Irreducible representations of the rotation group in terms of the axis and angle of rotation

ANNLiLS

OF

PHYSICS:

Irreducible

42, 343-346

(1967)

Representations of the Axis

II. Orthogonality Representations

of the and Angle

LuboraforU,*

in Terms

Relations between Matrix Elements and of Rotations in the Parameter Space H. E.

Lincoln

Rotation Group of Rotation

Uassuchusettn

Institute

&fOSES

of Technology,

Lexington,

Massachztsetfs

09173

In a previous paper the irreducible representations of the rotation group were given in terms of a parametrization of the rotation through the axis and angle of rotation. In the present paper we give the orthogonality rclat,iolls between the matrices of two such irreducible representations. We also show how the infinitesimal generators of the rotation group act in the parameter spacfs and show how finite rotations in the parameter space are to be described. Thus the previous paper and the present one can be used to replace completely the usual theory of irreducible representations in terms of the Euler angles by t:he far more convenient ones in terms of the angle and asis of rotation.

I.

INTRODUCTION

BND

SUMMARY

In Ref. (1), which hereafter WC shall call Part I, we gnve Dhe irreducible represent~ations of the rotation group in t,erms of t.he nnglc of rotation p, which we always take to be positive, and the axis of rotation which is given by the unit vector A. In the present paper we shall give the orthogonalit-y relations between the matrices belonging to different representations. We shall also show how the infinitesimal generators and finit,e rotations act in the parameter space given by the axis and angle of rotation. Thus Part I and the present paper can be used to replace completely the t’heory of t’he representations of the rotation group in terms of the Euler angles. Let. us introduce the polar angles 8, q of the unit vet tor j, by A, = sin 0 sin ‘p,

X, = sin 0 cos y,

A, = COY0.

(1.1)

Let us define R$ ,,,i p) by R;,,,(p)

=

where p == $. Then the principal * Operated

with

support

from

the

(I,)’

1 exp

[ipl-

J]I ,)/I),

(1.2)

result, of the present p:qcr is the orthogonality I;. H. Advanced 343

Research

Projects

Agency.

344

MOSlCS

Let us nom introduce a Hilbert space of single-valued cornples functions f( cl) defined for p = 1p 1 having the range from 0 to 2a and the polar angles 0 and cpof the unit vector 31 = (p/p) defined by ( 1.1) having the ranges 0 to T and 0 to 2a, respectively. Let us define the norm of functions in this space by

.i 2a

T I’ s 2a

dp sin” g

0

cl0 sin 0 o hlS(l!)

0

I2

and a corresponding inner produce for two functions of the space. Further, let Vi = a/dp; . Then let US define the operators L = {L, , Lz , L3} and M = {111, M2, Af3) which act on suitably differentiable functions in this space by

In terms of the polar coordinates LB = i cos 0 5 i

of ~1,these operators

- + cot $ sin e t0 - k $ , 4 i

&J

L+ = L1 + iL2 = ,ieip sin 0 2 1

- i

I

.{

sin 192 + i a/J I

cosO$-~cot$sinB~O+Jj$

M+ = Ml + iMz = -iei’

i

[

I>ap’ 1ae I>ap

cot 0 + i cot $ csc 8 -C d

(1.5)

1 - i cot s co8 0 2 + i

jK3 = -i

a 1ae

1 + i cot 5 cos e + i

L- = Ll - iIJ2 = ieP

take the form

1

cot 0 - i cot 5 csc 0 A!- .

>

,

1a0 I>ap

1 - i cot g cos 0 2

1 -- 2 cot 0 - i cot g csc 8 2

,

(1.6)

IRREDUCIBLE

HEPHESENTATIONS

OF

THE

ROTATION

GROW’.

It is readily shown that the operators Li aud dl iare Hermitiau product, which is used. Furthermore, they satisfy the conunutation [Iif1 ) Mz] = ?Ms ( cyc. ) ,

[LI , L2] = iL3 (cyc.),

It is easily shown tJhatI the operators

[Li ,

II

with the iuuer relations = 0.

Afj]

TV(p) = exp (@.M)

(IS)

for auy vector @are unitary in the space of functiousJ( p). Our second result shows how the operators Li and ilPi act ou Rk,,(

,,A d = -,dRi,

L+R$

,&j

L-c, Rhl ,dcl)

(1.7)

U( @) and IV( @) defined by

CT(@) = exp (i@*L),

L&k

315

p) :

,A d,

= -[ij

- 111’+ 1)C.j+ dN1’2R!nlwn(~1,

= -[(j

+ 111’+ i)(j

- djl’~%l,~+l,m(

(1.9)

v).

J/,R$ ,m( p) = /uR$ ,,( p), ( 1.10)

Front 1:1.9 i nud (, 1.10) L’R$,,,(

cl) = M’R$

,,,,(p) = j(j + l)Ri,,,,(~+).

(1.11)

Let us uovv consider a subspsce of the space of functious f( p) which cousist’s of functions which vanish identically for p > r. Then functions g(p) belonging to this subspace are fur&ions on the rotation group since the argument ~1of such functions are in a clue-to-one correspondence wit,h the rokttiou R(p). Let us further restrict t,he vector e of (1.8) so that fi = 1Q 1 is in tl le range 0 to 7r and the polar angles of @have the sxnc range as the corrcspunding angles of v. Then for every rotation R( Q) we have au operator 6-( @) and TY( 0). We cm uo\v state our final result of the preseut paper: C’(L3Mcl)

w)g(d

= gh~j,

where ~1’and pN are obtmained from the following

multiplication

WV’)

= dd,

= R(-e)R(v),

Rh”)

= R(v)R(@).

(1.12) rule for rotations: (1.1%)

The operators C(g) thus constitute a unitary representation of the rotatiorl group when acting on the space of functions g(p) Likewise the operators W( 0) constitute a different unitary representation of the rotation group.

346

MOSES II. PROOFS

We shall prove (1.9) and (1.10) first,. Equations ( 1.9) nre proved by direct substitution using (1.2) and Eqs. ( 1.5) of the present paper and Eqs. ( 1.3)) (2.5) and (2.7) of Part I. Equations ( 1.10) are proved similarly by replacing Ri, ,,,( p) by [R~,,J ( - p)]*. The orthogonality relations (1.3) are proved in the following way. First it is shown that the integral on the left-hand side of (1.3) is zero if VL does not equal r~, in the usual way, by letting M, operator first on RfLl, and then using the hermit,icity of A13 to let it operate on R$, . Equat’ing the expressions and using the first of (1.10) leads to the desired result. In a similar way one shows that the right-hand side of (1.3) vanishes when j does not equal k or when 11~’does not equal IL’. Thus we have justified the Kronecker deltas on the right-hand side of ( 1.3). Thus, to finish the proof of ( 1.3), we need only evaluate the integral on the left when k = j, 7~ = nl, and n’ = ~1’. It is readily shown that this integral is independcnt of the value of 1~ and 1)~‘. Hence we set 11~’ = j, 111= -j. The resulting integral is very easy to evaluate and gives us our result. Equation ( 1.12) can be proved in a variety of ways but, for the sake of brevity, we shall not give a proof in the present paper. RECEIVED:

October 5, 1966 REFERENCE

1. H. E.

blOSES,

Ann.

Phys.

37, 224 (1966).