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Multiple analysis of single graviton state functions Carswell, Robert Francis 1965

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MULTIPOLE ANALYSIS OF SINGLE GRAVITON STATE FUNCTIONS by ROBERT FRANCIS CARSWELL B.Sc.(Hons), University of Otago, 1963. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1965. In presenting th i s thes i s in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree l y a va i l ab l e fo r reference and study. I fur ther agree that per-mission for extensive copying of th i s thes i s for scho la r l y purposes may be granted by the Head of my Department or by h i s representatives,, It i s understood that copying or p u b l i -cat ion of th i s thes i s for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT The statefunctions of massless p a r t i c l e s described by a tensor f i e l d are c l a s s i f i e d by expressing them i n terms of the eigenstates of the operator of angular momentum. The general tensor statefunction can also be separated into func-tions of d i f f e r e n t p a r i t y . By i d e n t i f y i n g the graviton as a special case obtained by imposing certain a u x i l i a r y condi-tio n s , f a m i l i a r from the c l a s s i c a l theory of gr a v i t a t i o n , one ar r i v e s at a multipole analysis of single graviton state-functions. Employing standard composition methods one can use these r e s u l t s to arrive at selection rules governing the decay of objects into two or more gravitons. i i i TABLE OF CONTENTS ABSTRACT i i ACKNOWLEDGMENTS i v I. INTRODUCTION AND SUMMARY 1 II. MOTIVATION FOR CONSIDERING A TENSOR FIELD A III. TENSOR SPHERICAL HARMONICS 9 IV. ORTHOGONALITY CONDITIONS AND PARITY 14 V. SELECTION RULES GOVERNING THE DECAY OF POSITRONIUM INTO TWO GRAVITONS 21 APPENDIX A 25 APPENDIX B 26 BIBLIOGRAPHY 23 ACKN OWLEDGEMENTS I should l i k e to thank Dr. F. A. Kaempffer f o r suggesting t h i s t o p i c and f o r h i s v a l u a b l e advice d u r i n g the course o f the r e s e a r c h . I am indebted to the Canadian Commonwealth S c h o l a r s h i p Committee f o r the f i n a n c i a l a s s i s t a n c e which made my stay i n Canada p o s s i b l e . 1 I. INTRODUCTION AND SUMMARY 1. Introduction Multipole analysis i s a powerful tool in electromagnetic theory as i t uses, i n general method, language fashioned a f t e r c l a s s i c a l precedent. It i s p a r t i c u l a r l y useful i n obtaining selection rules f o r the decay of atomic states into two or more photons. The photon i s considered as a special case of the solu-t i o n of the wave equation f o r a vector f i e l d plus an a u x i l -i a r y condition (Lorentz condition) which can be interpreted as a t r a n s v e r s a l i t y condition. For well known reasons one considers i n quantum mechanics generalised photon states, including l o n g i t u d i n a l and timelike p o l a r i s a t i o n s which do not occur as free p a r t i c l e s , and eliminates these a f t e r the complete c l a s s i f i c a t i o n of a l l f o u r - p o l a r i s a t i o n states has been affected. This raises the question whether one cannot proceed in a s i m i l a r fashion to tre a t the case of the g r a v i t a t i o n a l f i e l d . Here again, the c l a s s i c a l theory invokes only transverse gravitons, corresponding to p a r t i c l e s of spin two, by the im-posi t i o n of a u x i l i a r y conditions on the solution of the wave equation f o r a tensor f i e l d . The analogy with quantum electrodynamics suggests to f i r s t of a l l treat the general solutions of the wave equation f o r a tensor f i e l d , then to make the corresponding multipole 2 analysis, and then to impose the a u x i l i a r y conditions only a f t e r t h i s has been accomplished. 2. Summary In Chapter II the suggestion i s made that, in seeking a g r a v i t a t i o n a l theory, the best approach i s probably one involving the use of a rank two tensor to describe the f i e l d . This leads to the consideration of a quantum mechanical ten-sor f i e l d as a possible basis f o r a g r a v i t a t i o n a l theory. Using the l i n e a r approximation to the f i e l d equations f o r a vacuum i n the Eins t e i n theory expressions i d e n t i c a l to those describing a p a r t i c l e of zero mass and spin two are obtained. Chapter III i s devoted to obtaining the eigenstates of the angular momentum operator f o r a tensor f i e l d by combining spin one states using the Clebsch-Gordan formulae, and then using the same formulae to combine the spin and o r b i t a l angular momentum eigenstates. The functions obtained are call e d tensor spherical harmonics by an analogy with the spin one case. In Chapter IV, certain a u x i l i a r y conditions are imposed on the general state function f o r a spin s, and i n the case of spin two, the functions obtained when the spin i s p a r a l l e l to the motion of the p a r t i c l e are in accord with those of Zhirnov and Shirokov (Zhirnov and Shirokov 1957). The anal-y s i s i s carr i e d through f o r a l l cases, and the parity of each of the functions obtained i s evaluated. Using the work of Zhirnov and Shirokov (Zhirnov and Shirokov 1957), i n Chapter V a b r i e f discussion i s given of s e l e c t i o n rules governing the decay of positronium into two gravitons with, f o r comparison, the selection rules for decay into two photons. I I . MOTIVATION FOR CONSIDERING A TENSOR FIELD When considering the f i e l d seen by a f r e e l y f a l l i n g ob-server under the g r a v i t a t i o n a l a t t r a c t i o n of a mass point (or s p h e r i c a l l y uniform mass d i s t r i b u t i o n ) , the quadrupole nature of g r a v i t a t i o n becomes evident. We can examine the stresses i n a body A, with centre of mass 0, f a l l i n g f r e e l y under the influence of a massive body E , and f i n d that the forces F are i n the directions shown. Thus we may construct f i e l d l i n e s f o r the force as seen by an observer at 0 as shown, and these are seen to be those corres-ponding to a quadrupole f i e l d . To describe t h i s f i e l d , a three-dimensional tensor of second rank i s required, so we conclude that in seeking a theory of gr a v i t a t i o n the most f r u i t f u l approach i s l i k e l y to be one using a tensor theory. If one thinks i n terms of Quantum Mechanics, one can now ask the obvious question: does a quantum theory of p a r t i c l e s described by a tensor f i e l d give a tenable g r a v i t a t i o n a l theory? The answer to t h i s question i s yes, i f we place some 5 r e s t r i c t i o n s on the p a r t i c l e s allowed. Pauli and Fierz (Pauli and Fi e r z 1939) showed that the equations f o r the description of massless p a r t i c l e s of spin two, with the a u x i l i a r y condi-t i o n of Lorentz invariance, are i d e n t i c a l to those f o r the l i n e a r approximation to Einstein's theory f o r a vacuum. These p a r t i c l e s are c a l l e d gravitons. Thus we are led to considering the statefunctions of massless p a r t i c l e s describ-ed by a tensor f i e l d i n the work that follows. For an empty space, the f i e l d equations of general r e l a t i v -i t y may be written as #</ = 0 i.j £[ ' .2 .3 .4} (2.1) where T^iJ i s the R i c c i tensor, A comma here denotes p a r t i a l d i f f e r e n t i a t i o n with respect to the appropriate co-ordinate component, and the *y are the C h r i s t o f f e l symbols of the second kind; namely r*ij = i 3 k i ( 9 * i . j * 9 * j . i '9ijj) where g« j i s the metric tensor. In the l i n e a r approximation to the theory, the metric tensor i s assumed to depart l i t t l e from that of Minkowski space 9 i j * ' ) i j + - y > + (2 .3) where r).. = Q'J s d i a g ( 1 ,1,1,-1) and #y , a"J are assumed to be small quantities with continuous second p a r t i a l derivatives. Then, Using these, (2.1) becomes *k 9Kh9*L(fy.* * r u i -y^M*.- 4 > - 4,/) - 1 fa <*r&M.j +fa ~ %*8 = 0 (2.5) Selecting only those terms to f i r s t order of -}f or i t s der-i v a t i v e s gives the equation TV*.? * >kij - - - J [ i v « • *s>J - ° u - 6 ) By the assumed continuity of Xlkju. YliJK * %A,K/ (2.7) Tir^ <- Yij,«, - - fy.u) = o (2.4) This becomes These equations may now be written i n Minkowskian co-ord-inates. (2.£) becomes tjU)j "fys* s ° . ( 2 - u ) and (2.10) becomes *">ti " ^°  U,12) or, more f u l l y and so and so where (2.13) The equations (2.11), and hence (2.12), are also s a t i s -f i e d by 4>»j = * + /\>,< (2.14) where A i i s an a r b i t r a r y vector. To see t h i s , substitute <^ tj for Yij in the l e f t hand side of equation (2.11). Then = ° (2.15) by the assumed continuity of Ai'jK£ . I f the requirement that 4>ii - O (2.16) i s imposed, the gauge i s r e s t r i c t e d only by the following equation Ai,i = 0 (2.17) a s <f>ii = + z A i , i a n d ¥n The additional requirement that 4>;K,K = ° ( 2 - L 8 > implies so A l j t ( K = 0 (2.19) Then (2.11) becomes, by the symmetry of <^ tj , and the two con-d i t i o n s (2.16) and (2.Id) Thus (j>,'fc must s a t i s f y the wave equation 0<^>^0, and the supplementary conditions that ^ i * - o , and that &iK should be symmetric and have vanishing trace. The gauge i s then r e s t r i c t e d by the conditions ?d<=o and OA-O, The solu-tions 4>iK then correspond to the transverse-transverse graviton states obtained i n Chapter IV. I I I . TENSOR SPHERICAL HARMONICS The component operators of angular momentum, denoted by 3", t > T 3 s a t i s f y the commutation re l a t i o n s TJt 3/ = i^uCm « T m (3.1) where each index has the range 1 to 3, and repeated indices are summed over i n t h i s range. 6 K i n i i s the f u l l y a n t i -symmetric unit tensor of rank three in three dimensions. There exi s t (e.g. Kaempffer 1965) simultaneous eigen-states of J * 2 * J K X a n d T 3 , denoted by |j,ro) , s a t i s f y i n g = j(jH)Jj,tn> j * [ o , i , i , j Considering f i r s t the spin one case, i . e . the case where j = I , the states Ji,ro)and the operators can be repre-sented i n the following way h\ with " • ° > = l ! ) "•-•>*(?) \o 0 1/ and J " * - 2 /(3.3) These can now be transformed to a new representation, under the unitary transformation Si - UTiU* g(m) =tllvm> (3.4) Then the eigenstates of S , S3, denoted here by Jjfrn), s a t i s f y S * § M - 2%M , 5 3 | H - w | M (3.5) 1 0 In matrix form, these quantities are * M • -Ail S(«) 5*> - k \ t S ' -2 (i ? 2 ) \0 0 it to o i s x - [ q o o '3 . ( ? i g (3.6) Then %0) corresponds to right hand, jf(-0to l e f t hand c i r c u l a r p o l a r i s a t i o n , and %(o) to longitudinal p o l a r i s a t i o n (Akhiezer and B e r e s t e t s k i i 1 9 6 2 ) . The states r e s u l t i n g from the addition of two such spin . one states are obtained in the normal way, these f i n a l states having angular momentum j- Z, I or 0 . (3 .7) These can be written more conveniently as 3*3 matrices (a,j) such that so Then ( 3.d) corresponds to X*j«n) - -MtfiOSffl) • g&gpO] ( 3 . 9 ) corresponds to j=( • X'W" fei-gp>»g<»J corresponds to and s i m i l a r l y obtain *!/°> = v f fe^SjW - ^ o f j - H ) Note that the f i v e states belonging to j * 2 are symmetric with zero trace; t h e j » I t r i p l e t of states i s antisymmetric; and the j * o state i s symmetric. It i s e a s i l y v e r i f i e d that ^ These new statefunctions are isomorphic to the eigen-( 3 . 1 1 ) ( 3 . 1 2 ) ( 3 . 1 3 ) ( 3 . 1 4 ) states of the operator 53 -ii o li s3 o o o 5 j i JO -C O -( o o o o o\ i 0 0 0 - 4 0 . 0 0 0 o o o o O -< o o o t o o o - c e o o o o i o i o o o o © o o i o o o o o o o o o o o 0 e - i o o 0 0 0 0 0 ( 0 0 o O O O O O O O O J 3 , « /;/: /,/ >,l l.i 2,' W 1.1 hi 2,1 2,i S.' hi ( 3 . 1 5 ) Let <p,<\\ Sij \ r,s> be the element of Scj in the p, <j ; r,s Hi place, with the matrix S,j labelled as indicated above. Then the equation (suppressing the angular momentum) becomes t t <p,9}SjrS>Xrs(<") - m%pM (3.16) Using this notation, the orbital angular momentum can be written as <M)Lijlr,s> - - i S p r ^ ^ P i ^ ; -Pi|p;) where the symbols have their usual meanings, and the total angular momentum operator 7tj - L;j *S,j • The eigenstates of the orbital angular momentum operator are the spherical harmonics Vg^fe) > as is well known, where n i s a unit vector in the 0 , ^ direction. 'Z,mC9^ A7T J sm-Q \s,n& dej S ' " ° ( 3 > 1 7 ) At this point i t becomes convenient to introduce the concept of "tensor spherical harmonics" in complete analogy with the "vector spherical harmonics" for the spin one case. They can be defined as the eigenstates of CT,2 obtained by combining the eigenstates of L / a and Stl in the usual man-ner. These are where j i s the total angular momentum number; JL the orbital part; S € (o,i,z] the spin part; m the projection of j on the z axis; 13 and m s the projection of S on the z axis. Given J ^ S , there are Z$+l such tensor spherical harmonics, those belonging to The most general state of energy a/, spin £ w i l l there-fore be a l i n e a r combination of Zs+I tensor spherical har-monics, corresponding to the ^ s + ' possible values of It, and may therefore be written n t*J+* 1 ( 3 . 1 9 ) For l a t e r use, the normalisation condition on the tensor spherical harmonics i s written out e x p l i c i t l y * irs.sh,iMvl*-'mi"-M*-'m*)C(j,*',r'.«<-'.~,-~;~:) -J / J 7 <3-20) 1 4 IV. ORTHOGONALITY CONDITIONS AND PARITY In the case where the s p i n s - E , the most general state i s ( 4 . 1 ) For a massless p a r t i c l e Lorentz invariance requires that the angular momentum and the motion be p a r a l l e l (Wigner 1 9 5 7 ) so the t r a n s v e r s a l i t y condition I <w,3.~;*L) = o (4.2) i s imposed. The states s a t i s f y i n g t h i s condition are c a l l e d transverse-transverse states as the expectation value of the spin com-ponent i n the 9,<f d i r e c t i o n i s then ±2. The states giving t\ f o r t h i s component are c a l l e d longitudinal-transverse states; i f the value i s zero, then the state i s l o n g i t u d i n a l - l o n g i -tudinal (Appendix A). The vector r\ can be represented by i t s components nfu) defined by -/ n = T. nfr) ( 4 . 3 ) Then, i n the representation of the previously given, one finds f o r the ny*) i n terms of the cartesian components 41,- S i n 0 c o s y /y\x= Sin&£m(^ , ^yi3*CosB Mo) = in3 = cosB Hf-0 = TfC-M. + iO,) - ^ s m ^ ' f ( 4 . 4 ) Note also that 15 ^ £P(±/)£P(*) - o Z I^)Sp(o) = 1 (4 . 5 ) Thus the t r a n s v e r s a l i t y condition written out i n f u l l i s fyz L J p ? % A. =• O (4 .6) J§q(f) > M<f°^ ' jSf(-~0 a r e components of an orthogonal system of vectors so the c o e f f i c i e n t s of each must be zero. ^ - k c(j, A — ',>)%£} ox& +271«jJ,z^l»,*)y<p,r,<le-ir] ~0 (4-7) ^ r ^ / a , ^ ^ - ^ ) ^ g i s / ^ - ^ J - O ( 4 . 9 ) We have the i d e n t i t i e s (Bethe and Salpe'ter 1 9 5 7 ) ( 4 . 1 0 ) Substituting ( 4 . 1 0 ) in ( 4 . 7 ) , one obtains = O ( 4 . 1 1 ) Using (4.S) and ( 4 . 9 ) similar and consistent expressions for the CLf may be obtained. The calculation outlined below i s written out in Appendix B. 17 Substituting the values of the Clebsch-Gordan c o e f f i c i e n t s (Condon and Shortley 1935) i n ( 4 . 1 1 ) , and r e c a l l i n g that the V (1) are orthonormal functions so that the c o e f f i c i e n t s of each must be zero, we obtain for the Ci < S the following relations 1/ ar> (H ' / l - ^ . , 0 - ) V l (4.12) Hence =sf^'^j^^^^y^.j-.,-;?)] (4.13) and whereM andNare normalising factors. The y (j, I,*M;were shown i n the previous section to be orthonormal functions so, using t h i s f a c t , obtain % N^N0Xli-r^)+ ( # » • - ; * ) u - i 5 > (4.14) and l a f o r the states s a t i s f y i n g the condition (i+.2), the trans-verse-transverse states. Other conditions may be imposed to y i e l d a complete set of orthonormal functions, the lon g i t u d i n a l - l o n g i t u d i n a l and the longitudinal-transverse polarisations, corresponding respectively to the conditions tfp1r^V*|. =o and €-p<^r ^ c/^tH^t'0 From 6 p i r ty£r (w,jo*;ij) = Q , we obtain and 16) From 603 , , H A ^ 1 -0> there i s only one possible solution For the s = I case a si m i l a r analysis can be performed. The most general state here, i s as in the previous section. 19 Here the t r a n s v e r s e s t a t e s are c h a r a c t e r i s e d by the equation .€p,r np}Jv- 0 , and the l o n g i t u d i n a l by n^^O, as y («,J>;a) i s a skew-symmetric t e n s o r . The tr a n s v e r s e modes are %(<->,•),*• ;*f - (^)y^,r^;y) + Iffif^wm;*) (4.22) and the l o n g i t u d i n a l In the case 5 = 0 , the most g e n e r a l s t a t e i s the o n l y s t a t e 3/P 1( f eM'4 , , ' ,s) = (4.24) The p a r i t y o p e r a t o r TT, when a c t i n g on a v e c t o r J£ g i v e s ff£--£and f o r a s p h e r i c a l harmonic -W'^S* (4.25) So TTXS(^J) - %'(**,) (4.26) Using these, the p a r i t y o f each o f the s t a t e s o b t a i n e d above may be i n v e s t i g a t e d , remembering t h a t Then 20 and - C-')1 y^C^j.*.;^) (4.27) Thus the statefunctions obtained f o r each t r a n s v e r s a l i t y condition are separated into states of d i f f e r i n g p a r i t y . These re s u l t s are summarised f o r ready reference i n the tabular form below, where the Cl^ i s the c o e f f i c i e n t of the tensor spherical harmonic i n t n e l i n e a r combi-nation Z) _ as i n (3.19), with the a u x i l i a r y £ r l condition shown. Spin ^uxi'iary Concb'Hon °5 J - ' p 2 • ° 0 0 O O 0 (-0j O UJ-M/ O O (-/' O 0 O 0 1 0 O \ 0 0 O 0 O 0 0 -US 0 0 — 0 0 / 0 0 21 V. SELECTION RULES GOVERNING THE DECAY OF POSITRONIUM INTO  TWO GRAVITONS "Since one graviton i s described by a tensor of second rank, the two graviton state can be represented by a tensor of fourth rank with J3 , r e f e r r i n g to the r e l a t i v e momentum of the two par-t i c l e s , the only variable i n the centre of mass system. Angular momentum eigenfunctions f o r t h i s case are obtained by the use of the Clebsch-Gordan formula with the ^ i < f ( * Vobtained i n Chapter I I I . These are %lmnhL.«fi) - ZdxL.s.wriX^xlc (5-1) where KtJ ' I C ( S , » , 2 / / ^ , ) X > 3 ( > (5.2) Here X i s the t o t a l angular momentum of the two gravitons; L the o r b i t a l angular momentum; M the projection of "J on the z axis; 5 the spin of the two-graviton system; and the z component of S« The general state with angular momentum T, z component M, i s then Y«,„M") - £ e ^ % , M ( * . l - . « . S ) (5.3) with L J± I) 3± 2 and 5 » 0, \, 2,3, If. Using the orthogonality requirement that n*t,itm„G.")mO (5-4) 2 2 f o r a massless system, Zhirnov and Shirokov (Zhirnov and Shirokov 1 9 5 7 ) determined the c o e f f i c i e n t s p u i , and a r r i v -ed at the following table giving the number of possible states of the system f o r a given p a r i t y and given value of J . Even States odd o 1 1 I 0 0 2 1 1 3 0 0 2 1 2*+l >S 1 0 This may be compared with the similar table f o r the pos-s i b l e two-photon states (Kaempffer 1 9 6 5 ) . 7 even Shxf-es od<f Shahs 0 1 1 t 0 0 2 2 I 3 1 0 Z / 2nH>$ 1 0 Hence we may construct a combined table giving selec-t i o n rules f o r the decay of positronium into two gravitons or two photons, under the conservation of angular momentum and of p a r i t y . Selection rules with respect to charge conjugality C a r e given also, noting that C * f o r m photon decay, and +1 f o r r> gravitons. This i s because under charge conjugation the e l e c t r i c current density j i s transformed to - -j , whereas mass remains invariant. Hence a single photon source changes sign underC, where the single graviton source g p V does not. The general res u l t follows by induc-t i o n . Poiirronium SraYe J Pccoy gravirons | by coneen P . T . T , C photons /atton op p , x T , e '5. o - i +/ Allowed Allowed Allowed Allowed 3 P. o + 1 Allouieol A/lowed Allowed Allowed 'P, + 1 -I Forbidden Forbidden forbidden Forbidden / -1 Forbidden Forbidden! forbidden Forbidden J 4 1 + 1 Forbidden Allowed 1 Forbidden Allowed '0, 2 f 1 + 1 Allowed Allowed I Allowed Allowed 2 +1 Allowed Allowed Allowed Allowed 2 + 1 - J /Wowed" forbidden Alhvoed forbidden % 3 -1 -I Forbidden Forbidden Forbidden Forbidden 3 + J -1 Forbidden Forbidden A/lowed forbidden % 3 -1 -r l Forbidden Allowed Forbidden Allowed For 3">4the selection rules are i d e n t i c a l for decay into two photons or two gravitons. From the table i t i s evident that the f i r s t s i g n i f i c a n t difference between decay into two photons and two gravitons occurs f o r the ^ D3 and ^Q^ states of positronium. I f angular momentum, parity, and charge conjugation are conserved, how-ever, then any state which can decay into two gravitons can decay into two photons. A test of C for other positronium states requires decay into an odd number of gravitons or photons, as then the charge conjugation number d i f f e r s i n the two cases. At present nobody knows how to make positronium decay into two photons from any but the ground state, so experimen-t a l v e r i f i c a t i o n remains beyond reach. 25 Appendix A. GRAVITON SPIN COMPONENTS IN THE DIRECTION OF MOTION The equation o f motion f o r a p a r t i c l e may be w r i t t e n as (P.S) 4> - m£-<f> ( A D where 5 i s the s p i n o p e r a t o r ; P the momentum o f the p a r t i c l e ; E i t s energy; and m the component o f s p i n i n the P. d i r e c t i o n . I f <$> i s to be con s i d e r e d as a te n s o r , t h i s may be r e w r i t t e n as k £ i j « < p 1 l s y i r s > - mE<)>P1 (A2) u s i n g the n o t a t i o n o f Chapter I I I . But Hence (A2) becomes IPiti-rrhs + i P i ^ S + r ^ * - " l E + r s ( A 4 ) I t e r a t i n g t h i s equation y i e l d s I f the c o n d i t i o n s Pid>ij - O , are now imposed, corresponding to ( 4 . 2 ) , (A5) reduces to 4ptf>; - 2<$>r$ U6) For a massless p a r t i c l e - p* r f l , thus we have the requirement t h a t Im{=2.. Hence the component o f s p i n i n the d i r e c t i o n o f motion must be ±2. 26 Appendix B. RELATIONS BETWEEN THE COEFFICIENTS a» FOR THE TRANSVERSE-TRANSVERSE STATES In accordance with the plan set forth on page 17, we write out the c o e f f i c i e n t of Y i n (4.11). Then ^ ^ ^ f e f e f =o (B1) Rearranging and substituting i n the values of the Clebsch-Gordan c o e f f i c i e n t s >2 (B2) On cancelling common factors obtain (B3) 27 (B4) • • a j i 7 7 * ^ 0 / <V <• J+z) The other equations (4.12) may be obtained by considering the coefficients of Y and / fe) to give respectively '4. -BIBLIOGRAPHY Akhiezer, A.I., and Berestetskii, V.B.? 1962, "Elements of Quantum Electrodynamics", Oldbourne Press. Bethe, H.A., and Salpeter, E.E., 1957, Handbuch Der Physik XXXV, Ed. S.Flugge, Springer-Verlag. Condon, E.U., and Shortley, G.H., 1935, "The Theory of Atomic Spectra", Cambridge. Kaempffer, F.A., 1965, "Concepts in Quantum Mechanics", Academic Press. Pauli, W., and Fierz, M., 1939, Proc. Roy. Soc. (London) A173. 211. Wigner, E.P., 1957, Rev. Mod. Phys. 2£, 255. Zhirnov, V.A., and Shirokov, Iu.M., 1957, Soviet Physics JETP 2 , 840. 

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