UBC Theses and Dissertations

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UBC Theses and Dissertations

A consideration of the possibilities of constructing composite particles of zero mass Rowe, Edward George Peter 1959

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A CONSIDERATION OF THE P O S S I B I L I T I E S OF CONSTRUCTING COMPOSITE PARTICLES OF ZERO MASS by EDWARD GEORGE PETER ROWE B . S c , M c G i l l U n i v e r s i t y , 1958. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f P h y s i c s a c c e p t t h i s t h e s i s a s 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 September, 1959. i i ABSTRACT Th i s t h e s i s i n v e s t i g a t e s the p o s s i b i l i t y o f c o n s t r u c t -i n g t h e o r i e s of composite p a r t i c l e s of zero mass. C o n s i d e r a -t i o n i s g i v e n p r i m a r i l y to the t r a n s f o r m a t i o n p r o p e r t i e s of s t a t e v e c t o r s and o p e r a t o r s used f o r the d e s c r i p t i o n of mass-l e s s elementary p a r t i c l e s under Lorentz t r a n s f o r m a t i o n of the c o o r d i n a t e system. A method o r i g i n a l l y used by Wigner i s developed more e x p l i c i t l y . The t r a n s f o r m a t i o n p r o p e r t i e s of b i l i n e a r combinations of s t a t e f u n c t i o n s and of o p e r a t o r s are c o n s i d e r e d i n some d e t a i l . I t i s found that the one ex-ample of a composite p a r t i c l e theory extant i n the l i t e r a t u r e has i n c o r r e c 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 . No example of a c o r r e c t l y t r a n s f o r m i n g theory i s g i v e n . In presenting t h i s thesis i n p a r t i a l fulfilment 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 this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this 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 The University of B r i t i s h Columbia, Vancouver £, Canada. i i i TABLE OF CONTENTS ABSTRACT i i ACKNOWLEDGMENTS iv I. INTRODUCTION AND SUMMARY 1 1. Introduction 1 2. Summary 2 I I . GENERAL TRANSFORMATION PROPERTIES OF STATE VECTORS AND OPERATORS IN QUANTUM MECHANICS 4 1. General Remarks 4 2. Transformations of State Function for One P a r t i c l e Theories 6 3. Transformations of An n i h i l a t i o n and Creation Operators i n F i e l d Theories 9 I I I . UNITARY IRREDUCIBLE REPRESENTATIONS OF THE INHOMO-GENEOUS LORENTZ GROUP 10 1. Statement of the Problem and Preliminaries 10 2. I n i t i a l Decomposition of the Types of Repre-sentation 12 3. Reduction of Representations to those of a L i t t l e Group 14 IV. TRANSFORMATION PROPERTIES OF BILINEAR COMBINATIONS OF GENERATING SPACE ELEMENTS OF IRREDUCIBLE REPRE-SENTATIONS 19 1. Introduction of the Momentum Eigenfunction Basis and the Direct Product Representation 19 2. Correctly Transforming Set Involving an Equality A ) = 8( * A p', A" ) 21 3. Correctly Transforming Set Involving an Equality ft. (jtfAp', X))~ ? U AP', A )) 22 V. ATTEMPT TO SIMULTANEOUSLY SATISFY COMMUTATION RELATIONS AND TRANSFORMATION PROPERTIES FOR COMPO-SITE PARTICLES OF ZERO MASS 28 1. Introduction 28 2. Statement of the Problem 29 3. Attempted Solution to the Problem 31 BIBLIOGRAPHY 36 ACKNOWLEDGMENTS I should l i k e t o thank Dr. F. A. Kaempffer f o r su g g e s t i n g t h i s problem and f o r h i s v a l u a b l e a d v i c e d u r i n g the course of the r e s e a r c h . I am g r a t e f u l t o Dr. W. Opech-owski and Dr. P. R a s t a l l f o r h e l p f u l d i s c u s s i o n s . I am indebted to the N a t i o n a l Research C o u n c i l o f Canada f o r f i n a n c i a l a s s i s t a n c e i n the form of a Bursa r y . 1 I . INTRODUCTION AND SUMMARY 1 . I n t r o d u c t i o n T h r o u g h o u t t h e deve lopment o f p a r t i c l e p h y s i c s , i t has been a c h e r i s h e d hope o f p h y s i c i s t s 1 , t h a t t h e v a r i o u s e l e m e n -t a r y p a r t i c l e s known m i g h t i n f a c t be d i f f e r e n t s t a t e s o f one v e r y e l e m e n t a r y p a r t i c l e , o r t h a t t h e y m i g h t be s i m p l e compo -s i t i o n s o f a s m a l l number o f v e r y e l e m e n t a r y p a r t i c l e s . T h i s hope has l e a d , i n t h e f i r s t p l a c e , t o a n a t t e m p t t o d e s c r i b e a l l t h e e l e m e n t a r y s p i n 1/2 f e r m i o n s a s n e u t r i n o s d i f f e r i n g o n l y i n t h e i r d r e s s , t h a t i s i n t h e i r e l e m e n t a r y i n t e r a c t i o n s . I n t h i s a t t e m p t , t h e b a r e masses o f t h e e l e m e n t a r y p a r t i c l e s a r e t a k e n t o be z e r o , and t h e d r e s s e d m a s s e s , t h e o b s e r v e d m a s s e s , a r e hoped t o a r i s e f r o m s e l f e n e r g y c a l c u l a t i o n s . None o f t h e s e a t t e m p t s s o f a r has met w i t h any s u c c e s s . The s e c o n d m e t h o d , t h a t o f s i m p l e c o m p o s i t i o n s , was one p o p u l a r i n t h e decade f o l l o w i n g 1 9 3 0 . The o b j e c t i v e o f w o r k -e r s u s i n g t h i s method was a n e u t r i n o t h e o r y o f l i g h t . V a r i o u s p h y s i c a l p i c t u r e s f o r t h e c o m p o s i t i o n s a t t e m p t e d were p r o p o s e d ; de B r o g l i e p i c t u r e s a p h o t o n o f f i x e d momentum K , a s a n e u -t r i n o and a n a n t i n e u t r i n o whose momentum v e c t o r s were p a r a l l e l t o K ; J o r d a n s u g g e s t e d p i c t u r e s f o r t h e e m i s s i o n o f p h o t o n s w h i c h i n v o l v e d e i t h e r t h e e m i s s i o n o f n e u t r i n o and a n t i n e u t r i n o , o r a change i n s t a t e o f one n e u t r i n o o r a n t i n e u t r i n o . I n t h e s e l a t t e r p i c t u r e s t h e r e i s no d i r e c t i n t e r a c t i o n b e t w e e n t h e n e u -t r i n o and a n t i n e u t r i n o } r a t h e r t h e e m i s s i o n and a b s o r p t i o n o f 2 t h e two p a r t i c l e s a r e c o r r e l a t e d . In 1938 P r y c e ( P r y c e 1938) e f f e c t i v e l y h a l t e d work on t h e n e u t r i n o t h e o r i e s o f l i g h t by p o i n t i n g o u t t h a t t h e t h e o r i e s p r o p o s e d up t o t h a t t i m e were n o t i n v a r i a n t under s p a t i a l r o t a t i o n s . P r y c e g i v e s a l i s t o f t h e works o f de B r o g l i e , J o r d a n , K r o n i g i n h i s p a p e r . The o n l y p o s s i b i l i t y f o r a t h e o r y o f c o m p o s i t e m a s s l e s s p a r t i c l e s w h i c h s u r v i v e d P r y c e * s c a t a c l y s m i c p a p e r was one p r o p o s e d by B o r n and Nagendra N a t h w h i c h c l a i m e d t o p r o v i d e a method f o r c o m b i n i n g n e u t r i n o and a n t i n e u t r i n o c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s t o f o r m s p i n z e r o c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . U n f o r t u n a t e l y , t h e work o f B o r n and Nagendra N a t h was e s s e n t i a l l y one d i m e n s i o n a l ; t h e o b j e c t i o n r a i s e d by P r y c e c o n c e r n e d o n l y s p a t i a l r o t a t i o n s o f t h e c o o r d i n a t e s y s t e m ; C a s e , who g e n e r a l i z e d t h e work o f B o r n and Nagendra N a t h , was s a t i s -f i e d t o o b t a i n a t h e o r y i n v a r i a n t under s p a t i a l r o t a t i o n s o n l y . To t h i s w r i t e r ' s k n o w l e d g e , no w o r k e r i n t h e t h e o r y o f c o m p o s i t e p a r t i c l e s has p r o p o s e d a t h e o r y i n v a r i a n t u n d e r a l l p r o p e r , i n -homogeneous L o r e n t z t r a n s f o r m a t i o n s . The o b j e c t i v e o f t h i s t h e s i s i s t o p r o v i d e some m a c h i n e r y f o r t h e t e s t i n g 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 o f c o m p o s i t e p a r t i c l e t h e o r i e s u n d e r a r -b i t r a r y L o r e n t z t r a n s f o r m a t i o n s , and t o i n v e s t i g a t e t h 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 o p e r a t o r s p r o v i d e d by B o r n and Nagendra N a t h ; i t i s f o u n d t h a t t h i s l a t t e r t h e o r y i s n o t i n -v a r i a n t u n d e r a r b i t r a r y L o r e n t z t r a n s f o r m a t i o n s . 2 . Summary In c h a p t e r I I , two t y p e s o f t r a n s f o r m a t i o n o f s t a t e 3 functions and operators under Lorentz transformation of the coordinate system are explained, the Heisenberg type and the Schrodinger type. General expressions are given for trans-formed vectors, momentum and spin operators, and state func-tions for single p a r t i c l e theories, and for creation and anni-h i l a t i o n operators i n f i e l d theories. Chapter III summarizes the famous paper of Wigner (Wig-ner 1939) giving the i r r e d u c i b l e representations of the inhomo-geneous Lorentz group. Its great importance l i e s i n the phys-i c a l assumption which i s used throughout t h i s thesis: The state functions describing the elementary p a r t i c l e s found i n nature generate i r r e d u c i b l e representations of the inhomogeneous Lor-entz group. Two method are given i n chapter IV for f i n d i n g b i l i n e a r combinations of the generating functions of the r e -presentations for massless p a r t i c l e s which are themselves generating functions of i r r e d u c i b l e representations. Finding such combinations indicates the p o s s i b i l i t y of combining mass-less p a r t i c l e s to form other massless elementary p a r t i c l e s . Chapter V gives the transformation properties of the creation and a n n i h i l a t i o n operators for the massless p a r t i c l e s whose represenations were found i n chapter I I I . It also presents a summary of the Born-Nagendra Nath s o l u t i o n , and points out that i t does not transform c o r r e c t l y . To t h i s writer's knowledge, there are no theories of massless composite p a r t i c l e s which are not objectionable on the grounds of incor-rect transformation properties. 4 I I . GENERAL TRANSFORMATION PROPERTIES OF STATE VECTORS AND  OPERATORS IN QUANTUM MECHANICS 1. General Remarks In quantum mechanics, the states of a physical system are described i n terms of state vectors ( y , vectors i n a Hilbert space . Physical observables are represented by operators A , which are l i n e a r , Hermitian, and defined for the whole of . Comparison i s made with experiment by means of the expectation values< I A l ^ . An observer has always i n mind a p a r t i c u l a r Lorentz frame K , and so more properly a l a b e l K should be a f f i x e d to the vectors \ ) K , and operators A* , denoting the frame of the observer. It i s then an im-portant problem how the r e s u l t s of experiments made by obser-vors i n two Lorentz frames K and K may be compared. To a c e r t a i n extent the transformation law connecting operators and vectors used by observers i n K and k. i s a r b i t -rary. One demands, however, that t r a n s i t i o n p r o b a b i l i t i e s be i conserved; i f l') K'and ll> K<are vectors used i n K to describe physical situations which i n K are described by 10* and 1 , then (II.1) One may so choose (Wigner 1939, Bargmann 1954) m u l t i p l i c a t i v e factors i n the vectors that (II.1) i s replaced by K < < " U > K . - K < ' I * > K (II.2) Two types of transformation for which (II.2) i s true w i l l be outlined: the Schrodinger type, and the Heisenberg type (Schweber's usage, Schweber 1955). 5 Schrodinger Type of Transformation <? This type of transformation is defined by stating that i the observers in K and K use the same set of operators. If the operator A K has the same dynamical significance for the observer in K as the operator A has for the observer i in K , then the statement is A* = A*. . (u .3) Two examples w i l l c l a r i f y the notion of "same dynamical signi-ficance". If An is the operator representing linear momen-tum along the X* axis in K , An'is the operator represen-ting linear momentum along the X-axis in K ; the two di r -ections are not the same i f K* is reached from K by a Lorentz transformation which includes a spatial rotation. If &K is the operator which in a f i e l d theory creates a parti-cle with momentum p measured in K and a certain spin state, BK' is the operator which creates a particle with momentum p measured in K and the same spin state relative to k . If I )K and | )H>are the vectors used by observers in l< and K to describe the same physical situation then the connection between the two is I >«• - U ( L ) I > K (ii.4) where, in order that (II.2) hold,U(L)must be unitary, and L i s the Lorentz transformation mapping K into K . \J(\~) is further restricted by the demand that i t form a representation 6 o f t h e p r o p e r L o r e n t z g r o u p . T h i s i d e a w i l l be e l a b o r a t e d and t h e p o s s i b l e r e p r e s e n t a t i o n s w i l l be f o u n d i n c h a p t e r I I I . H e i s e n b e r g Type o f T r a n s f o r m a t i o n T h i s t y p e o f t r a n s f o r m a t i o n i s d e f i n e d by s t a t i n g t h a t t t h e o b s e r v e r s i n K and K u s e t h e same v e c t o r t o d e s c r i b e t h e same p h y s i c a l s i t u a t i o n : I \ " 1 ( I I . 5 ) T h i s e v i d e n t l y may be t a k e n t o be a c o r r e s p o n d e n c e by a u n i t -a r y t r a n s f o r m a t i o n by LXl) on t h e s e t s o f s t a t e s and o p e r a t o r s f o u n d f o r t h e S c h r o d i n g e r t y p e o f t r a n s f o r m a t i o n . Then t h e r e l a t i o n between t h e o p e r a t o r s w i t h t h e same d y n a m i c a l s i g n i -f i c a n c e i n K and K i s AK- = u \ - r LA* oa) ( n . 6 ) I n v i e w o f ( I I . 5 ) , ( I I . 2 ) h o l d s i d e n t i c a l l y f o r t h i s t y p e o f t r a n s f o r m a t i o n . 2 . T r a n s f o r m a t i o n s o f S t a t e F u n c t i o n s f o r One P a r t i c l e T h e o r i e s S i n c e s t a t e f u n c t i o n s a r e u s e d f o r most c a l c u l a t i o n s i n one p a r t i c l e t h e o r i e s , and s i n c e t h e c l a s s i c a l d e r i v a t i o n o f t h e r e p r e s e n t a t i o n s o f t h e L o r e n t z g r o u p (Wigner 1939) ^employs s t a t e f u n c t i o n s f o r i t s d e v e l o p m e n t , i t i s o f i n t e r e s t t o s e e how t h e t r a n s f o r m a t i o n o f s t a t e v e c t o r s d e t e r m i n e s t h a t o f s t a t e f u n c t i o n s . I n t h e f o l l o w i n g some o f t h e r e s u l t s o f Wigner ( t o be summar i zed more c o m p l e t e l y i n c h a p t e r I I I ) w i l l be u s e d . F o r t h e d e s c r i p t i o n o f a s i n g l e e l e m e n t a r y p a r t i c l e t h e H i l b e r t space \ may be spanned by the set of simultaneous eigenvectors of a four-dimensional momentum operator P , and a spin operator 51 . An observer i n K spans "ty with the set [lP^\j> eigenvectors of PK and £,fc •, an observer i n K spans T^ r with the set ^  IP ^>K'|, eigenvectors of P*'and The expansions i n terms of eigenvectors of | ) K a n d l ) K > , describing the same physical s i t u a t i o n involving a single elementary par-t i c l e can be written (Wigner 1939) I>K » $ IMV p l p b , . (II.7a) i v * i i M ^ ' r i p i v . < f ^ V - \ i ^ i V f i f b / ^ ( f b (n.7b) where a summation convention has been employed f o r the spin variable ^ . The state functions i n V- and K are V'K and . One might write i n terms of these |V')„for | >K and | VV'*°r I V • i If one wishes to connect k and K by means of the Sch-rodinger type of transformation then from (1X3) P. - P.. , 5L K - C„. , and consequently From (II.4) one may now connect equations (II.7): i>„• = u i v - W . r ' A u i p i w p * > . <"•»> Physically i f a p a r t i c l e has i n K a d e f i n i t e momentum P and a spin state defined by ^ , i t w i l l be described i n K* by a li n e a r combination of vectors with a d e f i n i t e momentumUo 8 and various spin states. While t h i s statement i s a common one, i t contains i m p l i c i t assumptions about the r o l e of observers i n a quantum mechanical system (Everett 1957). This may be expressed ua)|ft>R = W L ) P p K a ( * p A \ t • ( i i . 9 ) One may now rewrite (II.8) or, using the fac t that IfM'e)l }(> i s an invariant d i f f e r e n t i a l , From the l a s t equation and (II.7b) one may deduce, since ^lp\>«| form a complete set of vectors VvCp \ ) = p'L)j-v ^  (u-"p ^ ) • (Iiao) The work of chapter III w i l l be large l y concerned with the de-termination of the matrices Q to y i e l d i r r e d u c i b l e represen-tations LJ . It i s int e r e s t i n g to note that the state function ^ ( p ^ ) i n K corresponding to the state function V'.'G'bin K* i s the same whether one has used a Schrodinger or Heisenberg type of transformation. This follows by observing that the state function for the Heisenberg type of transformation i s while that for the Schrodinger type i s the two are the same. 9 The t r a n s f o r m a t i o n s o u t l i n e d above a r e t r a n s f o r m a t i o n s o f t h e e n t i r e s t a t e f u n c t i o n ; t h e y a r e t o be s h a r p l y d i s t i n -g u i s h e d f r o m t r a n s f o r m a t i o n s o f s t a t e f u n c t i o n s a t a f i x e d s p a c e - t i m e p o i n t by n o n - u n i t a r y , f i n i t e - d i m e n s i o n a l r e p r e s e n t a -t i o n s o f t h e homogeneous L o r e n t z g r o u p ( f o r e x a m p l e , t h e t r a n s -f o r m a t i o n o f t h e D i r a c s t a t e f u n c t i o n , D i r a c 1 9 5 8 ) . 3 . T r a n s f o r m a t i o n s o f A n n i h i l a t i o n and C r e a t i o n O p e r a t o r s  i n F i e l d , T h e o r i e s 1 I t i s u s u a l i n f i e l d t h e o r e t i c a l work; t o u s e t h e H e i s e n -b e r g t y p e o f t r a n s f o r m a t i o n t o compare t h e d e s c r i p t i o n s o f o b -s e r v e r s i n two f r a m e s k and k' . The 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 c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s a r e e s p e c i a l l y i m p o r t a n t . L e t 0-(p^)„and d.(f ^ ) « b e t h e c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r p a r t i c l e s i n s t a t e s l f ^ > u s e d by an o b s e r v e r i n K . The o p e r a t o r s h a v i n g t h e same d y n a m i c a l s i g n i f i c a n c e f o r a n o b s e r v e r i n K a r e a n d By d e f i n i t i o n , i f I O) d e n o t e s t h e vacuum s t a t e ( w h i c h w i l l be t a k e n t o be t h e same f o r o b s e r v e r s i n K and K ' ) , a-'Cfb, t°> - If \ >» . From t h e r e l a t i o n d e r i v e d s i m p l y f r o m ( I I . 9 ) and t h e a s s u m p t i o n one has ' s y i e l d i n g S i m i l a r l y . ^1>.' = Q (f.^|, l ( u ' 'n), . ( n . 1 2 b ) The r e l a t i o n s (11.12) are the basis for the work of chapter V. 10 III. UNITARY IRREDUCIBLE REPRESENTATIONS OF THE INHOMOGENOUS  LORENTZ GROUP 1. Statement of the Problem and Preliminaries The unitary, irreducible representations of the inhomo-geneous Lorentz group have been found by Wigner (Wigner 1939) whose work has. been appended by Bargmann (Bargmann 1957); more recently a less mathematical derivation has been given by Shir-okov (Shirokov 1958). The summary to follow parallels the treatment of Wigner. Concern w i l l be restricted to the proper, inhomogeneous Lorentz group; that i s . transformations of coordinates L = ( A , « . ) , such that and A H „ > ° , I A A J * I . Here <) r t V=Oif >+V , °j^= -<};; = \ . The product of two Lorentz transformations ( A , a ) a n d ( H A) is ( A , 0 ( M » = ( A M , 4, + A b r ) . f, In order that the transformations of state functions A / / ¥ , = U( A,0 V< w i l l form a representation (more exactly, a ray representation), one requires that y f A , * ) U ( H J l ) s t ) ( A,h l« Jt) U ( A H ) C . + A \ 0 (in.2) in which CJ is a number of modulus unity so that (II.l) is satisfied. It has been shown (Wigner 1939, Bargmann 1954) that a l l physically permissible representations are equivalent to those for which toS±l. in addition to (III.2) one requires of U(Aj«.)that i t be unitary; that i s = <«M<P> = ( UV*, U<P) , ( I I I > 3 ) and that i t be i r r e d u c i b l e . In t h i s summary proofs of the i r -r e d u c i b i l i t y of the representations found w i l l not be given; they may be found i n the work of Mackey (Maekey 1955). One may decompose the representative and then the requirement f o r the components, corresponding to (III.2), i s T ( 0 T ( b ) = T («. +b) (ILI.5a) <k(*\)TM = T ( A a ) «A ( A ) (III.5b) fll(A) •i(M) = i ol(AM) . (III.5c) A group which has a p a r t i c u l a r l y close r e l a t i o n to the homogeneous Lorentz group i s the two-dimensional unimodular group of matrices with complex matrix elements. It deteimines transformations of coordinates by To each transformation of the form (III.6) there corresponds a Lorentz transformation; i n f a c t , the transformations by unimod-ular matrices form a two-valued representation of the homogen-eous Lorentz group. A l l physically permissible representations of the unimodular group are equivalent to true representations; that i s , i f A and Y\ are transformations of the unimodular group whose corresponding Lorentz transformations are A t T*^  ( I I. 5 c ) may be replaced by <U A U ( f i ) = A ( X ) . (III . 5 d ) It w i l l occasionally be convenient i n the following to consider representations of the unimodular group i n order to deal with true representations. 2 . I n i t i a l Decomposition of the Types of Representation  Form of Representatives The representation of the t r a n s l a t i o n operators T C O <p ( f)) = J». ^ ( n ) , ( i n . 7 ) where f'*- - f^  ^ v , i s a unitary representation with r e s -pect to the scalar product It corresponds to the natural representation i n configuration r-l space TU)4>C* [ )= V* ( » - * , ^ ) where • K The t r a n s l a t i o n a l elements of a l l the representations to be considered here have the form ( I I 1 . 7 ) . For every proper, homogeneous, Lorentz transformation ' * , define the operator by P ( A ) cp(Ft) = <p(A - f \ ) . 9 ) One has then . A-' ? a (III.8 ) and showing that P C A ) T ( O L ) = T ( A O P ( A ) , since A " f » f • A «. . In view of (III. 5b) , el( A ) PC A ) " commutes with a l l and hence with every function of P . Therefore can be written M U y . Q ( r . A ) P<A) , ( I I I 1 0 ) i n which GKf, A )operates only on the v a r i a b l e ^ . Decomposition of Types of Representation Since the set of exponentials forms a complete set of functions, one may approximate the operation of m u l t i p l i c a t i o n with any function "T^p)by V) - £ c„ T O O <p(P \) and i f one chooses ^(p)=-H Af)for a l l A , then the operation of m u l t i p l i c a t i o n commutes with a l l theT60and P(A) . If <jp(f\") belongs to a set of functions generating an i r r e d u c i b l e repre-sentation, then by Schur*s lemma,\(?^: t****; or 4<p)^(pV) r C < P ( p V ) Thus C p f p \ ) can be d i f f e r e n t from zero only for a set {fcVjof momenta f o r any two elements p . ^  of which there i s a proper, homogeneous Lorentz transformation ACfiO such thatA(p,^)p« V By choosing ^.(p)spp , an i n i t i a l decomposition of the types of representations i s found: class (1) (la) P P - fX- >© , pM > (lb) P P * ? L ' > O , PM < O 14 class (2) class (4) class (3) O. PH < O Only the representations of classes (1) and (2) have received physical i n t e r p r e t a t i o n . The representations of class (3) would correspond to p a r t i c l e s of imaginary mass; those of cl a s s (4) would correspond to p a r t i c l e s of no momentum and no energy. i s three dimensional and one may take p, Pt p, as independent v a r i a b l e s . One may write (II1.8) With the scalar product defined as i n (III.11), P ( A ) i s unitary. 3. Reduction of Representations to those of a L i t t l e Group The matrices OC P, A ) of equation (III.10) are subject to two demands: they must be unitary with respect to the scalar product (III.11); they must have such a form that (III.5c) i s s a t i s f i e d . The f i r s t demand implies For classes (1) and (2), the v a r i a b i l i t y domain of p (III.12) the second implies 15 since <p'(f\) = QCf .W)^ 9 C H " , p 1 t ) This then y i e l d s Q ( ? , A ) ^ L Q ( A > , M ) ^ - t Q ( P . A H ) ^ ( i l l . 1 3 ) which indicates that the matrices 6Kf, A) do not alone constitute a representation of the homogenous Lorentz group because of the i r p dependence. However the matrices (jj(PiA)do form a rep-resentation of the subgroup of homogeneous Lorentz transforma-tions which hold a s p e c i f i c f\» fi x e d , the l i t t l e group. At t h i s point i t i s convenient to consider the group of unimodular, two-dimensional matrices rather than the homo-geneous Lorentz group. Let a ^ 0 be chosen and consider the subgroup of unimodular matrices^A - j for which the correspond-ing Lorentz transformations A hold P0 fixed Afw= p0 . For this l i t t l e group, one has instead of (11.13), a true repre-sentation according to ( I l l . g d ) : For every p choose a unimodular matrix aC(f) so that <tf(f) Pd - p , and i n addition ot(p) i s continuous i n p with <<(?<>)-1. It i s possible (Wigner 1939) to choose the Cp(p\)so that Q(?,*(f>) = \ , and as a consequence d(*Cff')<P(p-\) = c p ( P h (III.14a) a i S ( f ) ) . ( f i ) « ^CPob . d n . i 4 b ) If A i s an a r b i t r a r y unimodular matrix, one may decora-pose i t A = £ ( p ) £ ( f , A ) £ ( A - ' p ) " ' ( I I I . 1 5 ) where p i f»A ) s oi(p> A oC ( A" p) belongs to the l i t t l e group: cLtpV* A ct ( A " > ) p 0 - o C ( p ) " A A"'p = <<Cp)"'f - p f t . Using t h i s decomposition, with the help of equations (III.1 4 ) , A ( X ) * ( f V > « <A ( 2cp)U(jgf) <A(2<A- ,I>-'J (pcpb i l ( A > < ! p ( f \ ) = Q ( f . , i f ) t > <P(A"'f 7) , ( i n . 1 6 ) where QCf 0 ||3)is an element of the matrix representation of the l i t t l e group. (III.1 6 ) defines e x p l i c i t l y a representation of the whole unimodular group i n terms of a representation of the l i t t l e unimodular group. The single-valued representations of the unimodular group i n general induce double-valued repre-sentations of the homogeneous Lorentz group; corresponding to the Lorentz transformation A . there are two elements df the unimodular group A and - A . For the representations to be used here, Corresponding to an i r r e d u c i b l e repre-sentation of the unimodular group, one has an i r r e d u c i b l e re-presentation of the Lorentz group (Mackey 1955) . Attention w i l l not be r e s t r i c t e d to class ^ r e p r e s e n -tations, representations used for the description of p a r t i c l e s with zero mass. It i s quite easy to see (Wigner 1939) that the elements of the unimodular l i t t l e group for^0=(°c>l ') have T H E F O R M /• K+L o \ r*ith<•»:,>-•'**\ 17 where X and ^ are r e a l , a n d O l l S ^ H T T . The l i t t l e group i s the E u c l i d e a n group i n two dimensions; "t(*«j) and S( |J)refer r e s -p e c t i v e l y to the t r a n s l a t i o n s and r o t a t i o n s of the E u c l i d e a n group; the Lore n t z t r a n s f o r m a t i o n c o r r e s p o n d i n g t o i s a r o t a t i o n about the s p a t i a l p a r t o f P% of magnitude fi ; here w i l l be r e s t r i c t e d o £: ji t ATT, and the r e l a t i o n £(|5+*w)*t £((.)will be used. The r e p r e s e n t a t i o n s o f the E u c l i d e a n group used f o r the d e s c r i p t i o n of p a r t i c l e s o f f i n i t e s p i n a r e those f o r which the t r a n s l a t i o n a l p a r t a re mapped onto the i d e n t i t y op-e r a t o r . One then e f f e c t i v e l y d e a l s w i t h the r o t a t i o n group i n two dimensions whose i r r e d u c i b l e r e p r e s e n t a t i o n s are J I where S i s a p o s i t i v e o r neg a t i v e i n t e g e r or h a l f i n t e g e r and fi i s the angle of r o t a t i o n . The n o t a t i o n w i l l be used to de-note the angle of r o t a t i o n fi , O i £*H determined by the r o -t a t i o n a l p a r t of the l i t t l e group element fi . The a b s o l u t e magnitude I Si i s i n t e r p r e t e d as the s p i n o f the p a r t i c l e des-c r i b e d by the r e p r e s e n t a t i o n ; the s i g n determines whether the s p i n i s o r i e n t e d p a r a l l e l or a n t i p a r a l l e l t o the s p a t i a l p a r t of the momentum v e c t o r p (Newton 1958). The r e p r e s e n t a t i o n used f o r the d e s c r i p t i o n o f p a r t i -c l e s of ze r o mass and s p i n S may now e x p l i c i t l y be w r i t t e n down: ;p.* : s €(2(pf 'A *c A"'P>) U<A.O «P(p) * t * UL ^ ( A ^ p ) , (III.18) where the v a r i a b l e \ has been dropped s i n c e the r e p r e s e n t a -t i o n of the l i t t l e group used i s one-dimensional. The work of the following two chapters w i l l be concerned only with rep-resentations of t h i s type; i t w i l l not require a d e f i n i t e r e -a l i z a t i o n of the matrices ot( P) . Fronsdal, (Fronsdal 1 9 5 9 ) however, has found a r e a l i z a t i o n of the «*(p)'s and written equation ( I I I . 1 8 ) i n terms of i t for the case of i n f i n i t e s i m a l Lorentz transformations. 19 IV. TRANSFORMATION PROPERTIES OF BILINEAR COMBINATIONS OF GENERATING SPACE ELEMENTS OF IRREDUCIBLE REPRESENTATIONS 1. I n t r o d u c t i o n of the Momentum E i g e n f u n c t i o n B a s i s and  the D i r e c t Product Repre s e n t a t i o n I t i s the purpose of t h i s chapter t o c o n s i d e r b i l i n e a r combinations of s t a t e f u n c t i o n s g e n e r a t i n g i r r e d u c i b l e r e p r e -s e n t a t i o n s of c l a s s (2) of the inhomogeneous L o r e n t z group wit h a view to f i n d i n g s e t s of such combinations which them-s e l v e s transform a c c o r d i n g to an i r r e d u c i b l e r e p r e s e n t a t i o n of c l a s s ( 2 ) . That s e t s w i t h t h i s p r o p e r t y may be found i s p h y s i c a l l y i n t e r p r e t e d as i n d i c a t i v e of the p o s s i b i l i t y t h a t , a t l e a s t i n t h e o r i e s w i t h a f i x e d number of p a r t i c l e s , comp-o s i t e p a r t i c l e s may e x i s t ; t h a t i s , one may regard a combina-t i o n of elementary p a r t i c l e s as i t s e l f an elementary p a r t i c l e . Shirokov (Shirokov 1959)has d e r i v e d a decomposition of the d i r e c t product of two i r r e d u c i b l e r e p r e s e n t a t i o n s used f o r the d e s c r i p t i o n of massive p a r t i c l e s by another method. I t i s convenient f o r the f o l l o w i n g to have a method of l a b e l l i n g the v a r i o u s s t a t e f u n c t i o n s under c o n s i d e r a t i o n . T h i s i s most e a s i l y done by u s i n g s t a t e f u n c t i o n s which are e i g e n f u n c t i o n s of the momentum op e r a t o r f ; one may then i l a b e l the s t a t e f u n c t i o n s by the f o u r - v e c t o r e i g e n v a l u e P , Cp(f, p) ; and the f o l l o w i n g e q u a t i o n h o l d s : f < P ( ? , P ' ) = r <*>(?,?'<> (iv.i) T h i s i s e q u i v a l e n t to the c a r r i e r space b a s i s used by Lomont (Lomont 1959) which d i a g o r r a l i z e s the t r a n s l a t i o n o p e r a t o r s "TC*). One notes that cp(A f, P ) i s an e i g e n f u n c t i o n o f momen-tum w i t h eigenvalue A p1 . To see this, observe t h a t from (IV.1) f o l l o w s A"'p <p( A"'f, f )* f'qpU'Vif) by r e l a b e l l i n g v a r i a b l e s ; then ptpCA'p.p') - A P' <o(A~'?, f') . T h i s permits one to de-fLne a l l < p ( P » ^ f o ) a f t e r a s i n g l e <f> ( p. f o) has been g i v e n by 9 ( P » A f l ) = 9 ( A" f . ( I V.2) A n o t a t i o n a l s i m p l i f i c a t i o n i s gained by su p p r e s s i n g the v a r i -able f and w r i t i n g Cpfpifor (pCp, f*K Let Cp(p ) and V'C^')be the s t a t e f u n c t i o n s g e n e r a t i n g z e r o mass r e p r e s e n t a t i o n s d e f i n e d by the s p i n s S , , the l i t t l e group v e c t o r s f0 , %\o, and the unimodular m a t r i c e s et(f), and TC^) , such that * ( ? ) f a * f , andT(^ )<|a* <\ . U s i n g equa-t i o n (III.18) with (IV.1), the r e p r e s e n t a t i o n s a r e : LHA.O <pCf > = t ^ CpCAf) ( i v .3a) , , ^ '* AV* A)) V U , * ) ^ ) • t ^ - ^ (A* ' ) , (iv.Sb) whereyJ(Ap', A ) - l u p )*A *(p' ) and S( A '^ (A ) s T( A<\ )* A T(^) . The d i r e c t product r e p r e s e n t a t i o n c o n s i s t s of the s e t J U(A,«0x VCA,c) ^  of o p e r a t o r s d e f i n e d on the s e t {<pCf>* V^V>| of f u n c t i o n s o f two v a r i a b l e s . The r e p r e s e n t a t i o n i s [U( A,«.) i VCA,ro]I<p(P',< ^ (f,] ^ U<A,*.) <«>Cp') x VCA,«.) Y^CV) c ' A ( ? W > - « . «ls*C?fA?.A 'mflJCAi IA))] = ± -* Jl <pCAp)«y<A^.(IV .4 ) One now asks f o r a s e t of l i n e a r combinations o f members of the s e t ^ <pfp')*^C '^jj which w i l l transform a c c o r d i n g t o an i r r e d u c -i b l e z e r o mass r e p r e s e n t a t i o n o f the Lo r e n t z group. One may 21 f i n d such sets i n two ways., one, with r e s t r i c t e d a p p l i c a t i o n , involving an equality of the typefi ( A p , A ) s A/fc/lp, A ) , the other, more general, involving an equality of the type ^ ( £ ( A p , A > ) 2. Correctly Transforming Set Involving an Equality  /SW.A) x | ( X A P ' , A ) The various representations (IV.3b) found by using d i f -ferent l i t t l e group vectors ^ o , and d i f f e r e n t unimodular ma-t r i c e s Y(f) , are related by unitary transformations (Wigner 1939). By using a unitary transformation, i f necessary, one may ensure that the representation defined by (IV.3b) i s r e -lated to that defined by (IV.3a) according to <Ua > ?• ' YCXV * * C p ) , (IV.5) where A ( A t o , X * - i ) i s a scalar which i s , of course, r e a l . As a consequence, £(Af, A ) = S C " K A y , A ) ( I V 6 ) i s an i d e n t i t y . In order to be quite d e f i n i t e , the assumption i s made that the representations (IV.3) are both p o s i t i v e energy representations, that i s , of class (2a). In t h i s case 7\ i s a posit i v e number. The changes required when either or both representations are of class (2b) are obvious. i It i s clear that the space spanned by the set of func-tions [ cp(f') * 4>(\p') a l l f' of class (2a) | = i s i n v a r i -ant under the transformations (IV.4). This follows from the fact that the set contains <jp(Ap')*V^Af') for a l l proper homo-geneous Lorentz transformations A , i n f a c t , on6 may show quite simply that the representation defined on th i s set i s an ir r e d u c i b l e , mass zero representation with spin S + . Such a representation w i l l have the form W(A,«0<t><t ) = t JL 4> ( A t ) , ( I V 7 ) i n which, without loss of generality, one may take "t0= G+'X)f0 , and ^CO+>)f) = oC ( f ) . As a consequence (IV.6) i s enlarged to ^ £CCi+?0 Af . S (x Af, A ) = A f, A ) . (IV.6a) If one e x p l i c i t l y writes down the correspondence 4 > ( o + * ) f ' ) *-* <pcp') x V'fxf') , he sees that the representation U(A,*0 * V (A , « . ) r e s t r i c t e d to operate on the set $ ( x)is precisely the representation W(A,<^, JL. > f I where (IV.6a) i s used with ^ - f i n the comparison of equations (IV.4) and (IV.7). The physical content of t h i s r e s u l t i s that i t i s possible, i n a one p a r t i c l e theory with-out i n t e r a c t i o n , to combine two massless p a r t i c l e s of spins S,** and with p a r a l l e l momenta to form a single massless p a r t i c l e of spin S + T . The method i s susceptible to the objection that the parameter % has been introduced unnaturally and that more than one set of l i t t l e group vectors and corresponding unimodular matrices has been used. Both these objections are removed i n the second method whose requirement d£( j!) = £(S)is less stringent. 3. Correctly Transforming Sets Involving an Equality For the determination of the sets to be found here, i t i s necessary to prove the following statement: It i s possible to choose the unimodular matrices *C(p) with the property o^(p)p0=p , i n such a way that i s independent of A , >>0. For definiteness, consideration i s r e s t r i c t e d to momen-tum vectors of class (2a) (implying >O ). Consider the set of momentum vectors defined by the set of a l l vectors A i n 3-space normalized to + \ . Define odCp) for this set so that i t i s continuous within the set. For A > O def ine 1 (*) so that (IV.8) Z U ) P. = A po If f6 = (e> o I I) , one may take Z(?0in the form / I o O O z(»> - ° 1 l o o \ O O with , , r + & T = *x It i s easy to see that Z(S^Z ( / O = Z C V ° (IV.9) Now for P i n the set J C ^ ^ n , »)j , define Sir x p) = £ < p ) zC * ) (IV.10) for then, £ f) p 6 = <*C f) *Xf6 = % p , as required. It i s further clear that whether or not ? belongs to the set ^ (Q I )j , (IV.10) h o l d s ^ f o r i f p i s an ar b i t r a r y momentum vector, p = /A<\ for some yu and some °| i n the set \ then £ ( x p ) = £ < V " 0 - 5 ( ^ ) 2 ^ ^ ) = X ( « \ )Z ( / . )2Cx)« icp) Z(>). I t i s f o r t h e < l ( p ) j u s t d e f i n e d t h a t the a s s e r t i o n above may be proven. I f o£(tr)is any unimodular matrix w i t h the p r o p e r t y i(»)f.sir, then i t i s r e l a t e d t o d.( V) d e f i n e d above by *'fr) • tfc*0^ where ^ i s a member of the l i t t l e group, t h a t i s * 9 p r e s e r v e s f0 . To see t h i s , w r i t e and from * < v ) p 0 - V get oCC*) v * f » s « « v ) o(CO f 0 . In p a r t i c -u l a r A «£(f; has the pro p e r t y t h a t A *cp)p 0= A p , so that one may w r i t e * ( A " p ) A"' £ < p ) Tcp.A) where ( C f, A} i s a member of the l i t t l e group, and ^ ( H ' p ) : *(A"p } Z c M = A" £(p)c c z ( X ) . ( iv .11) As a consequence of (IV.11), i t f o l l o w s t h a t A ^ ( U P ) = Z ( > ) «tCf) A A * ( p ) tCf .A) Z C A ) = Z ^ ) " T(f.A) 2 (IV.12) One may now decompose CC\°, A)into a t r a n s l a t i o n a l p a r t and a r o t a t i o n a l p a r t a c c o r d i n g t o equa t i o n (III.17) One may e a s i l y see th a t Z ( \ ^ has the form ~ / a o \ 2 ( » ( o a ) w i t h <*• - A . Since £ f c(p,A) i s a l s o a dia g o n a l m a t r i x , i t com-mutes with Z ( x) » as a r e s u l t r^xp)"X <(>A">)= zc\)x 7r(t,A)zM y c?tA) . ( iv.13) i s a l s o a t r a n s l a t i o n a l element of the l i t t l e group j s i n c e , from (II 1.17), <^r^f»M has the form ( r - ( 0 . ) ' one may see t h i s by m a t r i x m u l t i p l i c a t i o n Using t h i s r e s u l t , one may deduce from (IV.13) « ^ X p r A ; ( ^ A - > l ) ^ ( ^ ( ? , A 1 ) , ( I Y # 1 4 ) which i s independent of A » the a s s e r t i o n r e q u i r e d to be pro-ven. The more n a t u r a l c h o i c e i n the r e p r e s e n t a t i o n s (IV.3), e£(f ) sV (<\ ) and hence ^ ( A f , A )= S(Ap, A ) , may now be taken. These r e p r e s e n t a t i o n s now become U(A.O <P(p) * t * * d p ( A P ) (iv.15a) where <L(fi ( A f . A ) ) . K ( 5 ( A f ' A 3. <?0 = * ( r , j A f ' , A » . One asks, as b e f o r e , f o r a s e t ^  whose members are l i n e a r combinations of the s e t of f u n c t i o n s of two v a r i a b l e s C^£>(p'i and whose members tran s f o r m a c c o r d i n g t o a mass z e r o , s p i n S+v-r e p r e s e n t a t i o n of the proper, inhomogeneous L o r e n t z group. The most ge n e r a l l i n e a r combination i s which must s a t i s f y W ( I , < 0 - + (IV.17a) V < A . o > * C V ) , ± v " - ' ^ ^ ^ ^ ^ ( A t ' ) . ( I v . 1 7 b ) f.,.. . (IV. 16) 26 Of c o u r s e a n i n t e g r a l may r e p l a c e t h e summat ion i n ( I V . 1 6 ) o v e r a l l o r some o f t h e e i g e n v a l u e s f , *\ In t e r m s o f t h e d i r e c t p r o d u c t r e p r e s e n t a t i o n ( I V . 4 ) , t h e demands ( I V . 1 7 ) a r e w r i t t e n - CU'ifflpCf') * V ^ ' ) , ( I V . 1 8 a ) = * * l t s * o < c f f f A t , ^ ^ c ^ r W i ^ r , ) * ^ 0 • (iv. i 8 b ) S i n c e t h e members o f t h e s e t a r e l i n e a r l y i n d e p e n d e n t , one c a n s a t i s f y ( I V . 1 8 a ) o n l y by r e s t r i c t i n g t h e summat ion t o v a l u e s o f f and ^ s u c h t h a t • e q u i v a l e n t l y t h e r e s t r i c t i o n i s f f t ' l f . V - O i f p+«t * t . U s i n g ( I V . 1 4 ) one s e e s t h a t ( IV . 18b) may be s a t i s f i e d i f f* and ^' a r e m u l t i p l e s o f t' , t h a t i s f ' s ' X t ' , a n d ^ ' r / . t ' . and i n a d d i t i o n C(At;Af,= ftf). W i t h t h e r e s t r i c t i o n s s t a t e d , a l i n e a r c o m b i n a t i o n o f t h e f o r m ( I V . 1 6 ) w h i c h s a t i s f i e s ( I V . 1 7 ) may be w r i t t e n S i n c e t h e r e s t r i c t i o n s p u t on t h e l i n e a r c o m b i n a t i o n i n o r d e r t o s a t i s f y ( I V . 1 8 b ) were no t p r o v e n t o be n e c e s s a r y , one c a n n o t be c e r t a i n t h a t t h e e x p r e s s i o n ( I V . 1 9 ) i s t h e most g e n e r a l ; p h y s i c a l l y h o w e v e r , i t i s r e a s o n a b l e t h a t ( I V . 1 9 ) i s t h e most g e n e r a l e x p r e s s i o n . ( I V . 1 9 ) i s an e x p r e s s i o n c o n t a i n i n g s t a t e f u n c t i o n s d e s c r i b i n g p a r t i c l e s w i t h p a r a l l e l momentum v e c t o r s o n l y ; s i n c e the s p i n s o f massless p a r t i c l e s a r e p a r a l l e l or a n t i p a r a l l e l t o t h e i r momentum, i f the l i n e a r combination i n (IV.19) co n t a i n e d s t a t e f u n c t i o n s Cf or V* whose momentum vec-t o r s were not p a r a l l e l t o "t , the decomposition i n t o i r r e d -u c i b l e p a r t s would c o n t a i n terms r e f e r r i n g t o p a r t i c l e s of s p i n s other than S+v . Note t h a t i n (IV.19), an e x p l i c i t de-pendence on t (other than the t r i v i a l s c a l a r = 0 )can-not be admitted i n CC\y-) . I t i s a d i f f i c u l t y of t h i s s o r t which denies c o r r e c 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 o the a n n i h i -l a t i o n , c r e a t i o n o p e r a t o r c o n s t r u c t i o n d e s c r i b e d by Case (Case 1957) and to be o u t l i n e d i n chapter V. 28 V. ATTEMPT TO SIMULTANEOUSLY SATISFY COMMUTATION RELATIONS AND  TRANSFORMATION PROPERTIES FOR COMPOSITE PARTICLES OF ZERO MASS 1. I n t r o d u c t i o n Once one has accepted the formalism o f o p e r a t o r s and s t a t e v e c t o r s , he may c o n s t r u c t a theory o f elementary p a r t i -c l e s without i n t e r a c t i o n by s t a t i n g a Lagrangian, a f u n c t i o n of f i e l d v a r i a b l e s and t h e i r c o o r d i n a t e d e r i v a t i v e s , the use of which i n a v a r i a t i o n a l p r i n c i p l e (Schwinger 1951) leads t o a c o n s t r u c t i o n of momentum, angular momentum, and charge opera-t o r s . These o p e r a t o r s may be expressed i n terms of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s and they may be used to d e s c r i b e a system of f r e e elementary p a r t i c l e s . T h i s i s , o f c o u r s e , a r a d i c a l a b s t r a c t i o n from the p h y s i c a l world i n which one obser-ves p a r t i c l e s i n con s t a n t i n t e r a c t i o n with o t h e r s ; however the idea of 'free p a r t i c l e s ' r i s a concept which has l e a d to u s e f u l r e s u l t s i n the p a s t , and i s l i k e l y to do so f o r some time i n the f u t u r e . For the c o n s t r u c t i o n o f a theory of composite p a r t i c l e s , a theory of elementary p a r t i c l e s composed of (say, two) oth e r types of elementary p a r t i c l e s , one would ask f o r a p r e s c r i p t i o n f o r the combination o f the two f i e l d f u n c t i o n s i n t o a s i n g l e f i e l d f u n c t i o n , and of the two Lagrangians i n t o a s i n g l e Lagr-angian. As a p r e l i m i n a r y , one might ask f o r the combination of the two s e t s o f c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s i n t o a s i n g l e s e t . Here the most n a t u r a l combination, b i l i n e a r combi-n a t i o n s , w i l l be used. One r e q u i r e s o f the o p e r a t o r s of the combined s e t that they obey commutation r e l a t i o n s of the type 29 (boson or fermion) of composite p a r t i c l e envisaged, and th a t they transform, under L o r e n t z t r a n s f o r m a t i o n , a c c o r d i n g t o the law demanded by the p r o p e r t i e s of the composite p a r t i c l e . The purpose of t h i s chapter i s t o p o i n t out that the o n l y " s o l u t i o n " t o t h i s problem which has remained i n the l i t e r a t u r e does not trans f o r m c o r r e c t l y . The " s o l u t i o n " here r e f e r r e d to i s one i n which one attempts to form the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r a s p i n z e r o p a r t i c l e from the c r e a t i o n and a n n i -h i l a t i o n o p e r a t o r s of a fermion and i t s a n t i p a r t i c l e ; the r e -s u l t was o r i g i n a l l y o b t a i n e d by Born and Nagendra Nath (Born and Nagendra Nath 1936), and has r e c e n t l y been r e s t a t e d by Case (Case 1957). 2. Statement o f the Problem TC L e t &>(») and ftCf) be the c r e a t i o n and a n n i h i l a t i o n oper-a t o r s f o r massless p a r t i c l e s o f h a l f - i n t e g e r s p i n i S l \ l e t b ^ f ) from those d e s c r i b e d by the op e r a t o r s <L , i n that t h e i r s p i n i s d i r e c t e d p a r a l l e l or a n t i p a r a l l e l t o t h e i r momentum a c c o r d -i n g as the s p i n of the p a r t i c l e s CL i s d i r e c t e d a n t i p a r a l l e l o r p a r a l l e l to t h e i r momentum', t h i s i s r e f l e c t e d i n a d i f f e r -ence i n the s i g n of S i n the t r a n s f o r m a t i o n p r o p e r t i e s . Both p a r t i c l e s belong t o the same c l a s s of r e p r e s e n t a t i o n s which w i l l be taken to be c l a s s (2a). and bCp) be the cor r e s p o n d i n g o p e r a t o r s f o r the a n t i p a r t i c l e s . The p a r t i c l e s d e s c r i b e d by the op e r a t o r s t , a r e d i f f e r e n t By a comparison o f equations (11.10) and (III.18) one may deduce from equations (11.12) the t r a n s f o r m a t i o n p r o p e r t i e s 30 of &fpl,«.Cp) and IsVp) , Wp) : U ( A , O a ( p ) U ( W « t -* -* Q - U f ) ( v l a ) IM,*) Wp)U(A.*) = 1 < ^ b (A ? ) (V.2a) ,OW P ) (JKO - + ? ' V A r ) . (v.2b) U ( A The t r a n s f o r m a t i o n s f o r s p i n z e r o p a r t i c l e o p e r a t o r s a r e : U C A . c u ) $ * C K ) U ( A , « 0 " = t * *" i*(A-'ic) (V.3a) u ( U ) h « > u a , . r = * • ( v - 3 b ) The commutation r e l a t i o n s s a t i s f i e d by the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s of the s p i n I S l f e r m i o n s q. and ^  a r e : L*<p>,*<f*>]4 = Ifcti. Iti>]+*[{(f)Mi>h* tCffj.lTcrJ*o cv.4b> U«<f>.W>V W r > . b V i ] 4 = Lc*p),Uf;]4= L^cp),^(p',]/c>4(v.4c) The commutation r e l a t i o n s which must be s a t i s f i e d by the corres-(V.5a) l U o A ( K ) ] . = [ V ^ , ^ *')]- = ° . ( V ' 5 b ) S ponding o p e r a t o r s f o r the s p i n z e r o boson £ , are The problem i s t o f i n d b i l i n e a r e x p r e s s i o n s o f the op-v 1 * 1 e r a t o r s , , « , *a , which s a t i s f y equations (V.3) and (V.5). 3. Attempted S o l u t i o n t o the Problem  Tra n s f o r m a t i o n p r o p e r t i e s By the same method as was used t o a r r i v e at equ a t i o n (IV.19) one may f i n d t h r e e types of b i l i n e a r e x p r e s s i o n having the c o r r e c 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 : i i ' y C b C x ) d x t ((>-«) O (V.6b) ^ c ( k ) - \ X U x O a ( n * x ) K ) } (v.6c) o i n which l i m i t i n g processes t o a v o i d the p o i n t s \ f o r which the arguments o f any of the o p e r a t o r s would be 0 , a r e assumed; the l i m i t may have any f i x e d v a l u e . One may, with Case, i n t r o d u c e the n o t a t i o n a l s i m p l i f i -c a t i o n : «(-n = b ( f ) t ftC-f),Up) p (V.7) of c l a s s (2a). The commutation r e l a t i o n s (V.4) may now be con-densed t o [ a C f ) , « f t p > ] « . r J C p - p ' ) ( v.8«) L ' t**Cf), «4 . * C f ' ) ] 4 = 0 ( V.8b) i n which P may be a momentum v e c t o r of e i t h e r c l a s s (2a) or c l a s s (2b). With t h i s n o t a t i o n and the d e f i n i t i o n CfO=C«.(>) f o r O < > ( 06 • Q ( \ ) =-C\,( x) f o r - - a O < - « , C (*)= -Cc( x) f o r — l < X < C> , one may w r i t e ( ^ + * i e > C * > = \ J , C ( > ) 4 ( A r ) t ( ( ^<)<c) (V.9) i n which agai n l i m i t i n g processes are assumed to a v o i d p o i n t s f o r which the arguments of op e r a t o r s would v a n i s h . The expres-s i o n (V.9) transforms c o r r e c t l y as a s p i n z e r o a n n i h i l a t i o n op-e r a t o r ; i n the f o l l o w i n g , an attempt w i l l be made to s a t i s f y the commutation r e l a t i o n s (V . 5 ) w i t h an e x p r e s s i o n of the form K I P ! ( 0 = \ * A C U ) a(w) oXf v + o O (v.io) -*//» t h i s i s i d e n t i c a l i n form w i t h (V.9) i f = . . Under a Lo r e n t z t r a n s f o r m a t i o n (A,<0 (V.10) transforms u(A,o't(ou(Alo",'=" t ^ ; k ' £ " | ( A - , O which s a t i s f i e s ( V . 3 b ) o n l y i f \LVL) = A*V) . I f a proposed so-l u t i o n of the form ( V . 1 0 ) does not have^(«) ~ \(A'**), i t i s i n c o r -r e c t i n that i t does not transform p r o p e r l y . Commutation r e l a t i o n s In order t o s a t i s f y the commutation r e l a t i o n s (V . 5 ) by the method d e s c r i b e d by Case, i t i s necessary t o i n t r o d u c e a number of p h y s i c a l assumptions which are d e f i n i t e l y not of a r e l a t i v i s t i c c h a r a c t e r . One must agree to c o n s i d e r o n l y compo-s i t e p a r t i c l e s whose e n e r g i e s a re restricted/»< * < Q < ; of c o u r s e , to make such a r e s t r i c t i o n and s t i l l admit c o o r d i n a t e t r a n s f o r m a t i o n s one must have i n mind some i n i t i a l frame o f r e f e r e n c e i n which the r e s t r i c t i o n i s t i g h t e r ^ < ^ < * M < Q < Q< - Jc, and then agree t o r e s t r i c t a t t e n t i o n to t h i s frame and ot h e r s which may be reached from i t by L o r e n t z t r a n s f o r m a t i o n s which do not send a momentum v e c t o r with energy i n the i n t e r v a l p , Q to one wit h energy o u t s i d e the i n t e r v a l p , Q . T h i s then assumes t h a t i n a t e r r e s t i a l frame of r e f e r e n c e , composite p a r t i c l e s of e i t h e r a r b i t r a r i l y h i g h or a r b i t r a r i l y low e n e r g i e s cannot e x i s t , and f u r t h e r , t h a t one cannot g i v e t o an observer an a r b i t r a r i l y h i g h v e l o c i t y w i t h r e s p e c t to the t e r r e s t i a l frame. One must a l s o assume, wit h Case, t h a t the p a r t i c l e and a n t i p a r t i c l e s t a t e s found i n nature are empty f o r s u f f i c i -e n t l y h i g h e n e r g i e s : (?) | > = O , P+>Q < * o v >•% / 1 P ( V . l l ) The computations are s i m p l i f i e d by u s i n g a box normal-i z a t i o n ; (V.10) may be w r i t t e n as an energy i n t e g r a l u s i n g KH>^ > w i t h ( V . l l ) -ft. w i t h the d i s c r e t e analogue of the form ^ K 5 ) = \MK'^ L *. ii,i\)±UK+*J*)C(*«Uj (v.i3) -a i n which the n o t a t i o n It., £ means the f o u r v e c t o r whose d i r -e c t i o n i s 2 and whose energy i s ICH . The commutation r e -l a t i o n s (V.5) and (V.8) are r e p l a c e d w i t h ones u s i n g 5 symbols r a t h e r than £ f u n c t i o n s . In a s t r a i g h t f o r w a r d manner, u s i n g the commutation r e l a t i o n s one may show t h a t the r e l a t i o n s (V.5b) (with d i s c r e t e $ symbol) h o l d f o r ^ d e f i n e d by (V.13) i f C ( i n the f o l l o w i n g i t i s assumed t h a t C * l . Using (V.8) a g a i n , w i t h the e x p r e s s i o n (V.13), one has f o r the commutator [ s ^ * \ \ + S ( ^ J J . P U ' ) & < ! K + * J * ' ) . (v.14) I f n t n ' , then [ ^ (\*M, ^ ( t ^ D')] ~ O as r e q u i r e d ; i f Q. - n', then £(? H n (F.,Y) = SfPHlP!,) , and one may r e w r i t e (V.14) +• J c P s - i ) c a P s - * - * H - M * ) \ 4(o4**'>ic< vc';1 . < v - 1 5 ) By a change of v a r i a b l e i n the f i r s t summation o f (V.15), one may w r i t e ft+rM U «- -ft Using equations ( V . l l ) and the p h y s i c a l assumptions s t a t e d above i t , one sees t h a t the a p p l i c a t i o n of the second summa-t i o n on the r i g h t hand s i d e of (V.16) to a v e c t o r r e p r e s e n t i n g a p o s s i b l e p h y s i c a l s t a t e r e s u l t s i n z e r o . The c o n t r i b u t i o n of the t h i r d summation may be found by u s i n g the commutation r e l a t i o n s to w r i t e the summand S ( K M I < M ) + a(< PM+KM-^ISJCLVP.,*^. Once a g a i n employing ( V . l l ) , one sees t h a t the r e s u l t s of a p p l y i n g t h i s sum or - = d d c H . V t o a p h y s i c a l s t a t e ' are the same. As a consequence [ tK*).V^!)] - 4 ( O T ( I C W K C K , 5 ( K M J K O which i s i d e n t i c a l w i t h the r e q u i r e d r e s u l t i f 4 ^ - • T h i s i s the r e s u l t quoted by Case. The f a c t that the -f ( >*) r e q u i r e d to s a t i s f y the commuta-3 5 t i o n r e l a t i o n s does not have the p r o p e r t y \ (tc) Z-^C^'K) means t h a t the e x p r e s s i o n (V.10) u s i n g r t h i s 4^*)does not t r a n s f o r m c o r r e c t l y . U n f o r t u n a t e l y , t h e problem o f t h e p o s s i b i l i t y o f c o n s t r u c t i n g t h e o r i e s o f c o m p o s i t e m a s s l e s s p a r t i c l e s has not been answered. N e i t h e r has a c o r r e c t l y t r a n s f o r m i n g b i l i n e a r c o m b i n a t i o n been g i v e n w h i c h s a t i s f i e s the commutation r e l a -t i o n s , nor has i t been shown t h a t t h i s i s i n g e n e r a l not p o s s -i b l e . 36 BIBLIOGRAPHY Bargmann, V., 1947, Annals of Math. 48, 568. , 1954, Annals of Math. 59., 1. Born M. and Nagendra Nath, N.S., 1936, Proc. Indian Acad. S c i . A3_, 318. Case, K.M., 1957, Phys. Rev. 106, 1316. D i r a c , P.A.M., 1958, ' P r i n c i p l e s of Quantum Mechanics', 4th E d i t i o n , Oxford. E v e r e t t , H. , 1957, Rev. Mod. Phys. 29, 454. F r o n s d a l , C , 1959, Phys. Rev. 113. 1367. Lomont, J.S., 1959, ' A p p l i c a t i o n s o f F i n i t e Groups', Academic Press. Mackey, G.W., 1955, 'Theory of Group R e p r e s e n t a t i o n s ' , Dept. of Math., U n i v e r s i t y of Chicago. Newton, T.D., 1958, 'Representations of the Inhomogeneous Lor e n t z Group', AECL No. 731. Pryce, M.H.L., 1938, Proc. Roy. Soc. (London) A165. 247. Schweber, S.S., Bethe, H.A., and de Hoffman, F., 1955, 'Mesons and F i e l d s ' V o l . 1: F i e l d s , Row, Peterson and Company. Schwinger, J . , 1951, Phys. Rev. 82, 914. Shirokov, Iu.M., 1958, S o v i e t P h y s i c s JETP 6, 664, 919, 929. , 1959, S o v i e t P h y s i c s JETP 8, 703. Wigner, E.P., 1939, Annals of Math. 40, 149. I 

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