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The possible connection between certain universal symmetry operations and baryon and lepton conservation. Robertson, Dale Alexander 1963

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THE POSSIBLE CONNECTION BETWEEN CERTAIN UNIVERSAL SYMMETRY OPERATIONS AND BARYON AND LEPTON CONSERVATION by DALE ALEXANDER ROBERTSON B.Sc•, Carleton University, 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH: COLUMBIA A p r i l , 1963 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of Bri t i sh Columbia, I agree that the Library shal l make i t freely available for reference and study, I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for f inancial gain shall not be allowed without my written permission. Dale Robertson Department of Physics  The University of Br i t i sh Columbia, Vancouver 8, Canada. Date A p r i l 1963 ABSTRACT The aim of t h i s work i s to propose a possible explanation of certain conservation laws which hold i n the reactions among elementary p a r t i c l e s . The laws i n question are those of conservation of baryon number and conservation of lepton numbers. These are additive quantum numbers which take the values +1, - 1 , or 0 , for single p a r t i c l e s . At the present time, these laws must be con-sidered as empirical conservation laws, whose o r i g i n i s not known. Certain universal symmetry operations must be repre-sented by anti-unitary operators whose square i s - I . The existence of such an operator leads to a super-selection r u l e . The Hil b e r t space i s decomposed into two orthogonal subspaces, with no observables having matrix elements connecting the two subspaces,; In the presence of a super-selection r u l e , an operator can be constructed, which i s an observable whose eigenvalues are conserved, and whose eigenvalues have the properties of an additive quantum number. One has one such operator f o r the baryons, and two f o r the leptons. A consistent c l a s s i f i c a t i o n of the elementary p a r t i c l e s i s worked out, which allows the conservation laws to be explained as a r e s u l t of super-selection r u l e s , provided that the masses of the p a r t i c l e s have suitable transformation properties;. - V -ACKNOWLEDGEMENTS I should l i k e to express my sincere thanks to Dr . F . A . Kaempffer, for o r i g i n a l l y suggesting t h i s top ic and for providing valuable guidance at every stage of the work. I should also l i k e to thank the Nat ional Besearch Council for f i n a n c i a l a i d , i n the form of a studentship. - i i i -TABLE OF CONTENTS PAGE ABSTRACT i i ACKNOWLEDGEMENTS v CHAPTER I: INTRODUCTION I.a The Importance of Symmetry 1 I.b The Representation of Symmetry Operators 2 . I . b . i Unitary Symmetry Operators 3 I . b . i i Anti-Unitary Symmetry Operators 4 I . c The Operations of Combined Inversion $ and Reversa.1 of Motion ' I . c . i Combined Inversion 6 I. c . i i Reversal of Motion 12 CHAPTER I I : EXPLICIT. REPRESENTATION OF THE OPERATORS OF TIME REVERSAL AND COMBINED INVERSION I I . a Time Reversal 14 I I . a . i The Impossibility of Representing 14 Time Reversal by a Unitary Operator . I l . a . i i E x p l i c i t Construction of the 17 Operator © I L . a i i i i Value, of ©* for Angular Momentum 19 States II.b Representation of Combined Inversion 21. f o r P a r t i c l e s of Spin h Described by Dirac's Equation I l . b . i Representation of the Parity 21 Operation I l . b . i i Representation of the Operator of 24 P a r t i c l e Conjugation I I . b . i i i Representation of the Operation 26 of Combined Inversion - i v -I l . b . i v CHAPTER I I I : Remarks on P a r t i c l e Conjugation and Combined Inversion 28 I I I . a I l l . b I I I . c CHAPTER IV: IV. a IY.b CHAPTER V: V.a V.b PLATES Plate I APPENDIX I GENERAL PROPERTIES OF ANTI-UNITARY OPERATORS WHOSE SQUARE IS - I Degeneracy of States Introduction of Another Two Dimensions into the State Vector Space Removal of the Degeneracy THE EMPIRICAL CONSERVATION LAWS OF BARYON NUMBER AND LEPTON NUMBERS Baryon Number Lepton Numbers THE POSSIBLE CONNECTION BETWEEN THE CONSERVATION OF BARYON AND LEPTON NUMBERS AND THE SYMMETRY OPERATIONS OF COMBINED INVERSION AND REVERSAL OF MOTION Conservation of Lepton Numbers Conservation of Baryon Number 3.1 34 36. 39 41 following page Time Reversal Invariance i n C l a s s i c a l Mechanics APPENDIX I I The Customary Representation of the Angular Momentum Operators APPENDIX I I I Table of Baryons BIBLIOGRAPHY 45 43 7 55 56 58 ( 1 ) CHAPTER I : INTRODUCTION I . a The Importance of Symmetry The symmetry properties of a phys ica l system are of great value i n understanding the behaviour of the system. They are independent of any de ta i l ed theory, and thus may be of value i f no sa t i s fac tory de ta i l ed theory i s known. Symmetry propert ies can i n fact serve as useful guides i n the formulation of a theory. For example the demand that a ce r t a in symmetry be preserved, serves to l i m i t the possible choices of i n t e r ac t ion terms i n a Hamiltonian. Symmetry properties also lead to se lec t ion ru l e s , which fo rb id cer ta in processes. A ce r ta in process may be forbidden because i t f a i l s to preserve a p a r t i c u l a r symmetry. For example, there i s the se lec t ion ru le that a p a r t i c l e of spin 1 and a de f in i t e pa r i t y cannot decay in to two photons, (provided the relevant i n t e r ac t i on i s p a r i t y conserving). (2) I .b The Representation of Symmetry Operators In general , symmetry operations are represented i n quantum mechanics by operators, S>T--- • These operators produce from a given s ta te , the corresponding state r e s u l -t i n g from applying the symmetry operat ion. For example, i f S i s bhe operator which represents a ce r t a in ro t a t ion the state produced by ro ta t ing the coordinates. The measurable predic t ions of quantum mechanics can be expressed i n terms of the expectation values of operators, and the moduli of the sca lar products of state vectors . Consequently, a transformation which leaves these quan-t i t i e s invar ian t produces no phys ica l change. Thus the transformation representing a ce r t a in symmetry, must s a t i s fy the condit ions (3) I . b . i Unitary Symmetry Operators The most f a m i l i a r type of transformation which f u l f i l l s the requirements ( I . b . l ) and ( I .b.2), i s a un i ta ry transformation. This i s a transformation produced by a un i ta ry operator U . Unitary operators are defined by the requirement that ( I . b . i . l ) \J ZZ U I f the states are transformed by the ru le |b>= U/b> and the operators by S = US IT , then such a transformation preserves laoth the expectation values of the operators, and the values of the scalar products. Indeed one has <s'> =<b'|s'|b'> = <ub|usir'|ub> = <j.|u+uscr'u|b> = <b|s)b> = <s> <bV> = <Ub|Ua> = <b|U+U|a> = <b|a> I t should be noted that the preservation of the value of the sca lar product i s a stronger condi t ion than i s demanded by ( I .b.2), and thus the condi t ion that an operator be uni ta ry i s a su f f i c i en t condi t ion for i t to s a t i s f y ( I . b . l ) and ( I .b.2), but i t i s not a necessary cond i t ion . I . b . i i An t i -Uni ta ry Symmetry Operators The condi t ions ( I . b . l ) and (I.b.2) could also be s a t i s f i e d by an operator with the proper t ies ( I . b . i i . l ) < A b M c > = < £ | b > = < b | c > ( i . b . i i . 2 ) A [f l*>> + rl$] z y*Alb> + ^ /c> Such operators are termed a n t i - u n i t a r y . I f K represents the operation of complex conjugation, then A K i s un i ta ry , and i t fo l lows that an t i -un i t a ry operators may be wr i t t en i n the; form ( i . b . i i . 3 ) A = T K where T i s a un i t a ry operator. I f the an t i -un i t a ry operator has an inverse , then t h i s inverse may be wr i t ten i n the form K T ' and hence the inverse of an a n t i -uni ta ry operator i s a lso a n t i - u n i t a r y . The question of whether a ce r ta in symmetry i s to be represented by a un i ta ry operator, or an an t i -un i t a ry operator must be decided from case to case. Wigner (I960) has discussed spec i f i c c r i t e r i a which may be appl ied to determine which p o s s i b i l i t y holds i n a given case. 15) I . c The Operations of Combined Inversion and Reversal of Motion The purpose of t h i s work i s to discuss cer ta in pro-per t ies of two symmetry operations which are bel ieved to be un ive r sa l symmetry operations. These are the operations of combined invers ion and reversa l of motion, denoted by 12, and O r e spec t ive ly . Both of these operations may be sa id to have c l a s s i c a l analogues, as c l a s s i c a l mechanics and e l e c t r i c i t y posess analogous symmetries,provided one al lows for the operation of p a r t i c l e and a n t i - p a r t i c l e interchange as a possible symmetry i n the c l a s s i c a l case. This l a s t p rovis ion i s necessary to cover the case of combined i n v e r s i o n . Each of these operations i s a d iscre te symmetry oper-a t ion whose square should be equivalent to the i d e n t i t y , because of the phys ica l nature of the symmetry operation which i s invo lved . I t should be noted that i n t h i s con-text equivalent does not imply that the square of the operation i s the i d e n t i t y . This d i s t i n c t i o n a r i ses from the w e l l known freedom to mul t ip ly a l l state vectors by an a rb i t r a ry phase factor of modulus u n i t y . (61 I . c . l Combined Inversion The operation of combined invers ion i s a symmetry-operation re la ted to the invariance of an object under invers ion of coordinates. This symmetry has been known to ex i s t i n c l a s s i c a l physics for some time, but the consequences of i t are not as s t r i k i n g as they are i n the quantum mechanical case. The transformation of invers ion of coordinates may be expressed a n a l y t i c a l l y as ( I . c . i . l ) t -» t'= t Two types of vectors may be dis t inguished by t h e i r beha-viour under t h i s transformation. The components of vectors of the f i r s t type, usua l ly c a l l e d polar vectors , reverse s ign under t h i s transformation. That i s the components of the given vector i n the primed coordinates are the nega-t i v e s of;:the corresponding components of the same vector i n the unprimed system. This i s the behaviour of, for example, the pos i t i on vector £ , and of vectors formed from r by time d i f f e r e n t i a t i o n , such as v e l o c i t y v, and acce lera t ion a . I f mass i s assumed to be invar ian t under the above transformation, then Newton's second law w i l l be invar ian t provided that forces are assigned the same t rans-formation property. The second type of behaviour for vectors under the transformation of invers ion i s exemplified by vectors (7) which are cross products of vectors of the previous type. The cross product may be defined by the equations (AxB)^ = A 2 B 3 - A B2 e t c . and hence the components of such vectors are the same i n the two systems. Such vectors are u sua l ly termed a x i a l vec tors . Examples from mechanics include torque, L = rxF , and angular momentum, J = r x £ . The invariance of the equations of c l a s s i c a l mechan-i c s under invers ion of coordinates w i l l then be assured i f any one equation contains vectors of only one type . I t i s e a s i l y seen that t h i s i s i n fact the case, e .g . J = rxF£* The invariance of c l a s s i c a l electromagnetism under invers ion of coordinates may be assured by su i tab le assignments of transformation propert ies to the various quant i t ies i n the equations. Here the s i t u a t i o n i s somewhat more sub t l e . Indeed one might be tempted to think that such invariance does not hold when one considers, for exam-ple , such features as r i g h t hand ru les e t c . , which appear to depend upon the screw sense of the coordinates. In t h i s context, the behaviour of a compass needle s i tuated near a wire i s worthy of considerat ion, (See diagram fo l lowing page 7) Suppose that the compass needle i s so a l igned that with no current f lowing i n the wire , i t remains p a r a l l e l to the w i r e . The experimental set-up would appear to be symmetrical about the plane containing the wire and the compass needle, and thus one might expect the needle to react l i k e Buridan 's ass between equal bales of hay,that i s no de f l ec t ion would occur when the current flows* Of course when one performs the experiment i t i s known that a de f l ec t ion occurs. I f one examines the r e f l e c t i o n of the process i n a mi r ror , one sees that i n the mirror the; needle appears to def lect i n the opposite sense. (Re-f l e c t i o n i n a mirror i s equivalent to the invers ion of one coordinate a x i s , and may be obtained by fo l lowing the transformation of invers ion ( I » c . i . l ) by a sui table ro t a -t i o n . Thus provided there i s invariance under ro t a t i on of coordinates, invariance under invers ion of coordinates i s equivalent to invariance under mirror r e f l e c t i o n . ) Thus the above process does not appear to be invar ian t under i n v e r s i o n . The defect i n the preceding argument i s that no con-s idera t ion has been given to the transformation properties of the phys ica l q u a n t i t i e s , e l e c t r i c charge and magnetic poles,which are invo lved . Indeed from the form of Maxwell*s equations one can see that e l e c t r i c and magnetic quant i t ies of the same m u l t i p o l a r i t y have opposite t rans-formation propert ies under i nve r s ion . In p a r t i c u l a r e i ther the e l e c t r i c charges or the magnetic poles must change sign under invers ion while the remaining quanti ty i s i n v a r i a n t . Thus i n the example of the wire and compass needle, i n the mir ror image e i ther the d i r ec t i on of the current or the d i r e c t i o n oif the compass needle must be reversed, and the mir ror image of the process i s again a possible process* (9) Thus one can conclude that c l a s s i c a l electrodynamics i s invariant under inversion of coordinates, provided that appropriate transformation properties are.assigned to the* physical quantities involved. Although the transformation property of a geometrical e n t i t y such as, the position vector r can be determined a p r i o r i , the transformation properties of physical quantities must be determined from experimental evidence. The above example of the magnet and the wire shows the necessity of assigning opposite transformation properties to e l e c t r i c charges and magnetic poles. Maxwell's equations show that s i m i l i a r statements apply to for example,E.and B,etc. One i s however unable to decide d e f i n i t e l y which quantities remain invariant under inversion and which ones change sign under inversion. This r e s u l t appears to be true i n general i n c l a s s i c a l electrodynamics. That i s the classical.theory possesses such a high degree of symmetry that either possible assignment of transformation properties would preserve the invariance of the theory under inversion. P r i o r to 1957, the customary convention was to regard e l e c t r i c charge as being invariant under inversion, while magnetic poles were assumed to change sign. As w i l l be seen l a t e r , the r e s u l t s of the now w e l l known experiments on (3 - decay, (Wu et a l . , 1957), and other experiments on processes which proceed by means of the weak inte r a c t i o n s , indicate that a modification of t h i s point of view i s i n order. To see t h i s , i t i s f i r s t necessary to discuss the symmetry opera-( 1 G ) t i o n re la ted to invers ion of coordinates i n quantum mechan-i c s . In quantum mechanics, an operator FT , termed the pa r i ty operator, i s introduced. The d e f i n i t i o n of t h i s operator i s such that when ac t ing upon a ^ - f u n c t i o n . of argument r , i t produces the same function with argument - r . I f the Hamiltonian commutes with t h i s operator, then i t s expectation value, termed the p a r i t y , w i l l , be conser-ved. For the strong and electromagnetic in t e rac t ions , there i s good evidence for the conservation of p a r i t y . The absence of react ions forbidden by pa r i t y conservation ind ica tes that any pa r i t y v i o l a t i o n can occur only as a small cor rec t ion to the strong and electromagnetic i n t e r -ac t ions . For the weak in te rac t ions however, t h i s does not h o l d . The weak in te rac t ions do not conserve p a r i t y . For example, the experiment of Wu et a l . showed that there i s a de f in i t e co r re la t ion between the d i r ec t i on of the angular momentum of the nucleus, and the d i r e c t i o n of emission of the (3 p a r t i c l e , i n one pa r t i cu l a r case of /9 decay. This demonstrates the presence of a term propor-t i o n a l to J,*JP, i n the expression for the p robab i l i t y of emission of the e lec t ron i n a given d i r e c t i o n . This term i s a pseudoscalar, that i s i t changes s ign under invers ion of coordinates. Fermions of spin | are bel ieved to be described by the Dirac equation. This equation admits of an add i t i ona l symmetry operation, termed p a r t i c l e conjugation, which (11) can be in terpreted as producing a change from matter to ant i -mat ter . I t appears that the strong and electromag-ne t ic in te rac t ions are also invar ian t under t h i s t ransfor-mation, while the weak in te rac t ions are not. However, as was f i r s t pointed out by Landau, (1957), the weak i n t e r -act ions are invar ian t under the combination of these two operations. The combination of the pa r i t y operation with the operation of coordinate invers ion w i l l henceforth be termed the operation of combined i n v e r s i o n . Combined invers ion i s , i n view of the preceding remarks, a univer-s a l symmetry operation, which holds for a l l i n t e r ac t i ons . For example, the neutrino and ant i -neutr ino d i f f e r from each other i n t h e i r handedness. The neutrino has i t s spin and momentum a n t i - p a r a l l e l , while the ant i -neutr ino has i t s spin and momentum p a r a l l e l . Applying the pa r i t y oper-a t ion to a neutrino reverses the momentum, but leaves the spin unchanged, since spin i s assumed to behave>>like o r b i -t a l angular momentum. Hence the object r e s u l t i n g from applying the pa r i ty operation to a neutrino would be a neutrino with spin and momentum p a r a l l e l . However no such p a r t i c l e i s known. The r e su l t of applying the operation of combined invers ion would be an ant i -neutr ino with p a r a l l e l spin and momentum, which i s a possible s ta te . One can summarize by saying that the inver ted image of a p a r t i c l e i s the corresponding a n t i - p a r t i c l e , and that the change from a right-handed system to a left-handed one involves the change from matter to ant i -mat ter . (12) I . c . i i Reversal of Motion The operation of reversa l of motion, or time reve r sa l , i s bel ieved to be a second un iversa l symmetry operat ion. Again t h i s type of symmetry was o r i g i n a l l y not iced i n c l a s s i c a l mechanics. In c l a s s i c a l mechanics, the concept of reversa l of motion symmetry i s to be understood as implying that for a given process, one can exh ib i t a cor r -esponding time, reversed process,which i s a lso a possible process. The time reversed process i s obtained from the o r i g i n a l process by applying a set of ru les which corres-pond to reversing a l l motions. The fact that t h i s type of symmetry holds , i n c l a s s i c a l mechanics, can be shown from the equations of motion, i n Hamiltonian form,in a s t r a igh t -forward manner. (Appendix 1) This notion should not be confused with the thermodynamic concepts of r e v e r s i b i l i t y and revers ib le process. In quantum mechanics, r eve r sa l of motion i s represen-ted by an operator 0 . A l l in te rac t ions appear to be invar ian t under t h i s operat ion. I t turns out that t h i s operator must be taken as an t i -un i t a ry , and that as a r e su l t the experimental consequences are not as d i r ec t as they are fo r a uni ta ry operator such as p a r i t y . Time reversa l invariance leads to what may be ca l l ed "the p r i n -c i p l e of r e c i p r o c i t y " which re la tes the amplitude for a given process, to the amplitude for the corresponding process i n which the states are replaced by t h e i r time reversed s tates , and the sequence of i n i t i a l and f i n a l (13) states i s interchanged. Under cer ta in condi t ions , t h i s leads to the p r i n c i p l e of de ta i led balance, which re la tes the p r o b a b i l i t i e s for a process and the inverse process,, i n which only the sequence of i n i t i a l and f i n a l states i s interchanged. I t should be emphasized that the p r i n c i p l e of de ta i l ed balance does not fo l low i n general from i n v a r -iance under reversa l of motion. (14) CHAPTER I I : EXPLICIT REPRESENTATION OF THE O P E 1 A T I O N S OF TIME REVERSAL AND COMBINED INVERSION I I . a Time R e v e r s a l I l . a . i The I m p o s s i b i l i t y of Representing Time Reversal by a U n i t a r y Operator As was f i r s t pointed out by Wigner ( 1 9 3 2 ) , i t i s necessary t o use an a n t i - u n i t a r y operator t o represent time r e v e r s a l . To demonstrate t h i s , consider a hypo-t h e t i c a l o b j ect which i s c h a r a c t e r i z e d by the p r o p e r t i e s of p o s i t i o n , energy-momentum, angular momentum, and no ot h e r s . I t i s c l e a r t h a t t h i s assumption i s an a r t i f i c i a l one, and th a t a l l known elementary o b j e c t s have other p r o p e r t i e s , such as charge, baryon number, e t c . The behaviour of such a d d i t i o n a l a t t r i b u t e s under r e v e r s a l of motion w i l l be considered l a t e r . For such an o b j e c t , the s t a t e of reversed motion must have the f o l l o w i n g p r o p e r t i e s : The expectation values of the p o s i t i o n coordinates must be the same as the c o r r e s -ponding expectation values i n the o r i g i n a l s t a t e , while the e x p e c t a t i o n values of the l i n e a r and angular momenta must be the negatives of the corresponding values i n the o r i g i n a l s t a t e . These requirements are deriv e d by analogy w i t h c l a s s i c a l mechanics, except f o r i n t r i n s i c angular momentum, which does not have a-' c l a s s i c a l analogue. Thus one must p o s t u l a t e t h a t i n t r i n s i c angular momentum behaves (15) l i k e o r b i t a l angular momentum under reversa l of motion. Thus the operator © , which represents reversa l of motion, must have the fo l lowing properties: ' For any state j o ^ , one must have ( i i . a . i . l ) < Q ' > = < e o . l 5 / e ^ z <&./Q/a>= These requirements are su f f i c i en t to show that G cannot be a un i ta ry operator. I f i t were, then one would have <^o.)P|ea> = <CL\ G"'P G|a> - - < ^ | P|a> with s i m i l i a r r e l a t i ons for Q and J . Since was an a r b i t r a r y s ta te , these conditions could only be s a t i s f i e d i f e~'P e - - B ( i i . a . i . 2 ) e"' Q e - a e"Je = However, i f the operators are transformed i n t h i s way, the fundamental commutation r e l a t ions PQ - Q P - 7 J ( I I . a . i . 3 ) i / o i J X J = I J are not i nva r i an t , as one side changes s i g n . This defect does not appear i f an an t i -un i t a ry operator 0 n ~[~K ^s used. In t h i s case, the same commutation r e l a t i ons given (16) by ( I l . a . i . 2 ) , must h o l d between 0 and P j Q ^ J , a l though a s l i g h t l y d i f f e r e n t p roo f i s r e q u i r e d , which u t i l i z e s the f a c t t h a t @ ' i s a l s o an a n t i - u n i t a r y o p e r a t o r . One has . \ 0 " ' f e - - p Because o f the complex c o n j u g a t i o n i n v o l v e d , the form o f the fundamental commutation r u l e s between P and Q, , and between the angular momentum o p e r a t o r s , i s p r e s e r v e d . (17) l l . a . i i E x p l i c i t C o n s t r u c t i o n o f the Operator © The opera tor © can be c o n s t r u c t e d by impos ing , one by one, the c o n d i t i o n s demanded i n ( I I . a . i . 2 ) « As f a r as the coord ina te s and l i n e a r momenta are concerned, s i n c e Q. may be represented by 4 and JP by-x' i , the opera tor T commutes w i t h both & n a %q • Thus i t can be taken as hav ing no e f f e c t upon the c o o r d i n a t e s . I n the customary r e p r e s e n t a t i o n f o r the angular momen-tum m a t r i c e s , (Edmonds 1957, Appendix 2 ) , the m a t r i c e s J, and J 3 have o n l y r e a l m a t r i x e lements , w h i l e J 2 has o n l y imaginary m a t r i x e lements . Thus i n imposing the c o n d i t i o n s ( I I . a . i . 2 ) , the o p e r a t i o n o f complex c o n j u g a t i o n can be c a r r i e d out immedia te ly , l e a v i n g f o r the operator T the f o l l o w i n g c o n d i t i o n s . T J , T " ' = - J, ( I l . a . i i . l ) T J2 T ' — J? TJ 3 T- ' = - J 3 I t i s i n t u i t i v e l y c l e a r t h a t t h i s t r a n s f o r m a t i o n can be e f f e c t e d by the m a t r i x r e p r e s e n t i n g a r o t a t i o n by 7C about the YL a x i s , namely (Fano and Racah 1959) ( l l . a . i i . 2 ) T= e / 7 r J a * 1 + + ^ 0 % . . . . 21 m T h i s c o n j e c t u r e i s e a s i l y v e r i f i e d , u s i n g the r e s u l t ( I l £ a . . i i . 3 ) e l A B e ' A : 8 + i [A>B] + (J? JA,{A, B|+ • (IS) and the commutation r e l a t i ons of the angular momentum matrices, JxJ = i J . (Here [j4>Bj stands for the commutator,[/\iB] = ^B-S>l .) (19) I I . a . i i i Value of 0 f o r Angular Momentum States From the expression for the matrix ~J~ , ( I I . a . i i . 2 ) , one can deduce the important, r e s u l t that ( H . a . i i i . l ) _ r F i r s t of a l l i t follows from ( I I . a . i i . 2 ) that a l l the matrix elements of ~T are r e a l . Hence e*= TKTK = T T * K / T ~ T 2 Since the eigenvalues of any component of angular momentum are the 2 j+l values, j , j - 1 , . . . , - j , any compon-ent can be brought to diagonal form by a suitable trans-formation, with the elements of the diagonal being the values, j , j - l , . . . , - j . For J2 i n p a r t i c u l a r , there exists a matrix 5 such that r. J ( I I . a . i i i . 2 ) 5 XS"= J - J - J . Then one has and hence Se;"Ji5---(20) and hence D e 5 -Thus e * J z e 1 = e S - i and therefore ez= T ~ - e + I This proof appears to depend rather heav i ly upon the pa r t i cu l a r representation used fo r the angular momentum matrices, e spec i a l l y upon^the fact that two of the matrices can be chosen to have only r e a l matrix elements, while the t h i r d has only imaginary matrix elements. However i f another representation i s employed, the new matrices w i l l be connected with the present ones by a transformation of the form J [ = U J U H . Then the operator UOU~' has the properties demanded of the operator for r eve r sa l of motion i n ( I I . a . i.2), since e ' j ' e ' - ' - u e u " u j u " u e - ' u = - U j u - ' - - j ' The r e l a t ions ( I l . a . i i . l ) do depend upon the choice of representat ion, and need no longer hold , but one s t i l l has (21) II.b Representation of Combined Inversion for P a r t i c l e s of spin h described by Dirac's Equation I l . b . i Representation of the Parity Operation J~f The Dirac equation may be written as ( I l . b . i . l ) (/ * V^-yn)yr= O Here the are a set of matrices defined e s s e n t i a l l y by t h e i r anti-commutation r e l a t i o n s f"' ( i i . b . i . 2 ) r^K + y+r* = 2. f , iMV= --f One common e x p l i c i t representation of these matrices i s the Feynman representation, <n.b.i.3> r<- ( m •> '"gff) The operator JJ i s formed as a d i r e c t product of operators i n coordinate space and i n the four dimensional s p i n - c h i r a l i t y space spanned by the four components of the 7^*-function. In general, a transformation of coordinates i s accompanied by a transformation upon the ^  -function, of the form ( i i . b . i . 4 ) y-rx) —> y>(x') - A Y^M with the matrix f[ being determined by the transformation. In the case of coordinate inversion, (Il.b.i.5) x'=X< 7 X ; ' - ^ X ^ = ~*i> /=/>2,3 (22) one can determine the matrix A by demanding that the form of the equation be preserved. One has (i & % - Y - (i iv.Vx, -' Jr- h - "») ^  ^  Thus the form of the equation w i l l be preserved provided J\ s a t i s f i e s ; (n.b.i.6) A'*A= * » A"iA= - ) f From ( I I . b . i . 2 ) , i t i s clear that a suitable choice f or ^  i s ( I I . b . i . 7 ) A = where ^ i s a numerical f a c t o r . Thus the pa r i t y operation i s represented by the oper-ator (II .b.i.8) Tf - ? *4 n» where TJp i s the par i t y operator i n coordinate space, so defined that when i t acts upon a " ^ - f u n c t i o n of argument r, the re s u l t i s the same function with argument - r . The possible values of may be determined by imposing the condition that the square of the pa r i t y operation be the i d e n t i t y . Here a s l i g h t complication arises because of the transformation properties of the ^  -functions. The (23) functions transform under ro t a t ion i n such a way that the operation of ro t a t ion by 2J7" produces a change of s ign . Since t h i s operation must be considered to be the i d e n t i t y , one can only demand that Thus which i s termed the i n t r i n s i c pa r i ty of the p a r t i c l e i s permitted to take the values +1, - 1 , + i , or - i . The convention that ^ i s r e a l w i l l be adopted here. a (24) I l . b . i i Representation of the Operation of P a r t i c l e Conjugation The Dirac equation admits of a symmetry transformation termed p a r t i c l e conjugation, which may be in terpre ted as — t the change from matter to ant i -mat ter . I f ^ * and ^ = "f" % are solut ions to the Dirac equation and the adjoint equation, (f «. 3 / ^ - »,) V = O ( I l . b . i i . l ) . _ _ _ then one can define new solut ions "\fs and "f" (where f-f ) as fo l lows: j -(n .b. i i.2) -f*'- c ^ J ^ : C " , f The matrix C i s required to sa t i s fy the condit ions c f = C"' ( I l . b . i i . 3 ) . T These new solut ions have the spec ia l propert ies that the Lagrangian of the free f i e l d , and the energy-momentum vector are unchanged, while the current vector changes s i g n . (Bogoliubov and Shirkov, 1959, sect ion 12.3) Although the matrix C i s un i ta ry , the operation of p a r t i c l e conjugation i s not necessar i ly a un i ta ry operation, since C does not connect ^ and , but rather "*p* and"y. In f ac t , i t w i l l be seen that the operation of p a r t i c l e conjugation i s an an t i -un i t a ry operat ion. The propert ies of the new so lu t ions , ( I I . b . i i . 2 ) , which j u s t i f y the descr ip t ion of the transformation as p a r t i c l e conjugation, and the condit ions imposed upon the (25) matrix C do not depend upon the pa r t i cu l a r representation employed for the t matr ices . In the representation used i n t h i s work, ( I l . b . i . 3 ) , one has so that a su i table choice for C i s ( I I . b . i i . 4 ) C = 2 ^2 To demonstrate the an t i -un i t a ry nature of the t rans -formation of p a r t i c l e conjugation, i t i s necessary to wri te the transformation i n the form *)^ " 3 P "fr • One has Y r = c f T - c(Vfr<)Tz c * i Y * = i K K f r k V ^ i K K T Thus the transformation of p a r t i c l e conjugation i s repre-sented by the operator ( I I . b . i i.5) V- * *2 /C Since itf-] i s a un i ta ry operator, i t fol lows that P i s an an t i -un i t a ry operator and that the transformation of p a r t i c l e conjugation i s an an t i -un i t a ry transformation, ( 2 6 ) I l . b . i i i Representation of the Operation of Combined Inversion The operation of combined inversion, 5^  , i s repre-sented by the product of the operations of p a r t i c l e con-jugation, and pa r i t y . Thus i t i s given by ( I l . b . i i i . l ) x=J7P= 7^/rp/yaK For t h i s operator one has Thus p a r t i c l e s described by the Difac equation have two symmetry operators associated with them which are a n t i -unitary and whose square i s - I , provided that the p a r t i c l e s do not have other att r i b u t e s whose behaviour i s such that i n the corresponding subspace the operator also has i t s square - I . In t h i s case the operator i n the combined space w i l l have i t s square +1. To show t h i s e x p l i c i t l y , suppose that each variable i s dichotomic, so that one can write Ar=[Za* . A"'-till* Each of these operators has the square - I . The operator i n the combined space w i l l be represented by the Kronecker product of these matrices. »• 1 o o o7 By inspection the square of t h i s operator i s +1 • (27) I t should be noted that the operators P and 17 a n t i -coinmute, T IT = ~ TT P , so that the above d e f i n i t i o n of 2 contains an a rb i t r a ry choice of s i g n . However t h i s choice does not change any of the conclusions about A • (28) I l . b . i v Remarks on P a r t i c l e Conjugation and Combined Inversion The operation which has been termed p a r t i c l e conjuga-t i o n was o r i g i n a l l y termed charge conjugation, and was defined f o r charged p a r t i c l e s only. However i t i s possible and indeed desirable to generalize t h i s operation to i n -clude a l l p a r t i c l e s . In the case of neutral p a r t i c l e s , some p a r t i c l e s have d i s t i n c t a n t i - p a r t i c l e s , which d i f f e r i n other attributes than charge, such as strangeness, and other neutral p a r t i c l e s appear to be i d e n t i c a l with t h e i r a n t i - p a r t i c l e s . The neutral p a r t i c l e s with d i s t i n c t a n t i -p a r t i c l e s include the J\. p a r t i c l e , the neutral kaon, and the neutron. The photon and the neutral pion are examples of p a r t i c l e s which appear to be i d e n t i c a l with t h e i r a n t i -p a r t i c l e s . I f the operation of p a r t i c l e conjugation can be extended to include a l l p a r t i c l e s , then selection rules can be derived f o r interactions invariant under t h i s operation. Since the charge operator anti-commutes with the operator of p a r t i c l e conjugation, i t i s not possible to have a state of d e f i n i t e non-zero charge which i s invariant I under the operation of p a r t i c l e conjugation. However t h i s may be possible for a state of zero charge. Thus, positro-nium f o r example, i s invariant under p a r t i c l e conjugation, w i t h ( i i . b . i v | i ) r | p°^°»<«^> = * <;*H Cl i s termed the conjugality of the state. The equation (29) ( I l . b . i v . l ) i s not an eigenvalue equation i n the usual sense, since eigenvalues cannot be defined f o r a n t i -unitary operators. Indeed, i f one has ( I I . b . i v . 2 ) A M> = A/<0 then, on introducing a phase factor <X } one has (n.b.iv.3) A - * 3 and i n general A i s d i f f e r e n t from A . A positronium state can be written as a d i r e c t product of an electron state with a positron state, that i s as a product of an electron state with a state produced from an electron state by the operation of p a r t i c l e conjugation. Because of the anti-unitary nature o f the operation of p a r t i c l e conjugation, i f the electron state i s m u l t i p l i e d by a phase f a c t o r ° C of modulus unity, then the positron state must be m u l t i p l i e d by the complex conjugate o c , so that there i s no o v e r a l l change i n the positronium state and the property of invariance i s preserved. The demand that the electromagnetic i n t e r a c t i o n term \*JM , be invariant under p a r t i c l e conjugation, i s s a t i s f i e d i f Aytt changes sign, since JM changes sign under t h i s transformation. This enables one to attach a meaning to p a r t i c l e conjugation f o r the photon. The above require-ment i s equivalent t o the s t i p u l a t i o n that the photon have an i n t r i n s i c odd conjugality, and that an n-photon state have a conjugality of ( - l ) n . This assumption can be tested by examining the selection rules which r e s u l t , governing the decay of positronium, (Wolfenstein and (30) Ravenhall, 1953; Pais and Jost, 1952; Jauch and Rohrlich,-1955)• Since the p a r i t y operation i s unitary, i t follows from the preceding considerations that the operation of combined inversion i s also an anti-unitary operation. (31) CHAPTER I I I : GENERAL PROPERTIES OF ANTI-UNITARY OPERATORS WHOSE SQUARE IS -I III.a Degeneracy of States I f A i s an anti-unitary operator which s a t i s f i e s A**-I, then the transformed state A l ^ i s always l i n e a r l y independent of the state | , and i n fac t orthogonal to i t . (III.a.1) <AW> = tfbMAfc = -<^|b> hence (III.a.2) <$b|b> = O Thus provided the Hamiltonian commutes with A there i s always a two-fold degeneracy, at lea s t , which can be attributed to the corresponding symmetry. One has H I C J > = co I co} (III.a.3) H A j o > - G * A I C J > <£AC*>I CO> - O An example of such a degeneracy appears i n the l i t e r -ature under the name of Kramers' Theorem, (Kramers, 1930; Klein , 1950; Landau and L i f s h i t z , 1959). This theorem i s usually stated as follows: For atoms or ions with an odd number of electrons, there i s always a two-fold degeneracy of the energy l e v e l s , which cannot be removed by an exter-nal e l e c t r i c f i e l d , (but which may be removed by an exter-n a l magnetic f i e l d ) . Since electrons have i n t r i n s i c angular momentum g, i t follows from the rules f o r addition of angular momentum that the t o t a l angular momentum of the system i s h a l f i n t e g r a l , and thus t h i s degeneracy can be attr i b u t e d to rev e r s a l of motion symmetry. The d i s t i n c t i o n between (32) e l e c t r i c and magnetic f i e l d s a r i s e s because o f t h e o p p o s i t e t r a n s f o r m a t i o n p r o p e r t i e s o f t h e s e q u a n t i t i e s . I t i s easy t o see from M a x w e l l 1 s e q u a t i o n s t h a t one o f t h e q u a n t i t i e s E o r B changes s i g n under t h e t r a n s f o r m a t i o n , t i n t o - t , w h i l e t h e o t h e r one remains i n v a r i a n t . Because o f t h e h i g h symmetry o f M a x w e l l ' s e q u a t i o n s , i t i s not p o s s i b l e t o d e c i d e f r o m them a l o n e which one o f t h e f i e l d s i s i n v a r i a n t , and which one changes s i g n under t h e t r a n s f o r -m a t i o n o f t i m e i n v e r s i o n . The e l e c t r i c f i e l d E, i s u s u a l l y c o n s i d e r e d t o be t h e i n v a r i a n t q u a n t i t y , w h i l e t h e magnetic f i e l d B, i s assumed t o change s i g n . T h i s assignment o f t r a n s f o r m a t i o n p r o p e r t i e s i s c o n s i s t e n t w i t h a l l e v i d e n c e . One argument i n f a v o u r o f t h i s a ssignment stems f r o m t h e c o n s i d e r a t i o n o f t h e p o s s i b l e t r a n s f o r m a t i o n b e h a v i o u r o f t h e n e u t r i n o . N e u t r i n o s and a n t i - n e u t r i n o s d i f f e r i n t h e r e l a t i v e o r i e n t a t i o n s o f t h e i r s p i n and momentum. Because t h e n e u t r i n o i s a m a s s l e s s p a r t i c l e , t h e s p i n can o n l y be p a r a l l e l o r a n t i - p a r a l l e l t o t h e momentum. N e u t r i n o s have t h e i r s p i n and momentum a n t i - p a r a l l e l , w h i l e a n t i - n e u t r i n o s have t h e i r s p i n and momentum p a r a l l e l . A p p l y i n g t h e o p e r -a t i o n o f r e v e r s a l o f m o t i o n t o a n e u t r i n o r e v e r s e s b o t h t h e s p i n and t h e momentum, and hence t h e r e s u l t i n g o b j e c t i s s t i l l a n e u t r i n o . Thus r e v e r s a l o f m o t i o n does not change m a t t e r i n t o a n t i - m a t t e r i n t h i s c a s e . I f one demands t h a t t h i s b e h a v i o u r be f o l l o w e d by charged p a r t i c l e s , t h e n s i n c e t h e a n t i - p a r t i c l e has t h e o p p o s i t e e l e c t r i c charge t o t h e p a r t i c l e , i t f o l l o w s t h a t e l e c t r i c charge i s (33) unaffected by reversal of motion. Then the equations ( I I I .a.4) etiv D = f (III.a.5) c W / / = ± + * £ / d t show that the e l e c t r i c f i e l d s must transform i n the same way as the charges, and the magnetic f i e l d s must transform i n the opposite way. (34) I l l . b Introduction of Another Two Dimensions Into the State Vector Space The existence of an anti-unitary operator A , whose square i s - I , leads to a degeneracy i f the Hamiltonian commutes with the operator A • Furthermore, corresponding to any complete set of vectors J \>j} , there i s a second complete set Ajb^  , which cannot be obtained from the f i r s t set by a unitary transformation. This s i t u a t i o n can be accomodated by l a b e l l i n g the vectors with suitable phases, as follows i b » > = i A \ t y ( m . b . i ) ,,,;>> In the space spanned by the vectors ib'> = Ib> * [.] (III.b.2) „ ib">= ib>» m the operator A can be represented as follows ( i n.b.3) A= TK, T-- T„x r° ;'] , |B>- W To represents the dynamical e f f e c t of the operator. For example, i n the case of rev e r s a l of motion, TD transforms t into - t , P into -P., etc. Dynamical operators now take the form (III.b.4) B = Bp * (o | J Thus the dimension of the Hi l b e r t space has been doubled. The two subspaces are connected only by the operator A » which i s not a dynamical observable. A l l dynamical observables are diagonal i n the dichotomic space, and as a (35) r e s u l t , t r ans i t i ons from one subspace to the other cannot occur, e i ther spontaneously, or as a r e su l t of a dynamical i n t e r a c t i o n . Because a l l operators other than the operator / \ are diagonal i n the dichotomic space, and are represented by the i d e n t i t y matrix i n that space, the commutation r e l a t i ons between operators are unal tered. In p a r t i c u l a r , the Hamiltonian i s now wr i t ten as ( I I I . b . 5 ) H = HD * fo ?] I f the dynamical operator Bp i n ( I I I . b . 4 ) commutes withHj>> then one has B given by ( I I I . b . 4 ) commuting with H given by ( I I I . b . 5 ) so that any conservation laws of a dynamical o r i g i n are unaffected by the in t roduct ion of the extra degree of freedom represented by the dichotomic space here introduced. (36) I I I . c Removal of the Degeneracy The degeneracy previously discussed, between the states |b^and Aj^i would be removed i f there existed an operator of the general form ( i i i . c . i ) F = C x This operator would dis t i n g u i s h the d i f f e r e n t subspaces, as they belong to d i f f e r e n t eigenvalues of the operator. I t w i l l be shown that such an operator can be constructed, and that t h i s operator i s an observable, whose eigenvalues are -posi t i v e and negative integers, and which leads to an additive quantum number f o r n-particle states. One may describe the s i t u a t i o n occuring as a r e s u l t of the presence of an anti-unitary operator whose square i s - I , namely the existence of two subspaces, with a l l vectors i n one subspace being orthogonal to a l l the vectors i n the other subspace, and no observables having matrix elements connecting the subspaces, by saying that a super-selection rule e x i s t s between the subspaces. Since there i s complete degeneracy between the subspaces, one must add to the quantities T , which would have formed a complete set of quantum numbers, i n the absence of the super-selection r u l e , a dichotomic variable -f , taking the values +1 or -1, which distinguishes between the subspaces. The understan-ding i s that X and "f now form a complete set of quantum numbers. A state containing Y) p a r t i c l e s , characterized by the quantum numbers % f , can be written as a direct (37) product, (III.c . 2 ) \Y),x,p=: |»,r>x|f> . r ' /~ L|J Here i s a state containing Y\ p a r t i c l e s characterized by the quantum numbers f , which can be further decom-posed, into the direct product of an occupation state with a state characterized by X . (iii.c.3) I n,r> - / " > x jr> Then a state containing V)t p a r t i c l e s characterized by the quantum numbers % > i, , ^ p a r t i c l e s characterized oyfltf etc., may be written as (III.c.4) I' • ri f.~> = \n'^xi*?)*lnir?)tl$~ For the p a r t i c l e s characterized by the quantum numbers , one can introduce the operator of p a r t i c l e number , which has the property (ni.c.5) N x - K>^ >- nj Consider the operator FT*, defined as (iu.0.6) F r < = A/, x [ 0 °] which when operating upon the state (III.c . 2 ) gives (in.c.7) F|^r.^>= Then the operator F defined as F" ^ . ^Trj , i s the desired operator, whose eigenvalues are conserved, and (33) which leads to an additive quantum number. Applying p to the state (III.c.4) gives ( i i i . c . 8 ) P | . J r . v. rx 4..>> - x k *il*W«> Thus any state with d e f i n i t e numbers of p a r t i c l e s i s an eigenstate of F and the corresponding eigenvalue i s simply the number of p a r t i c l e s i n the subspace characterized by •f~+\ minus the number i n the subspace characterized byf=-l. Furthermore, i t can be seen that F commutes with a l l dynamical observables, since a l l dynamical observables commute with the number operator, and are diagonal i n the two-dimensional space associated with "f • In p a r t i c u l a r , F w i l l commute with the Hamiltonian, and thus i t s ex-pectation value w i l l not change with time, and hence represents a conserved quantity. F represents an additive quantum number, which takes the values +1 or -1, for s i n g l e - p a r t i c l e states. These properties of F w i l l be exploited to explain the existence of the quantum numbers termed the baryon and lepton numbers, which at the moment must be regarded as empirical quantum numbers, whose deeper significance i s unknown. These at t r i b u t e s of the elementary p a r t i c l e s have properties s i m i l i a r to those of the quantity F . Thus i t i s tempting to suggest that these quantum numbers owe t h e i r existence to certain super-selection r u l e s , a r i s i n g from symmetries which must be represented by anti-unitary operators whose square i s - I . (39) CHAPTER I V : THE EMPIRICAL CONSERVATION LAWS OF BARYON NUMBER AND LEPTON NUMBERS I V . a Baryon Number The c l a s s o f elementary p a r t i c l e s known as the baryons c o n t a i n s the p r o t o n , the n e u t r o n , and a l l h e a v i e r p a r t i -c l e s . (See t a b l e o f baryons App. 3) The baryons may be f u r t h e r grouped i n t o f o u r subgroups, which may be termed charge m u l t i p l e t s , and which comprise groups o f p a r t i c l e s o f d i f f e r e n t charge, whose masses are c l o s e t o g e t h e r . These f o u r subgroups are termed the nuc leons , and the A > , and 32 , hyperons . The j u s t i f i c a t i o n f o r t h i s c l a s s i -f i c a t i o n i s the charge independence of the s t rong i n t e r -a c t i o n s , and w i l l be d i s cus sed more f u l l y i n s e c t i o n V . b . l o f t h i s work. The l i g h t e s t baryon , the p r o t o n , appears to be an a b s o l u t e l y s t a b l e p a r t i c l e . More p r e c i s e l y , searches f o r p o s s i b l e decays o f the p r o t o n , (Backenstoss et a l , I960; Heines et a l , 1958), show t h a t a lower l i m i t f o r the l i f e -t ime of the proton aga in s t decay by any mode, i s a p p r o x i -23 mately 4x10 y e a r s . T h i s f a c t i s s u r p r i s i n g , s i n c e one can envisage many p o s s i b l e modes o f decay which would not v i o l a t e any known c o n s e r v a t i o n l aws , such as conserva t ion o f charge, c o n s e r v a t i o n of energy, e t c . For example the decay modes p e + + JJ ° ( iv . a .D p ^ e + + e + + e -would be p o s s i b l e w i t h o u t v i o l a t i n g the above c o n s e r v a t i o n l a w s . ( 4 0 ) T h o s e b a r y o n s w h i c h a r e m o r e m a s s i v e t h a n t h e p r o t o n a r e i n g e n e r a l u n s t a b l e b u t a l l t h e o b s e r v e d d e c a y p r o -c e s s e s c o n t a i n o n e b a r y o n i n t h e i n i t i a l s t a t e , a n d o n e b a r y o n i n t h e f i n a l s t a t e . S o m e t y p i c a l d e c a y s i n c l u d e n p 4 e" •¥ T7 A°-s> P + 77 ~ (IV.a.2) P + 7 T ° O b s e r v e d r e a c t i o n s s h a r e t h i s p r o p e r t y t h a t t h e n u m b e r o f b a r y o n s i n t h e i n i t i a l s t a t e i s t h e s a m e a s t h e n u m b e r i n t h e f i n a l s t a t e . F o r e x a m p l e , t h e r e i s t h e p r o c e s s (iv.a.3) 77 + cl n + n T h e s e f a c t s l e a d t o t h e c o n c e p t o f t h e b a r y o n n u m b e r . E a c h o f t h e b a r y o n s i s a s s i g n e d a b a r y o n n u m b e r , w h i c h i s g i v e n t h e v a l u e +1 f o r t h e p a r t i c l e , a n d -1 f o r t h e a n t i - p a r t i c l e . ( S e e t a b l e o f b a r y o n s A p p . 3) T h i s n u m b e r i s t h e n a c o n -s e r v e d q u a n t i t y , a n d i s a n a d d i t i v e q u a n t u m n u m b e r . A t t h e p r e s e n t t i m e , t h i s i s a n e m p i r i c a l c o n s e r v a t i o n l a w , w h o s e d y n a m i c a l o r i g i n i s u n k n o w n . (41) IV.b Lepton Numbers A s i m i l i a r pair of additive quantum numbers can be introduced f o r the class of p a r t i c l e s known as the leptons. The p a r t i c l e s included i n t h i s class are the electron and positron, the posit i v e and negative muons, and the n e u t r i -nos and anti-neutrinos. O r i g i n a l l y , i n the lack of e v i -dence to the contrary, i t was usually presumed that the neutrinos associated with pion and muon decay were the same as those associated with -decay* A single lepton number could be assigned to the p a r t i c l e s , i n accordance with the following t a b l e . TABLE I M e" V L, + 1 -I + 1 -/ The observed reactions are consistent with the conservation of t h i s number, and some unobserved processes are forbidden because they do not conserve t h i s number. For example, the muons decay according to the equations (IV.b.l) /U~ e~ + V + v (IV.b.2) M+ -» e + + -O + V and the (3 -decay of a free neutron takes place i n the following manner. (IV.b.3) A — * P + e~ + I t i s e a s i l y seen that these processes conserve the lepton number. One deficiency of t h i s scheme i s the f a i l u r e to (42) f o r b i d the decay of the muon by the channel (IV.b.4) M.~ -> e" ~h * which i s allowed by a l l dynamical conservation laws, such as energy, angular momentum etc., but which has not been observed. The discovery that there are i n fact two types of neutrinos, (Danby et a l . f 1962) may c l a r i f y the s i t u a t i o n . I t was observed that the neutrinos produced i n the decay of pions, according to the equations (IV.b.5) 77"*-^ V (HU),6) /7*- /«•' + V do not produce the reactions (IV.b .7) "Z' + H P + €T (IV.b.8) -j7 p H + e*~ These reactions should be possible i f the neutron decays according to (IV.b.3), and the neutrinos produced i n the pion decay are the same as those produced i n /3 -decay. The neutrinos r e s u l t i n g from the decay of the pions were observed to produce the reactions (IV.b.9) V+Y\ -=? p . - h MT (iv.b.io) i7+ p -> n + These r e s u l t s suggest that the neutrinos involved i n the various reactions must be di f f e r e n t i n some way. The single lepton number L, provides no means of distinguishing between the two neutrinos. I t i s possible to introduce a second lepton number, (Horn, 1962), which i s also conserved, (43) and whose values are given, along with those of the f i r s t lepton number, i n Table I I . TABLE II X4T e" e + if At + 1 -1. -hi -/ + 1 -/ t/ U -/ + 1 -H -/ -/ -/ The o r i g i n a l lepton number i s retained, so that a l l previous selection rules stemming from the conservation of L, are retained. The decay of the pion must be taken as (IV.b.ll) 77-"*-^ X ^ - f -(iv.b.12) rr~- -+ TT^ and the decay of the muon as (iv.b.13) ^*~* —> e™ •+ -*u< •+ to s a t i s f y the requirements of conservation of L, and L 2 . The introduction of the second lepton number forbids the reactions, (iv.b.14) t<c< •+ " " / - ^ P + e" (iv.b.15) ^ while permitting (IV.b.16) (IV.b.17) The reactions analogous to the above, but with the other types of neutrinos involved, have the opposite selec-t i o n r u l e s . That i s (IV.b.18) V e + M P+ eT' « •+ P - / ^ n-f e" -t- P ~ > + (IV.b.19) n + e (44) are allowed reactions and ( I V . b . 2 0 ) ^ + * » " ^ P ( I V . b . 2 1 ) z?e + P n-hsU* are forbidden. The reactions ( I V . b . 2 2 ) ^ C - - C - f -T ( i y . b . 2 3 ) e ± + er+ e 7 are also forbidden by the introduction of , but are allowed by the conservation of £ ( . These reactions are not observed, although they do not v i o l a t e any dynamical conservation laws, such as angular momentum, energy, etc. Decays of a muon into two neutrinos and three electrons or positrons are allowed i f the proper types of neutrinos are chosen. The decays (IV.b.24) s K * € H- e\e~ 4 V E +7% (iv.b.25) su- e - +e~+e+ + +V« conserve both L # and L- • (45) CHAPTER V: THE POSSIBLE CONNECTION BETWEEN THE CONSER.i VATION OF BARYON AND LEPTON NUMBERS AND THE SYMMETRY OPERATIONS OF COMBINED INVERSION AND REVERSAL OF MOTION V.a Conservation of Lepton Numbers For the leptons, one has two conserved additive quantum numbers, and two anti-unitary operators whose square i s - I . Therefore one may be able to explain the conservation of the two leptonJnumbers as a r e s u l t of the super-selection rules associated with the operators. In considering such an explanation, the following considerations must be kept i n mind: I f a certain additive quantum nupber arises from a given symmetry operation S say, then the single p a r t i -cle states jay and S/<^ must have opposite values of the quantum number. This can be seen from the discussion of section I l l . b , as the symmetry operator i n question i s the only operator which i s not diagonal i n the dichotomic space which i s introduced as a r e s u l t of the super-selection r u l e . No d i f f i c u l t i e s a r i s e i n attempting to explain the conservation of L, as a r i s i n g from the symmetry operation of combined inversion. The operation of combined inversion contains the operation of p a r t i c l e conjugation, and the above requirement i s s a t i s f i e d i f the p a r t i c l e and a n t i -p a r t i c l e have opposite values of L, . This demand i s c l e a r l y s a t i s f i e d , as a glance at Table I I w i l l v e r i f y . I t would also be possible to suggest that i t i s the second lepton (46) number, L., , that o r ig ina tes from the operation of combined invers ion , but the behaviour of the neutrinos suggests that i t i s r eve r sa l of motion which leads to the conservation of L 2 . As has been noted e a r l i e r , neutrinos d i f f e r from a n t i -neutrinos by the r e l a t i v e o r i en ta t ion of t h e i r spin and momentum. A p a r t i c l e such as the neutr ino, which has i t s spin and momentum a n t i - p a r a l l e l i s termed a l e f t handed p a r t i c l e , while the an t i -neu t r ino , which has i t s spin and momentum p a r a l l e l , i s termed a r i gh t handed p a r t i c l e • Since the operation of reversa l of motion reverses both the spin and the momentum, the handedness i s unchanged!, and thus a neutrino state must transform into another neutrino state under reversa l of motion. As can be seen from Table I I , these states w i l l then-have the same value of L, , and hence i t i s impossible to expla in the conser-va t ion of L, as a r e su l t o f r eve r sa l of motion symmetry. This object ion does not a r i se i n the case of the second lepton number. The can be considered to be transformed in to t h e , which has an opposite value of , by the operation of reversa l o f motion. S i m i l i a r l y the xT,and fe are connected by the operation of r eve r sa l of motion. The massive leptons may be treated i n the same way. One cannot regard the e lec t ron and posi t ron as being connected by reversa l of motion, as t h i s would not be consistent with the conclusions reached for the neutr inos, i n p a r t i c u l a r the conclusion that r eve r sa l of motion does not produce (47) matter from a n t i - m a t t e r , o r v i c e v e r s a . Thus one i s f o r c e d to regard the nega t ive muon as the t ime rever sed s t a te o f the e l e c t r o n , and the p o s i t i v e muon as the t ime rever sed s t a t e o f the p o s i t r o n . The mass d i f f e r e n c e between the e l e c t r o n and the muon would appear t o be a d i f f i c u l t y a t f i r s t s i g h t . However the o r i g i n and s i g -n i f i g a n c e o f t h i s d i f f e r e n c e i s an open q u e s t i o n . For the above suggest ion to be p o s s i b l e , a l l t h a t i s r e q u i r e d i s tha t the masses have a s u i t a b l e t r a n s f o r m a t i o n p r o p e r t y . The e l e c t r o n mass must be transformed i n t o the muon mass and v i c e v e r s a , under r e v e r s a l of m o t i o n . The e l e c t r o n mass must thus t r a n s f o r m as (v.a.i) e »»e e"'= and the muon mass must t r a n s f o r m as (V.a .2 ) e ^ 6>~'rz W e (48) V.b Conservation of Baryon Number A complication ar i ses i n the case of the baryons. The baryons have an add i t i ona l property, namely the i s o s p i n , whose transformation propert ies under symmetry operations must be considered. The i sosp in was o r i g i n a l l y introduced to describe the experimentally observed charge independence of the strong i n t e r ac t i ons . Because of t h i s independence, i t i s pos s i -ble to consider the proton and neutron as d i f ferent states of the same object , when discussing these i n t e r ac t i ons . A new dichotomic va r i ab l e , Tj , i s introduced, wi th proton and neutron states being eigenstates of 7^ > corresponding to the eigenvalues +1 and -1, r e spec t i ve ly . The analogy with the desc r ip t ion of spin £ suggests the in t roduct ion of three i sosp in operators, which are i d e n t i c a l i n form to the Pau l i spin matrices, (Fermi, 1962). This formalism can be extended to objects such as the pion, which appear i n three charge states, which are assigned i sosp in 1, with corresponding i so sp in operators, of the same form as the angular momentum matrices for spin 1,being introduced. Although i sosp in was o r i g i n a l l y introduced as a formal device, i t turns out to have dynamical s i gn i f i c ance . In p a r t i c u l a r , i f one adopts ru les for the addi t ion of i sosp in s i m i l i a r to those for angular momenta, the various states of the proton and p i o n , . f o r example, can be c l a s s i f i e d according to the t o t a l i s o s p i n , which i s conserved, i n (49) the strong i n t e r a c t i o n s . The strong in te rac t ions then depend upon the t o t a l i s o s p i n , but not on the t h i r d com-ponent, which i s re la ted to the charge. As as example of the u t i l i t y of the concept of i s o s p i n , the numerical r a t i o s of the various cross sections for pion-proton sca t te r ing at low energies, can be explained i n terms of the Clebsch-Gordan coe f f i c i en t s , (Brueckner, 1952) . The proton -7T + state i s a pure i sosp in s tate , with T= 3/2 , andTj=*/2 . The pro ton-7T~ state i s a mixture of i sosp in s ta tes , with the r e l a t i v e amplitudes of the two cont r ibut ing states being determined by the Clebsch-Gordan c o e f f i c i e n t s . The c o n t r i -buting s t a t e s are the state with T* 3/ tand Vs~ ~'/2 , and the state with Ts i/x , and fz~-'/2» I f one assumes that the sca-t t e r i n g depends only on the t o t a l i s o s p i n , then two am-pl i tudes can be introduced, which govern the sca t te r ing for the cases T - % a n d T * J £ , r e spec t ive ly . The r a t i o s of the cross sections fo l low d i r e c t l y from the Glebsh-Gordan coe f f i c i en t s , i f the scat ter ing amplitude for the7=%case i s large compared to the amplitude for the 7» '/2 case. The t h i r d component of the i sosp in i s r e la ted to the e l e c t r i c charge by the formula (v.b.i) a = T 3 - t - B / 2 - » - s/z Here Q i s the charge, the t h i r d component of the i s o -sp in , B the baryon number, and S the strangeness, (see the table of baryons, App, 3 ) . At the present t ime, t h i s must be considered as an empi r ica l r e l a t i o n , whose o r i g i n (50) i s unknown. E l e c t r i c charge and baryon number change sign under p a r t i c l e conjugation, so that the same behaviour must be assigned to Tj and 5 i f t h i s r e l ationship i s to be preserved under p a r t i c l e conjugation. I t i s not possi-ble to conclude anything d i r e c t l y about the behavior of the other components of the isospin, but i t i s consistent to assume that they also change sign under p a r t i c l e con-jugation. Assuming t h i s to be the case, i f the p a r t i c l e conjugation operator i s written as (v.b.2) r= r».x t h e n F L ^ r e m u s t s a t i s f y <V- b-3) T -r T P - = O I f P i s anti-unitary, and- contains the operation of com-plex conjugation, then one can show, i n the same manner as for angular momentum, that Vy^xrx/o^ce i s given by (V.b.4) P- . - <>I7RTXL~ and hence that {Pio***^- +X Uo,i,~ i.e. 71, A kyrerom (V.b.5) (P^o^y-x - J t= '/z . Vz - - tft - 'hfitnn i Thus provided that the parity operation has no effect i n isospace, one cannot have £ = - I f o r a l l the baryons, and hence i t i s impossible to explain baryon number as a conservation law a r i s i n g from 2 ;. A l l the baryons have spin i , and thus one would expect them to be described by the Dirac equation with the appro-priate mass value. I f t h i s i s the case, i f the operator of combined inversion i s written i n the form (51.) (V . D . 6 ) S - ^ j ) * 2T spin- X ^ - " V o * » * < * > then one has, f o r a l l the baryons (V.b.7) /Ep X T+*~>- \ * - _ J as has been shown previously. Thus the complete operator of combined inversion s a t i s f i e s £ ^ - 3 -j-J" — rtMcleans, ^ Jty/oer^ MS (V.b.S) Correspondingly, the operator of p a r t i c l e conjugation s a t i s f i e s Thus P--X f o r the baryons with non-zero hypercharge, and r*= + I f o r the baryons with zero hypercharge. Thus i t may be possible to understand the conservation of hypercharge i n strong and electromagnetic interactions as a r e s u l t of the approximate symmetry operation of p a r t i c l e conjugation. P a r t i c l e conjugation i s not a symmetry operation for the weak interactions, which do not conserve hypercharge. As f a r as the baryon number i s concerned, the remain-ing p o s s i b i l i t y i s that i t may be possible to explain the conservation of baryon number i n terms of a super-selection rule connected with reversal of motion. For t h i s to be possible, i t i s necessary to associate with every baryon a corresponding baryon with the opposite baryon number, which can be i d e n t i f i e d as the time reversed state. I t i s necessary that the charge of these two states be the same, to.be consistent with the conclusions reached concerning (52) the transformation properties of charge under reversal of motion. This transformation property of e l e c t r i c charge suggests that the t h i r d component of isos p i n i s invariant under reversal of motion. This i s the simplest way to preserve the relationship between the charge, the baryon number, and the strangeness, (V.h.l) • This i s easier to see i f the r e l a t i o n s h i p i s written i n the equivalent form (y.b.io) Q = T 3 + y/2 where / i s the hyper charge, defined as y= S+B . In t h i s form, a l l quantities can be taken as being invariant under reversal of motion. I t would be possible to have e l e c t r i c charge invariant under reversal of motion, with-out demanding that the t h i r d component of isos p i n be invariant, but only by demanding complicated and a r t i f i c i a l transformation properties f o r the strangness or hypercharge. Again, as i n the case of combined inversion, one i s able to determine the behaviour of by i t s r e l a t i o n s h i p to the charge, but i s unable to conclude anything d i r e c t l y about the other components of the i s o s p i n . The easiest thing to do i s to assume that they have the same behaviour as T3 • Thus isospin will.»be taken as being invariant under reversal of motion. The preceding transformation properties of charge, baryon number, and isospin, are s u f f i c i e n t to determine the transformation properties of the various baryons under reversal of motion. The proton, for example, must be transformed into a p a r t i c l e which has baryon number -1, (53) and for which T and 7 j hare the values i, and +£, respec-t i v e l y . From the table (App. 3) one can see that the £ T i s an acceptable choice. S i m i l i a r l y , the neutron must be transformed in to a p a r t i c l e with baryon number - 1 , and T* '/x » 7* 3 = • The 5p i s the only p a r t i c l e which meets these requirements. One may proceed i n t h i s manner, to assign transformation propert ies to a l l the baryons, which are consistent with the requirements of invariance of e l e c t r i c charge and i s o s p i n . The resu l t s o f such assignments may be indica ted schematically as fo l lows . e | P > = / s - > © / « > = elA'y = |A°> Written out more e x p l i c i t l y , the reversa l of motion operator i s now represented as (v.b.i2) e= © D * 1 * * * * * , *f° In accordance with the conventions used i n sect ion I l l . b , a proton state i s wr i t ten as (V.b . l3 ) \f> = | ' > > x f j ] x Q and an 3~ state as (v.b.i4, \T->= The baryon number space has been i d e n t i f i e d with the extra (54) dichotomic space introduced to accomodate the super-se lec t ion r u l e . As i n the case of the massive leptons, the masses of the p a r t i c l e s must have sui table transformation propert ies under reversa l of motion. The proton mass must be t rans-formed in to the mass of the , the neutron mass must transform in to the mass of the ^ ° , the S mass must be transformed in to the mass of the £*" , e t c . (55) APPENDIX I Time Reversal Invariance i n C l a s s i c a l Mechanics The concept of time reversa l invariance i n c l a s s i c a l mechanics may be formulated as fo l lows . For any given possible process, there i s a corresponding possible process which may be considered as the time reversed process. The time reversed process i s obtained from the o r i g i n a l process by reversing the l i n e a r and angular momenta, while leaving the coordinates i n v a r i a n t . This type of symmetry may be shown to hold by w r i t i n g the equations of motion i n Hamil-tonian form. # ( A . I . I ) -dq. " r<- 9 ZPC~ Z* I f yx(t)&nd 9 t - = f4(t) , are a set of solut ions to these equations, then time reversa l invariance w i l l be assured i f the functions % ( ~ * ) a n d lf>(-tfa.re s o l u -t ions to the equations (A. I .2 ) S f j " ^ * *<< with H*=f H(P'*t')» This fol lows s t raightforwardly from the equations ( A . I . I ) . The key point of the argument i s that the Hamiltonian i s a quadratic, and hence even funct ion of the generalized momenta. ( A . i . 3 ) HC-P**) = H(P,t) so that one has a ? ; dtA at " dtl r*L ') ~ r< (56) APPENDIX I I The Customary Representation of the Angular Momentum Operators The angular momentum operators are defined e s s e n t i a l l y by t h e i r commutation r e l a t i o n s , ( A . I I . l ) J X J - ^ ' J ( y)n - l) This non-commutattivity of the components of angular momen-tum impl i e s , i n t e r a l i a , that only one of the matrices may be diagonal i n a p a r t i c u l a r representat ion. I t can be shown from ( A . I I . l ) that the matrix (A.II .2) J*5 j £ - f J * + j j " commutes with each component of angular momentum. I t i s customary to choose a representation i n which J and J ^ a r e i n diagonal form. Matr ix representations for the angular momentum operators may be found of any dimension. I f the dimension of the matrices, N, i s wr i t ten as N= 2j+l, where j may assume i n t e g r a l or h a l f i n t e g r a l values, i t i s found that the possible eigenvalues of any component of angular mom-entum are the 2j+l values, j , j - 1 , . . . , - j . The represen-t a t i o n i s so chosen that J ^ , i s i n diagonal form with the elements of the diagonal decreasing monot ical ly , as one proceeds along the main diagonal . Thus J ^ . takes the form ( A . I I . 3 ) £ = (57) I t i s further convenient to l a b e l the rows and columns by the 2j+l values which appear along the diagonal of . Thus the top row w i l l be termed the j t h row, the second w i l l be l a b e l l e d as the ( j - l ) t h row, etc. The columns w i l l be l a b e l l e d i n the same manner, proceeding from l e f t to r i g h t . A general matrix, element i n , f o r e x a m p l e , , w i l l be denoted by ^ ^ J x ! yy,'/s • With a suitable phase convention, the non-vanishing matrix: elements of and J y are i • (A.II.4) A l l other matrix elements are zero:. (58) APPENDIX I I I Table of Baryons lilt to + t CO IH e 1 O + f** lU • (I* N 0 0 *** ™ 0 V s 0 T i . a I IS; 0 0 0 0 s . i IS 0 0 0-f> ». ft w <3 3 S to. i 00 T.i R r 1 U 6 la 0 I N 1 t W 0 0 0 0 to £ 0 X . L •*. <: e 0 0 + 0 0 CQ c 0 «/ •f-o 0 + >• w d S a . + 00 a 1 I (59) BIBLIOGRAPHY Backenstoss G.K., Fraunfelder H., Hyams B.D., Koester J.L., Marin P.O., Nuovo Cimento 16, 749, (I960) Bogoliubov N.N. and Shirkov D.V., Introduction to the Theory of Quantized Fie l d s , Interscience Publishers, (1959) Brueckner K.A., Phys. Rev., 86, 106, (1952) Danby G., G a i l l a r d J.M., Goulianos K., Loderman L.M., Mistry N., Schwartz M., Steinberger J . , Phys. Rev. Lett. £, 36 (1962) Edmonds A.R., Angular Momentum i n Quantum Mechanics, Princeton University Press, (1957) Fano U. and Racah G., Irreducible Tensorial Sets, Academic Press, (1959) Fermi E., Notes on Quantum Mechanics, University of Chicago Press, (1962) Horn D., Phys. Lett., 2, 303, (1962) Jauch J.M. and Rohrlich F., The Theory of Photons and Electrons, Addison-Wesley, (1955) (60) K l e i n M.J., Am. Jour. Phys., 20, 65 , (1952) Kramers H., Proc. Amst. Acad., |£, 959, (1930) Landau L., Nucl. Phys. 1, 127, (1957) Landau L., and L i f s h i t z E.M., Quantum Mechanics, Pergammon Press, (1959) Pais A., and Jost R., Phys. Rev., 8J7_, 871, (1952) Reines F., Cowan C.L., and Kruse H..W., Phys. Rev., 109. 609, (1957) Wigner E.P., Gott. Nachr., ^1, 546, (1932) Wigner E.P., Jour. Math. Phys., 1, 409, (i960) Wolfenstein L., and Ravenhall D.G., Phys. Rev., 8_8_, 279, (1952) Wu CS., Ambler E., Hayward R.W., Hoppes D.D., Hudson R.P., Phys. Rev., 10^, 1413, (1957) 

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