REFERENCEIC/67/50

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

CLEBSCH - GORDAN COEFFICIENTSFOR THE LORENTZ GROUP - I

PRINCIPAL SERIES

R. L. ANDERSON

R. RACZKA

M. A. RASHIDAND

P. WINTERNITZ

1967PIAZZA OBERDAN

TRIESTE

10/67/50

CENTRE FOR THEORETICAL PHYSICS

TRIESTE

CLEBSCH-GOHDAET COEFFICIENTS FOR TEE LOHESTTZ GROUP

I - PRINCIPAL SERIES

R.L.R.

M.A

AndersonRaczka

, Rashid

and

P e Winternitz

ERRATA

Several lines were unfortunately left out when formulae were

written.

Please replaoe formula (28) by:

5« A,-rtA,ITU>

ddf

-1-

A *

(28)

Please replaoe formula (35)

r^,

The CleTjaoh-Oordan ooeffioient in (37) is

The 9-J symbol in (40) should reads

k - Tt 2. i

and the two subsequent formulae are;

- {

and

IC/67/5O

INTERNATIONAL ATOMIC ENERGX AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CLEBSCH-GOKDAKT COEFFICIENTS FOR THE LORENTZ GROUP

I - PRINCIPAL SERIES

R . L . Ander son *

R. Raczka **

M.A. Rashid ***

and

P. Winternitz *** #

TRIESTE

July 1967

* National Soience Foundation Post-doctoral Fellow,

** On leave of absence from the Insti tute of Buclear

Researoh, Warsaw, Poland.

*** On leave of ataenoe from the Pakistan Atonic Energy

Coniaission, Lahore, Pakistan.

**** On leave of absence from Joint Institute of Nuclear

Research, Dubna, U.S.S.R.jand from Institute of

Nuclear Research, Prague, Czechoslovakia.

ABSTRACT

Explicit expression for the Ciebach-Gordajra coefficient for

the -unitary representations of the principal series for the SL(2,C}

group is obtained. Their orthogonality and. symmetry properties

are disoussed. Special oases of physical interest are dealt with

in detail.

-1-

QLEBSCH-GOEDAI COEFFICIENTS FOS THE LOBEHTZ GROUP

I -PRINCIPAL SERI3S

IHTEODUGOJIOir

The knowledge of the SL(2,C) Clebsch-Gor&an coefficients is of

fundamental importance in such areas of elementary particle physics

as "the analysis of angular distributions in relativistic scatteringl) 2)

theory , the theory of relativistic wave equations J and the problem

of the construction of invariant vertex functions for finite or in-

finite partiole multiplets .

The problem of reducing the 'tensor products of two irreducible

unitary representations has been completely solved by ETADIARK ' in a

series of papers.' However, the calculation of the explicit form

of the Clebsoh-Gordan coefficients has "been carried out only in

oertain special oases by DOLGI2TOV et al.^\ BISIACCHI and FRONSDAL6)

and BA1OERG7'.

We present here the solution to the problem of the C.G.

coefficients for arbitrary unitary representations of the principal

series. The oontents of the paper are given in the table of

contents.

CONTENTS

INTRODUCTION

1 . INTEGRAL REPRESENTATION FOR THE CLE23CH-G0RDAN COEFFICIENT

IN TERMS OF SL(2,C) TRANSFORMATION MATRICES.'

Si Derivation.

Orthogonality and completeness relations.

Explioit expression for-the SL(2,C) transformation matrices.

2 . THE CLEBSCH-GORDAN COEFFICIENT.

General express ion .

£ 2 S impl i f ica t ion when V , V-i » % s a t i s f y a t r i a n g u l a r r e l a t i o n ,

3 . CLEBSCH-GORDAtf COEFFICIENT FOR THE COUPLING OF DEGENERATE

REPRESENTATIONS.

4. CALCULATING NUMBERS.

5. RECURSION FORMULAE AND ANALYTIC CONTINUATION.

6. SYMMETRY PROPERTIES.

7. CONCLUSIONS.

-3-

1. INTEGRAL REPRESENTATION FOR TEE CLEBSCH-GORDM

COEFFICIENT Iff TEEKS OP SL(2,C) TRANSFORMATION MATRICES

Derivation

In the case of a finite or compaot group G, knowing the

matrix elements D (g) of an irreduoible unitary representation D"\

we oan construct the operator• a \

(n fixed)

T. (1)

which has the property

ZI (2)

Above M" = d/V\ where d = dimension of D and

h m "volmce" of Gt A denotes the set of invariant numbers which

characterize the given irreducible representation and m(n) represents

•the set of non-invariant numbers which determine the basis vectors

j A ; m]]> , dy.(g) is an invariant measure on the group space and

T->, is a -unitary, generally reducible, representation in some

Hilbert space H. Owing to the property (2) the set of unnormalized

veotors

.x ,AX 6 H n fixed (3)

spans the carrier space H of the irreducible unitary represent-

ation D A .

In the derivation of eq. (2) we have only used the

invarianoe of the measure dju-(g) on Q and the group properties

of the matrices JD (g) . The above construction of the basis

veotorfj (3) can therefore be extended also for every locally

oompaot group for which the invariant measure dul g) and the

matrices J 2)(g)i are known.

In the case of the Lorentz group, the explicit form of

the matrices {l>V^(g)] has been calculated recently "by various

authors"'. Thus we can construct explicitly the operators (l) and

the basis vectors (3) and utilize them for obtaining the Clebsch-

Oordan coefficients.

In fact, let us oonsider the tensor product of two

irreducible representations D ' ' 1 and D *•**- (V. integers or half-

integers - »<&£ n. - oo, i = 1»2) which are realized in Hilbert

spaoes H <1"i , H^^1 , and spanned Taj the seti of oanonical basis

veotors^Vj ^ ^ Mi yl and jjv2 j?2 J2 M ? / ^ respectively.

Due to ITAIMAHK's r e s u l t s we a l ready know tha t the decomposition

of the t enso r product B " ! J I ( ^ ) D i J t has the fol lowing form:

CO

if

where the summation extends over all V s suoh that

v + v- + Vp » integer (5)

In the t enso r product spaoe H » H ^ ' ' © H 1 2 ve can

oonstruot two s e t s of orthogonal b a s i s v e c t o r s . The f i r s t one

oons i s t s of the Kroneoker product of the o r i g i n a l b a s i s veotors

(6)

while the second one contains the basis veotors

v y J M J • v 1 f ^ j>4 > ( 7 )

- 5 -

which span an irreducible space H J5 contained in the direct integral

of the Hilbert spaces corresponding to the decomposition (4).

In what follows we shall omit indices v± P^^fz. in "fc3ie

ket vector (7).

The normalization property of the basis vectors (6) and

(7) are

Mlf

Wand

O ? JM 1 V'?'JV> - -^—4 ^ ^ Wv (9)'W r r

In ITaimark• s notations, the veotor v p J M^> corresponds

to the function fN^( E ) "belonging to the Eiltert space H(C) of

functions f(2 ) with the domain C being the complex plane (Ref. 8)

p. 162).

The connection between the basis vectors (6) and (7) oan be

aohieved with the help of the operators (l). We have

JM> -

(10)

In this formula for the state %f^ \^-^I \ vr &.J2_

we may take any veotor (6) from the tensor produot space

H 1"1 @> H r ' z . IT i s some normalization coefficient . The operator

TQ aots on the basis veotor (6) as

T ft

(11)

Using invariance of the measure dju(g) and group properties of thematrices D " (g) we can easily check that the basis vectors (10)have oorreotttitsnsforiaatiQn proper t ies .

The Clebsch-Gor&an coefficients are the matrix elements

of the t rans i t ion matrix between the basis vectors (6) and (7)#

Ut i l iz ing (10) and ( l l ) we get

M

(12)

In order to find the normalization coefficient

we oaloulate the square of a basis veotor

(10). From the group properties of the matrioes Dvf* (g) and the

normalization (9.) we find that

-7-

(13)

Note that the normalization (S) i s chosen so as to canoel the faotor

AV/> - • — — T (U)

4f

•which appears in the integration defining normalization of the

V^ (g) functions.JM , J ' M1

The group element g oan "be represented in the form

g- u , ( > t ^ ; O ) g(a) Wt (?, ,6,fa ) (15)

where C1 and U^ "belong to SU(2) and g(a) is an element of the one-

parameter non-oompaot subgroup of SL(2,C). The decomposition (15)

induces the following for the D ^(g) matrices:

v ? , t min£J,J')

(16)

Referred to the parametrization (15) the invariant measure on

SL(2,C) has the form

-8-

- d(oos if/ ) dY[ d j , d ( o o s G ) d £ s inh 2 a d a ( l ? )

Using formulas (12), (13), (16) and (17), carrying out the elementary

integrations over the variables V» "'7 , V' , ^ , 9_,s we arrive at

\ V 1?1 J 1 M 1 J K J>

32TT

(2J + 1)

x f L- Vf (a) ]>V^ (a) L^ft. (a) sinh2 a d

(18)

The last integration will be carried out in Seo. 2

Orthogonality and completeness relations

The Clebsch-Gordan coefficients as defined hy (12) figure

in the decompoeition

cO

(19)

•where the oanonical "bases are normalized according to (8) and (9).

Calculating the norms of "both sides in (19) we obtain

the orthgonality relation

(20)

Using (20) it is easy to oheck that the formula inverse

to (21) is

>*

(21)

Combining (19) and (21), we obtain

JM

' M' >

which is completeness.

-10-

A3) Explicit expression for the SL(2,C) transformation matrioes

In the following we shall need the explicit form of the

transformation matrioes D ?(a). For our purpose the most convenient9d")

one is essentially that of DUC and HIBU ' except that we haveo \

adopted the canonical basis as defined "by 1TAIMAEK ^

(23)

where

O

and

v f _ _

The ohoice of the phase CX T is such that the veotors

7 j> J H y form a oanonical basis in the sense of EAJMAEK ' i.e.,

the operation of all generators on jv o J M > is completely prescribed,

-11-

2 . THE CLEBSCH-GOEDM COEFFICIENT

§1) General expression^

In th i s section we carry out the integrat ion appearing

in the formula (18) for the C.G. coeff ic ients . Substi tuting (23)—2a

in (18) and introducing a new variable x = 1 - e we reduce the

integral in (l8) to a sum of terms of the type

,_4 d

j X) x-

To perform this integration, we develop each of the

hypergeonetrio functions into a power series. These series converge

uniformly for x < "1 and we can perform the integration term by

term from 0 to 1 - £ and take the limit € 0 . Sinoe this limit

exists, Abel's theorem on the value of a function on the boundary

of its region of convergence ensures that the obtained value is the

correct answer. Performing the integration (see e.g. Eef, 13), p. 284)

we obtain

, M,, TJ.M,. I JM >< a/*/ , Tj K I r W

I. OIV^

r'- »fc),^

(25)

'The sums with respeot to A's, 4fs and d'a are finite,

their limits being

t a n

min(J± f J^) i - 1,2

(0, - X - v) '<$ d « min(J -A , J - v) e t c

The sums over n , n. , n2 are from 0 to <=o and are convergent.

We have used the notation

( a ) -v 'nP(a+n)

Let us remark that the triple infinite sumn n, rip

figuring in (25) and all subsequent expressions can be re- .

arranged in such a way.that two of the sums can be expressed as

generalized hypergeometric functions of unit argument. The

Clebsoh-Gordan coefficient oan thus be expressed using a single

infinite and many finite sums over a product of P functions,

a terminating F, function and an J&V, function. Sinoe this does

not simplify oaloulations, we shall not give the explicit

-13-

expression here.

The number of fini-fce sums in formula (25) can be

reduced to at most four by a suitable choice of the superfluous

parameters J! , Mf. Using the symmetry relations (discussed in

Sec. 6) we oan always arrange the coefficient to be such that

Choosing

(26)

Jl " V l ' J2 " V2

we eliminate the sums over d' and d' and can perform the d- , doL t d x c.

summations with the help of

Thus ve obtain

-14-

0+:/

J

. fi.,:. ,

where

(T-V)l (J^M!(v+3v)J(y-^» (29)

pand B -was defined in eq. (24).

JJ» X dd»

In order to fix the C.G. coefficient completely it is

still necessary to ohoose the value of J' and the phase^for.

instance lay taking (see eq.(26))

J' - va - v 2 (26-0

and defining the coefficient

1 P i V l V l •• V 2 f 2 V2 ^ V2)| V f (Vl -

to be real and pos i t ive .

- 1 5 -

2) Simplification when v , v^ , "tf 2 satisfy a triangular

relation

If the invariant parameters "v , -v. , Vg satisfy the

relation

Vl "

we oan put (of. 26'•)

J' - v (26'»)

in eq,, (28). Thus achieving a farther simplification:

2.

(30)

The d , d' summations have completely disappeared

-16-

3. CLEBSCE-GOBDAN COEFFICIENT FOR THE COUPLING OF

REPRESENTATIONS

If we oonsider the case V -

reduces to

0, eq., (30)

J,, M1 , 0 p z J^ M 0 J3 J M

*

\ I?-

f ( H '

p , 0 j ^ 00 0 j) 0 0 )

re-f ( j - iPi/fc+ WQ v!., ;

In this case the "normalization Clehsoh-Gordan

coefficient" is specially simple and can "be oaloulated explicitly.

In fact, from eq. (31) "by manipulating the triple sum

f, oo 9 ofj,oo j o f

(32)

Inserting (32) into (3l) we obtain

- 1 7 -

In the evaluation (32) we have assumed positive phase

for the ooefficient.

From the presenoe of <(^ 0 , • Jg 0 j J o)> in the above

formula (33) i t is olear that, the only non-zero Clebsch-Gordan

ooeffioients are the ones with

J- + J« + J •» even integer

in this special oase.

So far we have made only a partial comparison of (33)

with the results of DOLGIUOV^'and DOMOKOS K This comparison

shows that our expressions (31) - (33) agree with their results.

-18-

4. CALCULATING LUMBERS

For ooncrete calculations of specifio Clebsoh-Cordan

coefficients it may prove advantageous to use a slightly different

approaoh.

So far we have made use of D ? matrices for the SL(2,C)

group expressed in terms of hypergeoir.etric functions. However,

they can also be expressed in terms of•elementary functions. Indeed,

using an integral representation for the hypergeometric function,

we have

X

** dd'

\ e •* /

^ . . — — — — — ,

Putting x » w ^ — and expanding the positive integer powers

of (l - e x ) and ( x - ea) into (finite) binomial series, we

obtain (this is a slight modification of the results in Ref. 9c))

)

(34)

where

- (-0 '

-19-

Using this D-matrix to calculate the Clebsch-Gor&an

coefficient, we get immediately

M2 I V f J M > < Vi ?1 *1 ' ^2 ?z J2 H |V f ™2 I V f J M > < Vi ?1 *1 ' ^2 ?z J2 H2

j J M >

Z <3>

(35)

The integrals in (35) are, in general, divergent and in

ezplioit calculations, they must be combined in such a way that the

divergences cancel.

Using (35) it is possible to calculate several simplest

Clebsch-Gordan coefficients for eaoh set of invariant parameters

v and p . Afterwards, using the obtained values and the recursion

relations of Sec. 5 it will be possible to calculate arbitrary

Clebsch-Gordan coefficients.

Let us demonstrate the application by two examples;

a) < Of, oo, *U°<>I Of06}

(36)

agreeing vi"ch (32) obtained previously by a different method.

b) A less trivial ClebGoh-Gordan coefficient which, can be computed

performing tedious ,though elementary, calculations is

4 * ,

- i joo >

or

R 4 4 t 4 R, 4 - 4 | 4 j > o o>

. f {H-p,l3f i+tf

(37)

In both cases we have ohosen the CW3 coefficients to be real

and positive.

Incidentally, the coupling of representation V = •§- , 0 = 0

is of special interest, since this and the representation \> = 0 ,

y = i/2 belonging to the supplementary series, are the only unitary

irreduoible repreaentations of the SL(2,C) group, the basis functions8

of which satisfy invariant wave equations J (these are the Majoranarepresentations),

-21-

5. RECURSION FORMULAE A M ANALYTIC COHTIMJATIOtf

Recursion relations for the SL(2,C) Clebsch-C-ordan

coaffioients can "be obtained from the knowledge of the action of

the generators H. and F. ^ on an arbitrary canonical basis

funotion. The procedure is to apply H. and F. separately to

eq. (21), expand the l.h.s. of the result by using ,(2l) again, and

then equate the coefficients of the orthonormal basis functions

| v £>„ J, M J v 2 p Jo p / * There its an important well-known

simplification which occurs, namely that the application of the

generators H. of the oompact subgroup SU(2) leads to recursion

relation*whose solution up to a multiplicative factor is the SU(2)

Clebeoh-Gordan coefficient. With the usual choice of phases for

the SU(2) Clebsch-Gordan coefficients we can therefore • faotorize

the SL(2,C) coefficients as follows:

X ^PiJ1 '^2 Pz J2

(33)

Applying the above procedure, we arrive at the following

recursion relations for X (v J-, f -tfp p J£ t V P J ) :

-22-

&o<!CD

.o r>-'r

•3fi v - ^

/v?

r.

r<

*4

M

TT -H

7*

H

i s .

A.1

r-

J4

— >H

•f

»•

V

V

t* •

CJ.

M

c

-r

zl

A .

oo

4M

oHj

oH

'H(aCD

" • *

-4-

73"

M

*zf-

V

- 2 3 -

73-

MJX

V

The above are a system of three recursion relations

satisfied "by the X functions. In order to use them as adjuncts

to the analytio approach, we note that we can eliminate the X*

functions with arguments containing J ' + 1 and CP ' + 1. This

results in a recursion relation which allows us to increase the

value of the argument J. Similarly we can obtain two other

recursion relations whioh enable increases in J ' or J*1 '. Every

time the relation involves at most 5 terms. Suppressing the

arguments v. p . , one of these looks like

J+l) + 0( 2X*(J 1J 2J) + 0(3 X ^ - l , J2, J)

+ K 4 X*(j, , J2-l , J) + tX5 X*(j^ , J2 , j-l) -.0

For the oases where v , v^ , v 2 form a triangle (this includes the

degenerate case), we evidently require only one starting value

namely the C.G. coefficient with J, => v,, J« = v 2, J • V,to generate

all the other C.G. coefficients. Thus the recursion relations do^

in these caseSj define the problem up to a "normalization coefficient"

which represents a special C.G. coefficient. This starting C.G.

coefficient can be chosen to be real and positive.

However, we can verify that, as expected, in other

cases also we require just one "normalization coefficient" to fix

all for fixed v , "V,, v 2 . Of course there is some arbitrariness

here in its choice and no canonical choice seems to be evident

exoept for two of the J values selected as the minimum. This

cannot be done in an arbitrary fashion. Indeed if

then ve can take the normalization coefficient to be the one

corresponding to

-24-

' 1 . 7 1

o r Vl » J

The above difficulty i-ras reflected in our inability

to do two of the d summations in the formula (28) for the C.G.

coefficient whenever v , v.. , v ? did not form a triangle.

H"ow "we discuss the important question of whether the

solution to our problem can be obtained "by "analytically continuing"

the corresponding C.G. coefficient for the finite-dimensional

(non—unitary) representations. Indeed it is well known that for

the finite-dimensional cases v,

fc + «

( 2 J , + l ) ( f

(-1) Z ((2J1

J M

n

(40)

•where | r is a 9-J symbol. Here the conventions in the book15)

by De-SEALIT ' have been employed. The connection between v , Pand ^ , n is

fe-n

ioj » ( + n) +1

-25-

The recursion relations given above are indeed "analytic

continuation" of the ones for the finita-dimensional case. They can

be obtained by replacing the invariant numbers P< , n characterizing

the finite-dimonsional irreducible representations by

z 12 -2-- ~z i® i

respectively. Thus we can define a new 9J-syffibol with complex

parameters whioh will rigorously satisfy the recursion relations.

One can, for example, express it in terms of analytically continued

Racah coefficients with complex parameters

Y

(41)

In the above equation, a's and b's are complex while'J's

are non-negative integers or half-integers. Thus ~Y summation is

indeed finite ', In terms of this 9-J symbol we oan easily express

the C.G. coefficient up to a normalization factor. This factor,

as emphasized before,, is a special C.G. coefficient and oan be

computed from the formulas in (28) or (35).

-26-

6. SYMMETRY PROPERTIES

Formula (18) determines the product of two Clebsoh-Gordan

coefficients. An arbitrary Clebsch-Gordan coefficient can "be obxained

"by specifying the values of the redundant parameters X'l-I etc. in a

convenient manner, calculating the square modulus of this "normalization

coefficient" from (18) and defining the overall phase for each set of

invariant parameters "V p.. , v ? P ? , v p . Dividing (18) by this

chosen coefficient we obtain a formula for the general coefficient

< Y. f, J1 M-, , Y2 pz J2 Mg | V p J M y , So far, except for the

cases like v.= v. =» v 2 a 0 » V, =• v^ = %- , v = 0 we have not been

able to reduce, in general, the normalization coefficient to a single

term. Thus we cannot establish the phase factors in the symmetry

relations and we shall only consider the relations between the moduli

of various Clebsch-Gordan coefficients.

The simplest symmetry relations of these coefficients

can be obtained directly from the properties of the D ' matrioes

(eq.3. (2 24))» Up to phase factors connected with the quantities

Ql Jwhioh are not important for the present purposes, these relations

are given in Ref.

It is easy to establish the following:

v f J M>

J-B ,

2 ?2 J^'l-V-fJ

2)^f J K>

(42)

-27-

Thase are of oourse "the dost o~bvious independent

relations. Applying some of than .repeatedly, we can also obtain

relations of the type

V2 ?2

-V -

It is clear that we have only found the simplest symmetry

relations and that the symmetry properties of 8L(2,C) Clebsch-Gordan

coefficients will ba muoh richer. Symmetries similar to those dis-

oovered by HEGGE ; for the SU"(2) group should also exist here, but

so far no attempt to disoover them has been made.

7. CONCLUSIONS

We have oonsidered the problem of the Clebsoh-Gordan

coefficients for the SL(2tC) group and have obtained general

expression for these coefficients for the principal series re-

presentations. The merit of our derivation is its rigour. We have

been able to achieve it from the knowledge of the transformation

matrices. Previous attempts by Dolginov et_al. for calculating

speoial oases were based on the use of recursion relations and the

observation that Pano functions satisfied the same recursion relations.

Effectively this was an analytic continuation approach. We have also

arrived at ooncrete results on analytio continuation from the finite-

dimensional case .

Our methods can "be applied for the calculation ofClebsch-

Gordan coefficients in all cases where multiplicity does not appear.

In particular, one might apply them for the couplings between principal

and supplementary, supplementary and supplementary, or unitary and

finite-dimensional non-unitary representations. This programme will

be the subjeot of future publications in which we shall also return

in detail to the problems of analytic continuation.

-28-

ACKNOWLEDGMENTS

We are indebted to Professors Abdus Salam and P. Budini

and to the IAEA for hospitality extended to us at the International

Centre for Theoretical Physios, Trieste. We are grateful to Professor

Abdus Salain also for suggesting the problem and for continued

encouragement. We are grateful /or very profitable discussions with

Dr. John Strathdee.

-29-

H3EEHE1TCES

1 ) a) M. TOLLER, C3RN preprint Th-750 (1967). This

oontains an extensive l i s t of references and

summarizes author's "work on the subject.

b) J . F . BOYCE, R. DELBOURGO, AHDUS SALAM and J . STRATHBEE,

ICTP, Tr ies te , preprint IC/61/9 (1967).

o) 1ST. Ya. VILEflKnr and Ta. A, SMOROXCTSKT, Zb. Sksperim.

i Teor. F i z . 35,, 794 (1958) (Soviet Phys., - JBTP

1£, 1209 (1964)).

2) See e . g . , C . FROffSDAL, UCLA, prepr in t -Oct . 1966.

3) See e . g . , C. FEOHSDAL and R. WHITS, Phys. Sev. 1^1, 1287 (1966).

4) M.A. HAIMAEK, Trudy Moakov. Mat. Obsc. _8, 121 (1959);

- % 237 (I960)} JLO, 181 (1961)5 English t r ans l a t i on :

Amer. Math. Soo. Trans l , Series 2, Vol. 36. (1964)

pages 101-229.

5) A.Z. DOLGIJOV and IJST. TOPTYGIIT, Zh. Eksperim. i Teor. ? i a .

35., 794 (1958); Soviet Phys. - JETP Bt 550 (1959);

Ibid, Zh. Ekeperim. i Teor. F i z . _3_7 1441 (l959)j Soviet Phys.

- CG3TF 10, 1022 (l96O)j

A.Z. DOLGINOV and AJJ. MOSKALEV, Zh. Eksperim. i Theor. F i z .

3X, 1697 (1959); Soviet Phys. - JEPT 10, 1202 ( I960) .

6) G. BISIACCEI and C. HL02JSDAL, Uuovo Cimento 41, 35 (1965).

7) P.G. BA1BEEG, Clarendon Lab ., Oxford, preprint 196 (1966).

8) M.A. HAIMAEK, "Linear Representations of the Lorentz group",

Pergamon Press , London (1964).

9) a) LJD. E-SKIU, I zv . Vysshikh Uohebnykh Zavedeni, Matematika

§j (25) , 179-184 (1961).

b) ' S. STROM, Arkiv fur Fysik 2£, 467 (1965).

o) A. SCIARRIKO and M. TOLLER, Hota In t ema no . 108

Universi ta d i Roma (1966).

d) DAO WAFG DUC and UGTHW VAU HIEU, Lubna preprint - JIITR

P-2777.

-30-

Ibid: Annals Ins t i tu t Henri Poincare S_, IT (19°7).

e) I.A. VEPJ3IEV, L.A. DADASHEV, Ins t i tu te cf Theoretical

and Experimental Physics, preprint no. 476 (i960).

10) See ;e.g. ;K. KJTOPP, "Theory and application of infini te

series(t2nd English edition, Blackio, London (1564).

11) I . S . GRADSHTEYU and I J i . HTZIIIK, "Tables of in-ce.graio,

series and products", Academic Press, JTew York.

12) G. DOKOKOS, University of California, Berkeley, preprint l /o? .

13) H. and P. are the generators of the SL(2,C) group. Theirmatrix elements are expl ici t ly knovn. v;o are u t i l i z ing

A. and C. as given "oy 1TABUHX (Eef. S) above).J <3

14) R. DSLBOUHGO, J . STRATHD33 and AHDUS SALAE, ICTP, TTieste

prepr int IQ/SI/21, sea also Ref. .7).

This expression i s equivalent to the one given in

JJSI. GSL'PASD, E.A.KI1IL0S and Z.Ya. SHAPIRO, "Bapresanta-Sio-.-.s

of the rotation and Lorenxg ;_rroupg and their ar;plic:,'c.icns"T

Pergamon Press, Oxford (1963).

15) A. de-SEALIT and I . TALKI, "Iluclear Shell Theory", Academic

Press, Hew York (1963).

16) 9—J symbol with complex parameters i s knotm as ?ar.o functions.

Their explici t expression can "be seen in H. KATSUiTOBU and

H. TAKE3E, Progr. Theoret. Phys ..(Kyoxo) 1A, 1^39 (1955).

17) T. EEGGE, JTuovo Cimento _1£, 545 (1956).

- 3 1 -

Available from the Office of the Scientific Information and Documentation Officer,

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