IC/67/72

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

QUASIPOTENTIAL APPROACHAND THE EXPANSION

IN RELATIVISTIC SPHERICAL FUNCTIONS

V. G. KADYSHEVSKY

R. M. MIR-KASIMOVAND

N. B. SKACHKOV

1967PIAZZA OBERDAN

TRIESTE

IC/67/72

lJiLL ATOI-HC 2E~]RGY AG3STCT

CSKTR3 FOR THEOEHPIC-L PHYSICS

QUASIPOTELTTIAL APPROACH A1TL TH3 3XPA2TSICCT

III RELATIVISTIC SPHERICAL PUirCTIOirs1"

V.G

H.LT. Kir-ICasim

N.3. Slcachkov***

October 1967

; o "1'ruovo Cimento"

* On leave of absence frca Joint Inotituto forNuclear Research, Du"ona, 11331";.

*" Joint Inst i tute for 1'uclear Research, Dubiia, U3S?..

Saratov State University, Saratov, Uoo?;,

ABSTRACT

The expansion in matrix elements of unitary

representations of the Lorents groups is applied in the

relativistic quasipotential scattering theory, A relativistic

analogue of the three-dimensional configuration space is

found. The formalism developed is very close to the non-

relativistic one* It allows us to hope that in the frame-

work of this approach, an adequate relativistio generalis-

ation of Regge's ideas can "be found.

QUASlPOTEiiTIAL APPROACH AtfD THE EXPASTSIOE

IN RELATIVISTIC SPHERICAL

1, INTRODUCTION

In the last ton years decompositions of relativistic amplitudes

and wave functions with respect to complete systems of functions connected

with unitary infinite-dimensaotial representations of the Lorsnts group

have often been used /l~137» '̂Ho purpose of the present paper is to

apply this type of expansions in the framework of the quasipotontial

approach (QPA) that was developed several years ago "by Lo.'-nov a d

Let us remember t ha t the bas ic idea of the CJ?A s t a r t i n g from

the four-dimensional formalism of quantum f i e l d theory and us ing a l l the

information about the analytical properties of the rel&tivistic scattering

amplitude, to introduce a three-dimensional description of the relativistic

two-particle system in terms of Schrodinger and Lippman—Schwinger type

equations**^ :

*̂ Analogous investigations have been undertaken earlier /~L5,l6j,A part of the results obtained "oy us is close to those ofRefs/15,167.

**) All variables in eqs.(l . l) and (l,2) are in tire centreof mass system. Further> the masses of the particles areconsidered to be identical and equal to m both in the relativisticand non-relativistic cases.

The quantity V(p,"q;E ) called quasipotential, is in general a

complex function of momenta and of energy. In the weak coupling case i t

can "be constructed with the help of perturbation theory.

The QPA ha3 been successfully applied to the investigation of the

asymptotic "behaviour of the scattering amplitude T and of the bound

state problem ^17-267.

A specific feature of the formalism of the QPA is the appearance of

three-dimensional kinematical factors, which have relativistic origin.

For example, in eq. (1.2) the integration over the three-dimensional k~

space is performed with the volume element

(1.3)

which is the Lorentz-invariant measure on the hyperboloid

For this reason it seems natural in the framework of the (3? A to

use a system of functions whioh is connected with the motion group of

the hyperboloid (l«4) (the1 Lorentz group), and is a oomplete and

orthogonal system on this surfaoe *v * In the spinlesa case, within which

we shall limit our considerations, such a system of functions has "been first

introduced by Shapiro £ 1

*) It is well known that the upper half of the hyperboloid (1.4)can be considered as one of the realizations of the Lohachevskyspace, and the Lorenta transformations as the motion group of thisspaoe. We shall often use this terminology.

-2-

— /[ _ .4. Y~ W V

0-5)

e+0 p(1.6)

(1-7)

In tlie following the inunctions (1*5) vil l play an important

role and their "basic properties will "be analysed in detail. Here we

only notice that the Shapiro transformation -which uses the "basis (1.5

plays the role of the Fourier transformation in the three-dimensional

Lobachevsky space j^jj\ This is the reason why, in Sec, 2, a nur/bor

of questions connected with the geometry and the motion group of this

spac© will "be considered.

2. THE LIOlvlEfJOTUM SPAC2 OP A PARTICLE AS A L03ACHUVSKI SPACE

Usually, speaking about the connection of the r e l a t i v i t y theory

with the Lobachevsky geometry, one means t ha t the r e l a t i v i a t i c velocity.

space, i s a three-dimensional space with constant (negative ) curvature

(see for instance / 2 8 - 3 l / ) . As mentioned above, the upper hal f of the

hyperboloid (1 .4) embedded in to the four-dimensional momentum space, can

be considered as a model of such a space. I t i s c l e a r , tha t many co-

ordinate systems may be introduced on t h i s surface and in -his way

different forms of the element of length , volume element, La~.xace operator,

etc. ,may be obtained. For ins tance , i t i s often convenient to U3e the

"spher ica l" co-ordinates

In other cases, to pararaetrire the Lobachevslcy space (1,4 ), it is useful

to project the hyperboloid on some Euclidean hyperplane and to assign to

the points (l«4 ) the Cartesian co-ordinates of their projections. In this

case, the part of the hyperplane on which the hyperboloid is mapped, can

also be regarded as on©'of the Lobachevsky space models-.

In partioular, the. components of the particle velocity

are the Cartesian oo-ordinates of the projections of the hyperl>oloid

points on the tangent plan© kQ =. m, the projection "being performed with

the help of rays going out from the origin of the co-ordinate systes

(0,<?)*) . As Ik/k^l^ 1 , the corresponding nodel of Lob ache vsky space

is the inner part of the Euclidean "ball v1* =, 1 with metric

and volume element

dL?

(A-^f (2.4)

Alternatively, the surface (1.4) may "be parametrized by tho

components of the momentum k. Indeed, these quantities are the

Cartesian co-ordinates of the hyperplane kQ = 0, onto which tho

hyperboloid is projected from the point (<=o,'5). Uow the entire threo-

dimonsional k-spaco is a model of the Lohachevsky space with metric

*) In geometry the co-ordinates (2.2) are called homogeneous.

and volume element

Thus the velocity and the momentum of a particle with mass m

corresponds to different co-ordinate systems in the same Lobachevsky

spaoe (1.4). The relation

K =

gives the transformation law from the one co-ordinate system to the

other *>* The essential difference between these two paranetrizations

of the hyperboloid (1.4) is that the metric of the velocity space,

defined by the relation (2.3), is independent of the mass m and is

universal for all particles, while the metric (2*5) of the curved

momentum space depends on m and, therefore, this space corresponds

to ono particle only.

(The remarks we made are self-evident hut they serve to clarify

the sense of the further analysis. We intend to consistently apply the

idea that in the relativistic case the momentum space of a particle with

mass m is a Lobachevsky space with metric (2.5). Within the framework

of $?A the relativisiic two-body problem is reduced to the problem of the

behaviour of one relativistic particle in a quasipotential field. This

will give us the possibility to make effective use of geometrical

'arguments and, as a result, to maker the exposition more compact and

closer to the non-relativistic theory.

*J In the non-relativistio (Euclidean) limit this transformationis reduced to a simple change of scale.

Let ~\L/(~p) be the wave function of a p a r t i c l e with spin 0,

mass m and momentum p , which i s a vec to r in Lobachevsky space . The

question a r i s e s oi1 how to define the opera tors of the angular momentum L—>

and the oo-ord ina te x* The group of motions of Lobachevsky space - thoLorentz group-oontains 0(3) as a subgroup and the opera tor L has the

ordinary form

U « 4" L:'?•

(2.3)

In the non-relativistic theory the co-ordinates x are generators of

translations in the Euclidean p-epace. In the curved p-space the

proper Lorenta transformations A:* play the role of translations and

we could try to introduce the operators 5c as generators of these

transformations *) •

It is not difficult to show that

(2.9)

Therefore.

X =(2.10)

*) In a similar manner the co-ordinate operator in the theory of cnar.-t iaed space-time was defined ^32-3^7. (In t h i s connection see also^367) The notation ( + ) for the shift operation in spacesjtrithconstant curvature (eo, (2.9)) we have borrowed from ~ " '^~~f

- 7 -

The operator (2#10) is hermitioa. in the metric

r

However, its components do not commute amongst themselves: and cannot "be

simultaneously reduced to diagonal form.

Let us consider now one of the Casimir operators *) of the

Lorenta group

x - ^ L

and let i t be equal to

f*< A_ iU ~ "M8- "*" r (2.12)

with

As i t is well known /38/ , the relations (2,12) select the so-called principal series of the unitary representations of tho Lorontagroup.

The functions (l»5) used in the Shapiro transformation arematrix elements of these representations, thus, eq.(2.12) connects thesquare of the co-ordinate operator (2.10) to the parameter v appearingin (l#5)« Obviously we can also write

i - [A + tC^+'Ol (2.13)

*) She second Casimir operator L • x is identical to zero in thespinless case.

-8-

From (2.13) we can conclude that, if in the decomposition of the wave

function ~lj/ ( p ) in terms of matrix elements of the Lorentz group only

the principal series occurs, the spectrum of 5c is limited from

"below

t = l** !

(2.14)

This result is in agreement with the conclusion of Newton and ¥igncr

that in the relativistic case the particle cannot be localized in the spaco

with an accuraoy "better than l/m . Let us notice that if X is the

relative co-ordinate of two particles then the inequality (2«14) Gives a

lower limit to the relative distance. In other words, such a phenomenon

as "collapse" or "falling of the particle on the centren(_AOj is

automatically excluded from consideration in the present approach*1') .

However, further, we shall deal with the quantity r^ instead of x* *

This can "be justified in the following manner. First, in virtue of

(2*10) and (2.12), Xz and r have the same non-relativistic limit

\ •> 1

where p is the radial part of the non-relativistic co-ordinate. Second,

T is the Casimir operator of the group of motions of curved p-cpace,2 > -*•

just as p is the Cassimir operator of the motion group of flat p-apace.Therefore, the range of T is independent of X(compare1 with the range

of x / and the entire relativistic formalism "becomes very cirnilar to thenon-relativistic one. However, a question arises: I.f the parameter r

plays the role of a relativistic distance,, is i t possible to define the

corresponding direction, i .e . , to construct the vector r' ? In this

connection let us consider the Shapiro transformation and find i ts non-

relativistic limit

*) It is interesting to notice that the "forbidden" region ix^iC Vfrom the group theoretical point of view, corresponds -tothe supplementary series of unitary representatiens of the Lorentgroup.

-S-

(2.16)on-* co

Taking into account (1.6) and (l.7)» i* m&y ^e said that the

functions f(ppT,r) are the "plane waves" in Looachevsky space, and that the

Shapiro transformation is the analogue of the Fourier transformation [ZJJr.

The vector r = rn in 5(pjii,r) appears as a variable canonical

conjugated to the momentum p. Therefore, the unit vector vi =. (

j is exactly the angular part -which we wanted to find.

Prom the four-dimen'sional point of view n is the space part of the

isotropic 4-vector

Therefore, the new T-space has very specific Lorenta properties: the

modulus of the radius-vector ? is a relativistic invariant, and i ts

direction is transformed as the three-dimensional part of (2«17)»

Let us return now to the translations (2.9) and consider them

more carefully. It is clear that, in contrast to the Euclidean shifts,

the transformations (2.9) do not form a group. However, >eing elements of

the Lorentz group they have certain group properties. In particular,

^ . (2.18)

Evidently the volume element (2.6) is invariant with respect to the

tranformation (2.9)

(2.19)

This property of d£l k allows a convolution to be defined for functions on

Lob achevsky space

% 00 - ̂ k % (̂ ̂ C-^« f)) <*•*>)

It is evident that

*

Putting f (f)= SHf) and " ^ (p) •- Y Vp/ in (2.20) and taking into

account (2.18) and (2.21), we obtain

Hence,

(2.22)

The non—oomrra.c;c.tivity of k and p in the operation (-) in the

argument of the 6 -function is irrelevant. Let us note also the useful

relation

-11-

4ii

In "spherical" co-ordinates

f* =

\ 'S (2.24)

this equality "becomes

(2.25)

3 . QITASIPOTEHTIAL EQUATIONS AHD THE LOBACEEV3KY SPACE

The purpose of t h i s sec t ion i s t o wr i t e down qua3 ipo ten t ia l

equations for the s c a t t e r i n g amplitude and the wave function in terms

of opera tors and q u a n t i t i e s defined in the Lobachevsky space and a f t e r -

wards to apply the Shapiro t ransformat ion t o these equa t ions . However,

before doing t h i s , i t i s usefu l t o remember the corresponding non-

r e l a t i v i s t i o formulae, in order t o emphasize at every s tage the analogy

"between the r e l a t i v i s t i c and the n o n - r e l a t i v i s t i c c a s e s . As i s well known,

the Lippman-Schwinger equation for the n o n - r e l a t i v i s t i c off-energy s h e l l

s c a t t e r i n g amplitude A(p}ci) i s wr i t t en in the form

(3.1)

-12-

vhore

"V

and V(p*,l?) is the Fourier transform of the potential

(3.2)

If the potential is local,

then

v - ^ - j . r

In the oase of spherical symmetry we have,in addition,

o )

(3.4)

amplitude A(p,"3) is normalized to the elastic scattering differential

crosG-soction-

-13-

In the absence of absorption (im V=0), the unitary condition can be obtained

in the usual way from (3.4) -

i - S K I ^ 1 ^

T'h.© wave function of the continuous spectrum in the p—representation

be >rritten in the forra:

SY¥\r T C ^ h-»

which, together with (3*1), ^ives- tho Schrodin^er oq,uatior.

or

if

-—-> ^ —V

In the local case (3#3)j the integral term in (3.?) and (3.3)

LG evidently a convolution of the functions V and ''f in k'-space

- 1 4 -

If, in (3.3) and (3.7) we pass to the ^-representation with the help of

the relation

and of (3.2),we obtain correspondingly

(3.11)

? J ^ ^ ; (3.12)

where

^f J L ^k. ,

fL1_ e (3.13)v -• W

in the Greon function of eq. (3.11). Let us also mention the

relation

(3.14)

-15-

.- , . :.• \ -

ITow lot us pass to the partial wave analysis. The "basic formulae to;;hich we shall refer in the following are:

i ) The plane wave expansion and related relations

(3.15)

2

o

) Wave function expansion

' (3.19)

-16-

^

(3.21)

where Geq (*r,r ' ) is the partial wave Green function defined "by

and satisifies the equation

Let us return, to the relativistio casej i.e., to the quasi-

potential, scattering theory. Instead of eq.(1.2) for the scattering

amplitude ^(P*,CL)> we shall use the similar equation ^41,427 :

-17-

On the energy she l l , EIp- E D , the function T(i?,ci) coincides with

the invariant r e l a t i v i s t i c amplitude T ( s , t ) , which i s connected with

the e las t ic scat ter ing different ia l cross-3ection by:

! Z

(3*26)

A specifio feature of eq.(3.25) is that it is tractable "by means of

convenient "three-dimensional" diagram techniques /41j42/- the quasi-

potontial V "being defined by a certain clas3 of "irreducible" diagrams

(compare with the Bethe-Salpeter formalism). In particular, in second-

order perturbation theory

VM

(notice the "non-local" dopendsjice of V on tho raonenta).

Let us introduce, instead of T, a new amplitude A,

-16-

Then (3.26) "beoomos

(3.28)

which coincides with the normalization condition, of the non-

relativistic elastic scattering amplitude. To obtain an equation for

A froni eq.. (3.25) it is necessary, obviously, to extrapolate (3.27)

off the energy ahell. Let us 'rewrite (3.2?) in the form

and remember that according to the relation

2 2remains valid off the energy shell. Here a => (p. + p-) •= 43

2 2 ^

and c . = (q, + q_) = 43 are the Kandolstam variables

to incoming and outgoing particles. Hence, the extrapolation oz'

amplitude A(s,t) has the forra

A (f ~4) =

Uow raalcing the following substitution in (3.25)

A (3.29)

-19-

•we finally obtain

(3.31)

^1*(3J31) is a direct relativistic generalization

of the Lippman—Schwinger ©q.(3,l}#

In the case of a real quasipotential,, from (3*31) follows a

relation whioh coincides exaotly with the non-relativistio unitarity

condition (3*5)

I"/'

*tv

Ep

E.

If we write this equality in terras of the amplitude T

(seo (3.27)), it can be seen that it is equivalent to the relativistic

two—particle unitarity condition *) .

Starting from (3*31) and talcing into account the fact thr̂

*) Let us notice that the validity of the unitarity condition(3.32) when Im V =, 0 is one of the main merits of the QPA?14/.

- 2 0 -

OJt(

now we are dealing vith quantities! defined in Lo"bachevsky space, i t is

easy to obtain analogues of the non-relativistic relations that we listed

above.

First, let us introduce the relativistic wave function of the

continuous spectrum (see (3.6) and (2.22))

(3.33)

For this quantity we obtain from (3.31) (compare with (3.7) and (3.8))

(3.35)

Defining (compare with (3.2), (3.10) and (3.13))

(3-36)

-21-

Sifi..:v -TT i \ „ . _ ! • • ]

(3.37)

(3.3S)

f - r r i . , f s r ' ^ Ajr =. v~z A A T A <„% _ if*i A ^ „ •.. a _i i=-. A

From (3.34) we get (compare with (3.12))

(3.33)

The relations (3.14) are generalized in an evident way

,7

Lot U3 go now to the partial wave analysis. First, let us

notioe ^437 (compare with(3.15)) that

- 2 2 -

oo

It is not difficult to show that in the non-relativistio limit

m » 1, < 1

V

The functions -f. (ch*Xq,r)j like the spherical Bessel functions

(qr)i oan "be expressed in terms of elementary functions

(3.41)

- 2 3 -

I t is also easy'to checfc (compare with (3.18)) that.

(3.42)

Of (3.43)

Q

j i n complete analogy with (3.22) and (3.23) we have

i-o

c*2

G- (r r') = ^ '

(3.44)

Up to now we were generalising the non-relativistic scatteringtheory in the integral formulation. Now we shall find analogues of thecorresponding, differential equations^With the help of the recursionformulae for the spherical functions

&0 -

v

- 2 4 -

it is easy to see that the functions a, (chX}x) introduced in (3.40)

satisfy the following "differential" equation *) •

(3.45)

From here, talcing into account (3.4O), we obtain tha analogueof the free Schrb'dinger equation in tbe relat ivis t ic domain

(3.46)

V̂

In the non-relativistio limit,, (3«46) obviously reduces to (3 f l6).

From the derivation of (3.45) a11^ (3.46) i t is clear thatthese equations are recursion relations for the functions p. and X .For instance, (3.46) may be written in the form-

*) The "second solution" of GQ..(3.45), as is oaoy. to guess,is connected with the srtherical functions Q ' ^ " ^ l̂ chU)It j.s interesting to notice that both colutiona" satisfy thosame differential equation in the variable ch 7C • So we againhave oxx analogy with the non-relativistic oase. For equations ofthe type (3.45) wQ can define an analogue of the Wronskian and withi t s help construct Jost functions.

- 2 c -

(3.47)

The function ^ * C ^ ; T , ~n ) also satisfies (3.46) and (3.47). We

shall write down two more equations, without explanations since their

analogy with the non-relativistic formalism is quite evident

•a) The Schrbdinger equation with quasipotential

(3.43)

where y ( r ) a n d v(^»?i E ) a r e ^f ined. hy (3.36) and (3.37).>

"o)The equation for the partial wave Green function (3.44)

J

-26-

Thus we have convinced ourselves that the relativistio

quasipotential soatterlng theory oan be formulated as a generalization,

in the framework of the Lobachevsky spaoe geometry, of the non-relativistic

scattering theory with nonrlooal interaction. Naturally the question

arises, is i t possible to construct a relativistio generalization of

the theory with local interaction? The next section is devoted to the

investigation of this question.

•4. THE LOCAL QUASIPOEENTIAL CASE

It was pointed out in Ref•$-$ that if we require from

eq.(1.2) only that it gives the right physical scattering amplitude,

then the quasipotential V(p,k;E ) oan be chosen "local" in the sense

of the definition (3.3 )i

It turns out /I'jJ that the function (4*1) satisfies the following

spectral representation;*)

QO

(4.2)

*) To be rigorous, the relation (4*2) must be written separately forthe "even" and the "odd" parts of the quasipotential, correspondingto even and odd moments in the partial wave expansion of V •Moreover, in (4.2) subtractions should be made in general. How-ever, to avoid complicating the picture, we shall work withrepresentations of the type (4*2).

-27-

In other words, in this appraoch, the quasipotential is a suporpositionof non-relativistio Yukawa potentials with an energy dependent speotralfunotion*

A l i t t l e modification to the reasoning of [vj],. allows us to prove,under the same assumptions, that eqs.(l .2) and (3»25) lead to the rightphysioal amplitude with a quasipotential of the form

(4.3)

(see the last footnote).

Taking into account (2*23), we oan write the formulae (4.3)the following manner:

'VV\J

7 (4.4)

So we have built a quasipotential whioh is local in the sense ojthe Lohaoheveky geometry. Putting

(4-5)

and using (4*4) we obtain

-23-

where

(4.7)

It is clear that V(cr, r ) oan be considered as a relativistio analogue

of the Yukawa potential *' . Evidently

_-\Jcr r

How let us apply these results in the framework of the

formalism of Sec. 3. Prom (3.30) and (4.4) we have at first

(4.8)

Further,substituting (4.S) in (3.35) and introducing the new wave

function

we ohtain (compare with ^41,42/)

*) Let us emphasize the "faot that (4*7) is more singular than theusual Yukawa potential

V(cr, r )* '̂ v when r —>• 0 •

**) In relativistio notations 23 (23 - 2E ) > 4™ - t - u - a -P 4 P P

-29-

(4.9)

Taking into account the definition (2.20) of the convolution

in the Lobaohevsky space,

Due to the spherical symmetry of the function V [(p*(-)ic) J

its Fourier decomposition can be written in the form (see (4.5))

(f (->t; V ) V fr

>c

From here, applying the "addition theorem" for the

Appendix)

(4.10)

i*(? >

-30-

we have (compare vith (3.3))

V [(

Now we subs t i tu te ( 4 . 1 1 ) in eq. (4 .9)> in t roduce t he n o t a t i o n

and obta in

where the operator HQ is defined "by the equalities (3.46) and (3.47)

and V ( r , E ) is given "by the expressions (4«6) and (4»7)«

I t is clear that in (4.12) a decomposition in spherical

harmonics can he made that results in equations for the partial waves

(compare with (3.20) and (3.21)). A detailed analysis of eq.

(4.12), the asymptotics of i t s solutions, the properties of i ts speotrum,

etc.,will "be made in a subsequent paper.

5 , C01TCLUSI01T

The formalism developed in t h i s paper might f i n d a p p l i c a t i o n s

in the analysis of the asymptotic "behaviour of the relativistic scattering

amplitude and of the hound state problem in oonneotion with Regge ideas.

-31-

The fact that the equations obtained "by us have the form of

reoursion relations must not he discouraging. It is known from the

theory of special funotions that the recursion formulae are often as

effective for defining these functions as the differential equations.

Let us also notice that if, as in Sec, 4, our purpose is

to obtain from the quasipotential equation the scattering amplitude

which coinoides with the physical one only on the energy shell, then

in this equation, in principle, ve can modify . not only the quasi-

potential (the substitution V(p%lc;E ) >vC(?(-)£)2 , E ),. but

also the energy denominator* In particular^ if instead of (1.2)

and (3»25) we could consider the equation

(5.1)

+

where ch X and cbX . are defined according to (2.25)pq pk

and ^lAqi • • ^ - is a faotor necessary for the validity of the

vuiitarity oondition (S.32) when Im 7 B 0, Taking into account (3*41)

it is easy to show from (5»l) that, for example,for the wave function

of the S-state, a seoond order differential equation can be found.

-32-

ACKITOtfLEDGIOTTS

The authors express their gratitude to Professors

A .IT. Tavkhelidze, I.T. Todorov, Drs. A.V. Efremov, S.W. Faustov

and A.T. Filippov for valuable discussions on different aspects

of this work*

One of the authors (V.G.K.) wishes to thank Professors

A"bdus Salam and P. Budini and the IAEA for the hospitality at the

International Centre for Theoretical Physios» Trieste and also to

thank Professor C. Fronsdal and Dr. M, Matsev for their interest in

the work and valuable discussions.

- 3 3 -

APPENDIX

I . "Addition theorems" fo r t h e s-pherioal funct ions connected withthe LoTaachevsky space

L.I)

where

(the evaluation of the relation (A.l) is given, for instanoejin

Taking into aooount (3.40) and (3.41), we can write eq.(A.l) in the

form jZJ

tK -> >

From here, owing to the equality

-34-

follows the "addition theorem" (4.10).

Let us also give for purposes of reference an "addition

theorem" of a more general type than (A.l) (it can "be obtained from

(A.l) if the recursion formula (3.41) ia applied)

\/x

(A.4)

whexe are Oegenbauer polynomials.

II. In some calculations the following formulae could "be

usefult

-35-

_ ,

(A. 5)

where

• J2.0-,•e+i

(A.6)

-36-

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•39-

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

Internationa! Centre for Theoretical Physics, Piazza Oberdan 6, TRIESTE, Italy

9693