i n e q u i v a l e n t mappings from C 3 i n t o A. 38 We can d e f i n e 3 - c h a r g e s i n t e r m s o f t h i s p e r i o d i c i t y o f t h e m e t r i c , but p e r i o d i c i t y i s d e t e r m i n e d not by t h e p r o p e r t i e s of t h e s p a c e t i m e b ut by t h e n a t u r e o f t h e m a t r i x r e p r e s e n t a t i o n o f th e L o r e n t z - s i g n a t u r e m e t r i c s . In p r a c t i c a l t e r m s , t h i s means t h a t i t d e p e n d s .on o u r c h o i c e o f a t e t r a d f i e l d o v e r M. C l e a r l y 3 - c h a r g e s d e f i n e d i n t h i s way a r e not t h e geom e t r o d y n a m i c a n a l o g t o c h a r g e w h i c h we s e e k , but t h e y a r e of some i n t e r e s t and w i l l be b r i e f l y d i s c u s s e d below. As an example of s u c h " c h a r g e s " , we l o o k a) a t t h e c l o s e d s p h e r i c a l F r i e d m a n n u n i v e r s e ( i e : t h e R o b e r t s o n - W a l k e r m e t r i c ) and b) a t f l a t M i n k o w s k i s p a c e w i t h one p o i n t removed. B o t h have t h e same t o p o l o g y - t h a t of t h e 3 - s p h e r e c r o s s t h e r e a l l i n e ( t h e u n i t y of a l l 3 - s p h e r e s o f n o n - z e r o r a d i u s ) . On b o t h s p a c e s we w i l l c h o o s e as our 3 - c y c l e C 3 t h e 3 - s p h e r e of r a d i u s R c e n t e r e d a t t h e p o i n t removed f r o m t h e m a n i f o l d . E a c h has d e f i n e d a t a l l p o i n t s i t s own L o r e n t z - s i g n a t u r e m e t r i c , of w h i c h t h e d i r e c t i o n of t h e l i g h t c one i s one c h a r a c t e r i s t i c . F i g u r e 3.4, a) and b) s k e t c h e s ( w i t h two d i m e n s i o n s s u p p r e s s e d ) t h e geometry so f a r d e f i n e d on t h e s y s t e m . We now make t h e c h o i c e o f a t e t r a d s y s t e m f o r t h e m a n i f o l d . One c h o i c e i s t o a l i g n e a c h arm o f t h e t e t r a d i n t h e f o u r C a r t e s i a n - c o o r d i n a t e d i r e c t i o n s , a s s k e t c h e d i n f i g u r e 3.5, (a) and ( b ) . I t becomes c l e a r t h a t a s one i n t e g r a t e s a r o u n d C3 i n th e R o b e r t s o n - W a l k e r m e t r i c , t h e r e p r e s e n t a t i o n o f t h e m e t r i c w i l l i n v e r t i t s e l f , t h e n r e t u r n t o t h e o r i g i n a l o r i e n t a t i o n , 39 F i g u r e 3.4: T o p o l o g i c a l s t r u c t u r e of "the F r i e d m a n n and M i n k o w s k i m a n i f o l d s F i g u r e 3.5: C a r t e s i a n t e t r a d f i e l d a p p l i e d t o t h e m a n i f o l d s of f i g u r e 3.4 making c h a r g e a s s o c i a t e d w i t h C^ e q u a l t o u n i t y . The r e p r e s e n t a t i o n of t h e M i n k o w s k i m e t r i c w i l l s t a y c o n s t a n t o v e r t h e i n t e g r a t i o n , and so w i l l have a c h a r g e o f z e r o . A n o t h e r c h o i c e of t e t r a d f i e l d i s s k e t c h e d i n f i g u r e 3.6. The R o b e r t s o n - W a l k e r m e t r i c now y i e l d s a c h a r g e o f z e r o , w h i l e t h e M i n k o w s k i m e t r i c y i e l d s a c h a r g e o f one. T h i s d e m o n s t r a t e s how " c h a r g e s " d e f i n e d i n t h i s way a r e n o t o n l y d e p e n d e n t on t h e t o p o l o g y of t h e s p a c e , b u t a l s o on t h e c h o i c e of r e p r e s e n t a t i o n 40 of t h e m e t r i c . 41 C h a p t e r 4 R e s t r i c t i n g t h e M e t r i c t o Obey A F i e l d E q u a t i o n Once i t has been e s t a b l i s h e d t h a t no c h a r g e s c a n be c o n s t r u c t e d out of a t o t a l l y u n r e s t r i c t e d m e t r i c , i t i s l o g i c a l t o t r y t o c o n s t r u c t them o u t o f a m e t r i c w h i c h i s c o n s t r a i n e d t o obey some f i e l d e q u a t i o n . The c h o i c e o f s u c h a f i e l d e q u a t i o n i s r e l a t i v e l y e a s y - we use E i n s t e i n ' s : where T ^ i s c o n s t r a i n e d t o c o r r e s p o n d t o some r e a s o n a b l e p h y s i c a l d i s t r i b u t i o n of e n e r g y and m a t t e r ( p e r h a p s i t i s z e r o , a s i n empty s p a c e - t i m e , o r p e r h a p s i t i s c o n s t r a i n e d t o have a p o s i t i v e m a s s - d e n s i t y t e r m , e t c . ) The s u p e r i o r i t y of t h e s e e q u a t i o n s o v e r t h e i r " c o m p e t i t i o n " i s w i d e l y a c c e p t e d ( c f . M i s n e r T h o r n e and W h e e l e r , c h a p t e r 17.6) b o t h i n m a t h e m a t i c a l e l e g a n c e and i n t h e i r c o r r e s p o n d e n c e t o t h e p h y s i c a l o b s e r v a b l e s . We c h o o s e f o r t h i s work t o r e s t r i c t T^v t o be z e r o . T h a t i s , we assume a m e t r i c f i e l d w h i c h o b e y s : * V " l 3 ^ v R ~ 0 (4.1) w h i c h a r e t h e f i e l d e q u a t i o n s f o r empty s p a c e t i m e . 42 T h i s a s s u m p t i o n seems r e a s o n a b l e b e c a u s e by f a r t h e l a r g e s t volume o f t h e u n i v e r s e i s a t l e a s t empty o f " h a r d " m a t t e r , and t h o s e p a r t s w h i c h a r e n o t empty ( e g : t h e i n t e r i o r o f n u c l e i ) a r e not w e l l o b s e r v e d . I f we a r e t o d e f i n e a 2 - c h a r g e , f o r example, as b e l o n g i n g t o a s p e c i f i c p o i n t i n o u r 3-space we w i l l c e r t a i n l y a l w a y s be a b l e t o f i n d a 2 - s u r f a c e w h i c h i s e v e r y w h e r e embedded i n m a t t e r - f r e e s p a c e and w h i c h e n c l o s e s t h a t p o i n t . Any u s e f u l d e f i n i t i o n of 2 - c h a r g e , i t would seem, would need t o be n o n - z e r o on s u c h a 2 - s u r f a c e a t l e a s t . One p l a u s i b l e a l t e r n a t i v e i s t o d e f i n e p - c h a r g e i n t e r m s of t h a t p a r t o f c u r v a t u r e w h i c h r e s u l t s from e n e r g y r a d i a t i n g from a c e n t r a l body. The body would t h u s have p - c h a r g e s p e c i f i c a l l y b e c a u s e o f t h e n o n - e m p t i n e s s of t h e s p a c e t i m e s u r r o u n d i n g i t . Such a c o n c e p t i s not v e r y s a t i s f a c t o r y , however, b e c a u s e our e x p e r i e n c e t e l l s us t h a t t h e r a d i a n t e n e r g y a r o u n d o b j e c t s i s a t r a n s i e n t phenomenon, t o o e a s i l y a f f e c t e d by c h a n g e s c o m p l e t e l y i n t e r n a l t o t h e o b j e c t . F o r t h e r e m a i n d e r o f t h i s t h e s i s , t h e r e f o r e , t h e m e t r i c i s c o n s t r a i n e d t o obey E i n s t e i n ' s e q u a t i o n s f o r empty s p a c e t i m e . 43 C h a p t e r 5 S p i n - T e n s o r R e p r e s e n t a t i o n of T e n s o r s i n F l a t S pace-Time I t w i l l be shown i n t h e n e x t t h r e e c h a p t e r s how a m e t r i c c o n s t r a i n e d t o obey E i n s t e i n ' s f i e l d e q u a t i o n s c a n be more c o m p a c t l y r e p r e s e n t e d by s p i n - t e n s o r s t h a n by t e n s o r s . T h i s c h a p t e r r e v i e w s t h e f o r m a l i s m of s p i n - t e n s o r s , t h e i r t r a n s f o r m a t i o n p r o p e r t i e s , and t h e i r s y m m e t r i e s , i n a s p a c e - t i m e w i t h t h e M i n k o w s k i m e t r i c (^f*v ) • C h a p t e r s 6 and 7 d e s c r i b e how s p i n o r n o t a t i o n becomes p r o g r e s s i v e l y more e c o n o m i c a l t h a n t e n s o r n o t a t i o n as t h e m e t r i c i s c o n s t r a i n e d t o obey f i r s t E i n s t e i n ' s e q u a t i o n s and t h e n E i n s t e i n ' s e q u a t i o n s f o r empty s p a c e . T h i s economy o f n o t a t i o n g r e a t l y s i m p l i f i e s t h e computer a l g o r i t h m r e q u i r e d f o r t h e c a l c u l a t i o n s and p r e s e n t e d i n c h a p t e r e i g h t . B e c a u s e t h e s p i n - t e n s o r s a r e t o be u s e d t o r e p r e s e n t t h e g r a v i t a t i o n a l f i e l d ( w h i c h i s t r a d i t i o n a l l y r e p r e s e n t e d by t e n s o r s ) , p a r t i c u l a r e m p h a s i s has been p l a c e d on d i s c u s s i n g t h e i s o m o r p h i s m between t e n s o r s and s p i n - t e n s o r s . The a l g e b r a of s p i n - t e n s o r s i s by c o n s t r u c t i o n v e r y s i m i l a r t o t h e a l g e b r a o f t e n s o r s , w i t h s u c h f e a t u r e s as i m p l i e d summation, r a i s e d and l o w e r e d i n d i c e s , and t h e l i k e . F o r t h i s r e a s o n , some e l e m e n t s of s p i n - t e n s o r n o t a t i o n w h i c h a r e o b v i o u s a n a l o g i e s of t e n s o r n o t a t i o n a r e n o t e x p l a i n e d i n d e t a i l . F o r a more t h o r o u g h 44 d i s c u s s i o n of s p i n - t e n s o r s , t h e r e a d e r i s r e f e r r e d t o t h e r e f e r e n c e s < 2 2 > . 5.1 Example of a Second-Rank S p i n - T e n s o r As an i n t r o d u c t i o n , c o n s i d e r a v e c t o r ( 1 - t e n s o r ) V*4: (5.1 ) T h i s c a n be r e p r e s e n t e d as a " rank-2 s p i n - t e n s o r " whose m a t r i x r e p r e s e n t a t i o n ("||HAJJ") i s : H A8 v° + v 3 v' + iv 7 V1 + Lv7- v t t-v 3 (5.2) Note t h a t t h e m a t r i x (5.2) c a r r i e s t h e same i n f o r m a t i o n as t h e v e c t o r r e p r e s e n t a t i o n (5.1) and i s t h e g e n e r a l d e f i n i t i o n of t h e r a n k - 2 s p i n - t e n s o r c o r r e s p o n d i n g t o a v e c t o r V1*. The i n v e r s e o f e q u a t i o n (5.2) i s : V = ^ ( H , , + H w , H 1 1 + H l „ L ( H l l - H l l \ H l l - H „ ) (5.3) E q u a t i o n s (5.2) and (5.3) t o g e t h e r d e f i n e t h e i s o m o r p h i s m between 1 - t e n s o r s and r a n k - 2 s p i n - t e n s o r s . Note t h a t : 1) ||H^|| i s H e r m i t i a n (||HA6|| = ||H^ ||+ ) 2) The s i n g l e i n d e x of V*4 i s r e p r e s e n t e d by one d o t t e d p l u s one u n d o t t e d i n d e x i n ||HAB| • T h e s e i n d i c e s w i l l be d i s c u s s e d b elow. 45 3) C a p i t a l Roman l e t t e r s a r e us e d t o d e n o t e t h e i n d i c e s of ||Ha^  j| . 4) The i n d i c e s of |Ha^ || run from 1 t o 2 o n l y . 5) The d e t e r m i n a n t of ||HA^|J i s t h e i n v a r i a n t i n t e r v a l : | H r i | ^ ( V l - V , l - V l , - V ' 1 ) (5.4) T h e s e c h a r a c t e r i s t i c s , o r e x t e n s i o n s of them, w i l l be common t o s p i n - t e n s o r s of a l l o r d e r s . I t i s common n o t a t i o n a l p r a c t i c e n o t t o d i s t i n g u i s h between a s p i n - t e n s o r (||Ha^ ||) and i t s ( A , B ) t h component, (H Ag), d e n o t i n g them b o t h a s "H^ " . ( t h i s i s e n t i r e l y a n a l o g o u s t o t h e s i m i l a r n o t a t i o n a l p r a c t i c e w i t h t e n s o r s ) F o r t h e sake o f b r e v i t y , t h i s w i ' l l be t h e c a s e i n most o f what f o l l o w s , a l t h o u g h when c o n f u s i o n i s p o s s i b l e t h e above n o t a t i o n w i l l be u s e d . T r a n s l a t i o n from t e n s o r t o s p i n - t e n s o r s p a c e c an be w r i t t e n more c o n c i s e l y i n terms of t h e P a u l i s p i n m a t r i c e s : (5.5) E q u a t i o n (5,2) can now be w r i t t e n : (5.6) 46 where summation i s i m p l i e d o v e r t h e f o u r v a l u e s o f p . S i m i l a r l y , t h e i n v e r s e t r a n s f o r m a t i o n c an be w r i t t e n i n terms o f : w o - i \ (5.7) so t h a t i t has t h e form: V ' = > r H f t 6 (5.8) where summation i s i m p l i e d o v e r t h e two v a l u e s of e a c h o f A and 5.2 L o r e n t z T r a n s f o r m a t i o n s of S p i n o r s S p i n - t e n s o r s were made t h e shape t h a t t h e y a r e (5.2) so t h a t L o r e n t z t r a n s f o r m a t i o n s i n s p i n - t e n s o r s p a c e c o u l d be a c h i e v e d by u n i t a r y t r a n s f o r m a t i o n s on H e r m i t i a n m a t r i c e s . The L o r e n t z t r a n s f o r m a t i o n s "S" a r e t h o s e which l e a v e t h e i n v a r i a n t i n t e r v a l (5.4) unchanged: * B and o v e r t h e f o u r v a l u e s of (5.9) T h i s i m p l i e s t h a t : (5.10) 47 w h i c h means t h a t any p o s i t i v e l y - o r n e g a t i v e l y - u n i m o d u l a r 2X2 complex m a t r i x c o r r e s p o n d s t o a L o r e n t z t r a n s f o r m a t i o n . Note i n (5.9) t h a t t h e t r a n s f o r m a t i o n i n v o l v e s (S) once and (S^ = S * T ) o n c e . T h i s e x p l a i n s why one i n d e x o f H A | i s d o t t e d ; t h e d o t i n d i c a t e s t h a t t h e i n d e x t r a n s f o r m s w i t h S* r a t h e r t h a n w i t h S. S p i n t e n s o r s w i l l l a t e r be i n t r o d u c e d w i t h u n e q u a l o numbers o f d o t t e d and u n d o t t e d i n d i c e s ( e9 :^Afcco£ ^ i n d i c a t i n g t h a t t h e s p i n - t e n s o r s have d i f f e r e n t t r a n s f o r m a t i o n p r o p e r t i e s t h a n t h e one i n t h e above example. As an example of a L o r e n t z t r a n s f o r m a t i o n , c o n s i d e r i n d e t a i l t h e S - m a t r i x c o r r e s p o n d i n g t o r o t a t i o n s a b o u t t h e x - a x i s . T h i s can be w r i t t e n i n t e r m s of t h e 2X2 m a t r i x : O x S 6 = A cos ^£ 2 2 - U r n *bs C o S 2 Z (5.11) and i t s complex c o n j u g a t e : = ( « * A 6 ) * { 5 ' 1 2 ) (where © x i s t h e a n g l e o f r o t a t i o n a b o u t t h e x - a x i s ) . To change from one frame o f r e f e r e n c e , H f l S, t o a r o t a t e d frame of r e f e r e n c e H^g, one p e r f o r m s th e m a t r i x o p e r a t i o n : (5.13) 48 W i t h summation n o t a t i o n t h i s i s w r i t t e n : H ~ » = Q ~ A a . ® H ' (5.14) S t r a i g h t f o r w a r d c a l c u l a t i o n s w i t h a x ^ f t c o n f i r m t h a t t h e d e t e r m i n a n t of t h e r o t a t e d s p i n - t e n s o r i s unchanged and t h a t H Ag c o r r e s p o n d s t o t h e r o t a t e d v e c t o r a s r e q u i r e d . The r o t a t i o n m a t r i x a x f l f t c a n ^ e w r i t t e n more c o n c i s e l y , a l t h o u g h r a t h e r o b s c u r e l y , i n terms of ||*x|| ( f r o m e q u a t i o n 5.5) a s [ f ] : = « P ( - i I * * . ! | | r ) (5.15) U s i n g t h e same a b b r e v i a t e d n o t a t i o n , t h e t h r e e s p a t i a l r o t a t i o n s c an be w r i t t e n : a, (5.16) The t h r e e L o r e n t z b o o s t s i n t h e same way c a n be w r i t t e n a s : V I I = ~ P(- k j £ ) k = l , Z , 3 (5.17) where t h e v e l o c i t y p a r a m e t e r Y k i s d e f i n e d i n terms of t h e ( k ) t h component of t h e v e l o c i t y a s : [ f ] In f a c t , o xg A i s a r b i t r a r y up t o a f a c t o r ± e l * . T h i s a r b i t r a r i n e s s i s i m p o r t a n t i n o t h e r u s e s o f s p i n - t e n s o r s , but does, n o t e n t e r i n t o t h e g r a v i t a t i o n a l t h e o r y d i s c u s s e d h e r e . 49 \ 3 . t a A " 1 (^-) (5.18) where c i s t h e speed of l i g h t . 5.3 E x t e n s i o n t o H i g h e r O r d e r S p i n - T e n s o r s The e x t e n s i o n o f s p i n - t e n s o r f o r m a l i s m t o r e p r e s e n t s p i n -t e n s o r s of h i g h e r o r d e r i s v e r y d i r e c t . A r a n k - 2 t e n s o r T ^ i s r e p r e s e n t e d by a rank-4 s p i n - t e n s o r w i t h two d o t t e d and two u n d o t t e d i n d i c e s , a c c o r d i n g t o : W o = <r r A i ffvBoT'* < 5 - l 9 > A r a n k - n t e n s o r c o r r e s p o n d s t o a r a n k - 2 n s p i n - t e n s o r w i t h n d o t t e d i n d i c e s and n u n d o t t e d i n d i c e s . O d d - r a n k e d s p i n - t e n s o r s and o t h e r s p i n - t e n s o r s w i t h uneven mixes of d o t t e d and u n d o t t e d i n d i c e s can a l s o have p h y s i c a l meaning, examples of w h i c h a r e t h e g r a v i t a t i o n a l s p i n - t e n s o r s w h i c h w i l l be p r e s e n t e d i n c h a p t e r s 6 and 7. T h e s e c a n n o t be t r a n s l a t e d d i r e c t l y i n t o v e c t o r s or t e n s o r s u n l e s s i n c o m b i n a t i o n w i t h o t h e r s p i n - t e n s o r s . 50 5.4 The S p i n - T e n s o r " M e t r i c " To c o n t i n u e c o n s t r u c t i n g s p i n - t e n s o r s p a c e i n a manner a n a l o g o u s t o t e n s o r s p a c e , " m e t r i c " s p i n - t e n s o r s a r e r e q u i r e d . D e f i n e t h e s p i n - t e n s o r s £ f t S , £ • A B £ftB ' a n d ^AB s u c h t h a t t r a n s f o r m a t i o n s f r o m 6^ . t o 6~u P A G r w r i t t e n a s : AB ( e q u a t i o n s 5.5 and 5.7) can be , A B - K CAC rB° (5.20) w i t h i m p l i e d summation on t h e s p i n o r i n d i c e s . The r e q u i r e d m a t r i c e s a r e : AB AB AB - ('° o) -(-1 i ) (5.21 ) Note the f o l l o w i n g c h a r a c t e r i s t i c s of t h e £ s p i n - t e n s o r s : 1 ) £ * 6 = - £ B A 2) Summation i s a l w a y s from t h e s e c o n d i n d e x o f €. A B and £ A . ,but a l w a y s from t h e f i r s t i n d e x o f £ A B and £ A 8 • C o n s i d e r t h e s e e x a m p l e s : £ 8 C H Afc = " £ C 8 H A B = " H A C 51 3) C o n t r a c t i n g two £ s p i n - t e n s o r s g i v e s ; £ A 6 £ f t c = ^ c (5.22) So t h a t : C A B £ f t B = 2 (5.23) 4) T h e r e a r e no £ s p i n - t e n s o r s w i t h m i x e d i n d i c e s , so t h a t d o t t e d and u n d o t t e d i n d i c e s c a n n e v e r be c o n t r a c t e d w i t h e a c h o t h e r . In f l a t s p a c e - t i m e , t h e f o l l o w i n g i d e n t i t i e s a r e i m p l i e d by t h e d e f i n i t i o n s o f t h e ( a ) and ( £ ) m a t r i c e s and i l l u s t r a t e t h e p a r a l l e l s between £ and t h e M i n k o w s k i m e t r i c (5.24) VVv 8 + * » V r = V > £ f t 8 ( 5 - 2 6 ) The ( £ ) s p i n - t e n s o r s , l i k e t h e m e t r i c i n t e n s o r s p a c e , a r e u s e d t o c a l c u l a t e l e n g t h s , i n n e r p r o d u c t s , and c o n t r a c t i o n s of s p i n - t e n s o r s . The f o l l o w i n g e x a m ples i l l u s t r a t e e a c h of t h e s e , u s i n g and G A B £ ^ as p r o t o t y p e s p i n - t e n s o r s c o r r e s p o n d i n g t o t h e t e n s o r s and T ^ r e s p e c t i v e l y . 52 1 ) L e n g t h : 2) C o n t r a c t i o n : H , * G \ \ <=>V P T> V 3) I n n e r p r o d u c t : H c 6 G R C f e 6 = V r V v T - ' 5.5 The Complex C o n j u g a t e of a S p i n - T e n s o r C o n s i d e r a t e n s o r T^v an<3 i t s s p i n o r r e p r e s e n t a t i v e T A 6££ : T — _ Ac 60 -r (5.27) The complex c o n j u g a t e of T ^ i s : We r e q u i r e t h a t (5.28) be i n v a r i a n t under L o r e n t z t r a n s f o r m a t i o n s , and use t h e f a c t t h a t <*y*C i s H e r m i t i a n t o deduce t h a t : f T.a* * ) * = T (5.29) k 1 ABC©/ 1 COAB where T C O p ; 6 i s some a s - y e t - u n d e f i n e d s p i n - t e n s o r . I f T ^ v i s r e a l , t h e n T" C 0 A^ i s t h e same m a t r i x a s T C D f t ^ : 53 (TABCOV*=TCDAB = T C O F T B (5.30) By t h e same r e a s o n i n g as i n t h e above example, t h e complex c o n j u g a t e of any s p i n - t e n s o r w h i c h r e p r e s e n t s a r e a l s p i n - t e n s o r can be c a l c u l a t e d by: 1) r e p l a c i n g d o t s w i t h " u n - d o t s " and " u n - d o t s " w i t h d o t s . 2) r e a r r a n g i n g i n d i c e s so t h e d o t s a r e once a g a i n on t h e r i g h t - h a n d s i d e . To be c o n s i s t e n t , t h e complex c o n j u g a t e s of £ f t a and 6.c0 must d e f i n e d a s € f t B and £ c 0 r e s p e c t i v e l y . 5.6 Symmetry and A n t i s y m m e t r y i n S p i n - T e n s o r s D e f i n e a n o t a t i o n r e p r e s e n t i n g s y m m e t r i z e d ( i n d i c e s i n p a r e n t h e s e s ) and a n t i s y m m e t r i z e d [ i n d i c e s i n s q u a r e b r a c k e t s ] s p i n - t e n s o r s : T"CAB)C6 - ~z ( ~TAttab + T 0 A iii) (5.31) e t c . Any p a i r of i n d i c e s c an be b r o k e n i n t o s y mmetric and a n t i s y m m e t r i c p a r t s as f o l l o w s : ^ A B C D = ^ C A & C O + ^ C A B U O (5.32) 54 T h i s s i m p l e s e p a r a t i o n o f s y m m e t r i c and a n t i s y m m e t r i c p a r t s does no t e x t e n d t o t r i p l e t s o r l a r g e r numbers of i n d i c e s . A s t r i n g o f s y m m e t r i c i n d i c e s can be b r o k e n i n t o non-s y m m e t r i c p a r t s u s i n g t h e g e n e r a l e x p a n s i o n : + R 0 ( E A 6 0 + R E(A8CO)l ( 5' 3 3 ) T h i s w i l l be u s e d i n t h e computer a l g o r i t h m ( s e e a p p e n d i x A) t o c a l c u l a t e d e r i v a t i v e s of p u r e l y s y mmetric s p i n - t e n s o r s . Any p a i r of a n t i s y m m e t r i c i n d i c e s can be r e p l a c e d by an ( € ) and two c o n t r a c t e d i n d i c e s u s i n g t h e f o l l o w i n g scheme: T C A B 1 . =4-T E E . . . . CD • • • 2. E . . . CO ... T _ i . ~r E r LAB]... co... Z 1 E . ..cD...bft& (5.34) S p i n - t e n s o r s w h i c h r e p r e s e n t symmetric o r a n t i s y m m e t r i c t e n s o r s must have p a r t l y s y m m e t r i c and p a r t l y a n t i s y m m e t r i c i n d i c e s . Assume, as an example, a t e n s o r T ^ w h i c h c o r r e s p o n d s t o a s p i n - t e n s o r T A B ^ D : The s y m m e t r i c p a r t o f t h i s t e n s o r c o r r e s p o n d s t o t h e s p i n - t e n s o r 55 sum: \+-$) *=> T(AB)(C6) +• T ^ 1 L c p 3 ( 5 - 3 6 ) The a n t i s y m m e t r i c p a r t o f T ^ c o r r e s p o n d s i n t u r n t o t h e s p i n -t e n s o r sum: " W l ^ T I T C A ^ F ^ M + T F F U 6 ) C F T B] (5.37) Note t h a t i f one of t h e t e r m s i n t h e sum (5.37) was not p r e s e n t t h e t e n s o r T^^-j would no l o n g e r be m a n i f e s t l y r e a l - v a l u e d . T h e s e symmetry p r o p e r t i e s w i l l be p a r t i c u l a r l y u s e f u l when e s t a b l i s h i n g t h e more c o n c i s e n o t a t i o n f o r g r a v i t a t i o n a l s p i n -t e n s o r s t h a t i s p r e s e n t e d i n c h a p t e r 6. 5.7 D e r i v a t i v e s of S p i n - T e n s o r s D e f i n e t h e d e r i v a t i v e s o f s p i n - t e n s o r s i n t h e n a t u r a l way, t r e a t i n g t h e c o e f f i c i e n t of t h e d e r i v a t i v e e x a c t l y as t h e c o e f f i c i e n t s of t e n s o r s . F o r example, t r a n s l a t e s i n t o t h e s p i n - t e n s o r : " b * H A » - H * , c * - V ( s „ , \ t . » , , A <5.38> The comma i s u s e d t o d e n o t e o r d i n a r y ( n o n - c o v a r i a n t ) d i f f e r e n t i a t i o n . 56 C h a p t e r 6 S p i n - T e n s o r R e p r e s e n t a t i o n o f N o n - F l a t S p a c e - T i m e s In a n o n - f l a t s p a c e - t i m e , t h e M i n k o w s k i m e t r i c 1 s r e p l a c e d by t h e s y m m e t r i c m e t r i c t e n s o r g ^ , so t h a t t h e i n v a r i a n t l i n e e l e m e n t i s not s i m p l y ds = J d t * - d x l - d y 2 - d z 2 ' (as i t was b e f o r e ) . C l e a r l y t h e s p i n - t e n s o r f o r m a l i s m , w h i c h was s t u c t u r e d p a r t l y a r o u n d t h i s i n v a r i a n c e , w i l l have t o change so t h a t J H A ^ H A 6 w i l l e q u a l t h e new l i n e e l e m e n t . T h e r e a r e two ways t h i s c an be a c h i e v e d w i t h o u t l o s i n g a l l t h e s t r u c t u r e b u i l t up i n c h a p t e r 5: 1) T r a n s l a t e v e c t o r s i n t o s p i n - t e n s o r s i n s u c h a way t h a t t h e d e t e r m i n a n t c an s t i l l be u s e d t o c a l c u l a t e t h e p r o p e r l e n g t h . T h i s method makes t h e m a t r i c e s v a r y f r o m p o i n t t o p o i n t a s a r e s u l t of t h e c u r v a t u r e . 2) No l o n g e r compute t h e l e n g t h a s a d e t e r m i n a n t , but i n s t e a d use a r e v i s e d s p i n o r m e t r i c w h i c h i s d e p e n d e n t on t h e c u r v a t u r e . T h i s w o u l d l e a v e t h e ff^k unchanged but make £ A 6 v a r y f r o m p o i n t t o p o i n t as a r e s u l t of t h e c u r v a t u r e . P e n r o s e < 2 3 ) c h o o s e s t h e f i r s t o f t h e s e o p t i o n s , and t h i s w i l l a l s o be t h e a p p r o a c h t a k e n f o r t h e r e m a i n d e r o f t h i s p a p e r . We r e q u i r e , t h e r e f o r e , t h a t t h e € s p i n - t e n s o r s be g l o b a l i n v a r i a n t s e q u a l t o t h e c o n s t a n t m a t r i c e s d e f i n e d i n ( 5 . 2 1 ) . T h i s i m p l i e s t h a t : 57 ^ A f c ^ = ^ Ae-,^ ~0 (6.1) where £-A6;p i s t h e c o v a r i a n t d e r i v a t i v e of 6 f t B i n t h e d i r e c t i o n of t h e b a s i s v e c t o r x^,, as w i l l be d e f i n e d b elow. In o r d e r t o d e f i n e 0 " ^ ^ i n terms of t h e c u r v a t u r e , we r e q u i r e t h a t : A e t A RC _ .ftfc (6.2) ffr i sv + S c *r - ^ L w h i c h i s m e r e l y an e x t e n s i o n o f e q u a t i o n (5.26) i n t o c u r v e d s p a c e - t i m e . W i t h t h e above two r e q u i r e m e n t s , we can now b r i n g t h e whole a l g e b r a of s p i n - t e n s o r s , as summarized i n c h a p t e r 5, a l m o s t i n t a c t i n t o c u r v e d s p a c e - t i m e s . E v e r y w h e r e t h a t t h e r e i s \j»v we r e p l a c e i t w i t h g^v, a n c * e v e r y w h e r e a p p e a r s we use th e t h a t i s d e f i n e d i n (6.2) above. 6.1 C o v a r i a n t D e r i v a t i v e s In k e e p i n g w-ith t h e i n t e n t i o n of g i v i n g s p i n o r - s p a c e n o t a t i o n a p a r a l l e l s t r u c t u r e t o t e n s o r n o t a t i o n , t h e c o v a r i a n t d e r i v a t i v e of a s p i n - t e n s o r ( M A g f o r example) i s d e f i n e d i n te r m s of a " s p i n c o n n e c t i o n " i 6 o t , as i n t h e f o l l o w i n g example: The Greek i n d i c e s c a n be r e p l a c e d by two Roman i n d i c e s i n t h e p r e d i c t a b l e way: 58 In a l l t h a t f o l l o w s , t h e symbol i s r e s e r v e d f o r c o v a r i a n t d i f f e r e n t i a t i o n . C o v a r i a n t d i f f e r e n t i a t i o n i s a d e r i v a t i o n and hence o b e y s t h e p r o d u c t r u l e : ( * / X A ) ; v = (»,.%) A* + (6-5) Because AJA» v i s a t e n s o r , however, we know t h a t e a c h i n d e x c a n be t r a n s l a t e d i n t o s p i n o r s p a c e i n d e p e n d e n t l y : A ^ W / ^ A ^ ) ( 6- 6 ) E q u a t i o n s (6.5) and (6.6) a r e c o m p a t i b l e o n l y i f : A A . V = 0 <6-7> i*& rAB c o v a r i a n t d e r i v a t i v e s w h i c h a r e z e r o . B o t h " f u n d a m e n t a l " s p i n o r s (€. and ) have, t h e r e f o r e , The two c o n d i t i o n s , ( 6 . 1 ) and ( 6 . 7 ) , and t h e r e q u i r e m e n t t h a t I a\ be sym m e t r i c i n (p,>H, c o m p l e t e l y d e t e r m i n e t h e s p i n : n A c o n n e c t i o n s I g>j . The r e q u i r e d c o n n e c t i o n c o e f f i c i e n t s c a n be w r i t t e n a s ( 2 * } : 59 6.2 The C u r v a t u r e S p i n - T e n s o r s [ f ] The f i r s t s t e p i n d e f i n i n g t h e c u r v a t u r e s p i n - t e n s o r s i s t o d i r e c t l y t r a n s l a t e t h e Riemann t e n s o r R ^ « ; u s i n g t h e e y ^ s p i n o r s . To do so, t h e f o l l o w i n g s y m m e t r i e s o f R ^ vpr a r e kept i n mind: fA-v^c- ~ ~f^ vf*f o- = Rp-vrp — P*£j*vl[.{«"l (6.9) ^rh/e<a = 0 ( 6 . i o ) R h tve'JM = 0 (6-ancWi Identity) (6 -11) From (6.9) and (5.37) i t f o l l o w s t h a t t h e Riemann s p i n - t e n s o r c a n be w r i t t e n : &pV(V \ [^C^N)(R.^P ^rtN + R / a . . . . c f 1 ( 6 * 1 2 ) D e f i n i n g two new s p i n - t e n s o r s ( t h e " c u r v a t u r e s p i n - t e n s o r s " ) ^ABCO a n d ^MJco a P P r o P r i a t e l y , t h i s c a n be r e - w r i t t e n a s : [ f ] The d i s c u s s i o n i n t h i s s e c t i o n i s b a s e d on t h e d e v e l o p m e n t of t h e s u b j e c t by P e n r o s e ( 2 3 ) . 60 ^KKliL* € M - £ | n w +4lLM C^C^i] ( 6 ' 1 3 ) Note the following symmetries of the curvature spin-tensors: *V _ -y — *V ( 6 1 5 ) More can be discerned about "X^CQ by expanding the spin-tensor version of (6.10). After contracting the second and fourth indices (both dotted and undotted), and using (6.14) to subtract out the (fl terms, one finds that: *Y ^ £ — "Y ^ C (616) This statement that K^^IL* is equal to i t s complex conjugate implies that Xrnofc0 is real and hence can be expressed in terms of some real value X as: , 6 - , 7 > where X i s defined as: 61 \ = X 7 A* = 1 ^ . AS ( 6 . 1 8 ) The f i n a l s t e p i n t h e r e d u c t i o n o f t h e Riemann t e n s o r t o s p i n - t e n s o r form i s t o r e w r i t e 9C M c 0 i n t e r m s o f X and a f u l l y s y m m e t r i c t e n s o r VA6co as f o l l o w s : (6.19) E q u a t i o n (6.19) and i t s complex c o n j u g a t e s e r v e t o d e f i n e t h e c u r v a t u r e s p i n o r s 4 ,^B ) C 0 and • The Riemann t e n s o r has now been r e d u c e d t o t h e f o u r s p i n - t e n s o r s tf^ifa , ^\$CD ' ^A«'c6 ' a n d X , w h i c h p r o v i d e f u l l i n f o r m a t i o n a b o u t t h e c u r v a t u r e , w i t h a l l t h e s y m m e t r i e s i m p l i c i t i n t h e i r s t r u c t u r e . The s p i n - t e n s o r s c o r r e s p o n d i n g t o t h e c o n t r a c t e d Riemann t e n s o r R ^ and t h e c u r v a t u r e s c a l a r R can be c a l c u l a t e d d i r e c t l y f r o m ( 6 . 1 3 ) . They a r e r e s p e c t i v e l y : RMNAN = X HN -cPriNMNi (6.20) R= 4 A (6.21 ) 62 6.3 The B i a n c h i and R i c c i I d e n t i t i e s E x p a n s i o n of (6.11) i n t o s p i n - t e n s o r f o r m , u s i n g (6.13) and t h e s y m m e t r i e s of t h e c u r v a t u r e s p i n - t e n s o r s , y i e l d s t h e f o l l o w i n g i d e n t i t y : ^ ^ N C S + i ^ X e M R ^ ^ M A G ^ ' ^ ^ M W a ( 6 * 2 2 ) S e p a r a t i n g t h e r e a l and complex p a r t s of t h i s , one i s l e f t w i t h t h e B i a n c h i i d e n t i t y i n t h e f o r m of t h e two e q u a t i o n s : (6.23) The R i c c i i d e n t i t i e s r e l a t e t h e commutation of t h e c o v a r i a n t d e r i v a t i v e t o t h e c u r v a t u r e . F o r example, f o r any v e c t o r v f , T e n s o r s of h i g h e r o r d e r s obey s i m i l a r i d e n t i t i e s . The c o r r e s p o n d i n g i d e n t i t i e s f o r s p i n - t e n s o r s a r e d e r i v e d by P e n r o s e ( 2 3 ) and a r e q u o t e d below: 63 1) The o p e r a t o r ( ^ 0 F ^ E F +^EF^DF ) w o r k i n g on a s p i n -t e n s o r i . . ^ . . . of a r b i t r a r y rank can be r e p l a c e d by a s e r i e s of t e r m s a s f o l l o w s : ... B a) f o r e v e r y u n d o t t e d i n d e x (A) i n ^ " "" , add a t e r m o f t h e f o r m b) f o r e v e r y d o t t e d i n d e x (B) i n % ^ , add a t e r m o f t h e form (P . & t 2) The o p e r a t o r ( ^ E * ^ E 6 + ^ t G ^ 6 ' f ) w o r k i n g on a s p i n -t e n s o r j of a r b i t r a r y rank can be r e p l a c e d by a s e r i e s o f t e r m s a s f o l l o w s : a) f o r e v e r y u n d o t t e d i n d e x (C) i n ^ . . . c ° . , add a t e r m of t h e form b) f o r e v e r y d o t t e d i n d e x (D) i n ^ , add a t e r m of t h e form V s . - i ^ y . ^ -To i l l u s t r a t e t h e s e R i c c i i d e n t i t i e s , c o n s i d e r t h e f o l l o w i n g e x a m p l e s : 64 and: ( 0 " * i e i i , % . = O F V t f H , « F , * ( 6 - 2 6 ) 6.4 E i n s t e i n ' s E q u a t i o n s The t e n s o r s t a t e m e n t o f E i n s t e i n ' s e q u a t i o n s (4.1) t r a n s l a t e s d i r e c t l y i n t o s p i n - t e n s o r n o t a t i o n u s i n g (6.20) and ( 6 . 2 1 ) . The s p i n - t e n s o r e x p r e s s i o n i s : ~<fl -.-AC -RTTT •• (6.27) where T , ^ ^ i s t h e s p i n - t e n s o r f o r m o f t h e s t r e s s - e n e r g y t e n s o r In empty s p a c e - t i m e , ( T M N ^ = 0 ) , t h e sym m e t r i c and a n t i s y m m e t r i c p a r t s of (6.60) g i v e r e s p e c t i v e l y : O (6.28) X-0 (6.29) Thus i n empty s p a c e - t i m e t h e c o m p l e t e l y s y m m e t r i c , rank-4 s p i n -t e n s o r and i t s complex c o n j u g a t e H^c© c o m p l e t e l y d e s c r i b e t h e g e o m e t r y . The economy o f n o t a t i o n p r o m i s e d a t t h e b e g i n n i n g o f c h a p t e r 5 i s becoming e v i d e n t . 65 C h a p t e r 7 E x p a n s i o n o f P u r e l y G e o m e t r i c S p i n - T e n s o r s The f o l l o w i n g c h a p t e r e x p l o r e s how p u r e l y - g e o m e t r i c s p i n -t e n s o r s i n an empty E i n s t e i n s p a c e - t i m e ( t h a t i s , s p i n - t e n s o r s made up o f o n l y V A B c o and a n d t h e i r d e r i v a t i v e s ) can be e x p r e s s e d i n terms of a b a s i s s e t o f s p i n - t e n s o r s . S e c t i o n (7.1) s e t s up c l a s s e s of s p i n - t e n s o r s whose d e r i v a t i v e s a r e a l g e b r a i c a l l y r e l a t e d , and s e c t i o n (7.2) d e f i n e s a c o m p l e t e s e t of a l g e b r a i c a l l y i n d e p e n d e n t s p i n - t e n s o r s f o r empty E i n s t e i n s p a c e - t i m e s . The f i n a l s e c t i o n d i s c u s s e s how 0-, 1- and 2-t e n s o r s c a n be w r i t t e n i n terms of t h i s b a s i s s e t of s p i n -t e n s o r s . 7.1 C l a s s i f i c a t i o n of G e o m e t r i c S p i n - T e n s o r s P u r e l y g e o m e t r i c s p i n - t e n s o r s w i l l be c l a s s i f i e d a c c o r d i n g t o 1) t h e number of % v B t l i and »^\BCO s p i n - t e n s o r s and 2) t h e number o f d e r i v a t i v e s t h a t t h e y h a v e . F o l l o w i n g U n r u h < 3> , a te r m w i t h j c o v a r i a n t d e r i v a t i v e s and k c u r v a t u r e s p i n o r s i s s a i d t o be of o r d e r ( j , k ) . (A s p i n o r of o r d e r ( j , k ) w i l l be r e f e r r e d t o below as S ( j , k ) ) F o r example, t h e s p i n - t e n s o r term i s o f o r d e r ( 1 , 2 ) . 66 I f a p o l y n o m i a l can be w r i t t e n as t h e l i n e a r sum o f s p i n -t e n s o r s S ( j , k ) a c c o r d i n g t o t h e f o l l o w i n g scheme: (where ^ M a r e t h e c o e f f i c i e n t s ( p e r h a p s z e r o ) of e a c h t e r m ) , t h e n t h e p o l y n o m i a l i s s a i d t o be o f d e g r e e ( n ) . F o r example, a p o l y n o m i a l "{3}" of d e g r e e 3 i s t h e sum of terms of t h e f o l l o w i n g o r d e r s : F o r f u t u r e r e f e r e n c e , t h e e x p a n s i o n of p o l y n o m i a l s of d e g r e e 0 t h r o u g h 4 a r e shown below. {0} = <*. S ( 0 , 1 ) {1 } = <*.S( 1 , 1 ) {2} = « U S ( 2 , 1 ) + *,S(0,2) {3} = «C DS(3, 1 ) + * , S ( 1 , 2 ) {4} = <* 0S(4,1) + S ( 2 , 2 ) + * t S ( 0 f 3 ) • • • € t C • 1) (7.1) and c o u l d be, f o r example: + 8 ( i x 4 ¥* l M , )Yi6 M (7.3) 67 Note t h a t a l l o r d e r s a r e r e p r e s e n t e d , and t h a t t h e d e g r e e of e a c h o r d e r i s non-ambiguous. The above d e f i n i t i o n o f " d e g r e e " stems f r o m t h e a c t i o n s o f t h e R i c c i I d e n t i t i e s ( a s d e s c r i b e d i n s e c t i o n 6.3) w h i c h i d e n t i f y s p i n - t e n s o r s o f o r d e r ( j , k ) w i t h s p i n - t e n s o r s of o r d e r ( j - 2 , k + l ) . The d e f i n i t i o n e n s u r e s t h a t p o l y n o m i a l s of d i f f e r e n t d e g r e e a r e l i n e a r l y i n d e p e n d e n t , and t h a t t h e i r c o v a r i a n t d e r i v a t i v e s w i l l a l s o be l i n e a r l y i n d e p e n d e n t . In what f o l l o w s , o n l y g e o m e t r i c s p i n - t e n s o r s w h i c h can be w r i t t e n as l i n e a r sums o f p o l y n o m i a l s of d e g r e e (n) w i l l be s o u g h t . T h i s i n c l u d e s many p o s s i b l e s p i n - t e n s o r s , b e c a u s e terms w h i c h a r e n o t of t h i s f o r m can o f t e n be w r i t t e n i n t h i s form u s i n g T a y l o r s e r i e s . T h i s r e s t r i c t i o n s i m p l i f i e s t h e s e a r c h f o r c l o s e d d i f f e r e n t i a l f orms i n t h e f o l l o w i n g way. Remember t h a t t h e o b j e c t o f t h e s e a r c h i s t o f i n d a t e n s o r ^ whose a n t i -s y m m etric c o v a r i a n t d e r i v a t i v e T r, _ v a n i s h e s , and w h i c h can be r e p r e s e n t e d by a sum o f s p i n - t e n s o r p o l y n o m i a l s of t h e form {n} (n = 0,1...) as d e f i n e d a b o v e . B e c a u s e t h e s e p o l y n o m i a l s a r e l i n e a r l y i n d e p e n d e n t , t h e c o v a r i a n t d e r i v a t i v e o f e a c h of them must v a n i s h b e f o r e t h e c o v a r i a n t d e r i v a t i v e of -j can v a n i s h . The s e a r c h has t h u s been n a r r o w e d t o a s e a r c h f o r a s p i n - t e n s o r p o l y n o m i a l o f any d e g r e e , c o r r e s p o n d i n g t o an a n t i s y m m e t r i c t e n s o r , w h i c h has a v a n i s h i n g a n t i s y m m e t r i z e d d e r i v a t i v e . 68 7.2 A C o m p l e t e Set of G e o m e t r i c S p i n - T e n s o r s P e n r o s e 1 2 3 1 has shown t h a t a l l g e o m e t r i c s p i n - t e n s o r s can be c o n s t r u c t e d out o f components p i c k e d e x c l u s i v e l y f r o m t h e s e t o f a l l s y m m e t r i z e d d e r i v a t i v e s o f VF T 6 C 0 and H'ABC.B • T n e f o l l o w i n g s e c t i o n r e v i e w s h i s p r o o f and d e f i n e s t h e n o t a t i o n w h i c h w i l l be u s e d when r e f e r r i n g t o t h e s e b a s i c s p i n - t e n s o r s . In empty s p a c e - t i m e , where ^ A B j . 0 = X = 0 r t h e B i a n c h i and R i c c i i d e n t i t i e s t a k e on t h e somewhat s i m p l e r f o r m : °RP 'MNRS U (7.4) ^ W ^ P = 0 (7.6) ^ ^ ( ^ 0 ^ ) ^ = 0 (7.7)-^ ^ H ^ C ^ - ^ i ^ (7.8) w i t h l o g i c a l e x t e n s i o n s t o o p e r a t e on h i g h e r o r d e r s p i n - t e n s o r s . A l l of t h e above i d e n t i t i e s e x p r e s s c o n d i t i o n s o n l y on t h e p a r t s °f ^Ad. . . ^ M N V Q C O E w h i c h a r e s k ew-symmetric, and do n o t a f f e c t t e r m s ^ B ' . . . ^ H ^ ^ C D E F ) w ^ ^ c n n a v e c o m p l e t e l y symmetric i n d i c e s . 69 These i d e n t i t i e s imply that the antisymmetric part of any term ^ A 8 ^ B C - - ^ e F ^cuzz can be re-written as a series of terms which a l l have fewer derivatives. Although not immediately obvious, a l i t t l e pondering confirms that t h i s w i l l always be possible, and indeed Penrose ( 2 3 ) has quite c a r e f u l l y proven i t to be so. Now i f every spin-tensor term \ A v B v E W °A '' ' «E 'CHI3 can be written as: + (terms with fewer derivatives) then by a process of induction, a l l algebraic combinations of spin-tensors can be formed out of the sets of fully-symmetric derivatives of a n d ^A&cb • These sets: CP ... . \ (6 Q7 CDEF) v (B \ b\T)EFC-,H) (7.9) form a complete set of a l g e b r a i c a l l y independent spin-tensors. An abbreviated notation, introduced by Penrose ( 2 3 ) , w i l l be used for these spin-tensors. The }ft6 symbols are dropped and their indices tagged onto the Y symbols. Two examples of th i s new notation are: 70 a n d : V ^ o 6 ? * ^ = H 7 ^ " * " (7.11) A l t h o u g h t h e n o t a t i o n f o r t h i s c o m p l e t e s e t i s v e r y s i m p l e , o p e r a t i n g on i t s members c a n i n v o l v e v e r y c o m p l i c a t e d "book-k e e p i n g " t o keep t r a c k o f a l l t h e c o n t r a c t i o n s . F o r example, when one t a k e s t h e d e r i v a t i v e ^ P 1 N ( t f f t S C D E v : c ' H ), i t i s i m p o r t a n t t o remember t h a t t h e i n d i c e s o f t h e d e r i v a t i v e a r e n o t " a u t o m a t i c a l l y " f u l l y s y m m e t r i c w i t h t h e i n d i c e s of t h e s p i n -t e n s o r . To e x p r e s s t h e r e s u l t i n terms o f t h e b a s i s s e t (7.9) i n v o l v e s b r e a k i n g t h e s y m m e t r i c i n d i c e s i n t o t h e i r u n s y m m e t r i z e d form u s i n g ( 7 . 3 3 ) , t a k i n g t h e d e r i v a t i v e u s i n g t h e p r o d u c t r u l e , and t h e n s i m p l i f y i n g t h e a n t i s y m m e t r i c p a r t s u s i n g t h e R i c c i I d e n t i t i e s . The r e s u l t i s a s p i n - t e n s o r of t h e for m : + terms ( U o C " ^ ) / C;UT\ v (7.12) + t e r ~ s ( t e C ' ^ A . ) The c o m p l e x i t y of s u c h c a l c u l a t i o n s e s c a l a t e s v e r y r a p i d l y a s one g o e s t o s p i n - t e n s o r s o f h i g h e r and h i g h e r d e g r e e . 71 7.3 R e p r e s e n t i n g T e n s o r s a s P r o d u c t s o f B a s i s S p i n - T e n s o r s To r e p r e s e n t a t e n s o r o f ran k p i n terms o f t h e b a s i s s p i n -t e n s o r s ( 7 . 9 ) , one t a k e s t h e p r o d u c t of a number o f them and c o n t r a c t s i n d i c e s between t h e f a c t o r s . The number of non-c o n t r a c t e d ( " f r e e " ) i n d i c e s must c o r r e s p o n d t o t h e ran k o f t h e t e n s o r w h i c h i s r e p r e s e n t e d . F o r example t h e p r o d u c t W u;A6co (7.13) c o r r e s p o n d s t o a rank-0 t e n s o r , or s c a l a r . S i m i l a r l y , vi> S C D E w (7.14) ' X X ' G C O E c o r r e s p o n d s t o a t e n s o r of rank 1. H e r e , and i n what f o l l o w s , t h e l e t t e r s X, Y, Z, X, Y, and Z w i l l be u s e d e x c l u s i v e l y f o r t h e f r e e i n d i c e s , a s an a i d t o r e c o g n i t i o n . The f o l l o w i n g c h a r a c t e r i s t i c s of s u c h s p i n - t e n s o r s f o l l o w d i r e c t l y f r o m t h e d i s c u s s i o n of t h e p a s t t h r e e c h a p t e r s , and w i l l be u s e d f r e q u e n t l y i n t h e c a l c u l a t i o n s w h i c h a r e d e s c r i b e d i n c h a p t e r 8: 1) The c o n t r a c t i o n o f i n d i c e s w i t h i n a b a s i s s p i n -t e n s o r w i l l a l w a y s be z e r o : C iy A B C ° 6 _ i w ABCO E 0 E F • otr 2) The o r d e r i n w h i c h t h e f a c t o r s a r e w r i t t e n i s i r r e l e v a n t : Vf . 1 0 ABCO u/ABCO vj/ 'AGCDXX 1 — 1 TABCOX.X 72 3) E v e r y t i m e a c o n t r a c t i o n i s " i n v e r t e d " , ( t h a t i s , when two c o n t r a c t e d i n d i c e s a r e e x c h a n g e d ) a f a c t o r of (-1) must be a p p l i e d a s i n t h i s e x a m p l e: U » u f A B t O _ _ u/A n, BCD T A B £ O X * ' 1 S C O * * » f t 4) The i d e n t i t y : e n a b l e s terms w i t h m u l t i p l e f a c t o r s t o be r e - w r i t t e n as t h e sum of terms w i t h d i f f e r e n t c o n t r a c t i o n s . F o r example: 1±> U ^ E P G > I O A B C O W ; H _ ~ E P G , H T D • T A B C -= L u E P G t M , A B C H M » O 1 E F G H » D T I A B C _ 0> iwEFQH u ; A B C ^ 0 * E F G H 1 T o T f t e c A l l r a n k - 2 s p i n - t e n s o r s t o be d i s c u s s e d h e r e a r e r e a l and a n t i s y m m e t r i c , and so must t a k e t h e form ( 5 . 3 7 ) . One s u c h s p i n -t e n s o r m i g h t be f o r example: (7.16) In C h a p t e r 9 i t w i l l be n e c e s s a r y t o d e a l w i t h t h e d u a l s o f rank - 2 s p i n - t e n s o r s , w h i c h a r e d e f i n e d i n terms o f t h e s p i n -t e n s o r v e r s i o n of t h e L e v i - C i v i t a t e n s o r d e n s i t y £ o C ^ ^ . T h i s s p i n - t e n s o r c a n be w r i t t e n i n t e r m s o f t h e f u n d a m e n t a l s p i n o r s A B as ( 2 3 ) ; A B C O A B C O 73 where i i s t h e s q u a r e - r o o t o f -1. S p e c i f i c a l l y , c o n s i d e r a p r o t o t y p e r a n k - 2 s p i n - t e n s o r r e p r e s e n t i n g a r e a l t e n s o r T ^ : =ffr***V* T y v*y (7.18) From ( 5 . 3 6 ) , t h e r e a l and a n t i s y m m e t r i c p a r t o f t h i s t e n s o r i s r e p r e s e n t e d by t h e s p i n - t e n s o r : w h i c h i s r e a l and a n t i - s y m m s t r i c by c o n s t r u c t i o n . We use (7.17) t o c a l c u l a t e t h e d u a l o f t h i s a s f o l l o w s : The r e s u l t i s i n t e r e s t i n g : t h e d u a l of t h e r e a l , a n t i s y m m e t r i c p a r t of t h e s p i n - t e n s o r T x y ) ^ i s t h e i m a g i n a r y , a n t i s y m m e t r i c p a r t o f t h e same s p i n - t e n s o r . By t h e same a r g u m e n t s , t h e d u a l s o f r a n k - 3 and rank- 4 s p i n -t e n s o r s a r e f o u n d t o have a s i m i l a r f o r m . The d u a l of a 3-form must be of t h e fo r m : 74 and t h e d u a l o f a 4-form w i l l be: (>\->\) (7.23) By c o n t r a s t , r e a l rank-1 and r a n k - 0 s p i n - t e n s o r s w i l l be, r e s p e c t i v e l y , o f t h e f o r m s : (>| + y[) (7.25) U s i n g t h e s e , i t w i l l be p o s s i b l e t o t e s t rank-2 and r a n k - 3 s p i n -t e n s o r s f o r e x a c t n e s s w i t h o u t c a l c u l a t i n g a c o m p l e t e s e t of r a n k - t h r e e and rank-4 s p i n - t e n s o r s . 75 C h a p t e r 8 Computer S e a r c h f o r C l o s e d , N o n - E x a c t p-Forms T h i s c h a p t e r , t o g e t h e r w i t h t h e d o c u m e n t a t i o n i n c l u d e d i n a p p e n d i x A, d e s c r i b e s t h e d e v e l o p m e n t o f a c o m p u t e r - c o m p a t i b l e n o t a t i o n and a computer p r o g r a m w h i c h was u s e d t o c a l c u l a t e t h e s p i n - t e n s o r s and t h e i r d e r i v a t i v e s . 8.1 O v e r v i e w of t h e P r o j e c t The g o a l of t h i s r e s e a r c h i s t o s y s t e m a t i c a l l y s e a r c h f o r 1-, 2 - , and 3-charges t o as h i g h an o r d e r as p o s s i b l e . Unruh < 3 ) h a s s e a r c h e d up t o d e g r e e - 3 by "hand" c a l c u l a t i o n s and f o u n d no c h a r g e s , but b e c a u s e of t h e c o m p l e x i t y i n h e r e n t i n e x p r e s s i n g s p i n - t e n s o r s i n terms of t h e c o m p l e t e s e t d e s c r i b e d i n s e c t i o n ( 7 . 2 ) , even t h e s e d e g r e e - 3 c a l c u l a t i o n s were e x t r e m e l y a r d u o u s . I t q u i c k l y became e v i d e n t w h i l e f i r s t t r y i n g t o e x t e n d Unruh's work t h a t c a l c u l a t i n g t h e d e r i v a t i v e s by hand would not be s a t i s f a c t o r y . In p a r t i c u l a r , h a n d - c a l c u l a t i o n s were deemed u n f e a s i b l e b e c a u s e : 1) The amount o f t i m e r e q u i r e d would be u n r e a s o n a b l e . 2) The p o s s i b i l i t y f o r e r r o r would be t o o l a r g e , making t h e r e s u l t s u n t r u s t w o r t h y . 3) The m a g n i t u d e o f t h e j o b would make i t v e r y d i f f i c u l t t o c h e c k t h e c a l c u l a t i o n s a d e q u a t e l y . 76 Consideration was then given to using a computer' to generate the spin-tensors and to calculate t h e i r derivatives. A computer would offer a le v e l of r e l i a b i l i t y unattainable in hand-calculations, and would seem to be p a r t i c u l a r l y suited to some of the recursive processes involved in ca l c u l a t i n g the derivatives. Most computer languages are completely unsuitable for t h i s task, however, because i t consists almost solely of symbol-manipulation tasks whereas the "standard" programming languages are geared to arithmetic c a l c u l a t i o n s . To use FORTRAN, for example, would have entailed developing a whole system of subroutines for symbolic manipulation, many written at the assembler l e v e l . Use of the computer for the cal c u l a t i o n was feasible only because a high-level language, APL (A Programming Language), was used. APL offered very good matrix-manipulation c a p a b i l i t i e s , so that the strings of indices, stored as vectors, could be very simply combined, reordered, compared, or restructured. The result was that the basic operations of putting the contractions into ascending order, tr a n s l a t i n g the codes for spin-tensors into readable form, and the l i k e , could be performed by very compact APL "functions" (the counterparts of subroutines). These functions, and the driving program "AAA" are l i s t e d in Appendix B. One problem with the code being so compact that i t is very d i f f i c u l t to decipher by anyone but the author. For thi s reason, a thorough review of the algorithms upon which the code is based 77 i s p r e s e n t e d i n A p p e n d i x A. The f i r s t t a s k i n d e v e l o p i n g t h e computer p r o g r a m was t o d e v e l o p c o m p u t e r - c o m p a t i b l e n o t a t i o n f o r s p i n - t e n s o r s and a " c a n o n i c a l f o r m " so t h a t two e q u i v a l e n t s p i n - t e n s o r s would a l w a y s l o o k a l i k e . T h e s e two i n n o v a t i o n s a r e d i s c u s s e d i n some d e t a i l i n s e c t i o n s (8.2) and ( 8 . 3 ) . Once t h e p r o c e d u r e s and c o n v e n t i o n s had been d e v e l o p e d f o r h a n d l i n g t h e s p i n - t e n s o r s , code was w r i t t e n t o do t h e f o l l o w i n g t a s k s : 1) To g e n e r a t e a c o m p l e t e l i s t o f l i n e a r l y -i n d e p e n d e n t s p i n - t e n s o r t e r m s , o f d e g r e e s 1 t h r o u g h 8, c o r r e s p o n d i n g t o a l l p o s s i b l e terms i n t h e e x p a n s i o n s of a n t i s y m m e t r i c 0-, 1-, and 2-t e n s o r s [ t ] • 2) To c a l c u l a t e t h e a n t i s y m m e t r i z e d d e r i v a t i v e s of a l l r a n k - 0 and rank-1 s p i n - t e n s o r s i n t h e above l i s t and t o e x p r e s s them i n terms of s p i n - t e n s o r s from t h a t . l i s t . 3) To c a l c u l a t e t h e c o n t r a c t e d d e r i v a t i v e s of a l l rank-1 and rank-2 s p i n - t e n s o r s w h i c h a r e d u a l s c f ran k - 2 and rank-3 s p i n - t e n s o r s and t o e x p r e s s them i n t e r m s of t h e s p i n - t e n s o r s from t h e above l i s t . The p r o g r a m was d e v e l o p e d and run on t h e Amdahl 470 V/6 Model I I computer f a c i l i t y a t t h e U n i v e r s i t y of B r i t i s h C o l u m b i a . The r e s u l t s o f t h e c a l c u l a t i o n s a r e p r e s e n t e d and a n a l y s e d i n c h a p t e r n i n e . [ f ] In t h i s c h a p t e r , and i n t h e a p p e n d i c e s , a s p i n - t e n s o r "of r a n k - n " means a s p i n - t e n s o r w h i c h r e p r e s e n t s a t e n s o r o f r a n k - n . Note t h a t t h i s i s a d i f f e r e n t u sage from t h a t f o u n d i n c h a p t e r s 5, 6, and 7. 78 8•2 C o m p u t e r - C o m p a t i b l e N o t a t i o n and S u p p o r t F u n c t i o n s H e r e , i n t h e r e s t o f t h i s c h a p t e r , and i n A p p e n d i x A, t h e words " t e r m " , f a c t o r " , and "name" a r e u s e d as f o l l o w s : -"Term" i s us e d t o r e f e r t o a whole s p i n t e n s o r term, t h a t i s , a p r o d u c t o f y's and y 's w h i c h a r e c o n t r a c t e d i n a s p e c i f i c way. F o r example: i s a t e r m . p a r t i c u l a r *f or V . F o r ^ABco and H'^ EFXH a r e f a c t o r s . - t h e "Name" of a f a c t o r i d e n t i f i e s t h e f a c t o r (and t h u s t h e number of i n d i c e s i t h a s ) but n o t how i t i s c o n t r a c t e d . The name of a te r m i s made up of t h e names of i t s f a c t o r s . In t h i s p rogram, t h e o r d e r e d p a i r w h i c h i s t h e number o f d o t t e d and u n d o t t e d i n d i c e s s e r v e s as t h e name of a f a c t o r . F o r example: n/ftBcoeF /6\ T X y has t h e name ^j,] b e c a u s e i t has 6 d o t t e d and 2 u n d o t t e d i n d i c e s , W u;AB u/CD£F H T f t B t O T EFXH 1 Y - " F a c t o r " r e f e r s t o one example: The e s s e n t i a l p i e c e s o f i n f o r m a t i o n w h i c h t h e program u s e s t o s p e c i f y a s p i n t e n s o r a r e : 1) t h e name of t h e term ( t h i s i s s t o r e d i n t h e m a t r i x "NM"); 2) t h e c o n t r a c t i o n s of e a c h of t h e d o t t e d and u n d o t t e d i n d i c e s ( t h e s e a r e s t o r e d i n t h e m a t r i c e s TM and TM r e s p e c t i v e l y ) ; and 3) t h e c o e f f i c i e n t m u l t i p l y i n g t h e t e r m ( t h e n u m e r a t o r and d e n o m i n a t o r a r e s t o r e d i n t h e m a t r i x " C F F " ) . T h e s e m a t r i c e s c o u l d be, f o r example: CFF= [|] TM = U 3 3 3 3 S 3 J NM = [ ! 5 z ] (8.1) 79 and would r e p r e s e n t : i f V A (r/BCDEFG (8.2) 5 ' ABCO T EFC T * y E a c h s e t of m a t r i c e s CFF, TM, TM, and NM c o r r e s p o n d s t o e x a c t l y one s p i n - t e n s o r . To p r o v e t h i s , and t o d e f i n e t h e new n o t a t i o n , t h e f o l l o w i n g s t e p - b y - s t e p a l g o r i t h m i s p r e s e n t e d w h i c h e s t a b l i s h e s t h e p r o c e d u r e f o r t r a n s l a t i n g f r o m s t a n d a r d n o t a t i o n t o m a t r i x n o t a t i o n and back a g a i n . The example s p i n - t e n s o r w h i c h w i l l be u s e d i s : 5 *ABCO T XEFC, I Y 1) T r a n s l a t i o n o f CFF i s t r i v i a l - t h e t o p number i s t h e n u m e r a t o r and t h e b o t t o m number t h e d e n o m i n a t o r of t h e c o e f f i c i e n t of t h e s p i n - t e n s o r . In t h e example: CFF = ( | ) 2) " C a l c u l a t i o n " o f NM i s s t r a i g h t f o r w a r d from t h e above d e f i n i t i o n of t h e meaning of e a c h p a i r ( £ ). F o r t h e above example: NM = T 4 5 5"! L o 1 1J 3) To " c a l c u l a t e " TM and TM p r o c e e d as f o l l o w s : -a) W r i t e t h e s p i n - t e n s o r i n s t a n d a r d n o t a t i o n w i t h i t s f r e e i n d i c e s r a i s e d . F o r t h e example w h i c h i s b e i n g u s e d : ^ .L I f u ; A B X u ^ C D E F Y G 5 ' A B C O • E F G T -b) Wri_te t h e s p i n - t e n s o r as t h e c o n t r a c t i o n o f t h e V and H* f a c t o r s w i t h a s e r i e s o f and £ * e f a c t o r s . The example i s t h u s w r i t t e n : S ' ABCO l l J E F Q H 'RVTUVW C C f c t - C C C C , - c ) R e p l a c e e v e r y i n d e x o f t h e £ A B f a c t o r s w i t h a number e q u a l t o t h e number o f t h e f a c t o r t o w h i c h t h a t i n d e x i s c o n t r a c t e d . R e p l a c e f r e e i n d i c e s w i t h t h e number 24. L e a v e t h e d o t s o v e r t h e d o t t e d i n d i c e s as shown below: -d) D r o p t h e and H* symbols and w r i t e e a c h £ ' s n u m b e r - i n d i c e s as a v e r t i c a l p a i r , t h e f i r s t i n d e x on t h e bottom, as i n t h i s example: * * 3 3 3 3 3 3] L' » 24- I \ 2 I 2.4 i ] -e) S e p a r a t e out t h e u n d o t t e d numbers and c a l l t h i s m a t r i x "TM": 2 2 3 - 3 1 3 3"| » -2-4 I I o 2 2 2 4 ] - f ) S e p a r a t e out t h e d o t t e d numbers, s t r i p them of t h e i r d o t s , and c a l l t h i s m a t r i x "TM": TM = [1] 4) To t r a n s l a t e back from TM and TM i n t o s t a n d a r d n o t a t i o n , p r o c e d e as f o l l o w s , ( u s i n g t h e TM, TM, NM, and CFF w h i c h was u s e d as an example a b o v e ) . -a) W r i t e t h e symbols f and 4* a s s p e c i f i e d by t h e m a t r i x NM, and l e a v i n g room f o r t h e r e q u i r e d i n d i c e s t o be w r i t t e n i n . F o r t h e example, w r i t e : N M = [ t ? ? ] = > Y_„_Y : y --b) P r o c e e d i n g from l e f t t o r i g h t i n t h i s s t r i n g , p u t one l e t t e r ( f i r s t A, t h e n B, t h e n C, ... e t c . ) i n e a c h u n d o t t e d s l o t of e a c h f a c t o r . F o r t h e e x a m p l e: ¥ Y • v 81 - c ) r e p e a t t h i s f o r t h e d o t t e d i n d i c e s : -d) P r o g r e s s i v e l y r e p l a c e e v e r y number i n TM w i t h a l e t t e r ( f i r s t A, t h e n B,... e t c . ) , r e p l a c i n g a l l t h e numbers "1" f i r s t , t h e n t h e numbers "2", e t c . R e p l a c e t h e number 24 w i t h X, Y, o r Z as r e q u i r e d : T M _ f 2 2 2 3 3 3 3 3 "]_*> TEFG, TK.L MNl r n "L« > 2.4- » i 2 X 24-] l_A8 X C D H I Y J -e) Do t h e same f o r TM, but p u t a d o t o v e r e v e r y l e t t e r : - f ) Take e v e r y v e r t i c a l p a i r i n TM and TM and make i t t h e i n d i c e s of an £ f t S m a t r i x . Put t h e lo w e r e l e m e n t of t h e p a i r on t h e l e f t i n £* B: TM =» £ A e £ 8 F £ XCl €cz £0 K £H L e IM -g) C o n t r a c t t h e s e £ ' s w i t h t h e Y f a c t o r s d e r i v e d i n s t e p s b) and c) a b o v e . T h i s i s now s t a n d a r d n o t a t i o n : A&CD 'EFGHXA ' J|tLM N B t- fc 6 t t t , £ t . = U) mABX u/COHTYA ' A B C O ' H I A The same i n f o r m a t i o n as i s s t o r e d i n CFF, TM, TM, and NM c o u l d a l s o be s t o r e d i n t h e 2x40 a r r a y : ?. 1 1. 1 T> % "*> Z 0 0 0 0 0 0 0 2 . 0 0 0 0 0 0 0 0 0 0 0 , , . ,5" » 11*1 | 0 o o o o o o i o o o o o o o o o o o * * * ^ — . _ — T M TNI (8.3) D O O O o ] •v o o o + s s o o o o o o l o o oJvo > \ o o o o  OJ NM where column 1 i s r e s e r v e d f o r CFF, c o l u m n s 2 t o 16 f o r TM, co l u m n s 17 t o 31 f o r TM, and c o l u m n s 32 t o 40 f o r NM. Z e r o e s a r e u s e d t o f i l l i n t h e unused s p a c e s i n t h i s m a t r i x as shown i n t h e 82 above examp l e . Read TM and TM a s s e t s of v e r t i c a l p a i r s (*) where e a c h p a i r i n d i c a t e s a c o n t r a c t i o n between f a c t o r a and f a c t o r b. T h e s e p a i r s i n TM and TM c o r r e s p o n d d i r e c t l y t o t h e i n d i c e s of £ w m a t r i c e s a c c o r d i n g t o t h e above method o f t r a n s l a t i o n from s t a n d a r d n o t a t i o n t o t h e m a t r i x n o t a t i o n . N o te i n t h e example t h a t t h e i n d i c e s of f a c t o r number 1 a r e a l l s t o r e d as " 1 " s . T h i s f e a t u r e of t h e n o t a t i o n i m p l i c i t l y e n s u r e s t h a t t h e i n d i c e s a r e s y m m e t r i c . C o n t r a c t i o n s t o 24 h o l d t h e p l a c e f o r t h e f r e e i n d i c e s X and x. In t h e p r o g r a m s , a s p i n - t e n s o r i s u s u a l l y s t o r e d i n an a r r a y of t h e form shown i n ( 8 . 3 ) , b ut when c a l c u l a t i o n s a r e done, t h e 2 x 4 0 a r r a y i s s p l i t i n t o t h e f o u r a r r a y s (CFF, NM, TM, TM) by t h e f u n c t i o n ^UNRAVV ( f u n c t i o n B.1) t o f a c i l i t a t e t h e c a l c u l a t i o n s . Once t h e c a l c u l a t i o n s a r e c o m p l e t e , t h e terms a r e r e t u r n e d t o t h e 2 x 40 a r r a y f o r m a t u s i n g t h e f u n c t i o n VRERAVV ( f u n c t i o n B . 2 ) . The f u n c t i o n VWRITEV ( f u n c t i o n B.3) t r a n s l a t e s t h e a r r a y s i n t o t h e more p a l a t a b l e form u s e d t o p r e s e n t t h e r e s u l t s i n a p p e n d i x C. 83 8.3 C a n o n i c a l Form To e n s u r e t h a t t h e computer c o u l d a l w a y s know whether or not two s p i n - t e n s o r s were e q u a l , i t was n e c e s s a r y t o e s t a b l i s h a c a n o n i c a l f o r m s u c h t h a t t h e r e was e x a c t l y one c a n o n i c a l way t o w r i t e any g i v e n s p i n - t e n s o r . T h i s work i s i n c o m p l e t e , i n t h a t t h e " c a n o n i c a l f o r m " w h i c h was d e v e l o p e d , and i s d e s c r i b e d below, has not been p r o v e n t o be a c o m p l e t e c a n o n i c a l f o r m f o r s p i n - t e n s o r s of a l l o r d e r s . The r e q u i r e m e n t s f o r t h e c a n o n i c a l form a r e t h a t : 1) I t must n o t d i s c a r d a s " n o n - c a n o n i c a l " any s p i n -t e n s o r s w h i c h c a n n o t be w r i t t e n as a sum of o t h e r s p i n - t e n s o r s w h i c h a r e c a n o n i c a l . 2) I t must d i s c a r d a s " n o n - c a n o n i c a l " any s p i n -t e n s o r s w h i c h can be w r i t t e n a s a sum of o t h e r s p i n -t e n s o r s w h i c h a r e c a n o n i c a l . The f i r s t of t h e s e r e q u i r e m e n t s c a n be p r o v e n f o r t h e c a n o n i c a l form w h i c h i s d e s c r i b e d below. The s e c o n d has not been p r o v e n and i t i s p o s s i b l e t h a t s p i n - t e n s o r s o f h i g h d e g r e e s w i l l be l i n e a r l y d e p e n d e n t i n some way w h i c h t h i s c a n o n i c a l f o r m w i l l n ot " s e e " . T h i s i s a p o s s i b l e s u b j e c t of f u r t h e r r e s e a r c h . T h i s c a n o n i c a l f o r m may not p r o v i d e a " m i n i m a l " l i s t o f s p i n - t e n s o r s o f a c e r t a i n o r d e r , b u t i t does p r o v i d e a v e r y much s h o r t e n e d and " c o m p l e t e " l i s t . The l i s t s o f s p i n - t e n s o r s up t o d e g r e e s e v e n have been t h o r o u g h l y c h e c k e d and t h e r e a r e no s p i n -t e n s o r s i n t h e s e l i s t s w h i c h a r e n o t l i n e a r l y i n d e p e n d e n t . The c a n o n i c a l f o r m i s d e f i n e d by t h e f o l l o w i n g r u l e s , w h i c h a r e w r i t t e n i n t h e i r o r d e r o f p r i o r i t y . 84 [ I ] In s t a n d a r d n o t a t i o n , t h e o r d e r i n w h i c h t h e f a c t o r s a r e w r i t t e n i s a r b i t r a r y , so we d e f i n e a c o n v e n t i o n s u c h t h a t : a) f a c t o r s w i t h more d e r i v a t i v e s a r e t o t h e r i g h t and b) f a c t o r s w i t h H5 a r e t o t h e r i g h t o f s i m i l a r f a c t o r s w i t h f . F o r e x a m p le: ABCO ' COEF ' A 8 EF ' v o . * / and U» U) HjA&CO n; ABCOEF 6F (8.5) TASCO ' EF ABCOEF 1 ' a r e n o t c a n o n i c a l , but t h e f o l l o w i n g i s : ^ 6 c o ^ 6 % F A 6 ' ° M r A B C O E F 6 F (8-6) I t i s i m m e d i a t e l y c l e a r t h a t any s p i n - t e n s o r c an be w r i t t e n i n t h i s c a n o n i c a l f o r m . [ I I ] E a c h p a i r i n TM and TM c a n be v e r t i c a l l y i n v e r t e d , c h a n g i n g o n l y t h e s i g n of CFF ( t h i s c o r r e s p o n d s t o r e p l a c i n g £^B ^ £. 6 A ) , s o : a) e a c h column i s a l w a y s w r i t t e n w i t h t h e s m a l l e r numbers i n t h e t o p row. The o r d e r i n w h i c h t h e co l u m n s a r e w r i t t e n i s a r b i t r a r y , so b) t h e p a i r s a r e a r r a n g e d i n a s c e n d i n g o r d e r o f t h e t o p e l e m e n t o f e a c h p a i r . 85 Thus 2 i i l l ) 2.4 2 2 Z 2 2 (8.7) and » » I l 1 1 1 2 . I Z*\ (8.8) a r e n o n - c a n o n i c a l forms o f I 1 \ I I 2. 2. 2. 2. 1 (8.9) H e r e , t o o , i t i s c l e a r t h a t e v e r y s p i n - t e n s o r can be w r i t t e n a c c o r d i n g t o t h i s c a n o n i c a l f o r m . [ I l l ] In any t e r m w i t h two £ A 8 ' s , t h e p r o d u c t £ A , J £ C 0 can be r e p l a c e d a c c o r d i n g t o ( 7 . 1 5 ) . S p e c i f i c a l l y , i t i s p o s s i b l e t o ' r e p l a c e any t e r m where one c o n t r a c t i o n " e n c l o s e s " a n o t h e r , s u c h a s : (where t h e D - c o n t r a c t i o n " e n c l o s e s " t h e E - c o n t r a c t i o n b e c a u s e t h e E - c o n t r a c t i o n i s a c r o s s two f a c t o r s w h i c h a r e p o s i t i o n e d between t h e D - c o n t r a c t i ' o n ' s two f a c t o r s ) . Any t e r m s u c h as (8.10) can be r e p l a c e d w i t h two o t h e r t e r m s , one w i t h t h e c o n t r a c t i o n s d i s j o i n t , and t h e o t h e r w i t h them p a r t i a l l y d i s j o i n t . T h i s i s b e s t i l l u s t r a t e d by an example. The s p i n -T ABCO T E ' FG.H » (8.10) 86 t e n s o r "T" i n (8.10) c a n , a c c o r d i n g t o ( 7 . 1 5 ) , be r e p l a c e d by: T~r . R C OY" 8 C D>W, u4'" s u (8.,,) ABCO 1 T E F G H \> u/ABC 'ABCO ' E T>GH ~ ^ A B  1 c T An e x a m i n a t i o n o f (8.10) and (8.11) makes i t c l e a r t h a t no c o n t r a c t i o n s w i l l be e n c l o s e d a f t e r we have t r a n s f o r m e d from (8.10) t o (8.11) w h i c h were not e n c l o s e d b e f o r e t h a t t r a n s f o r m a t i o n . T h i s e n s u r e s t h a t t r a n s f o r m a t i o n s o f t h e above t y p e w i l l a l w a y s y i e l d s p i n - t e n s o r s w i t h fewer " e n c l o s e d " c o n t r a c t i o n s , so t h e p r o c e s s c a n be c o n t i n u e d u n t i l a l l e n c l o s e d c o n t r a c t i o n s a r e removed. We know, t h e r e f o r e , t h a t t h e r e need n e v e r be any terms where one c o n t r a c t i o n e n c l o s e s a n o t h e r . T h i s becomes t h e b a s i s of t h e c a n o n i c a l r u l e number I I I : a) No terms a r e a l l o w e d where one c o n t r a c t i o n e n c l o s e s a n o t h e r . T h i s e l e m e n t of t h e c a n o n i c a l f o r m t a k e s on a p a r t i c u l a r l y s i m p l e form i n t h e m a t r i x n o t a t i o n f o r s p i n - t e n s o r s : b) The b o t t o m row o f e v e r y m a t r i x TM o r TM must c o n s i s t of numbers w h i c h a r e i n a s c e n d i n g o r d e r f r o m l e f t t o r i g h t (remember t h a t t h e t o p row i s a l r e a d y i n a s c e n d i n g o r d e r b e c a u s e of r e q u i r e m e n t I I , a b o v e ) Thus T M . [ i l ' t «1 5 » »] (8.,2) i s n o t i n c a n o n i c a l f o r m but t h e f o l l o w i n g i s : 87 (8.13) -L » I ' » 3 3 t- Z 2 Z 4- 4-I t may not be i m m e d i a t e l y o b v i o u s t h a t a) and b) a r e e q u i v a l e n t . To see i t , c o n s i d e r j u s t two p a i r s o f a m a t r i x "TM": where f , g, h, and j r e p r e s e n t i n t e g e r s . By t h e p r e v i o u s c a n o n i c a l r e q u i r e m e n t s , ( f < g ) , ( h < j ) , and ( f < h ) . Assume t h a t r u l e b) i s b r o k e n i n t h i s TM, so ( g > j ) . T h i s means t h a t t h e c o n t r a c t i o n (^ ) has e n c l o s e d t h e c o n t r a c t i o n ( ) , so r u l e a) above has been b r o k e n . In o r d e r t o t r a n s l a t e a m a t r i x TM s u c h as (8.12) i n t o c a n o n i c a l form, we need a v e r s i o n of (7.15) i n t h i s m a t r i x n o t a t i o n . E q u a t i o n (7.15) i m p l i e s t h a t i n m a t r i x - n o t a t i o n , a m a t r i x of t h e form: (8.14) c a n be w r i t t e n as t h e sum of two m a t r i c e s of t h e fo r m : (8.15) No p r o o f of t h i s i s n e c e s s a r y - i t i s a s i m p l e t r a n s l a t i o n o f (7.15) i n t o m a t r i x f o r m a c c o r d i n g t o t h e d e f i n i t i o n of t h e m a t r i x n o t a t i o n g i v e n i n s e c t i o n 8.2. 88 [ I V ] Any two f a c t o r s , o r g r o u p s o f f a c t o r s , w i t h t h e same names can be f r e e l y i n t e r c h a n g e d . As a r e s u l t , an h i e r a r c h y i s e s t a b l i s h e d h e r e , t o o , s p e c i f y i n g t h a t t h e g r o u p o f f a c t o r s w i t h more c o n t r a c t i o n s t o i t s r i g h t (and o u t s i d e t h e two g r o u p s i n q u e s t i o n ) would be w r i t t e n t o t h e r i g h t . F o r example, T A B C O » T E p c i H T (8.16) would be w r i t t e n i n p r e f e r e n c e t o : <+> U> o/ABCO M/EF6 H ' AfcCD 'EFGH ' T (8.17) b e c a u s e o f t h i s r u l e as a p p l i e d t o t h e two c e n t r a l f a c t o r s . The f u n c t i o n V E X A D V V ( f u n c t i o n B.18) t e s t s whether t h e r e would be any a d v a n t a g e t o e x c h a n g i n g two s u c h s i m i l a r g r o u p s of f a c t o r s . C l e a r l y any g i v e n s p i n - t e n s o r c an be made t o comply w i t h t h i s r u l e of t h e c a n o n i c a l form, b e c a u s e t h e r u l e w i l l not r e j e c t a s p i n - t e n s o r u n l e s s i t can be shown e x p l i c i t l y t h a t t h e s p i n -t e n s o r c a n be r e - w r i t t e n i n a b e t t e r way. The above f o u r r u l e s f o r c a n o n i c a l form a r e n o t enough t o e n s u r e t h a t a l l s p i n - t e n s o r s g e n e r a t e d w i l l be l i n e a r l y i n d e p e n d e n t . As one p r o c e e d s t o h i g h e r and h i g h e r o r d e r , t h e i n t e r - r e l a t i o n s h i p s between s p i n - t e n s o r s become more s u b t l e . Some examples a r e g i v e n below. C o n s i d e r t h e 6 t h - d e g r e e s p i n - t e n s o r "T" 89 w h i c h i s c a n o n i c a l by t h e above r u l e s . When one e x c h a n g e s t h e f i r s t two f a c t o r s , however, i t becomes c l e a r t h a t T i s a n t i s y m m e t r i c i n [DE] T = -^ 6 c eT A 6 c„r% 6 l, r e F O H I t c a n t h e r e f o r e be w r i t t e n : (8.19) Which p r o v e s t h a t (8.18) and (8.20) a r e not l i n e a r l y i n d e p e n d e n t . A s e c o n d example i s t h e 6 t h - d e g r e e s p i n - t e n s o r "U": " • ' W i ^ W f , , " * 4 (8.2.) B e c a u s e of t h e s i n g l e c o n t r a c t i o n o v e r A, U must become n e g a t i v e upon exchange o f (BCD) w i t h ( E F G ) : U = " ^ A E r i ? * < 8 - 2 2 ) w h i c h c o n t r a d i c t s t h e r e q u i r e d symmetry o f t h e l a s t f a c t o r . The t e r m must, t h e r e f o r e , be z e r o . 90 D e f i n e , t h e r e f o r e , t h e f i f t h e l e m e n t of c a n o n i c a l form: [V] E a c h t e r m must be s u c h t h a t i f any two s i m i l a r f a c t o r s a r e e x c h a n g e d , t h e te r m w i l l be uncha n g e d . The p r o b l e m of d e t e r m i n i n g w h i c h terms were c a n o n i c a l i n th e f i f t h s e n s e above was h a n d l e d w i t h t h e APL f u n c t i o n s VSYM7 and VZTERMSv* ( f u n c t i o n s B.35 and B . 3 6 ) . The f u n c t i o n VSYMV c a l c u l a t e s e x p l i c i t l y t h e terms w h i c h r e s u l t f r o m e x c h a n g i n g two of t h e f a c t o r s . The f u n c t i o n V2TERMS"v a t t e m p t s a l l p o s s i b l e e x c h a n g e s o f p a i r s of f a c t o r s , and r e p l a c e s t h e o r i g i n a l t e r m w i t h t h e t e r m t o w h i c h i t i s p r o p o r t i o n a l ( p e r h a p s z e r o ) i f s u c h a t e r m r e s u l t s from t h e exchange. F o r example, ^Z TERMS v" would r e p l a c e : T = W O E H Y * B % G * T D E V ( 8 ' 2 3 ) w i t h : T = - L & = 4 - V . u;FT8C HyDEFC, # ( 8 24) L a t e r , when VZTERMSv" a n a l y s e s Q, w i l l i t t r y t o r e p l a c e i t w i t h T, and t h u s i n i t i a t e an e n d l e s s c y c l e ? The answer i s "no"; b e c a u s e o f t h e 4 t h r u l e of c a n o n i c a l f o r m ( a b o v e ) , t h e r e p l a c e m e n t s w i l l o n l y move i n one d i r e c t i o n . In e f f e c t , t h i s means t h a t when two terms a r e p r o p o r t i o n a l (and t h e c o n s t a n t of p r o p o r t i o n a l i t y w i l l a l w a y s be o n e - h a l f ) t h e l a r g e r one ( i n t h e 91 above example "Q"), w i l l a l w a y s be t h e c a n o n i c a l f o r m . The above f i v e r u l e s remove v e r y many of t h e l i n e a r l y -d e p e n d e n t s p i n - t e n s o r s , w h i c h makes t h e l e n g t h s o f t h e l i s t s of s p i n - t e n s o r s much s h o r t e r i n d e e d . The a u t h o r has c a r e f u l l y c h e c k e d t h e l i s t s up t o d e g r e e s e v e n and i s c o n v i n c e d t h a t t h e s e a r e m i n i m a l l i s t s . F o r h i g h e r o r d e r s i t must be a d m i t t e d t h a t new s y m m e t r i e s may become i m p o r t a n t w h i c h t h e above c a n o n i c a l r u l e s c a n n o t d i s c e r n . T h i s would be a s u b j e c t f o r f u r t h e r r e s e a r c h . 8.4 C a l c u l a t i o n of The D e r i v a t i v e s A d e r i v a t i v e s u c h a s : * s M l V 0 ^ \ t a ^ C D " & " ) ( 8 - 2 ' 5 ) i s more d i f f i c u l t t o c a l c u l a t e t h a n i t f i r s t a p p e a r s t o be. T h i s i s p r i n c i p a l l y b e c a u s e t h e i n d i c e s of t h e d e r i v a t i v e o p e r a t o r a r e n ot s y m m e t r i c w i t h t h e i n d i c e s of t h e f a c t o r upon w h i c h i t o p e r a t e s . The c a l c u l a t e d r e s u l t , however, must c o n s i s t o n l y of f a c t o r s w h i c h have f u l l y s y m m e t r i c i n d i c e s . The d e t a i l s o f t h e a l g o r i t h m w h i c h was u s e d t o c a l c u l a t e t h e s e d e r i v a t i v e s i s p r e s e n t e d i n A p p e n d i x A, s e c t i o n A.3. The f o l l o w i n g i s a v e r y b r i e f summary o f t h i s a l g o r i t h m . 92 1) The p r o d u c t r u l e i s a p p l i e d : ~ V O x i TABCO J T EFGH 1 ^COEFGH (8.26) ' ABCO I °XX ' EFG H / T + V <4>*& . ( \ U/ • ABCO T EFGH V OXX » CDEFGH^ 2) The f u l l y s y m metric p a r t o f e a c h of t h e above t e r m s a r e added t o g e t h e r a s t e r m s o f t h e d e r i v a t i v e . R = W . W AB COEFG H ^ TA6C0XX ' EFGH • + U ivAB U ; C O E F G » A L A B C D « E F & X X H T (8.27) * T A 6 t « T E F G H 1 X X + c r\ti$^w\n*et»"i t p a r t 3) The a n t i s y m m e t r i c p a r t of e a c h o f t h e s e terms i s c a l c u l a t e d and t h e r e s u l t i n g terms a r e added t o (8.27) t o c o m p l e t e t h e c a l c u l a t i o n of t h e d e r i v a t i v e . T h i s a n t i s y m m e t r i c p a r t i s c a l c u l a t e d by s p l i t t i n g e a c h of t h e H » A 6 C — W f a c t o r s i n t o i t s n o t -f u l l y - s y m m e t r i c components a c c o r d i n g t o t h e i d e n t i t y ( 5 . 3 3 ) . T h i s f r e e s c e r t a i n i n d i c e s i n t h e V f t ( 6 c " A ^ f a c t o r s t o be a n t i s y m m e t r i z e d w i t h t h e i n d i c e s o f t h e a p p l i e d d e r i v a t i v e : 93 4) i * \ ± \ C * \ (8.28) T h e s e a n t i s y m m e t r i z e d i n d i c e s of d e r i v a t i v e s a r e t h e n c o n t r a c t e d u s i n g t h e i d e n t i t y ( 5 . 3 4 ) : d [ K ^ C ] ' OEfC, ) Oj 0 ToEFC | t -XC F i n a l l y , t h e R i c c i i d e n t i t i e s ( s e e e q u a t i o n s (7.5) t o ( 7 . 8 ) ) a r e u s e d t o r e p l a c e e a c h ^ " i 7 * * w i t h a new f a c t o r f X H A B G r y ^ A B f k e e p i n g i n mind t h a t t h i s r e p l a c e m e n t w i l l i n t r o d u c e n o t - f u l l y - s y m m e t r i c i n d i c e s i n t o t h e f a c t o r w h i c h f o l l o w s t h e new f a c t o r . T h i s p r o c e s s i s r e p e a t e d u n t i l a l l a n t i s y m m e t r i c p a r t s of t h e t e r m s have been s i m p l i f i e d by t h e R i c c i i d e n t i t i e s , o r have been f o u n d t o be z e r o . The r e s u l t i n g l i s t o f t e r m s i n t h e d e r i v a t i v e i s c o n s o l i d a t e d by p u t t i n g e a c h member of i t i n t o c a n o n i c a l form and a d d i n g l i k e t e r m s . 94 C h a p t e r 9 R e s u l t s of Computer S e a r c h F o r P - C h a r q e s The f o l l o w i n g c h a p t e r p r e s e n t s t h e r e s u l t s o f t h e c omputer s e a r c h f o r 1-, 2-, and ' 3 - c h a r g e s . 9.1 C o m p l e t e l i s t s of I n d e p e n d e n t S p i n - T e n s o r s A p p e n d i x C c o n t a i n s c o m p l e t e l i s t s of e v e r y s p i n - t e n s o r t e r m from d e g r e e 1 t h r o u g h 9 w h i c h c o n f o r m s t o t h e c a n o n i c a l form d e s c r i b e d above. C a r e has been t a k e n t o e n s u r e t h a t e v e r y s p i n - t e n s o r l i s t e d up t o d e g r e e - 7 i s l i n e a r l y i n d e p e n d e n t . F o r d e g r e e s 8 and 9 t h i s c a n n o t be a s s e r t e d w i t h t h e same c o n f i d e n c e b e c a u s e t h e r e i s t h e p o s s i b i l i t y of two t e r m s b e i n g l i n e a r l y -d e p e n d e n t due t o some h i g h e r - o r d e r s y m m e t r i e s w h i c h were not d i s c e r n e d . C a r e has been t a k e n t o a v o i d t h i s p o s s i b i l i t y , but e x p e r i e n c e w i t h t h e l o w e r o r d e r s i n d i c a t e d t h a t e v e r y move t o h i g h e r o r d e r s made i t n e c e s s a r y t o add a r e f i n e m e n t t o t h e c a n o n i c a l form, so i t seem t o be f o o l i s h t o a s s e r t t h a t t h e p r e s e n t c a n o n i c a l form has no f l a w s a t h i g h e r d e g r e e s . The l i s t s o f d e g r e e 8 and 9 s h o u l d t h e r e f o r e be t a k e n as c o m p l e t e ( a l t h o u g h n o t n e c e s s a r i l y m i n i m a l ) l i s t s whose l e n g t h s g i v e an u p p e r b o u n d a r y t o t h e number o f i n d e p e n d e n t s p i n - t e n s o r t e r m s of t h o s e d e g r e e s . 95 An immediate o b s e r v a t i o n i s t h a t no t e r m s e x i s t where t h e rank and d e g r e e add t o an odd number. T h i s f o l l o w s d i r e c t l y from t h e r e q u i r e m e n t t h a t t h e t o t a l number of i n d i c e s i n a s p i n -t e n s o r t e r m be an even number. One s u r p r i s i n g r e s u l t o f t h e c a l c u l a t i o n s i s t h e s m a l l number o f terms of d e g r e e 4 and l e s s . T a b l e 9.1 l i s t s t h e r a n k s and d e g r e e s where no t e r m s were f o u n d . Rank D e g r e e 0 0 2 0 1 3 2 2 2 4 T a b l e 9.1 Rank and Degree of S p i n - T e n s o r s w i t h No N o n - z e r o Terms The g r a p h i n f i g u r e 9.1 shows t h e number o f i n d e p e n d e n t t e r m s o f e a c h rank as a f u n c t i o n of t h e i r d e g r e e . Rank-0, -1, and -2 s p i n - t e n s o r s a r e a r e a l w a y s t h e sum o f a p a i r o f c o n j u g a t e t e r ms from t h e l i s t s i n A p p e n d i x C. Rank-3 and rank-4 c a n be f o u n d from t h e l i s t s by s u b t r a c t i n g p a i r s o f c o n j u g a t e rank-1 or rank - 0 t e r m s . T h i s e x p l a i n s why t h e number of rank-3 s p i n - t e n s o r s d i f f e r s f r o m t h e number o f rank-1 s p i n - t e n s o r s , as i s v i s i b l e on t h e g r a p h . Terms from t h e l i s t w h i c h a r e t h e r e own 96 Si • zs A t i £ 7^ 20-J o- IS £ n z: 5^ . o -i V i Xl - *• ge *>) _ l-Z-i (Z-clnr-yts)'. SYMBOLS . • RcmW- 0 • Rowk-.l X • Remk-3 1 r 1. 2.31.3;....4 _ 7';; sr 6.;.*.7.;.?. Degree of S p ' m t e * s o r s — — i s : F i gure 9.1: Graph of the Number of Spin-Tensors of Each Degree 97 c o n j u g a t e s ( t h a t i s , terms w h i c h a r e r e a l ) and t erms w h i c h a r e (-1) t i m e s t h e i r c o n j u g a t e s ( t e r m s w h i c h a r e p u r e l y i m a g i n a r y ) w i l l i n one c a s e add t o z e r o and i n t h e o t h e r c a s e w i l l d o u b l e . Note t h a t , w h e r e v e r t h e number o f terms d e c r e a s e s a s t h e d e g r e e i n c r e a s e s , t h e r e w i l l be a c l o s e d s p i n - t e n s o r . T h i s c a n be u n d e r s t o o d most d i r e c t l y when one t h i n k s o f t h e d e r i v a t i v e o p e r a t i o n a s a m a t r i x t r a n s f o r m a t i o n f r o m t h e d e g r e e - n s p a c e t o t h e d e g r e e - ( n - M ) s p a c e ( t h i s i s t h e r e p r e s e n t a t i o n w h i c h w i l l be u s e d i n s e c t i o n 9.2). When t h e number of t e r m s d e c r e a s e s from l e f t t o r i g h t on t h e g r a p h , t h e c o r r e s p o n d i n g m a t r i x w i l l have more rows t h a n c o l u m n s , so some c o m b i n a t i o n o f t h e rows must add t o z e r o . T h i s i s o b s e r v e d t o be t h e c a s e 1 from d e g r e e - 5 t o r a n k - 2 d e g r e e - 6 s p i n - t e n s o r s . I t w i l l be shown i n s e c t i o n ( 9 . 3 ) , however, t h a t t h i s c l o s e d s p i n - t e n s o r t h a t i s f o u n d a t d e g r e e - 5 i s e x a c t and hence i s n o t a 1 - c h a r g e . No o t h e r d i p s i n t h e g r a p h a r e v i s i b l e f o r h i g h e r o r d e r s , and t h e t r e n d seems t o be t o w a r d r a p i d l y - i n c r e a s i n g numbers of terms a t t h e h i g h e r d e g r e e s . 9.2 T a b l e s of D e r i v a t i v e s of S p i n - T e n s o r Terms The d e r i v a t i v e o f e v e r y s p i n - t e n s o r t e r m l i s t e d i n A p p e n d i x C up t o d e g r e e - 6 was c a l c u l a t e d by t h e APL p r o g r a m t h a t i s d i s c u s s e d i n d e t a i l i n A p p e n d i c e s A and B. D e r i v a t i v e s o f . o r d e r 7 were a t t e m p t e d , but c a l l e d f o r more computer t i m e and s p a c e t h a n seemed w o r t h w h i l e . The r e s u l t s came f r o m t h e computer i n th e form shown i n t h e example o u t p u t l i s t shown i n f i g u r e 9.2. DEL OF :. 1/1 P S I ( A B C D ) P S I ( A B C D ) P S I ( E F G H ) P S I ( E F G H ) EQUALS: 12/1 P S I ( A B C D ) P S I ( A B E F ) P S I ( C D G H ) P S I ( E F G H X x ) "4/2 P S I ( A B C D ) P S I ( A B C D ) P S I ( E F G H ) P S I ( E F G H X x ) DEL OF : 1/1 p s i ( a b c d ) p s i ( a b c d ) p s i ( e f g h ) p s i ( e f g h ) EQUALS: 12/1 p s i ( a b e d ) p s i ( a b e f ) p s i ( c d g h ) p s i ( X e f g h x ) "4/2 p s i ( a b e d ) p s i ( a b e d ) p s i ( e f g h ) p s i ( X e f g h x ) F i g u r e 9.2: Example Computer P r i n t o u t o f Terms i n a D e r i v a t i v e 99 (The t e r m s l i s t e d i n t h i s o u t p u t a r e meant t o be summed t o g i v e t h e t o t a l d e r i v a t i v e ) . The d e r i v a t i v e s were t h e n t r a n s l a t e d i n t o t h e m a t r i x f o r m d i s c u s s e d below, f o r b r e v i t y and e a s e of m a n i p u l a t i o n . The t a b l e s w h i c h f o l l o w show, i n m a t r i x f o r m , t h e d e r i v a t i v e s o f s p i n - t e n s o r s i n terms of t h e b a s i s s e t o f s p i n -t e n s o r s o f t h e n e x t h i g h e r o r d e r . The c o e f f i c i e n t o f e a c h t e r m i n a d e r i v a t i v e i s l i s t e d under t h e Roman-numeral i d e n t i f i c a t i o n number f o r t h a t t erm, and t o t h e r i g h t of t h e i d e n t i f i c a t i o n number f o r t h e t e r m whose d e r i v a t i v e i s b e i n g f o u n d . F o r example, row 4 of t a b l e 9.4 i n d i c a t e s t h a t t h e d e r i v a t i v e of term (6/0/11 + 6 / 0 / I I * ) i s e q u a l t o : T h i s example a l s o c o r r e s p o n d s t o t h e computer p r i n t o u t shown i n f i g u r e 9.2. = ( 12)»( ( 7/1 / I I ) + ( 7/1/11 * ) ) + ( - 2 W ( 7/1 /111 ) + ( 7/ 1 /111 * ) ) w h i c h means i n s t a n d a r d n o t a t i o n t h a t : ( 8 . 1 ) The m a t r i c e s f o r t h e d e r i v a t i v e s a r e p r e s e n t e d on t h e f o l l o w i n g p a g e s . 100 A) Rank-0 S p i n - T e n s o r s : (3/1/1) +(3/1/1*) (2/0/1) +(2/0/1*) 2 T a b l e 9.2 D e r i v a t i v e s of Rank-0 Degree-2 S p i n - T e n s o r s (5/1/1) +(5/1/1*) ( 5 / 1 / I I ) +(5/1/11*) (4/0/1) +(4/0/1*) 3 0 (4 / 0 / 1 I ) + (4/0/11*) -2 2 T a b l e 9.3 D e r i v a t i v e s of Rank-0 Degree-4 S p i n - T e n s o r s (7/1/1)+ (7/1/1*) (7/1/11)+ (7/1/II*) (7/1/111)+ (7/1/III*) (7/l/IV)+ (7/1/IV*) (7/l/V)+ (7/1/V*) (7/1/VIJ+ (7/1/VI*) (7/1/VID+ (7/1/VII*) (7/l/VIII)+ (7/1/VIII*) "(7/l/IX)+ (7/1/IX*) (6/0/1)+ (6/0/1*) 0 4 0 0 0 0 0 0 0 (6/0/11)+ (6/0/II*) 0 12 -2 0 0 0 0 0 0 (6/0/111)+ (6/0/III*) 0 0 0 4 0 0 0 0 0 (6/0/IV)+ (6/0/IV*) -4/5 2 -6/5 0 1 0 0 0 2 (6/0/VJ+ (6/0/V*) 0 0 0 0 0 5 -9 1 2 T a b l e 9.4 Derivatives of Rank-0 Degree-6 Spin-Tensors B) Rank 1 S p i n - T e n s o r s (6/2/1) +(6/2/1*) (5/1/1) + ( 5 / 1 / 1 * ) 0 ( 5 / 1 / I I ) + ( 5 / 1 / 1 1 *) 1/4 T a b l e 9.5 D e r i v a t i v e s of Rank-1 D e g r e e - 5 S p i n - T e n s o r s C) Rank 2 S p i n - T e n s o r s ( 7/1 / 1 ) - ( 7 / 1 / 1 * ) ( 7 / 1 / I I ) - ( 7 / 1 / 1 1 *) ( 7 / 1 / 1 1 1 ) - ( 7 / 1 / I I I * ) (7/1/V) -(7/1/V*) (7 / 1 / V I ) - ( 7 / 1 / V I * ) ( 6 / 2 / 1 ) - ( 6 / 2 / 1 * ) 6/5 -3/5 1/5 "1/2 1 T a b l e 9.6 D e r i v a t i v e s of Rank-2 D e g r e e - 6 S p i n - T e n s o r s 103 D) Rank-3 S p i n - T e n s o r s (4/0/1) - ( 4 / 0 / 1 * ) ( 4 / 0/1I) - ( 4 / 0 / 1 I * ) (.3/1/1) - ( 3 / 1 / 1 * ) -12/5 -1 T a b l e 9.7 D e r i v a t i v e s o f Rank-3 Deg r e e - 3 S p i n t e n s o r s (6/0/1) -(6/0 / 1 * ) ( 6 / 0 / 1 I ) - ( 6 / 0 / 1 1 * ) ( 6 / 0 / I V ) -(6/0 / 1V*) (6/0/V) -(6/0/V*) (5/1/1) -(5/1 / 1 * ) 0 0 -2 - 1 ( 5 / 1 / I I ) - ( 5 / 1 / I I *) -12/5 4/5 -2 0 T a b l e 9.8 D e r i v a t i v e s of Rank-3 D e g r e e - 5 S p i n - T e n s o r s 1 04 9.3 R e s u l t s o f t h e S e a r c h F o r 1-Charqes A 1 - c h a r g e , i f i t e x i s t e d , w o u l d make i t s e l f m a n i f e s t i n a rank-1 s p i n - t e n s o r N X x w h i c h c o r r e s p o n d e d t o a r e a l 1 - t e n s o r ( i e : a v e c t o r ) . T h i s N x x w o u l d be c l o s e d ( s e e s e c t i o n 7 .3), w h i c h means t h a t t h e a n t i s y m m e t r i z e d d e r i v a t i v e o f i t i s z e r o : [ ^ v N x x - ^ X x N ^ ] = o A Y N%€ Y XvV R N / £ Y X = 0 (8.2) I t would a l s o n ot be e x a c t , w h i c h means t h a t f o r a l l 0-forms ^ : The s e a r c h f o r 1 - c h a r g e s p r o c e e d s as f o l l o w s : 1) E v e r y rank-1 s p i n - t e n s o r whose d e r i v a t i v e i s z e r o i s s e t a s i d e . T h e s e a r e t h e c l o s e d s p i n - t e n s o r s . 2) The d e r i v a t i v e s of a l l rank-0 s p i n - t e n s o r s a r e compared t o e a c h c l o s e d rank-1 s p i n - t e n s o r t o d e t e r m i n e i f any l i n e a r c o m b o n a t i o n of rank- 0 terms w i l l have a d e r i v a t i v e e q u a l t o t h e rank-1 t e r m . T h i s p r o c e s s i s c a r r i e d o u t by m a t r i x r e d u c t i o n on t h e m a t r i c e s of d e r i v a t i v e c o e f f i c i e n t s w h i c h were p r e s e n t e d i n t h e p r e v i o u s s e c t i o n . The c a l c u l a t i o n s i n d i c a t e t h a t t h e d e r i v a t i v e s of t h e rank-1 s p i n - t e n s o r o f d e g r e e - 3 ( ( 3 / 1 / I ) + ( 3 / 1 / I * ) ) a r e z e r o , w h i c h i s t o be e x p e c t e d b e c a u s e of t h e l a c k o f rank-2 s p i n -t e n s o r s o f d e g r e e - 4 . T h i s , and t a b l e ( 9 . 5 ) i n d i c a t e t h a t ( ( 5 / 1 / 1 . ) + ( 5 / 1 / 1 * )) and ( ( 3 / 1 / I ) + (3/3/1 * ) ) a r e t h e o n l y c l o s e d 105 rank-1 s p i n - t e n s o r s up t o o r d e r 5. A c o m p a r i s o n w i t h t a b l e s 9.2 and 9.3 r e v e a l s t h a t t h e y a r e b o t h e x a c t , however, and so a r e r e j e c t e d a s 1 - c h a r g e s . 9.4 R e s u l t s of t h e S e a r c h f o r 2 - C h a r g e s A 2 - c h a r g e would m a n i f e s t i t s e l f i n a r a n k - 2 s p i n - t e n s o r TXY>,< c o r r e s p o n d i n g t o a r e a l a n t i s y m m e t r i c 2 - t e n s o r " T ^ v - j " . The s p i n - t e n s o r T X v x <^ would be c l o s e d , meaning t h a t : We w i l l work i n t h e d u a l s p a c e ( t h e s p i n - t e n s o r d u a l - s p a c e was d i s c u s s e d i n s e c t i o n 7.3) and l o o k f o r a d u a l - s p a c e s p i n - t e n s o r A •i*YXY . T • • - i P . . . T * W * * ( 8 - 5 ) 1 XYXY ~ L & x r i W XY£W 1 w h i c h , i f T X V x r i s c l o s e d , w i l l obey: Txvxv must not be e x a c t , so we must c h e c k t o make s u r e t h a t t h e r e i s no rank-1 s p i n - t e n s o r N X x whose d e r i v a t i v e i s T x y x , j , : [>YY N*4 - l x i N Y v] * ( T A V X V - T w / e w ) V N x i (8.7) xx 1 06 The p r o c e d u r e h e r e i s e x a c t l y p a r a l l e l t o t h a t d e s c r i b e d above f o r l o o k i n g f o r 1 - c h a r g e s . The m a t r i c e s r e p r e s e n t i n g t h e d e r i v a t i v e s o f Rank-2 s p i n - t e n s o r s a r e e x a m i n e d i n s e a r c h o f c o m b i n a t i o n s whose d e r i v a t i v e s add t o z e r o . In t h i s c a s e , t h e answer i s s i m p l e - o n l y one s e c o n d - r a n k s p i n t e n s o r e x i s t s below o r d e r 8, and i t s d e r i v a t i v e i s not z e r o , as c a n be seen i n t a b l e 9.6. No 2 - c h a r g e s e x i s t , t h e r e f o r e , below o r d e r 8. 9.5 R e s u l t s o f t h e S e a r c h f o r 3-Charges F o r 3 - c h a r g e s , we work e n t i r e l y i n t h e d u a l s p a c e . The c l o s e d 3-form r e q u i r e d f o r a 1-charge w i l l be r e p r e s e n t e d by i t s A d u a l - s p a c e s p i n - t e n s o r U x x : A B e c a u se U x i i s t o be c l o s e d , we r e q u i r e t h a t t h e c o n t r a c t e d d e r i v a t i v e v a n i s h : >** = 0 (8.9) and b e c a u s e i t i s t o be n o n - e x a c t , we r e q u i r e t h a t t h e r e be no A A 2-form T X Y i ^ whose c o n t r a c t e d d e r i v a t i v e i s U X x : ^tv**[*x*V«-iV w , / ] VTxr„ (8.-0) 107 The r e q u i r e d d e r i v a t i v e s a r e p r e s e n t e d i n s e c t i o n 9.2, and t h e c a l c u l a t i o n s p r o c e e d a l o n g t h e same l i n e s as t h o s e f o r t h e 1- and 2-charge.s s o u g h t a b o v e . I t was f o u n d t h a t no l i n e a r c o m b i n a t i o n o f t e r m s w o u l d have z e r o d e r i v a t i v e , so t h e r e a r e no c l o s e d 3-forms of d e g r e e l e s s t h a n d e g r e e 7. 108 Summary A s p e c i f i c t y p e o f c o n s e r v e d q u a n t i t y , t h e " G e o m e t r o d y n a m i c a l A n a l o g t o E l e c t r i c C h a r g e " o r " p - c h a r g e " has been d e f i n e d and has been shown t o r e l y f o r i t s e x i s t e n c e upon t h e e x i s t e n c e of a c l o s e d n o n - e x a c t d i f f e r e n t i a b l e p - f o r m . A p r o o f , due t o Unruh, has been b r i e f l y r e v i e w e d w h i c h shows t h a t no p - c h a r g e c a n be d e f i n e d on a s p a c e - t i m e w i t h an u n r e s t r i c t e d m e t r i c . Many o p t i o n s were e x p l o r e d ' i n c h a p t e r 4, but i n a l l c a s e s i t was f o u n d t h a t t h e " c h a r g e s " depended upon s o m e t h i n g b e s i d e s t h e c u r v a t u r e and t o p o l o g y of t h e s p a c e - t i m e . In the. s e c o n d h a l f o f t h e t h e s i s , t h e m e t r i c was r e s t r i c t e d t o obey E i n s t e i n ' s e q u a t i o n s f o r empty s p a c e t o see whether t h e added s t r u c t u r e would make p - c h a r g e s p o s s i b l e . I t was shown i n c h a p t e r s 5, 6, and 7 t h a t b e c a u s e o f t h i s r e s t r i c t i o n , a l a r g e c l a s s o f g e o m e t r o d y n a m i c a l q u a n t i t i e s c a n be r e p r e s e n t e d v e r y e c o n o m i c a l l y i n terms of c o m b i n a t i o n s of P e n r o s e ' s i n d e p e n d e n t , f u l l y s y m m e t r i c s p i n - t e n s o r s . In o r d e r t o s e a r c h f o r p - c h a r g e s i n t h i s m e t r i c s p a c e , a computer p r o g r a m was w r i t t e n t o g e n e r a t e a l l p o s s i b l e s p i n -t e n s o r s and t o c a l c u l a t e ( s y m b o l i c a l l y ) t h e i r d e r i v a t i v e s . T h i s p r o g r a m was d e s c r i b e d i n s e c t i o n s 8.2 t h r o u g h 8.4 and i n A p p e n d i c e s A and B. T h i s p a r t o f t h e r e s e a r c h a c h i e v e d ' t h e 109 f o l l o w i n g : 1) A c o m p u t e r - c o m p a t i b l e n o t a t i o n f o r P e n r o s e ' s s p i n - t e n s o r s was d e v e l o p e d . 2) A c a n o n i c a l - f o r m f o r s p i n - t e n s o r t e r m s was d e f i n e d w h i c h c a n be u s e d as a s e t of r u l e s so t h a t a l l e q u i v a l e n t s p i n - t e n s o r s c a n be w r i t t e n i n e x a c t l y t h e same way. T h e r e i s room f o r f u r t h e r r e s e a r c h h e r e t o d e t e r m i n e i f t h e p r e s e n t c a n o n i c a l f o r m i s a d e q u a t e f o r s p i n - t e n s o r s of h i g h e r rank t h a n r a n k - 7 . 3) A l g o r i t h m s were d e v e l o p e d w h i c h a r e c a p a b l e o f c a l c u l a t i n g t h e d e r i v a t i v e of any s p i n - t e n s o r t e r m , i n t e r m s which a r e f u l l y i n d e p e n d e n t a t l e a s t up t o d e g r e e 7. The r e s u l t s of t h i s computer work a r e t h e c o m p l e t e l i s t s o f i n d e p e n d e n t s p i n - t e n s o r s i n A p p e n d i x C and t h e i r d e r i v a t i v e s , t a b u l a t e d i n s e c t i o n 9.2. I t was f o u n d t h a t no 1-, 2-, o r 3-c h a r g e s e x i s t f o r t e n s o r s up t o and i n c l u d i n g d e g r e e 6. F u r t h e r work c o u l d be done e x t e n d i n g t h e s e t a b l e s t o h i g h e r d e g r e e s of s p i n - t e n s o r s . A computer program would be needed f o r t h e t a s k , and i t c o u l d use t h e a l g o r i t h m s d e v e l o p e d h e r e , as t h e y were d e s i g n e d f o r s p i n - t e n s o r s of a r b i t r a r y d e g r e e . The c o s t s o f s u c h c a l c u l a t i o n s would be so h i g h , however, t h a t t h e r e i s some doubt as t o whether i t would be a w o r t h w h i l e a c t i v i t y , when t h e r e s u l t s o f t h i s r e s e a r c h g i v e no i n d i c a t i o n t h a t p - c h a r g e s w i l l be any more l i k e l y w i t h p - f o r m s o f h i g h e r d e g r e e s . 1 10 R e f e r e n c e s 1. M i s n e r T h o r n e and W h e e l e r , page 463. 2. See t h e i n t r o d u c t i o n t o Wheeler (1962) 3. Unruh (1971) Gen. R e l . t G r a v . 4. Unruh (1971) ( t h e s i s ) 5. W h e e l e r ( 1 9 6 2 ) , page 233 6. See, f o r example, Croom o r Hu. 7. Croom, pg 44 8. Croom, pp 16-19 9. F o r a d e f i n i t i o n o f t h e Q u o t i e n t Group, s e e f o r example Croom, pp 163-164 10. See, f o r example, Croom 11. F o r a p r o o f of deRham's th e o r e m s see Hu, pp 137 f f . T h i s p r e s e n t a t i o n of them a r e b a s e d on t h e d i s c u s s i o n by F l a n d e r s (1963) pp 66-68, and W h e e l e r (1962) pp 278-279. 12. C f : Hu; M i s n e r , T h o r n e , and W h e e l e r ; o r F l a n d e r s 13. Unruh, 1971, (Gen. R e l . & G r a v . ) pg 29 14. Unruh, 1971, ( t h e s i s ) pp 74, 75 15. F o r a d i s c u s s i o n of t e t r a d f i e l d s , see f o r example, M i s n e r , T h o r n e , and W h e e l e r , pp 1 6 9 f f 16. Unruh ( t h e s i s ) pg 75 17. i b i d pg 76 18. T h i s i s t h e i m p l i c a t i o n o f Unruh's d i s c u s s i o n on pages 77 and 78 o f h i s t h e s i s , a l t h o u g h he d o e s n o t s t a t e h i s r e s u l t s i n p r e c i s e l y t h i s way. 111 19. Croom, pg 125 20. Unruh ( t h e s i s ) pg 79 21. U n r u h ( t h e s i s ) pp 80-81 22. See Bade and J e h l e ( 1 9 5 3 ) , P e n r o s e (1960) and P i r a n i (1962) 23. P e n r o s e (1960) 24. Bade and J e h l e (1953) 25. M i s n e r , T h o r n e , and W h e e l e r , C h a p t e r 20 26. F l a n d e r s , pg 68 B i b l i o q r a p h y Bade, W.L. and J e h l e , H. "An I n t r o d u c t i o n t o S p i n o r s " , Rev. Mod. Ph y s . 25, 3 (1953) pp 714-728 B e r n s t e i n , H.J. and P h i l l i p s , A.V., " F i b e r B u n d l e s and Quantum T h e o r y " , S c i e n t i f i c A m e r i c a n , J u l y , 1981, pp 1 23-137 Croom, F.H., B a s i c C o n c e p t s o f A l g e b r a i c T o p o l o g y , S p r i n g e r - V e r l a g , New Y o r k , 1978 F l a n d e r s , H., D i f f e r e n t i a l Forms Academic P r e s s , New York , 1963 G e r o c h , R. e t a l ^ , "A Space-Time C a l c u l u s B a s e d on P a i r s of N u l l D i r e c t i o n s " , Math. Phys. J_4, 7, ( 1 973) pp 874-881 ' G i l m a n , L., and Rose, A . J . , APL, An I n t e r a c t i v e  A p p r o a c h W i l e y , New Y o r k , 1976 Hu, S.T., D i f f e r e n t i a b l e M a n i f o l d s H o l t , R i n e h a r t , & W i n s t o n , New Y o r k , 1969 H u s e m o l l e r , D., F i b e r B u n d l e s M c G r a w - H i l l , New Yor k , 1 966 M i s n e r , C.W., T h o r n e , K.S., and Wh e e l e r , J.A., G r a v i t a t i o n Freeman, San F r a n c i s c o , 1973 Newman, E., and P e n r o s e , R., "An A p p r o a c h t o G r a v i t a t i o n a l R a d i a t i o n by a Method of S p i n -C o e f f i c i e n t s " , J . Math. Phys. 3, 3 (1962), pp 566-578 P a p a p e t r o u , A., L e c t u r e s on G e n e r a l R e l a t i v i t y R e i d e l , D o r d r e c h t , 1974 P e n r o s e , R., "A S p i n o r A p p r o a c h To G e n e r a l R e l a t i v i t y " , Ann. Ph y s . J_0 (1960) pp 171-201 113 P i r a n i , F.A.E., " I n t r o d u c t i o n To G r a v i t a t i o n a l R a d i a t i o n T h e o r y " , L e c t u r e s on G e n e r a l R e l a t i v i t y ( B r a n d e i s Summer I n s t i t u t e i n T h e o r e t i c a l P h y s i c s ) P r e n t i s - H a l l , New J e r s e y , pp 305-326 P o r t e r , R.D., I n t r o d u c t i o n t o F i b e r B u n d l e s D e k k e r , New Y o r k , 1 977 Unruh, W.G., "Homotopy of t h e M e t r i c T e n s o r " , Ph.D. T h e s i s , P r i n c e t o n U n i v e r s i t y , 1971 Unruh, W.G., " E x c l u d e d P o s s i b i l i t i e s o f G e o m e t r o d y n a m i c a l A n a l o g t o E l e c t r i c C h a r g e " , Gen. R e l . & G r a v . 2, 1 ( 1 9 7 1 ) , pp 27-Unruh, W.G., L e c t u r e N o t e s ( u n p u b l i s h e d ) , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1981 W h e e l e r , J.A., G e o m e t r o d y n a m i c s Academic P r e s s , New Y o r k , 1962 1 1 4 A p p e n d i x A Computer A l g o r i t h m s G e n e r a t i n g S p i n t e n s o r s and D e r i v a t i v e s The a l g e b r a i n v o l v e d i n f i n d i n g t h e d e r i v a t i v e s o f h i g h -d e g r e e s p i n t e n s o r s i s l o n g and i n h e r e n t l y n o n - s y m m e t r i c . T h i s p u t s two b u r d e n s on t h e s y m b o l i c m a n i p u l a t i o n p r o g r a m d e v e l o p e d f o r t h e t a s k : 1) t h e code must be o f t e n s e l f - r e f e r e n c i n g and a b l e t o d e a l w i t h many s p e c i a l c a s e s . 2) c h e c k i n g t h e a n s w e r s , e s p e c i a l l y f o r h i g h e r o r d e r t e r m s , q u i c k l y becomes p r o h i b i t i v e l y d i f f i c u l t [ t ] . C o n f i d e n c e i n t h e r e s u l t s must, t h e r e f o r e , come a s much from a c a r e f u l a n a l y s i s o f t h e a l g o r i t h m employed as from i n d e p e n d e n t c a l c u l a t i o n s f o r c o m p a r i s o n . F o r t h i s r e a s o n , t h i s a p p e n d i x i s i n c l u d e d t o p r o v i d e an o u t l i n e of t h e c o m p l e t e program, e s p e c i a l l y f o c u s s i n g on t h e a l g o r i t h m u s e d t o c a l c u l a t e d e r i v a t i v e s . The c o m p l e t e code i s l i s t e d i n a p p e n d i x B, but b e c a u s e i t was w r i t t e n i n APL, w h i c h i s among l a n g u a g e s p a r t i c u l a r l y p r o n e t o b e i n g opaque t o e v e r y o n e but t h e programmer, i t c a n be e x p e c t e d t o e n l i g h t e n o n l y t h e m e t i c u l o u s who a r e f a m i l i a r w i t h APL and w i s h t o c h e c k d e t a i l s n o t d i s c u s s e d i n t h i s summary. Note t h e f o l l o w i n g a b o u t t h e d i s c u s s i o n w h i c h f o l l o w s : [ f ] One s u c c e s s f u l c h e c k i n g p r o c e d u r e u s e d was t o s t e p t h r o u g h t h e f i r s t few l o o p s o f t h e p r o g r a m w i t h a h i g h - o r d e r s p i n o r , c h e c k i n g v a l u e s a t e a c h s u c c e s s i v e s t e p . 1) Whenever p o s s i b l e , examples a r e g i v e n i n b o t h s t a n d a r d n o t a t i o n and i n t h e computer- n o t a t i o n . F a m i l i a r i t y w i t h t h e c o m p u t e r - n o t a t i o n w i l l be i m p o r t a n t f o r u n d e r s t a n d i n g t h e summaries of t h e a l g o r i t h m s . 2) F u n c t i o n s i n APL a l w a y s use o n l y one o r two a r g u m e n t s , and t h e s e a r e w r i t t e n b o t h b e f o r e and a f t e r t h e name o f t h e f u n c t i o n . T h u s : q :== n FUNCTION m i s a s t a t e m e n t t h a t " q " i s a s s i g n e d t h e v a l u e o f "FUNCTION", where FUNCTION has u s e d n and m as p a r a m e t e r s . 3) When examples a r e p r e s e n t e d , t h e 2 40 a r r a y s r e p r e s e n t i n g e a c h t e r m , s u c h a s : fl 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 5 26 2G 0 0 0 0 0 0 0 0 0 O 0 0 0 2 2 2 2 2 O 0 * • • 0 0 0 0 0 0 0 0 "1 1 o o o o o o ol 0 0 0 0 0 0 0 0 5 5 0 0 0 0 0 0 OJ a r e a b b r e v i a t e d by r e m o v i n g t h e z e r o s w h i c h s e p a r a t e t h e d i f f e r e n t components (CFF, TM, TM, NM). The above a r r a y would t h u s be w r i t t e n : IS life Z(, \ 1 I 1 2 Z | 5 5J 4) When p o s s i b l e , when a p a r t i c u l a r p i e c e of code i s b e i n g d e s c r i b e d , t h e l i n e number of t h a t c o d e i s w r i t t e n i n p a r e n t h e s e s a f t e r t h e d e s c r i p t i o n . 5) The examples w h i c h a r e g i v e n i n s e c t i o n 3 a r e p r o g r e s s i v e , so t h a t t h e r e s u l t s o f one example a r e u s e d as t h e i n p u t of t h e n e x t example. 6) APL f u n c t i o n s and t h e i r names a r e s e t a p a r t w i t h t r i a n g l e symbols ( " V " ) , j u s t a s t h e y a r e i n t h e computer l i s t i n g s i n A p p e n d i x B. The t r i a n g l e c a n be r e a d as a " f a t b r a c k e t " w h i c h i n d i c a t e s when one e n t e r s o r l e a v e s t h e c o n t r o l of t h e g i v e n f u n c t i o n . 1 16 A.1 The D r i v i n g Program The f u n c t i o n VAAAV ( f u n c t i o n B.5) g e n e r a t e s and p r i n t s a l l s p i n t e n s o r s and t h e i r d e r i v a t i v e s f o r a g i v e n rank and d e g r e e . The a l g o r i t h m u s e d i s summarized below: 7 AAA - c a l l V C REATEV t o g e n e r a t e l i s t "QQ" o f a l l i n d e p e n d e n t s p i n t e n s o r s . -DO f o r e a c h s p i n t e n s o r i n QQ: - c a l l VDERIVSV t o g e n e r a t e l i s t "0" o f t h e t e r m s i n t h e d e r i v a t i v e ( t h e r e a r e two ways t o t a k e t h e d e r i v a t i v e f o r rank 1 so t h e s e s t e p s a r e r e p e a t e d f o r t h e two c a s e s ) -use VORDALLV t o p u t e a c h t e r m i n c a n o n i c a l f o r m -add l i k e t e r m s u s i n g VCNSDTV, t h u s making t h e l i s t s h o r t e r - p r i n t t h e s p i n t e n s o r , t h e n t h e t e r m s i n t h e d e r i v a t i v e , u s i n g VWRITEV -RETURN . 2 The G e n e r a t i n g F u n c t i o n V C R E ATEV VCREATEV ( f u n c t i o n B.6) and i t s l o o p VCOMBINV ( f u n c t i c .7) g e n e r a t e a l l i n d e p e n d e n t s p i n t e n s o r s i n t h e f o l l o w i n g way: VCREATE - g e n e r a t e l i s t o f a l l p o s s i b l e names o f s p i n t e n s o r s ( i e : a l l p o s s i b l e c o m b i n a t i o n s o f f a c t o r s t o t h e g i v e n d e g r e e ) u s i n g VNAMESV: ( f u n c t i o n B.9) VNAMES -DO f o r e a c h o r d e r : 117 -use V P M U V t o make a l i s t DD o f a l l p o s s i b l e d i s t r i b u t i o n s o f d e r i v a t i v e s - f o r e a c h e l e m e n t o f DD use VNMSFNV ( f u n c t i o n C10) t o p u t a l l c o m b i n a t i o n s of V and ¥ b e s i d e t h e s e d e r i v a t i v e s -RETURN - d e l e t e from t h e l i s t a l l names where t h e r e a r e t o o few no n - s y m m e t r i c i n d i c e s t o a l l o w f u l l c o n t r a c t i o n V -DO f o r e a c h name: -use V PRTTYPEV ( f u n c t i o n B.12) t o g e n e r a t e a p r o t o t y p e l i s t o f numbers f o r TM -use V GENRTV ( f u n c t i o n B.14) t o c r e a t e a l i s t M1 of a l l p o s s i b l e c o m b i n a t i o n s of t h e s e numbers i n a r r a y s c o n f o r m i n g t o t h e s t a n d a r d f o r m d i s c u s s e d i n s e c t i o n 8.3. - r e p e a t t h e above two s t e p s i n t o a l i s t M2 f o r TM - t a k e t h e o u t e r p r o d u c t (VCR0SS3V, f u n c t i o n B.16) of M1 and M2, t o g e t a l l p o s s i b l e c o m b i n a t i o n s of TM and TM -add t o t h i s l i s t t h e CFF and NM t o y i e l d a c o m p l e t e l i s t o f s p i n t e n s o r s -RETURN -use VSPECORDV ( f u n c t i o n B.15) t o remove from t h e l i s t a l l s p i n - t e n s o r s w h i c h do not c o n f o r m t o p a r t 4 of t h e h i e r a r c h y d e s c r i b e d i n s e c t i o n 8.3. V 118 A. 3 C a l c u l a t i n g t h e d e r i v a t i v e s The f u n c t i o n V DERIVSV ( f u n c t i o n B.19) d e c i d e s how t o t a k e t h e d e r i v a t i v e o f a g i v e n s p i n - t e n s o r ( t h e form o f t h e d e r i v a t i v e w i l l depend upon, i t s r a n k ) . I t i n s e r t s t h e number 30 t o h o l d t h e p l a c e of t h e a p p l i e d d e r i v a t i v e so t h a t t h e e v e n t u a l c o n t r a c t i o n s w i l l be c o r r e c t f o r t h a t r a n k , t h e n c a l l s t h e f u n c t i o n V D E L O P V t o a p p l y t h e c h a i n - r u l e o v e r t h e whole t e r m . V D E R I V S V s u p p l i e s V D E L O P V w i t h i n f o r m a t i o n ( i n "KEY") as t o : 1) w h i c h f a c t o r s h o u l d t h e DEL o p e r a t o r s t a r t on ( i n t h i s c a s e t h e f i r s t f a c t o r ) 2) a t how many f a c t o r s f r o m t h e end s h o u l d t h e d e l o p e r a t o r s t o p ( i n t h i s c a s e do a l l f a c t o r s ) and 3) what number h o l d s t h e p l a c e o f t h e a p p l i e d d e r i v a t i v e . F o r example: Z dft(Y LTA8CDEFGH * (A. 1 ) c o r r e s p o n d s t o : i DE&W S \,i J 2 2. * | z i 2 2 . U 2-4-U?J v o i l I l \ i I M i l i 3olz 3\ r i \ i i t io \« » \ i i i i U 3 l + (l t> 3D) DEL09 Lzji Z 24 z*r \zi i z i -L aoU +) (A.2) G i v e n a te r m TT and c o d e d i n s t r u c t i o n s i n "KEY", t h e f u n c t i o n VDELOPV ( f u n c t i o n B.21) a p p l i e s t h e c h a i n r u l e and r e t u r n s t h e t e r m s i n t h e d e r i v a t i v e , "DLST", u s i n g t h e f o l l o w i n g a l g o r i t h m : V DELOP -DLST := { O H l i n e [ 4 ] ) -PL := K E Y [ 1 ] ( t h e ( P L ) t h f a c t o r i s f i r s t t o be a c t e d on by the D E L ) ( l i n e [ 6 ] ) - I F the P L t h f a c t o r i s not c o n t r a c t e d t o the a p p l i e d DEL ( c a l l the a p p l i e d DEL "DEL A") THEN: -add t o DLST a te r m where t h e D E L A i s f u l l y s y m m e t r i z e d w i t h t h e ( P L ) t h f a c t o r Q i n e s [ 1 0 - 1 2 ] ) E L S E : i f t h e P L t h f a c t o r i s c o n t r a c t e d t o D E L A , t h e n t h e r e i s no f u l l y s y m m e t r i c p a r t of t h e d e r i v a t i v e of t h a t f a c t o r , so s k i p t h e above p r o c e d u r e and c o n t i n u e : -Add t o DLST t h e terms r e s u l t i n g from t h e D E L A b e i n g n o t - f u l l y s y m m e t r i c w i t h t h e ( P L ) t h f a c t o r . C a l l t h e s e "NFST", "Not F u l l y Symmetric Terms", and use VINDELV t o f i n d t h e m . ( l i n e [ 1 4 ] ) -Add t o DLST t h e terms due t o t h e DEL* o p e r a t i n q on t h e ( P L + 1 ) t h f a c t o r ( l i n e [ 1 5 ] ) F o r e x a m p l e : z. 'ABCDEFGH 1 *' sr J~ Y , • *T v'S t > l V + NFS.T L U a ( v y A B c o i F G , H n « 1 + Lii> \ . ( u / A S G c o i f i H ^ C x r 120 i s r e p r e s e n t e d by: ( l O 30^ DELOP [_^ \ » l 2. I » \ » » I I 2 3 o \ i O 2 1 j i I 1 I I 1 I* L4\ 6 -f I I l l I I I -I i I I I T. » L2.|2-2-2_lz- 2 2 1 1 1 *•«• + ( 1 0 3 0 ) 2 + (l O IO) OELOlP [ i . | 2 1 « 2 \« I I l l I 2 3o' 2 2 30 j 2 2 2 1 2 1 14/ (A.4) 1 « 2 1 I I I I I I 2 3o 1 2. -jo | z i 2 2 2 2 24 Zt, The f u n c t i o n V l N D E L V ( f u n c t i o n B.23) f i n d s t h e s e NFST by u s i n g e q u a t i o n ( 5 . 3 3 ) . The s y m m e t r i z e d i n d i c e s of t h e ( P L ) t h f a c t o r a r e expanded i n t o a sum of terms where t h e f i r s t i n d i c e s ( t h a t i s , t h e i n d i c e s o f t h e f i r s t DEL, "DEL*/ 1, i n t h e f a c t o r ) a r e no l o n g e r s y m m e t r i z e d . V l N D E L V t h e n commutes t h e a p p l i e d d e r i v a t i v e s w i t h e a c h DEL ( and r e t u r n s t o LST t h e NFST r e s u l t i n g . V I N D E L LST := {0} -UPORIENT TT so a l l numbers "PL" a r e a l o n g t h e t o p row of TM and TM ( s e e f u n c t i o n B . 2 4 ) ( l i n e [ 1 2 ] ) - u p o r d e r a l l t h e s e p a i r s so t h e bo t t o m numbers a r e i n o r d e r ( t h i s s i m p l i f i e s g r o u p i n g them i n t o l i k e t e r m s ) ( l i n e s [ 1 3 - 1 6 ] ) - m u l t i p l y CFF by ( 1 / n ) , ( l i n e [ l 7 ] ) , where n= ( o t f t ) , and where oc and 0 a r e t h e number of o c c u r e n c e s of (PL) i n TM and TM r e s p e c t i v e l y . DO [ f ] f r o m A = 0 t o ex. DO [ t ] f r o m B = 0 t o p 121 - R e c a l l TM and TM - R o t a t e t h e P L - p a i r s i n TM and TM by amounts A and B so t h a t t h e DEL t t a k e s a d i f f e r e n t p a i r of i n d i c e s e a c h t i m e , ( l i n e s [ 2 8 - 2 9 ] ) -use t h e 7COMMUTEV f u n c t i o n ( f u n c t i o n B.25) t o add t o LST t h e t e r m s due t o commuting DEL w i t h e a c h of t h e s e s p e c i f i c DEL ' s . ( l i n e [ 3 3 ] ) -KK := NFST of t h e D E L A a c t i n g on ( a l l of f a c t o r PL e x c e p t D E L , ) . Use f u n c t i o n "VINDELV. ( l i n e s [ 3 4 - 3 7 ] ) -add t o LST t h e NFST of DEL, a c t i n g on KK. Use VDELOPV f u n c t i o n , ( l i n e [ 3 8 ] ) -RETURN -RETURN V F o r example, c o n s i d e r : t h e f i r s t t w e l v e terms o f w h i c h a r e e q u a l t o : [ f ] T h e s e c o u n t e r s a r e i n c r e m e n t e d by SP o r ED ( o f t e n >1) where t h e n e x t SP o r ED t e r m s w i l l be i d e n t i c a l ( t h u s s a v i n g r e p e t i t i v e c a l c u l a t i o n s ) . CFF i s i n c r e a s e d t o match t h i s . 122 I n computer n o t a t i o n , (A.5) e x pands i n t o t e r m s a s f o l l o w s ( o n l y t h e f i r s t t e r m i s e x p l i c i t l y e x p a n d e d b e l o w ) : + ( i o - i i) 0VS6FSatn (vac) + o t h e r t e r m s from t h e e x p a n s i o n where: and where VDVSOFSUMV, ( f u n c t i o n B . 2 0 ) , a p p l i e s VDELOPV t o e a c h t e r m i n KK. The f u n c t i o n VcOMMUTEV ( f u n c t i o n B.25) r e p l a c e s two commuting DEL o p e r a t o r s w i t h e i t h e r or *F , a c c o r d i n g t o t h e R i c c i i d e n t i t y . B ecause we a r e o n l y l o o k i n g f o r a n t i s y m m e t r i c p a r t s of t h e e x p r e s s i o n , we assume f i r s t t h a t t h e u n d o t t e d , t h e n t h a t t h e d o t t e d i n d i c e s o f t h e two DEL o p e r a t o r s a r e a n t i s y m m e t r i c . T h i s g i v e s a l l t h e a n t i s y m m e t r i c p a r t s of t h e e x p r e s s i o n . The f o l l o w i n g summarizes t h e a l g o r i t h m f o r t h e d o t t e d i n d i c e s o n l y : 123 7 COMMUTE ( w i t h l o o p VCOMMLPV) -NLST:= {0} ( t h e l i s t o f terms i s i n i t i a l i z e d ) - s p l i t NM, l e a v i n g a h o l e i n p o s i t i o n PL f o r t h e new f a c t o r ( l i n e [ 1 1 ] ) -renumber TM and TM i n a c c o r d a n c e w i t h t h e new p o s i t i o n of some f a c t o r s a s a r e s u l t o f t h e above s p l i t ( l i n e s [ 1 2 - T 5 ] ) - l o c a t e t h e c o n t r a c t i o n s t o t h e ( P L ) t h f a c t o r ( s t o r e t h i s i n f o r m a t i o n i n C1 and C2) ( l i n e s [ 1 6 - 1 7 ] ) -W := number o f u n d o t t e d i n d i c e s i n ( P L ) t h f a c t o r -CFF := CFF/2 -NM[PL] := (e) ( t h e new f a c t o r ' s name) - I F (DEL| and D E L ^ a r e n o t a n t i s y m m e t r i z e d on u n d o t t e d i n d i c e s ) , THEN DO [ $ ] A = 0 t o W - c o n t r a c t t h e d o t t e d i n d i c e s o f DEL^ and DEL t (we a r e h e r e c a l c u l a t i n g t h e p a r t of them w h i c h i s a n t i s y m m e t r i c i n t h e s e d o t t e d i n d i c e s ) ( V C O M M L P V l i n e s [ 1 4 - 2 3 ] ) - c o n t r a c t t h e new f a c t o r t o t h e f i r s t i n d e x o f t h e ( P L + 1 ) f a c t o r , a c c o r d i n g t o t h e R i c c i i d e n t i t y . (COMMLP l i n e s [ 2 4 - 2 7 ] ) - i f t h e (PL+1) f a c t o r has more t h a n one d e r i v a t i v e , c a l l VHIORDIV t o c a l c u l a t e t h e t e r m s where t h e f i r s t i n d e x of f a c t o r (PL+1) i s not f u l l y s y m m e t r i c w i t h f a c t o r ( P L + l ) . [ f ] [ $ ] A g a i n , we use SP t o i n c r e m e n t A by more t h a n 1 t o a v o i d r e p e t i t i v e c a l c u l a t i o n s . [ f ] F o r example, a f t e r a p p l y i n g t h e R i c c i i d e n t i t y , we might h a v e : . » A B C O <J ( E d C)l w h i c h i s e q u a l t o : W i t h o u t 7HI0RDV, t h e s e c o n d t e r m o f t h i s would be m i s s e d . 124 -RETURN -Repeat t h e above, a c t i n g on t h e d o t t e d i n d i c e s i n p l a c e o f u n d o t t e d , and v i c e - v e r s a . As an example, c o n s i d e r : 41. 6 OEP-&H fcyc a C f l - * ^ 1 X + i i l V A ' £ \ \* ^tSaOEFfiH 4 2 ' 8 OEFSH ^C-A «1H<Y « «.) 1 X t h e f i r s t t e r m of w h i c h i s e q u a l t o : _ J3-. I f IU A 6 (jMaCDEFM ~ S4- ' A6V DE FGj H ' ft N T * + uy u-'A & u/6Ncbe^H &4- ' f tSY0EF6H l ft. N ' X T h i s i s r e p r e s e n t e d i n t h e p r o g r a m a s : Cl o T>O) C O M M U T E , [li 1 ? } }A ^ z ^ 1 2 1 *°U 3 ~\ + (3 O "31 0 HloH-ftl f - H I 1 ^ 2 3 I 24 3 3 3 3 3 3 \2 4 2 l L*4 I 3| I ' I X 1 I i » | I I i-vl ^ * i j (A.9) (A.1 0 ) (A.11 ) F o r h i g h e r o r d e r s , when two d e r i v a t i v e s a r e removed from a f a c t o r by t h e R i c c i i d e n t i t y i n VCOMMUTEV, i t i s s t i l l p o s s i b l e t h a t t h e f a c t o r i s Not F u l l y Symmetric on t h e i n d i c e s o f t h e l e f t - m o s t d e r i v a t i v e . Such a f a c t o r w i l l be f u l l y s y m m e t r i c i n 125 e i t h e r d o t t e d o r u n d o t t e d ' i n d i c e s , d e p e n d i n g on w h i c h R i c c i i d e n t i t y i t came fr o m . VHI0RD1V and VHI0RD2V, r e s p e c t i v e l y , c a l c u l a t e t h e NFST f o r t h e s e two c a s e s . The f u n c t i o n s do t h i s by e x p a n d i n g t h e .term i n t o a sum o f t e r m s where t h e i n d i c e s o f t h e l e f t - m o s t d e r i v a t i v e , b o t h d o t t e d and u n d o t t e d , a r e not s y m m e t r i z e d w i t h t h e o t h e r i n d i c e s . They t h e n a p p l y t h i s f i r s t d e r i v a t i v e t o t h e b a l a n c e o f t h e f a c t o r u s i n g VINDELV. The s t e p s f o r VHI0RD1V a r e a s f o l l o w s : VHI0RD1 -remove one d e r i v a t i v e f r o m t h e name o f f a c t o r (PL) ( l i n e [ 1 0 ] ) - g r o u p t h e p a i r s i n TM so t h a t s i m i l a r c o n t r a c t i o n s t o (PL) a r e a d j a c e n t e a c h o t h e r , ( l i n e s [ 1 1 - 1 3 ] ) - s a v e t h i s v e r s i o n of t h e t e r m ( l i n e [ 1 6 ] ) -DO f o r e a c h d o t t e d i n d e x o f f a c t o r ( P L ) : - r e t r i e v e t h e s a v e d v e r s i o n of t h e t e r m - r o t a t e [ f ] t h e i n d i c e s i n (PL) so t h a t a new i n d e x comes f i r s t ( l i n e [ 2 1 ] ) - r e p l a c e t h e f i r s t c o n t r a c t i o n t o (PL) w i t h a c o n t r a c t i o n t o (DX), a p l a c e - h o l d e r f o r what w i l l now be c o n s i d e r e d t h e a p p l i e d d e r i v a t i v e , ( l i n e [ 2 4 ] ) - c a l l V I N D E L V t o t a k e t h e NFST of t h i s d e r i v a t i v e ( l i n e [ 2 5 ] ) and add t h e s e t e r m s t o t h e l i s t -RETURN V As an example o f t h i s , c o n s i d e r : [ t l H e r e , as b e f o r e , t h e r a t e o f r o t a t i o n w i l l depend on how many s i m i l a r p a i r s a r e i n TM 126 • • • • • • • . • (A. 12) ( = 0 b e c a u s e N and Q a r e f u l l y s y m m e t r i c ) In t h e pro g r a m , t h i s i s r e p r e s e n t e d by: T h i s c o m p l e t e s t h e summary of t h e f u n c t i o n s f o r c a l c u l a t i n g t h e d e r i v a t i v e s A.4 R e p l a c i n g Terms by S t a n d a r d - F o r m Terms Terms c a l c u l a t e d by V D E R I V S V a r e not i n c a n o n i c a l form, and t h e r e f o r e c a n n o t be add e d o r s u b t r a c t e d f r o m e a c h o t h e r . The f u n c t i o n VORDALLV t a k e s a l i s t "GR" o f s u c h t e r m s and r e p l a c e s i t by a l i s t "NGR" of e q u v a l e n t terms i n s t a n d a r d form, as f o l l o w s : VORDALL - i n i t i a l i z e NGR -DO f o r e a c h e l e m e n t o f GR: - o r d e r t h e f a c t o r s u s i n g VTERMORDV -use V P A I R O R D V t o i n v e r t any p a i r s i n TM or TM where t h e l a r g e r number i s on t o p . T h i s i n v o l v e s c h a n g i n g t h e s i g n of CFF. -use VHIERv" t o put b o t h t h e t o p and b o t t o m rows of TM i n t o a s c e n d i n g o r d e r as f o l l o w s : V H I E R -remove t h e v a l u e o f CFF ( c a l l e d "CF") from i n f r o n t of TM - r e o r d e r t h e p a i r s o f TM so t h e t o p ones a r e i n o r d e r ( l i n e s [ 8 - 9 ] ) -DO f o r e a c h n u m e r a l Q i n t h e t o p row: - u p - o r d e r a l l l o w e r n u m e r a l s w h i c h a r e p a i r e d ^ t o a Q-numeral ( l i n e s [ 1 7 - 1 9 ] ) -RETURN - e x i t i f t h e b o t t o m row i s i n o r d e r now - f i n d t h e f i r s t p l a c e where t h e b o t t o m row d e c r e a s e s f r o m r i g h t t o l e f t ( l i n e [ 2 5 ] ) - r e p l a c e t h e two o f f e n d i n g i n d i c e s , u s i n g t h e f i r s t p a r t of t h e sum i n ( 7 . 1 5 ) . C a l l t h i s t e r m " L " . ( l i n e s [ 2 8 - 2 9 ] ) -change t h e s i g n o f CFF and c a l l HIER t o put L i n o r d e r ( t h e r e may be o t h e r c o n t r a c t i o n s out of o r d e r ) . Put t h e r e s u l t i n g l i s t o f t e r m s i n t o t h e a n s w e r - l i s t , ( l i n e [ 3 1 ] ) - r e c a l l t h e o r i g i n a l o f f e n d i n g t e r m and r e p l a c e i t u s i n g t h e s e c o n d p a r t o f t h e sum i n ( 7 . 1 5 ) . - c a l l V H I E R V t o s e t t h i s t e r m i n o r d e r . Put t h e r e s u l t i n g l i s t o f terms i n t o t h e answer-l i s t , ( l i n e s [ 3 2 - 3 4 ] ) - r e p e a t t h e above, t h i s t i m e r e - o r d e r i n g t h e c o n t r a c t i o n s i n TM. - l a m i n a t e t h e r e s u l t i n g e x p a n s i o n s of TM and TM so NGR i s an a r r a y o f t h e f u l l e x t e r i o r p r o d u c t o f TM x TM. ( l i n e s [ 2 2 - 2 8 ] ) 128 c a l l V S P E C R E P V t o r e a r r a n g e g r o u p s of f a c t o r s i n o r d e r t o c o n f o r m t o t h e h i e r a r c h y d e s c r i b e d i n p a r t (4) o f s e c t i o n 8.3. V The f u n c t i o n V S PECREPV ( f u n c t i o n B.32) e s t a b l i s h e s t h e h i e r a r c h y by r e q u i r i n g t h a t i f two g r o u p s of f a c t o r s a r e s i m i l a r , t h e n t h e r i g h t - h a n d g r o u p s h o u l d have no l e s s numbers i n t h e t o p row of TM t h a n t h e l e f t - h a n d g r o u p . I f t h e y a r e e q u a l i n t h a t r e s p e c t , t h e n t h e r i g h t - h a n d g r o u p must have no l e s s numbers i n t h e t o p row o f TM t h a n t h e l e f t - h a n d g r o u p . I n e a c h c a s e , numbers c o n t r a c t e d between t h e two g r o u p s a r e e x c l u d e d from t h e c o u n t . V S P E C R E P V f i r s t f i n d s s u c h s i m i l a r g r o u p s ( u s i n g VCOMPLISTV, f u n c t i o n B . 3 3 ) , t h e n a s k s VEXADVV i f e x c h a n g i n g them would be a d v a n t a g e o u s ( l i n e [ 2 2 ] ) , and, f i n a l l y , e x c h a n g e s them ( l i n e s [ 2 3 - 2 4 ] ) i f t h e answer i s " y e s " . 129 A p p e n d i x B L i s t i n g o f t h e APL S o u r c e Codes L i s t e d on t h e f o l l o w i n g pages a r e t h e c o m p l e t e c o d e s f o r a l l t h e f u n c t i o n s u s e d i n t h e c a l c u l a t i o n s . The a l g o r i t h m s upon w h i c h t h e s e programs a r e b a s e d a r e d i s c u s s e d i n a p p e n d i x A. P l e a s e r e f e r t o t h e L i s t o f APL F u n c t i o n s a t t h e f r o n t o f t h e t h e s i s t o l o c a t e a s p e c i f i c F u n c t i o n . 1 30 <? U N R f i V F ' K S [1] C F F * £ J f P K G C£] ™ * E' 15 f F K G f f| 1 4 . F K .3 C3: I d * E 15 f F K G f o 15 I P K G C4] N M f £• -Q f P K S C5] T M f ( T M [1 ; ] *|T;, / T M C6] l y * c i t ! c i ; ] *o> •• i d C7] N M * <; r + ^ N M ) ^ i j ; . / N M Funct ion ( B . 1 ) - V U N R A V V • 7 R E R R I , ' [•] F K G * R E R R i - ' ; Z c n z * £ 15 P U CE'] F K G * C F F , C£] C£ 15 t ( T M , [ £ ] z> > C3] P K S * F K G , [ £ ] C£ 15 t C T M , c£] 2> ^ [?] <:E 9 T ('NM, f£] Z> ) Funct i on ( B . 2 ) - V RERAV V <? T E X T f W R I T E M T X | C F F ; N M ; T M ; T M ; L ; E ; R R R ; L T H ; r < ; E ; G G Cl] fl CE'] fl T R M H S L f l T E S N U M B E R C O D E S I N T O R E R D R B L - E F O R M C3] fl C4] • T E X T * Q 1 | j | j p 1 1 C5] L T H f ( p M T X ) [ 1] C6] I' * Ij C7] l _ O • F ; 1 C8] -Mj I F F O L T H C9] UNRRI,-' M T X [ D ; ; ] Cl ij] 1—* ( p N M ) [ £ ' ] Cl 1] G S f f ( + / N M ) C l £ ] R R R * < ; i_ , 5 6 ) p £ 3 [13] C14] E*-f l [15] Ij W R I T E L O D F TM C16] W R I T E L O D F TM [17] R R R * <; (;i_, 5;> p. • P S I <; ' > !• C£] F I L F [ R R R ] CIS] ^ ( ( N M [ j ; ] > N M [ g ; ] > X 1 ( p N M ) [p] > C19] R R R [ C i ; -, 5 ] * ,:; ( p G ) , 5 ;i p 1 > P S I . ( 1 CEO] F l R R f , R R R CE'l] R R R * ( f l R R ^ f l L F [ £ 3 ] > / R R R CEE'] F l R R * J J , R R R , ' > 1 CE'3] R R R * •;; • ij • y , , : > C F F [ 1 ; ] > , • / • , ( > C F F [ £ ; ] , 1 1 , R R R CE4] R R R * 1 I j l jp i - R R R , J fjClp CE5] T E X T * T E X T , [ J ] R R R CE6] -J-LODP 1 7 F u n c t i o n (B.3) - V W R I T E V 131 r? F L W R I T E L D D F ' T M J C [ I ] ^0 IFF 0=(PTM) [£] [£] c«>0 [ 3 ] L O D F : C * - C + l [4] -HJ IFF C> (pTM) [£] [5] 4EX IFF TM [ C . ; C ] > £ 3 C6] E*E+1 C7] fiR" [TM [ i ; c ] ; ] f G G f f t R R [ T M [ ] ; c] ; ] , E+FI_ [8] F l R R [TM [ £ ; C ] ; ] f G S f f t R R [TM [ ? ? C ] ; ] , E + F L [9] -> L. • • F [1 CQ E X ; F l R R [TM [ J ; C ] ; ] f G S f F l R R [TM [1 ; C ] ; ] , c'4 + F'L. [ I I ] * L D O P F u n c t i o n ( B . 4 ) - y W R I T E L O O P ^ <? RNK R f i f t D E S ; I,.'N; W«/N; H W H ; IH-'N; I_; S; T ; o ; E I E ; I Cl] if [£] if :*: :*: * r.3] ,T MR IN DR I VER FOR EACH RFlNK [4] if C5] ' R 1 , ( T R N K ) , ' D 1 ,(^BEG) C6D Hl^Nf ' M ' , 1,'N [7] WDl-'N*- 1 WD 1 , I 'N [8] jW'.-'W, 1 f RNK RLL DEG ' [9] jl''N, 1 fEE ' Cl i j ] E « - 1 1 Cl 1] JWIJVN , ••«>. <; ij j n ij> P* i Cl£] (f.E!E) [1] C13] Ij [14] L-D.OF: s*>s+i [15] ->DUT IFF S>l_ [16] T*-EE [ s ; ; ] [17] • ' - DEL. OF : 1 [18] j j H m - ' N , 1 *-1 , wru-'N, 1 •1 [ i ] a Icmo [19] D * l OOP ' 0 EEUfiL. s: ' !• 1 0ljp* C£0] jWDVN ,'«>>, WDl-'N , i C 1 ] 0 ' CSI 3 • 1 DER I VS T [££] • *-DRDftl_U • [£3] DfZTERMS D [£4] JJWDWN, 1 •> 1 , WDt-'N, 1 j [1] WRITE [£5] •*l_OOF IFF RNK^l [£6] a*>l OOP ' 0 D R - ' ? i I j l j p B [£7] JIWD^'N, 1 *r 1 , WIH-'N, i [1] ° ' [£8] D>£ DERIVS T [£9] • fORDFiLL • [30] •fZTERHS • [31] jWD^N, 1 *r 1 n WD '^N, [1] WRITE [3£] ^LQDF [33] OUT: ->|j F u n c t i o n ( B . 5 ) - V A A A ^ 1 32 <?CREATE [ • ] <7 O F U T f R N K C R E A T E D R D ; N M S J M J ; M £ ; L G ' [ £ 3 (f A. A * C R E A T E * . :*: . * C 3 ] l f T H I S S U E R O U T I H E C R E A T E S A L L . S F I H T E H S D R S [4] ,ii D F T H E G I l - ' E N R A N K A N D O R D E R [ 5 3 tf ' [ 6 3 rf [ 7 3 * CEO [ 9 3 4SKIF' I F F . X . £ I ( R N K + D R I i ) [ICO 1 R A N K F I N D O R D E R I N C D M F A T I E L E 1 [ 1 1 ] + 0 [ 1 £3 i *: D P U T f <; o £ 4 0> P Ci [ 1 3 ] N M S * - R N K N A M E S D R D [ 1 4 ] L S f ( p N M S ) [ 1 ] + 1 [ 1 5 ] L D D F i U G f L G - l [ 1 6 ] + O U T L I F F L G < |j [ 1 7 ] 4 T W D I F F R N K = £ [ 1 8 ] C Q M E I N ( R H K , R N K ) [ 1 9 ] - t L D D P [£ri] T W O : C O M E I H(2, ir:i [ £ 1 ] C P M E I N [ £ £ ] - t L D D F [ £ 3 ] D U T L : E E f O F ' U T 12.41 D P U T ^ Z T E R H S O F U T [ £ 5 ] ° F ' - ' T [ ; ; 1 3 + 1 [ £ 6 3 + 0 Function (B.6) -VCREATEV C O M E I H R ; M I ; M 3 [ 1 ] ( R O U T I N E S P E C I F I C TO 1 C R E A T E 1 ) [ £ 3 M J FG E H R T R [ 1 3 F R T T Y F E N M S [ L G ; J J ] [ 3 ] -M-nnp I F F ( p M j ) [ i ] = ij [ 4 3 MOfGENRT R [£•] F R T T Y F E NMS [ L G 5 £ j! 3 [FJ] ->LDOF I F F (pM£> [ 1 ] =|j [ 6 3 MI«.<<PMI> cn , £ 1 5 > t M l • C3D C f p M i > [ i ] , £• 1 5 > P 0 [ 7 3 M£*> < ( p M £ ) [I] , £ 1 5 > f M £ , [ 3 3 ( (;PM£;, [ 1 3 , £ 15>P0 [ 8 3 MI^MI C R O S S ; : M£ [ 9 3 M£«> < CpM 1 > [ 1 ] , £ 9 ; . p N M S [l _ G ; ; 3 C1 03 MI<K ,.; C p M i > [ i ] , £ 1 > P 1 > , [ 3 3 " 1 . C 3 3 M £ [ 1 1 3 H J f S F E C D R C M l [ 1 £ 3 • P U T f D P U T , [ 1 3 M l F u n c t i o n (B.7) - y c O M B I N V 1 33 <? n F U T f R H K F I L L D R D [1] E E f R N K CREFlTE DHD ce: O F ' U T f 1 D E G R E E 1 , O O R D ) , } fjijp ' ' C3] • F ' U T f 1 l l j l j p ( 1 J F?fiNK ' , ( T R H K ) , D F U T ) C4] D F U T f G F ' U T , [ 1 ] W R I T E EG! F u n c t i o n ( B . 8 ) - V A L L V <? NMif -RF iNK NFlMES D R D ; E S J S| L | F ; G j H i D D J E C l ] If C£] * CREFlTES Fl L I S T DF NFlMES Nx£ x M C3] n C4] s*r I : :ORD+I> *£ C5] C6] N M S * ( i; (j , Ci;, p f j C7] L . « H j C3] L • • F ; L * L + 1 C9] -Ki I F F L>S Cl U] DDf -DS [ £ J L J PMU ( £ p D S [ 1 ? L] > Cl 1] G!f- |j C l £ ] I NLODP : G!f-G! + J C13] •+LDQP I F F E > ( p D D ) [ J ] C14] Ef -NMSFN r 'D [E ! ; ] C15] F * ( CpB> C l ] . £ 9> C16] W *- < + ,-•E > + F [ 1 £ ] p R Fl N K > £ x f .,- E C17] Cl 83 . F C l ] «- <P»> C l ] [19] NMSfNMS, [ J ] ( F t ( E , [3] F p f i ; , [ £ 0 ] I NLDDP F u n c t i o n ( B . 9 ) - VNAMESV <? E f NMSFN D C l ] E * - Ij 0 F'O C£] F f <|V"*> +1 C3] I N L ; F * F - 1 C4] + DUT I FF F < |j C5] Ef- ( <4 |j;i PERM ( + . . . D = F;i > C R D S S p E Cb] -j-INL C7] DUT ; E f - E , [1 . 3] ( 4 - E ; . CS] E f-E+ ( p E ) p D F u n c t i o n ( B . 1 0 ) - y N M S F N V 134 •=7 Mf=tT4rWD PMU SMX}K [ I ] ->ZOUT IFF SMX[1]=0 [£'] -tPRCJUT IFF WIi=|j [3] K f l + l / S M X [4] MFlT> OH, WD) p|] [5] L O D F : K f K - l [6] *U I^F K<|j [ ? ] + lj I F F ( ( W D - l ) x K ) < S M X [ l ] - K [ g ] MFtTt-MFiT, [1] <; (WD- J ;i FMU ( ('SMX [ 1 ] - K ) , K ) ) C R D S S g ] 1 p K [Q] ->l_DDF [ i ij] ZDUT : MAT* \, WD;;i p. n [ I I ] +0 T i c ] D R D U T ; h f i T f |j |j p|j Function ( B . 1 1 ) - y P M U V V --TRfRHK FRTTYFE N n M ^ ^ C 1 ] L*>~ 1 t (WOM* |j;, / •, p N O H [£'] S T R ^ i j p i j [3] -T <!- Ij C4] L.ODF : J + I C5] -•OUT IFF J;> L [6] STRfSTR, ((NDM [ J]) [?] ->t_OOP [8] OUT;STRfSTR,RNKp£ 4 Funct ion (B. 12) - V P R T T Y P E V rr NWMf MJ C R O S S £ M C [1] -•ONE IFF ( p H J ) [1] = 0 [ £ ] •tTHD I FF ( p M £ ) [ 1 ] = ij [3] M l + <: >::pM£;:. 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' = £ ] + < T M [ £ ; C £ ] > t 4 T S [£ ; '=£] ] [17] C F F [ £ ; 1 ] f C F F [ £ ? 1 ] X ( p C 1 > x ( p C £ ) [13] H f l j [19] S P f f j [ £ ' ! ] ] SFll/f RERFli-' can L D O F F i : Ftt-Fi + S P [ £ £ ] -••OUT I F F F l>pCl [£3] E D f Ij [24] B f | j [£5] LDOF'B ; E f E + ED [£6] -tLDOPFt I F F E > p C £ [£7] U N R F H - - ' S F H - ' [£8] TM [J C l ] f TM [? 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C£] ] [143 CFF [ £ 5 1 ] f CFF [£; 1] X (pC£) [153 s P f o [163 S l-'f R E R A l , - ' [173 Df 0 [183 LOOP;DfD+SF [193 + 1] IFF D>pC£ [ £ 0 3 U N R A I.-' SV [ £ 1 3 TMf TM [ ; CeC£] [ £ £ 3 SPf + ,- (TM [£ 5 C £ ] =TM [£ ! C £ [1 ] ] *!' [ £ 3 3 C F F [ i ; 1 ] fCFF [ i ; 1 ] XSP [ £ 4 3 It![£;'=£[i]]fDX [ £ 5 3 NLSTf NLST, [J] KEY INDEL R E R A I.--' [ £ 6 3 4 L O O P F u n c t i o n (B.28) -<?HI0RD1V 1. 4 3 •=7 N G R * - O R D A L L G R ; Y ; 2i ; Z £ ; L G ; R J L ; C F F ; N M ; T M ; TM m n [8] if * : A: :*: O R D A L L * * * [33 If F'JTS E A C H M E M E E R O F ' GR 1 ( D I M N x £ x 4 0 ) I N T O [ 4 ] fl C A N O N I C A L F O R M [15] rt [£,] +HERE+ppGR [7] H E R E : GR*- ( 1 , p 6 R ) pGR [8] rt [ 9 ] rt [10] " ' « K i [ 1 1 ] LG*. (p(5R) [1] [ I ' d ] R J L * - H p |j [ 1 3 ] N G R * |j £ 4 | j p i j [ 1 4 ] 0 L O D P l Y + Y+l [ 1 5 > + O U T I F F Y > L G [ 1 6 ] U N R A V G R [ Y ; ; ] [ 1 7 ] T E R M O R D [ 1 8 ] T M * - F A I R O R D TM [ 1 9 ] T M t - F ' A I R O R D TM [ 8 0 ] 2 J f H I E R ( C F F , [ 8 ] T M ) [ 8 1 ] 2£ * - H I E R ( ( £ 1 p l ) , [ P ] T M ) [ £ £ ] 2 i * z j , [33 <<pzi> Cl £3 , < : 1 6 - C p 2 i > C33>>P0 [£3] 2£*-2£, [3] ((p2£) [ i £'] , (it.-':p2£) [ 3 ] ) )P O [ £ 4 ] 2i* - z i ' -ROSS3 28 [ 8 5 ] 2i [; ; i ] *-2i [; ; i ] v2i [; ; 17] [ 8 6 ] 2 1*. ( I 7 j t - , 3£) [3] 2 1 [ 8 7 ] N M * ( ( p 2 1 ) [ l £ ] , 9 ) p (£ 9> t " M , [ £ ] ( £ 9 P 0 > [ 8 8 ] NGRf-NGR, [1] 21, [3] NM [ 8 9 1 -4-LOOF [3lj] O U T ; N G R * - S F E C R E F N G R Function (B.29) -VORDALLV <7 T E R M O R D ; H [ 1 ] If 121 if F U T S N A M E S I N R I G H T O R D E R [3] if [ 4 ] H*- C 1 I j X + . ^ N M ) + N M [ £ ; ] - N M [ 1 ; ] [ 5 ] H * - 4 H [ 6 ] NM<S-NM[;H] [ 7 ] T M * - T M R E F L A C E ( p N M ) p H , , p H [8] T M > T M R E F L A C E ( p N M ) p H , •, p H Function (B.30) - VTERMORDV r? L I £ T * - H I E R T;E;L;II;W;CF;E;R [£] m 5H--'EN T ( E I T H E R TM D R T M ) W I T H T O F " O F C D N T R F t C T I D N [3] ft I H O R D E R , T H I S P U T S L O W E R D N E S I N O R D E R C4] fl C5] C F f £ 1 f T C6] T f f, 1 4.T [ ? ] -::r--T> [ £ ] [8] C " f 4 T [ i ; ] C9] T t - T [ ; E ] [ l l j ] 4 D K I F F x / H > 4 T [ £ • ; ] C l l ] K«-0 CIS] « < - " l t T [ i ; : [13] L D D P : E * - E + 1 [14] -+QUT I F F B!>R [15] UT[I;]=E [16] + L O O F I F F !] = + .,• L [17] r.f4L....T [ £ ; ] CIS] E f ( L / - , p L ) [1 ] -1 [1 SO T [ £ ; E + - , +...-L-] f T [ £ ; E + D ] [ £ Q ] 4 L O D P [81] fl CSS] fl T H E F O L L O W I N G U S E S e ( F l E ) 6 ( C D ) = e ( A C ) e ( E D ) + g ( D F ) ) e ( E C ) CSS] fl [£4] D U T i + Q K I F F X / W > + T [ £ ; ] [85] r'*-(aiT[£;]) < ( ~ U T r £ ; ] ) ) 1 4-w CS6] »<-i> C l ] + ( Ci 1) [87] L f T CSS] L . [ £ ; D ] f T [ i ; D [ £ ] ] , T [ £ ? D [ i ] ] CS9] t - C i ; * C £ ] ] f T [ £ ; D [ £ ] ] [ 3 0] E f £ i P ( ~ 1 X C F [ I ; i ] ) , C F [ £ ; i ] [31] L I S T f H I E R ( E , [ £ ] L ) [3£] L f T C33] i - [ £ ; D ] f T [ £ ; » [ £ l ] ] [34] L I S T f L I S T , [1] H I E R ( C F , [ £ ] L ) [35] + 0 [36] O K ; L I S T f ( (1 £) , ( p T ) [ £ ] + ! ) p C F , [ £ ] T F u n c t i o n (B.31 ) 1 45 <? N 6 R f S P E C R E F SRJ LGJ C L J ONM ; TMJ TM; C F F ; NM J F ; EJ F ; L 5 R J L [1] if [8] if T E S T S FiND E X C H F I N G E S COMPONENTS DF ERCH g DF 1 SR 1 SD [3] (f THEY F I T THE CFlNONICFlL FORM [4] fl [5] L G f ( p G R ) [1] [£3 R J L f l j p Q [7] DNMf 2 0 PU [8] [93 L O D P : D f i i + i [103 40UT I F F D>L6 [ H 3 UNRFU-' GR [I'j ; 3 [18] + S K I P I F F ( p N M ) [£3#(pDNM> [£] [133 4 H U R R Y IFFX,-• X ,- (NM=DNM> [143 SKIF;DNMfNM [153 C L f C O M P L I S T NM [ l b ] r<r ( P C L ) [13 [17] HURRY ; E f 0 [133 E D O F ; E f E + 1 [19] -4-L.DDF- I F F E > P [8 0] F f ( C L [ E ; 13 + i. C L [ E ; 21 [81] L f F + C L [ E ? £ ] [£8 ] + EDOF I F F F E X Fi D k--' L [83] TMfTM REPLFiCE (p, pXpF;, p F , L , L, F [84] T.MfTM R E P L A C E ( £ , 8>::pF;:> p F , L , L , F [85] G R f 6 R , [ 1 3 DRDFiLL ( 1 £ 4 |T;i pRERFH-' [8b] R J L f R J L i D [87] -4. L O D P. [88] OUT : NGR*, •.,. <; •, ( p s R ) [ 1 ] > e RJL> [ 13 GR F u n c t i o n ( B . 3 2 ) - V S P E C R E P V C L f C D M P L i s T N;H;X;Y [1] if [8] fl L I S T S E X C H F I N G E F l E L E P F i I R S I N N R H E ' N ' [3] fl I : ; X , Y ; I : x i s (POS OF 1ST M E M S DF 1ST GROUP [4] fl Y I S W I D T H DF T H E GROUPS [5] fl [ 6 ] (PN) [8] [7] C L f 0 2 P0 [8] X f 0 [9] L D D P 1 : X f x + i [1 0] •4-lj I F F X > W f £ ' [1 1] T>0 [18] L D D F ' 2 : Y f Y + 1 [13] -J-LDDP1 I F F ( Y + £ x X ) > W C14] + L O D F £ I F F ^ X / N [ ; Y + •, X ] = H [ j Y + X+YX3 [15] C L f C L , [ 1 3 ( Y,X> [16] 4 L D D F £ F u n c t i o n (B.33) - V C O M P L I S T V 1 46 <7CNSDT [ D ] r? <7 NGRfE C N S D T 6 R ; L J J ; K ; R J L J C F J M f £ T cn If [ £ ] if * :*: * CNSDT * * * [3] if A I T 'S A L L L I K E S F ' . T N S R S . A F T E R T H E ' E ' T H ONE, [4] If THUS SHORTENING T H E L I S T C5] If C6] L*. (PGR;;, [i] C7] RJL^opn [8] J*- (pGR) C9] K*-, L Cl U] LOOF; E*-E+J cm 4 L D D F - IFF E g R J L C18] -•OUT I F F E>L C13] S T f X.. -X,-' ( 0 |j J ) I (GR = JpGR f E ; ; ] > C14] c F * s T R G R [ ; ; n [15] M*-r,-CF [ J O ] [16] ' -F [; i ] * • ( M T C F [; 8] > X C F c; i ] [17] C F * . ( + / C F [ ; n::., M C18] GR [E; ; i ] * . C F [19] R J L f R J L , S T / K [8 0] RJL*- ( R J L ^ G ) / R J L [81] -+LDOP [88] OUT; NGR*- (.... (Kg RJL ) ) R G R F u n c t i o n (B.34) - VCNSDTy <7 N L S T f E X C H S Y M O T M J T M i ; T M P ; C F F ; T M ; T M ; N M cn N L S T * 1 £ 4 l j pOTM [8] OTM c8 5 1] *-8XOTH [£•; 1] C3] TM£*-TM1 fOTM C4] U N R A V TM£ C5] + |j I FF . , . = /NM [ 1 ; E X C H [ 1 ; ] ] C6] T M * - T M R E F L A C E E X C H C7] TMfTM R E F L A C E E X C H C8] TM|='*-RERAV C9] N L S T ^ O R D A L L T M £ C1 0] N L S T * N L £ T j [ 1 ] T M \ Cl 1] NLST*-1 CNEDT N L S T F u n c t i o n (B.35) - V S Y M V 147 r? N L S T f ZTERMS DLST j| A j| E $ L N ; TM j ; TM£;; F N ; R E P ; e r r ; i n ; cn •LST*1 CHSDT DLST TM ; N M ; S T ; !• i F j ; r>i F P [83 R* 0 [33 L O O P : A*A + i C4] INPDP:+DUT IFFi-Fl;. (pQLST) [ 1 3 > [53 * * 0 . [63 UNRAV DLST [RJ ? ] C7] I N L P ; £ * * + ! C8] -* L • O F' I F F (E+ l ) > (pNM) [ £ ] [93 TM \ *-OLST [R? ; ] [103 T M£f ( £ 8 PE:* i^+iy !• ( E ;+l) , E > SYM T M ] [113 4NEW I F F (pTMg) [13=0 [183 ST*-<;o 1 4 . T M I > X . = 8 3 1 Oj 0 1 4-TMg) [133 S T > S T [ I ; ; 1 3 X S T [ £ ; ; P J [143 + I HLP I F F ( + /ST ) * |j [153 M E W ; D L S T * D L S T , [ } ] TM£ C163 • L S T * ( R ^ i . (pDLST) [13 ;i .r^DL ST [173 • LST*1 CNSI 'T DLST' C183 J/INPDP [193 O U T ; N L S T f D L S T F u n c t i o n ( B . 3 6 ) - V Z T E R M S V 1 48 A p p e n d i x C C o m p l e t e L i s t o f S p i n t e n s o r s t o D e g r e e 8 What f o l l o w s i s a l i s t i n g of a l l 1 i n e a r l y - i n d e p e n d e n t s p i n t e n s o r t e r m s up t o and i n c l u d i n g t h o s e o f d e g r e e 8 and f o r r a n k s 0 ( s c a l a r s and 4 - f o r m s ) , 1 ( v e c t o r s and 3 - f o r m s ) , and 2 ( 2 - f o r m s ) . The n o t a t i o n u s e d i s b a s e d on t h a t o f P e n r o s e ( 2 3 ) as d e s c r i b e d i n c h a p t e r 7, but r e v i s e d t o t h e f o r m d e s c r i b e d below so t h e c o mputer c o u l d t y p e t h e c h a r a c t e r s . " p s i " and "PSI" r e p r e s e n t T* and Y r e s p e c t i v e l y . D o t t e d i n d i c e s a r e w r i t t e n i n m i n u s c u l e Roman l e t t e r s , u n d o t t e d i n d i c e s i n c a p i t a l s , w i t h t h e l e t t e r s X and x r e s e r v e d f o r t h e f r e e i n d i c e s . A l l o t h e r i n d i c e s a r e c o n t r a c t e d , t h e f i r s t l e t t e r w r i t t e n a l w a y s c o r r e s p o n d i n g t o l o w e r one i n t h e s t a n d a r d n o t a t i o n . F o r example: P S I ( A B C X ) p s i ( A d e f g h ) p s i ( B C d e f g h x ) (C. 1 ) r e p r e s e n t s : 7T> B C D E . F G H (C.2) ABcx * The components of r a n k - 2 s p i n - t e n s o r s must a l w a y s i n c l u d e a 149 f a c t o r o f £ x y o r £ w . In t h e f o l l o w i n g t a b l e s t h e s e £ f a c t o r s a r e n o t shown, but a r e t o be i n f e r r e d w h e r e v e r t h e y a r e r e q u i r e d i n o r d e r t o have two f r e e i n d i c e s , d o t t e d and u n d o t t e d . Thus p s i ( e f g h ) p s i ( e f g h ) P S I ( A B C D X i ) P S I ( A B C D X i ) ( c 3 ) r e p r e s e n t s t h e s p i n - t e n s o r : 1 EFGiH 1 ' F » 6 C D X I • Y XV The Roman n u m e r a l s w r i t t e n t o t h e r i g h t o f t h e s e t a b l e s w i l l be u s e d t o i d e n t i f y t h e t e r m s i n t h e t e x t . Terms w h i c h a r e c o m p l e x - c o n j u g a t e s h a r e t h e same n u m e r a l , but w i t h an a s t e r i x on one t o i n d i c a t e t h e c o n j u g a t e . PSI(ABCD)PSI(ABCD) (2/0/1) p s i ( a b e d ) p s i ( a b e d ) (2/0/1*) T a b l e C.1 : Rank 0 Degree 2 S p i n - t e n s o r Terms p s i ( a b c d ) p s i ( X a b c d x ) (3/1/1) PSI(ABCD)PSI(ABCDXx) (3/1/1*) T a b l e C.2 : Rank 1 Degree 3 S p i n - t e n s o r Terms P S I ( A B C D ) P S I ( A B E F ) P S I ( C D E F ) . . (4/0/1) p s i ( a b c d ) p s i ( a b e f ) p s i ( c d e f ) . . (4/0/1*) P S I ( A B C D E f ) P S I ( A B C D E f ) . . . . (4/0/11) p s i ( A b c d e f ) p s i ( A b c d e f ) . . . . (4/0/11*) T a b l e C.3 : Rank 0 Degree 4 S p i n - t e n s o r Terms 151 p s i ( A b c d e f ) p s i ( A X b c d e f x ) (5/1 / I I ) PSI (ABCDEf ) PSI (ABCDEXf x ) (5/ 1 / I I*) p s i ( a b c d ) p s i ( a b e f ) p s i ( X c d e f x ) (5/1/1) P S I ( A B C D ) P S I ( A B E F ) P S I ( C D E F X x ) (5/1/1*) T a b l e C.4 : Rank 1 Degree 5 S p i n - t e n s o r Terms PSI (ABCD)PSI(ABEF)PSI(CDGH)PSI(EFGH) . (6/0/1) PSI (ABCD)PSI(ABCD)PSI(EFGH)PSI(EFGH) . (6/0/11) PSI ( A B C D ) P S I ( A B C D ) p s i ( e f g h ) p s i ( e f g h ) . ( 6 / 0 / I I I / I I I * ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d g h ) p s i ( e f g h ) . (6/0/1*) p s i ( a b e d ) p s i ( a b e d ) p s i ( e f g h ) p s i ( e f g h ) . (6/0/11*) PSI (ABCD)PSI(ABEFGh)PSI(CDEFGh). . . . ( 6 / 0 / I V ) p s i ( b c d e ) p s i ( A b c f g h ) p s i ( A d e f g h ) . . . . ( 6 / 0 / I V * ) PSI (ABCDEFgh)PSI(ABCDEFgh) . (6/0/V) p s i ( A B c d e f g h ) p s i ( A B c d e f g h ) . (6/0/V*) T a b l e C.5 : Rank 0 Degree 6 S D i n - t e n s o r Terms P S I ( A B C D ) P S I ( A B C E F g ) P S I ( D E F X X g ) . . . (6/2/1) p s i ( b c d e ) p s i ( A b c d f g ) p s i ( A e f g x x ) . . . (6/2/1*) T a b l e C.6 : Rank 2 D e g r e e 6 S p i n - t e n s o r Terms 153 psi (abed)psi(abef)psi(cghx)psi(Xdefgh). (7/1 / 1 ) psi (abed)psi(abef)psi(cdgh)psi(Xefghx). (7/1/II) psi ! abed)psi(abed)psi(efgh)psi(Xefghx). (7/1/III) PSI (ABCD)PSI(ABEF)PSI(CGHX)PSI(DEFGHx). (7/1 / 1* ) PSI (ABCD)PSI(ABEF)PSI(CDGH)PSI(EFGHXx). (7/1/11 *) PSI (ABCD)PSI(ABCD)PSI(EFGH)PSI(EFGHXx). ( 7/1 / 1II*) PSI (ABCD)psi(efgh)psi(efgh)PSI(ABCDXx). (7/1/IV) PSI (ABCDEh)PSI(ABCFGh)PSI(DEFGXx) . . . (7/1/V) psi (Abcdef)psi(Abcdgh)psi(Xefghx) . . . (7/1/V*) psi (bcde)psi(Abfghx)psi(AXcdefgh) . . . (7/1/VI) psi (bcde)psi(Abefgh)psi(AXdefghx) . . . (7/1/VII) PSI (ABCX)psi(Adefgh)psi(BCdefghx) . . . (7/1/VIII) psi (fghx)PSI(ABCDEf)PSI(ABCDEXgh) . . . (7/1/VIII*) PSI (ABCD)PSI(AEFGXh)PSI(BCDEFGhx) . . . (7/1/VI*) PSI (ABCD)PSI(ABEFGh)PSI(CDEFGXhx) . . . (7/1/VII*) psi (ABcdefgh)psi(ABXcdefghx) (7/1/IX) PSI (ABCDEFgh)PSI(ABCDEFXghx) (7/1/IX*) PSI ( A B C D )PSI(ABCD)psi(efgh)psi(Xefghx). (7/1/IV*) Table C.7 : Rank 1 Degree 7 Spin-tensor Terms PSI (ABCD)PSI(ABEF)PSI ( C D G H ) P S I ( E F I J ) P S I (GHIJ) PSI (ABCD)PSI(ABEF)PSI ( C D E F ) P S I ( G H I J ) P S I (GHIJ) PSI (ABCD)PSI(ABCD)PSI ( E F G H ) P S I ( E F I J ) P S I (GHIJ) PSI (ABCD)PSI(ABEF)PSI ( C D E F ) p s i ( g h i j ) p s i ( g h i j ) PSI ( A B C D ) P S I ( A B C D ) p s i ( e f g h ) p s i ( e f i j ) p s i ( g h i j ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d g h ) p s i ( e f i j ) p s i ( g h i j ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d e f ) p s i ( g h i j ) p s i ( g h i j ) p s i I a b e d ) p s i ( a b e d ) p s i ( e f g h ) p s i ( e f i j ) p s i ( g h i j ) PSI (ABCD)PSI(EFGH)PSI ( A B C D I j ) P S I ( E F G H I j PSI (ABCD)PSI(ABEF)PSI ( C D G H I j ) P S I ( E F G H I j PSI (ABCD)PSI(ABCD)PSI ( E F G H I j ) P S I ( E F G H I j p s i ( f g h i ) p s i ( f g h i ) P S I ( A B C D E j ) P S I ( A B C D E j PSI ( A B C D ) P S I ( A B C D ) p s i ( E f g h i j ) p s i ( E f g h i j p s i ! b c d e ) p s i ( f g h i ) p s i ( A b c d e j ) p s i ( A f g h i j ps i ( b c d e ) p s i ( b e f g ) p s i [ A d e h i j ) p s i ( A f g h i j p s i [ b c d e ) p s i ( b c d e ) p s i ! A f g h i j ) p s i ( A f g h i j PSI ( A B C D ) p s i ( f g h i ) P S I ( A B C D E j ) p s i ( E f g h i j p s i ( A c d e f g ) p s i ( B c d h i j ) p s i ( A B e f g h i j ) PSI ( A B C D E i ) P S I ( A B F G H j ) P S I ( C D E F G H i j ) p s i ( g h i j ) P S I ( A B C D E F g h ) P S I ( A B C D E F i j ) PSI ( A B C D ) P S I ( A B E F G H i j ) P S I ( C D E F G H i j ) p s i ( c d e f ) p s i ( A B c d g h i j ) p s i ( A B e f g h i j ) PSI ( A B C D ) p s i ( A B e f g h i j ) p s i ( C D e f g h i j ) PSI (ABCDEFGhi j ) P S I ( A B C D E F G h i j ) p s i ( A B C d e f g h i j ) p s i ( A B C d e f g h i j ) T a b l e C.8 : Rank 0 D e g r e e 8 S p i n - t e n s o r Terms 155 PSI PSI p s i p s i PSI PSI PSI PSI p s i PSI p s i ps i PSI PSI PSI PSI p s i PSI p s i PSI p s i PSI ps i PSI ABCD)PSI(EFGH)PSI A B C D ) P S I ( A B E F ) P S I b c d e ) p s i ( f g h i ) p s i b c d e ) p s i ( b c f g ) p s i A B C D ) p s i ( e f g h ) P S I A B C D ) p s i ( e f g h ) P S I A B C D ) p s i ( f g h i ) P S I A B C D ) p s i ( f g h i ) P S I a b e d ) p s i ( a b e f ) p s i A B C D ) P S I ( A B E F ) P S I A b c d e f ) p s i ( X b c g h i A c d e f g ) p s i ( B c d h i x A B C X X d ) p s i ( A e f g h i ABCDEh)PSI(ABFGXi ABCDEi)PSI(ABFGHx ABCDEg)ps i ( F g h i x x g h i x ) P S I ( A B C D E F g h ABCD)PSI(ABCEFGhi c d e f ) p s i ( A B c d e g h i A B C D E i ) P S I ( F G H X X i ) C D E G H i ) P S I ( F G H X X i ) A b c d e f ) p s A d e f h i ) p s A B C X X i ) p s A B C D X i ) p s ABCDEx)ps [ A g h i x x ) I A g h i x x ) [ D e f g h i ) [ X e f g h i ) [ E f g h i x ) A B C D E f ) p s i ( E g h i x x ) c g h i ) p s i ( X X d e f g h i ) C GHI)PSI(DEFGHIxx) p s i ( A X d e f g h i ) p s i ( A B e f g h i x ) p s i ( B C d e f g h i ) P S I ( C D E F G X h i ) P S I ( C D E F G H i x ) P S I ( A B C D E F h i ) PSI(ABCDEF i x) P S I ( D E F G X X h i ) p s i ( A B f g h i x x ) p s i ( C X d e f g h i ) A B C X ) p s i ( A B d e f g h i b c d e ) p s i ( A b f g h i ) p s i ( A X X c d e f g h i ) A B C D ) p s i ( A e f g h i ) p s i ( B C D e f g h i x x ) f g h i ) P S I ( A B C D E f ) P S I ( A B C D E X X g h i ) A B C D ) P S I ( A E F G H i ) P S I ( B C D E F G H i x x ) T a b l e C.9 : Rank 2 D e g r e e 8 S p i n - t e n s o r Terms P S I ( A B C D ) P S I ( A B E F ) P S I ( C D E F ) p s i ( g h i j ) p s i ( X g h i j x ) P S I ( A B C D ) P S I ( A B C D ) p s i ( e f g h ) p s i ( e f i j ) p s i ( X g h i j x ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d g h ) p s i ( e i j x ) p s i ( X f g h i j ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d g h ) p s i ( e f i j ) p s i ( X g h i j x ) p s i ( a b e d ) p s i ( a b e f ) p s i ( c d e f ) p s i ( g h i j ) p s i ( X g h i j x ) p s i ( a b e d ) p s i ( a b e d ) p s i ( e f g h ) p s i ( e f i j ) p s i ( X g h i j x ) P S I ( A B C D ) P S I ( A B E F ) P S I ( C D G H ) P S I ( E I J X ) P S I ( F G H I J x ) P S I ( A B C D ) P S I ( A B E F ) P S I ( C D G H ) P S I ( E F I J ) P S I ( G H I J X x ) P S I ( A B C D ) P S I ( A B E F ) P S I ( C D E F ) P S I ( G H I J ) P S I ( G H I J X X ) P S I ( A B C D ) P S I ( A B C D ) P S I ( E F G H ) P S I ( E F I J ) P S I ( G H I J X X ) P S I ( A B C D ) P S I ( A B E F ) p s i ( g h i j ) p s i ( g h i j ) P S I ( C D E F X x ) P S I ( A B C D ) p s i ( e f g h ) p s i ( e f i j ) p s i ( g h i j ) P S I ( A B C D X x ) P S I ( A B C D ) P S I ( A B C E F j ) P S I ( D E G H I x ) P S I ( F G H I X j ) P S I ( A B C D ) P S I ( A B C D E j ) P S I ( E F G H I x ) P S I ( F G H I X j ) P S I ( A B C D ) P S I ( A B E F G j ) P S I ( C D E H I j ) P S I ( F G H I X x ) P S I ( A B C D ) P S I ( A B C E F j ) P S I ( D E G H I j ) P S I ( F G H I X x ) P S I ( A B C D ) P S I ( A B C D E j ) P S I ( E F G H I j ) P S I ( F G H I X x ) p s i ( f g h i ) P S I ( A B C D E j ) P S I ( A B C D X x ) p s i ( E f g h i j ) p s i ( f g h i ) P S I ( A B C D E f ) P S I ( A B C D X j ) p s i ( E g h i j x ) p s i ( f g h i ) P S I ( A B C D E f ) P S I ( A B C D E j ) p s i ( X g h i j x ) p s i ( b c d e ) p s i ( A b e f g h ) p s i ( A d e f i j ) p s i ( X g h i j x ) p s i ( b c d e ) p s i ( A b c d f g ) p s i ( A e h i j x ) p s i ( X f g h i j ) p s i ( b c d e ) p s i ( A b c d f g ) p s i ( A e f h i j ) p s i ( X g h i j x ) p s i ( b c d e ) p s i ( A b c d e f ) p s i ( A g h i j x ) p s i ( X f g h i j ) Rank - 1 D e g r e e - 9 c o n t ' d . p s i b c d e ) p s i A b c d e f ) p s i ( A f g h i j )ps i ( X g h i j x ) PSI ABCD)PSI ABCDEf)ps i ( E g h i j x ) p s i ( X f g h i j ) PSI ABCD)PSI A B C E X f ) p s i ( D f g h i j )ps i ( E g h i j x ) PSI ABCD)PSI 'ABCDEf)ps i ( E f g h i j )ps i ( X g h i j x ) PSI ABCD)PSI A B E X ) p s i ( C f g h i j ) p s i ( D E f g h i j x ) PSI (ABCD)PSI ( A B C D ) p s i ( E f g h i j ) p s i ( E X f g h i j x ) p s i ( b c d e ) p s i ( f g h i ) p s i ( A b e d j x ] p s i ( A X e f g h i j ) p s i ( b c d e ) p s i ! f g h i ) p s i ( A b c d e j I p s i ( A X f g h i j x ) p s i ( b c d e ) p s i ( b c f g ) p s i ( A d h i j x ] p s i ( A X e f g h i j ) p s i ( b c d e ) p s i ( b c f g ) p s i ( A d e h i j ) p s i I A X f g h i j x ) p s i ( b c d e ) p s i ( b c d e ) p s i ( A f g h i j] p s i ( A X f g h i j x ) PSI ( A B C X ) p s i ( d e f g ) p s i ( A d h i j x ) p s i ( B C e f g h i j ) PSI ( A B C X ) p s i ( d e f g ) p s i ( A d e h i j] p s i ( B C f g h i j x ) PSI ( A B C D ) p s i ( f g h i ) P S I ( ABCEXj) p s i ( D E f g h i j x ) PSI ( A B C D ) p s i ( f g h i ) P S I ( ABCDEj) p s i ( E X f g h i j x ) PSI ( A B C D ) p s i ( f g h i ) p s i ( E f g h j x ] PSI ( A B C D E X i j ) PSI ( A B C D ) p s i ( f g h i ) p s i ( E f g h i j) PSI ( ABCDEXjx) PSI (ABCD)PSI (EFGH)PSI< ABCIXj PSI I D E FGHIjx) PSI (ABCD)PSI (EFGH)PSII ABCDIj PSI E F G H I X j x ) PSI (ABCD)PSI (ABEF)PSI< CGHIXj >PSI DEFGHIjx) PSI (ABCD)PSI (ABEF)PSI CDGHIj >PSI (EFGHIXjx) PSI (ABCD)PSI (ABCD)PSI EFGHIj IPSI (EFGHIXjx) p s i ( f g h i ) p s i ( f g j x ) P S I (ABCDEh )PSI ( A B C D E X i j ) p s i ( f g h i ) p s i ( f g h i ) P S I (ABCDEj )PSI (ABCDEXjx) Rank - 1 D e g r e e - 9 c o n t ' 158 PSI PSI PSI PSI p s i PSI PSI p s i p s i PSI p s i PSI PSI PSI p s i p s i PSI PSI p s i p s i PSI PSI p s i PSI A B C D ) p s i ( h i j x ) P S I ( A E F G X h ) P S I ( B C D E F G i j ) A B C D ) p s i ( h i j x ) P S I ( A B E F G h ) P S I ( C D E F G X i j ) A B C X ) p s i ( d e f g ) p s i ( h i j x ) p s i ( A B C d e f g h i j ) A B C D ) P S I ( E F G X ) p s i ( h i j x ) P S I ( A B C D E F G h i j ) A g h i j x ) P S I A B C D E i ) P S I A B C D E i ) P S I A c d e f g ) p s A c d e f g ) p s ABCDXe)ps A c d e f g ) p s A B C D X e ) p s i A B C D E i ) P S I A B C D E g ) p s i c d e f ) p s i ( A B c g h i j x c d e f ) p s i ( A B c d g h i j A B C D ) p s i ( A X e f g h i j A B C D ) p s i ( A B e f g h i j g h i j ) P S I ( A B C D E F g x g h i j ) P S I ( A B C D E F g h A B C D ) P S I ( A E F G H X i j ABCDEFgh)PSI(BCDEFXi j ) A B C F G H j x ) P S I ( D E F G H X i j ) ABCFGHi j ) P S I ( D E F G H X j x ) A B c d h i j x ) p s i ( B X e f g h i j ) A B c d e h i j ) p s i ( B X f g h i j x ) A B e f g h i j ) p s i ( C D f g h i j x ) B c d h i j Af g h i j ABFGHj F g h i j x ) p s i ( A B X e f g h i j x ) ) p s i ( B C D e f g h i j x ) ) P S I ( C D E F G H X i j x ) ) P S I ( A B C D E F X h i j ) ) p s i ( A B X d e f g h i j ) ) p s i ( A B X e f g h i j x ) ) p s i ( B C D e f g h i j x ) ) p s i ( C D X e f g h i j x ) ) P S I ( A B C D E F X h i j ) ) P S I ( A B C D E F X i j x ) ) P S I ( B C D E F G H i j x ) ) P S I ( C D E F G H X i j x ) A B C D ) P S I ( A B E F G H i j A B C d e f g h i j ) p s i ( A B C X d e f g h i j x ) A B C D E F G h i j ) P S I ( A B C D E F G X h i j x ) T a b l e C.10 : Rank 1 D e g r e e 9 S p i n - t e n s o r Terms 129a LEAVES 130-V7 NOT FILMED; QUALITY TOO POOR TO REPRODUCE. 5 4 2 

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