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Symmetry properties of n-pion wave functions
ANNALS
OF PHYSICS:
Symmetry
26,
418441
(1964)
Properties
of n-Pion
Wave
Functions
M. GRYNBERG AND Z. KOBA* Institute
for
Nuclear
Research,
Warsaw,
Poland
Symmetry properties of the configurational wave functions of an arbitrary number of pions are examined in the nonrelativistic approximation. In Section II general behavior of solutions of the n-pion SchrBdinger equation is described in the case where the interaction Hamiltonian depends only on the over-all relative distance R, and a scheme of generalized partial wave classification is introduced by means of the grand angular momentum A and the total angular momentum J. Each of these subsets of the configurational wave functions forms an invariant subspace of the symmetry group 8, . Section III presents a systematic method of working out how many times and what kinds of irreducible representations of 8, are included in the representation acting on this space of wave functions specified by A and J. The results thus obtained for A I 4 are tabulated. I. INTRODUCTION
AND
SUMMARY
A number of pion-pion resonances have been found and it seems quite possible that in the near future other resonances consisting of a larger number of pions may be detected and a new region of “pion spectroscopy”’ may become of vital interest. In this connection it will be desirable to have a general kinematic scheme of describing the system of an arbitrary number of pions, in order that we can separate intrinsically dynamic properties of the latter from those which are independent of the detailed mechanism of the interaction. In the present work we should like to discuss symmetry properties of the n-pion wave functions. Since pions obey the Bose statistics, an n-pion wave function must remain unchanged under exchange of any pair of the pions. Such a totally symmetric wave function in the centre of mass system with definite values of the total energy, W, the total angular momentum J, its third component, Js , the parity, II, the total isospin, T, and the total charge, T3 , can be constructed by putting
@(Jf’, J, Jo , II; TTs) = C a&(W, id
Here fi( W, J, JB , II; sym.)
J, Ja , II; sym)Tj( T, Ts ; sym.)
denotes a configurational
Present address: University Institute for Theoretical 1 This name has been suggested by Dr. B. C. Magli6. 418
Physics,
wave function Copenhagen.
(1.1)
(function
SYMMETRY
OF
‘Il-PIONS
419
of coordinates or momenta) of n pions with definite values of W, J, J3 , H, as well as a definite symmetry property. By a “definite symmetry property” we mean here that this function should transform under any permutation of pions according to an irreducible representation of the symmetry group S, . When this representation has a dimension Q I 2, then the Q functions making up the bases of the representation are labelled by the index i. The second factor qj( T, TX ; sym.) represents, on the other hand, an isospin wave function of n pions with definite values of T and TZ , and with the same symmetry property as .$ . The index j has the same meaning for 7 as i has for & It is known that we can then find such a set of coefficients a;? , depending on explicit forms of the representation matrices, that the sum of the Q” products on the right hand side of (1.1) is invariant under any permutation of coordinates (or momenta) and charge of n pions simultaneously. Thus we obtain an over-all wave function which satisfies the Bose statistics. An arbitrary n-pion wave function with definite W, J, Ja , II, T, and T, , can be expressed as a superposition of the functions of the type ( 1.1)) if we have worked out a complete orthonormal set of wave functions for ti’s and ~,~‘s separately.* In this way we shall have obtained a complete orthonormal set of over-all wave functions. From ( 1.1 j we realize that the Bose statistics implies certain correlations between the charge properties and the space-time properties of the system. For instance, the charge branching ratios and the angular and momentum distributions of the decay products-when the n-pion system decays through strong interactions-are related to each other depending on the isospin, the total angular momentum (spin), and the parity of the system3 (3). The symmetry properties of the n-pion isospin wave functions4 were investigated in detail by Pais (4). In this note, therefore, we shall be concerned with the symmetry properties of the configurational functions. The relativistic effects are, in general quite essential in the n-pion problem.’ 2 In our previous works (1, 2) explicit expressions of 5%‘~ and vi’s for 3- and 4.pion systems are given in the nonrelativistic approximation. As is also seen from these examples, we need of course other indices, too; in order to specify them uniquely. 3 One of t,he simplest example of the correlation is the case of a neutral system of two pions. Depending on T = 2, 1, and 0, the branching ration ~0 G/T+ X- of the decay products is 2, 0, and f/i, respectively. For 2’ = 2 and 0, the relative angular momentum of the two pions is always even; for T = 1, the relative angular momentum is always odd. 4 Explicit forms of the 3-, 4-, and 5.pion isospin wave functions with definite 7 and definite symmetry are given in (I, 2, 6). 5 In a fully relativistic treatment of the problem, we can proceed, for instance, in t,he following way. First we construct a wave function of n pions, $ (W, J, Jc , T; kr , kz , . . . , k,) with definite values of W, J, JB , and T, but without any definite symmetry property, in t,erms of t,he momenta of n pions. $ will be found, for instance, by repeated application
420
GRYNBERG
AND
KOBA
Here we restrict ourselves, however, to certain particularly simplifying aspects of the n-pion problem which appear in the nonrelativistic approximation. In this limit we can namely define a nonnegative integer A as the eigenvalue’ of the magnitude of “grand angular momentum”’ introduced ’ by Smith (7). This value has a direct physical meaning as a generalized angular momentum. (See the examples given in Section 1I.C.) Moreover, it can be shown that the set of independent wave functions with a given value of A forms an invariant subspace of the symmetry group S, . Since the energy and the total angular momentum in the center-of-mass system are also invariant under any permutation of n pions, we can classify the wave functions by W, A, J, and JB (where A 2 J 2 1Ja 1 ), and each subset thus obtained makes again an invariant subspaceof S, . The main purpose of the present work is to find formulas for an arbitrary n, which show how many times and what kind of irreducible representations are included in the representation of S, acting on the subset of wave functions specified by W, A, J, and Ja . Actually they depend only on A and J. Generally speaking, not all the symmetries are included in a subset specified by A and J. This is the caseespecially with smaller value of A and A - J. On the other hand, not all the kinds of symmetry can appear, in general, in the functions v( T, T, ; sym) with specified T. (This does not depend on T3.) These circumstances lead, in view of the form of the over-all wave function (1) to certain selection rules, simpler examples of which are known individually and have been utilized. Although these selection rules are strictly valid only in the nonrelativistic limit, we expect that they will also help us in semiquantitative analysis at a not too high energy. Further, a similar mathematical procedure may be applied also in a fully relativistic treatment, although the physical meaning of the parameter is no longer so clear-cut as in the nonrelativistic case. In Section II we discussthe solution of n-pion Schriidinger equation. This part is not essentially new, but summarizes and generalizes known results (8, 9, 7, 1, 2, 3)) formulating them in a way adapted for our purpose and adding some of Wick’s procedure, which has been given explicitly in the case of three pions (6). Then we can make use of the known procedure of successive symmetrieation and antisymmetrization in order to obtain a function of a given definite symmetry. 6 To be precise, the square of the grand angular momentum tensor has eigenvalues A (A + 3n - 5). 7 In our previous work (1,s) we called il “effective angular momentum.” We have noticed, however, that this quantity had been systematically discussed by Smith (7) and others (except its relations to the symmetry properties, which are our main interest in (1, 2) and in the present work) and called by him “grand angular momentum.” So we shall follow his nomenclature.
SYMMETRY
421
OF 'WPIONS
examples. Section III presents a method of enumerating the possible kinds of symmetry appearing in the configurational wave functions when the latter are classified by the grand angular momentum A and the total angular momentum J. Explicit forms of wave functions ti in (l.l), with definite values of A, J, Ja , and lI, and with definite symmetry, will be given in a forthcoming work by one of US (;\I. G.j. II.
SOLUTION
A. ELIMINATIOX
OF
THE
n-PION
OF THE CENTER-•
SCHRODINGER
EQUATION
F-~/IASS ;\IOTION
Throughout this work we restrict ourselves to the nonrelativistic approximation and describe the system of n pions by a configurational wave function 9” . satisfying the SchrGdinger equation
& $ kj2 + V(x, , xz, . . . , x,,) ‘ko = wo\ko
(2.1)
with
In order to eliminate the center-of-mass motion we introduce a unitary transformation IT and go over to a new set of canonical variables, ri’s and pj’s.
ro ‘I
\
Xl
r1 I r2 = u “.” I ,
1:
!h-1 I
1&9
PO fkll Pl f/ pz = U I+ 1. li, 1 IPn-1,
(2.2)
Here we identify r. and POwith the center-of-mass coordinate and momentum, respectively.’ rO = L
(x1 + x2 + . . . + xn),
po =
+
d/n
n
(kl + k, + . . . + k,)
(2.3)
The rest of the variables, the relative coordinates, rl , rz , . . . , rnWl, and the relative momenta, pl , p2 , . . . , pn-l , can be defined in a variety of ways, each set being related to other ones through unitary transformations for the (n - 1) variables. The choice of a particular set depends of course on the physical prob8 These definition ventional formulation
of center of mass coordinates ones by factors &and l/d- 72, respectively. with respect to ri’s and pi’s.
and
momentum In this way
deviate from the conwe get a more symmetric
422
GRYNBERG
lem in which we are interested. definitions :
AND
KOBA
For example, one can make the following
simple
xi = s2 (XI - x21 r2 = &
1x1 + x2 - 2x3)
.. . (2.4) 1 {Xl
+
X2
+
’ ’ * +
Xj
-
jXj+l]
rj = dJj . . .
r,-l
=
dnci
_
1)
and the same transformation Another choice is: 11 +
{x1
x2 + - -. + xnwl -
+
(n -
1)x%]
from ki’s to pj’s.’
x2
+
* * *
+
Xj
-
jXj+l)
j = 1,2, * . ’ , h -
1
(2.5)
- h(Xh+l + .*- + xn>l
rh+i = dzc:+ 1) I(xh+l + -. . + Jh+z)- kb+z+1] 1 = 1,2, * . * , n - h -
1
and the same transformation for the momenta. This set of relative coordinates and momenta corresponds to a division of the whole system into two subsystems consisting of particles 1, 2, . . . , h, and particles h + 1, h + 2, . . . , n, respectively. The center-of-mass motion is eliminated by putting \k0 = exp (ipo.ro)Wrl where po is a constant
vector
representing
9 This corresponds to the following procedure. relative motion, then add to this two-particle motion of the latter with respect to the center forth.
, r2, . . . , r,-d the center-of-mass
(2.6) momentum,
and
First take two particles and specify their system a third one and specify the relative of mass of the two-particle system, and so
SYMMETRY
423
OF 'CPIONS
we get (2.7) with ps2=-v+-
.E+A!L+L (
w = wo-
&-$, )
22
21
(po%d
Here we have assumedthat V does not depend on r. , that is to say, no external field is present. B. SEPARATION ANGULAR"
OF THE PARTS
EQUATION
INTO
“GRAND
RADIAL'
AND
“GRAND
Let us define “grand radius,” R, and “grand relative momentum,” P, by n-1 iz
n-1 v-22 =
R2,
2
P:
=
(2.8)
P2.
and normalized relative coordinates and momenta by pi = ri/‘R,
ni = p/P
(2.9)
R and P represent, so to speak, an over-all relative distance and an over-all relative momentum of n particles, respectively, as is seen from the following relations.
(2.10)
In the following we confine ourselves to specifically simple caseswhere the interaction potent,ial V depends only on R. (Two examples will be given in the next subsection.) Then the Schrijdinger equation can be separated into a “grand radial” part, which depends only on R, and a “grand angular” part, which depends only on ~1, p2, . . . , en-1 (8). We put Wrl,r2,...,
r,-,)
= Z(R)Hy”-3’(~,
m, ... 9 en-1)
Here Hy n-3) is a harmonic polynomial of degree A in 3 (n it is homogeneousof degree A, so that Hii3”3’(aF;1 , a&, . . . , a&-~) = u*H~+~‘(&
(2.11)
1) variables (8) ;
, f2, . . . , i&-1)
(2.12)
424
GRYNBERG AND KOBA
and satisfies the (3n-3) -dimensional Laplacian equation,
(2.13) (RAHi3n-3)(~l,e2,
*** .en-dl
= 0
Then the Schrodinger equation (2.7) is reduced to an equation for the grand radial function Z(R) . d2Z im+
3JJ7 - &JT(R)
-
*(*
+i2n - 5)
Z = 0
(2.14)
>
The parameter A which appears in (2.11)-(2.14) is called (the eigenvalue of) the “grand angular momentum” (7).‘” This is a natural generalization of the relative orbital angular momentum of the two-body problem. C. GRAND RADIAL
FUNCTIONS:
Two EXAMPLES
In certain simple casesthe equation (2.14) can be explicitly solved. Two such examples are given in the following. It is seenthat they show features remarkably analogous to the well-known two-body cases. 1. System qf n Free Pions In this case V = 0, and the solution is given by ~+(3,-6,/2(PR)
+
bNA+C3n-&P@1
with P2 = 2/.lw,
a, b = con&.,
(2.16)
where J, and N, are the v-th order Bessel functions of the first and the second kind, respectively. If we require that no singularity is present at the origin, the second term must disappear and we get ZA(R> = R&
Jn+~3n-dPR)
(2.17‘)
which gives, when combined with normalized harmonic polynomials to be discussed below, a complete orthogonal set of n free particle wave functions normalized for continuous values of P. The physical meaning of the parameter A is now obvious: it represents the degree of penetration of n particles through the over-all centrifugal barrier. A = 0, 1, 2, *** ) correspond to generalized S, P, D, * . . , waves, respectively. lo See footnote
7.
SYMMETRY
OF
‘ai5
‘GPIONS
A quantitative measure of concentration of particles can be obtained by the relative probability of finding all the particles effectively within the over-all distance,” R 5 Ra . lrn / Z*(R)
1’ exp (-g)
lF4dR
2. System o;f n Pions Bound through Oscillator Potentials between Each Pair We assume,
or, with
the help of (2.10), v zz I/‘(R)
The eigensolution z dk
of (2.14) with .A)
= n?$R2
(2.19)
(2.19) is given by 2/&wR2)
(2.20)
The normalization
factor
= N. R” exp ( - $&“&R2)L~+C3’-s,12(
where Lka is the (generalized) N is given by
Laguerre
polynomial.”
l/Z
[r(k The corresponding W, = &i
+ A :!+(n
-
l))]”
>
(2.21)
eigenvalue of the energy is o(2k + A + j?$(n
= v% w{q •k x(n
- l)}
- l)] 4 = 2k + A = 0, 1, 2, . . .
(2’22)
Thus, for a specified energy level Q,the grand angular momentum A can take the values A = q, q - 2, q - 4, . . . , 1 or 0
(2.23)
eleI* In deriving (2.18) we have used IIrzi Sri = R3~ dR d& where & is the ‘kurface ment” of the 3(n - I)-dimensional hypersphere R = 1. The harmonic polynomials are so normalized that the integration over this surface gives 1. The function I, on the right hand side is the Bessel function of imaginary argument. I2 This function satisfies the following differential equation.
426
GRYNBERG AND KOBA
Equations (2.20)-( 2.23) represent a straightforward generalization of the three-dimensional oscillator (bound two-body system). D. GRAND ANGULAR FUNCTION; PROPERTIES OF THE HARMONIC POLYNOMIALS The harmonic polynomial Hk3%-‘) of degree A in 3(n - 1) variables, el , @2, . * . ) @n-l(c pi2 = l), has been defined by (2.12) and (2.13). In the simplest caseof two-body problem (n = 2)) it is reduced to a superposition of well-known surface harmonics,
Hi3’M
= ,=g* A(m) YA”h?l),
1el 1 = 1 (2.24)
where A(m)% are arbitrary coefficients with c 1A 1’ = 1, and we see that there are (2A + 1) linearly independent polynomials, specified by the value of m and here they are so chosen as to be orthogonal to each other and normalized on the unit sphere. Now we shall generalize these relations to arbitrary n. 1. Number of Independent Harmonic Polynomials There are (‘*
+ 3n - 5,
(A+3n-6)! A! (3n - 5)l
linearly independent harmonic polynomiaIs of degree A in 3 (n - 1) variables (8). In choosing them orthogonal to each other and normalized on the unit hypersphere, we find it convenient for our later purpose to specify them by means of the following set of indices: 11,
12,
..-
, L-1
;ml,
m2,
..a
,m,-I
; VI,
VZ,
*a-
, h-2
(2.26)
lj and mi represent respectively the magnitude and the third component of the angular momentum related to the relative coordinate ri and relative momentum pi . The physical meaning of vi’s will be clarified later. li and vI can take the nonnegative integral values which satisfy n-1 h
=
gl
n-2 li
+
2 gl vj
(2.27)
and rn( can take all the integral values within the limits -1i 6 mi s li
(2.28)
Enumeration of possible values of (2.26) with (2.27) and (2.28) leads to (2.25). 2. Decomposition of a Harmonic Polynomial In order to work out explicit forms of the above-mentioned orthonormal set of harmonic polynomials, we have to make use of a decomposition formula,
SYMMETRY
OF
H y3)(@,
@) . . . ) &-1)
X HYj’(el
7 ~2, . . . , ~j)HYT-33’-3)(ej+l,
=v*E”Ab,
427
‘WPIONS
A’, A”)N(j,
n - j -
1; A’, A”, v>
@jj+t, . *. 3 en-1 ) (2.29)
e3+1 + n-1 + ... + &-1 g @: ) &+2
where A’s are normalized satisfies
arbitrary A’
coefficients, +
A”
2v
+
v is a nonnegative
=
integer that (2.30)
A
and G, is the Jacobi polynomial13 of degree v. The formula (29) is written, for the purpose of later application, in a general form, so that it remains valid in a case CYL: pi2 # 1. The normalization factor N is given by [N(j,
n - j -
1; A’
A”
v)12
x
A’
A”
+
I’[v
f
+
=
(4v
+
2A’
(3% - 5)/2]
+
2A”
-I-
3n - 5)
I’[v + A” + (3n - 3j - 3)/2]
v!(I’[AN+(3n-3j-3)/2])2T
(
(2.31)
v+A’+$)
The fifth factor in (2.29), ( ~:ZI’ eie)“Gy, can be expressed as a homogeneous polynomial of degree v in (&+I + &+2 + e-e + &-I)
and
(et + ez” + ...
+ et),
thatistosay,ofdegree2vin]~I/,]p2j,~~~,I@,-11. A particular case of (2.29) is a recurrence formula suitable for application to the set of relative coordinates (2.4), Hi3n-3? el ,
e2
, . . . , en-d =
CA&-2,
L-1,
vn-2)N(n
-
2,
1;
L-2,
L-1,
X Hk3nn_-;@(el, e2, - * * , en-dH~i!,(en-l) X I3 The
Jacobi
Gv,-,(L-2
polynomial
+ L-1 + (3n - 5)/2, L-1 + 24; &-1) is defined
by
vn-2)
(2.32)
428
GRYNBERG
AND
ICOBA
with A,,
-I- L-1 + 2v,-n = An-1 = A
(2.33)
Another corollary to (2.29)) which is convenient for application to the set of relative coordinates (2.5)) where the whole system of n particles is decomposed first into two subsystems of h and (n - h) particles respectively, is obtained by using the formula twice.14 Hyn-3)(pl
, @2) . * * ) @n-l> = c
A(lh ) A’, A”, v, v’>
X N(n
-
2, 1; A’ + A” + 2v’,k,
x N(h
-
1, n - h -
x
(&(A’
+
A”
)( (1 - pt)“G,r(A
+
l,, +
v)
1; h’, A”, v’)
2v’ +
(h
-
5)/2,
+ A” + (3n - 8)/2,
h
+
%;
eh2>
A” -I- (3n - 3h-
<&+1+ *** +
3)/2;
@“n-1)/1 -
&>
(2.34)
Here v and v’ specify the way of dividing the grand radius, R (or the grand relative momentum, P) , into three parts: the relative distance (or the relative motion) of the two subsystems, and the internal distances (or internal motions) of the two subsystems themselves. lt, , A’, and A” are respectively the magnitudes of the grand angular momenta associated with these three motions.‘5 3. Orthonormal Set of Grand Angular Functions Specijied by A, li , mi , vi We can apply the recurrence relation (2.32) and (2.33) in succession, and finally arrive at a complete decomposition:
= zA(b,L, x
fly
n-3)(
e.. ,Zn--l,ml, Zl , - - - , L1
, ml
.e. ,m,-1, , . - . , m,-1
VI, *+* , vn-I)
, VI , * * f , vn-2
e1 ,
;
e2 3
* . . , en-d
(2.35)
I4 Equation (2.34) was used in the description of the 4-pion system, n = 4, h = 2 (9). 16 For the relative motion of the two subsystems, the grand angular momentum is of course reduced to the usual angular momentum.
SYMMETRY
OF
429
‘WPIONS
where16 H!p'(Zl
) . . . ,L
,rnl
) *-.
,m,-1
,Vl
, . . . ,vn-2
;@1,@2,
***
) en-1)
n-2
(2.36 XGvj
(
Aj+lj+l+(3j+1)/2,Zj+l+%;,
x ffi=l 1@iIZi YF
l2
+p2J’+1 2
. . . +
$+1
)
(fi)
with” Al Ai+1
=
11,
=
Ai
-I-
l&l
+ 215,
n-1
An-1 E A = glL
j = 1, 2, . . . , n -
+ 2Fl vi.
In this way we have worked out harmonic (3n - 4) indices,” Ii , mi , and vj ; they satisfy
polynomials orthonormal
s
. . ..vL2.p1,@2,
H
..-,Zk-l,ml’,
~!3n-3)(Z1',Zi,
2 (2.37)
n-2
--.,mL,s',
(2.36) specified by relations, .**,pn-l
4. Classification by A, J, and J3 We can apply further a well-known unitary transformation to the set (2.36) with the help of products 01 the Clebsch-Gordan coefficients, and obtain a new complete set of orthonormal harmonic polynomials which are specified by the total angular momentum and its third component, J and J3 , together with (n - 3) intermediate angular momenta, instead of m(‘s. rfi We have
used
an evident
I7 As is well known, Zi operators associated with construct operators whose these operators are given I* h is uniquely determined
relation,
and rni can be defined as the relative coordinates eigenvalues are just the in Smith’s work. when Z;‘s and Y~‘S are
eigenvalues of the angular momentum and momenta ri and pi We ran also parameters AC and Yj . Explicit forms of given.
430
GRYNBERG AND KOBA
(J,
Hk3"-3'
J3
; h
, 13 , 113 , 13 , 1123 , * * * , 623
1..
n-2
, k-1
, VI , . . . , h-2
;
el,ez,~~~,en-l)
= $a m,
C(J,
J3
; h , 12 , 112 , 13,
1123 , ”
’ , 1123 . . . n-2
, k-1
; li
, mi)
(2.39)
3
x H~“-3’(Z1,Z2,
...
,
1+I
, ml
, m2
, . . . , m,-1
, VI , . . . , h-2 @l,
...
;
, en-d
Notice that this transformation is relevant only to the last factor IIIyT(e
J1
20 J32
(2.40) -J
Each subset includes only a finite number of linearly independent polynomials, which can be made orthonormal by specifying by Zi’s, Zi, ...‘s. and vj’s. E. SYMMETRY OPERATION ON THE WAVE FUNCTIONS The representation matrices of the symmetry group S, , when acting on the original set of individual coordinates, z1 , x2 , . . . , xn (or momenta, kl , k2 , . . . , k,) , are obviously given, for example, by unitary matrices,
$02)
=
‘0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
‘*a** *** e-0
0’ 0 0 0 ’
,o
0 0 0
-**
1,
‘Q(123)
=
(0 10 0 0 0
0
1 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
*** *** -*a --. -**
0’ 0 0 0 0
,o
d 0 0 0
*-*
1,
(2.41)
When acting on our new set of coordinates (or m_omenta,(2.2))) the symmetry operations are represented by unitary matrices p !g = @jju-’ (2.42) Since the first member of the new set of variables (2.2)) r. or _po, is completely symmetric with respect to all the particles, all the matrices g must have the following form: 1 f 0 0 -a- 0 ___,_-----------0 I g= (2.43) 0 I 2-J
SYMMETRY
431
OF 'TL-PIONS
where T stands for an (n - 1)-dimensional representation of X, , which acts on the relative coordinates (or momenta). It has been shown in a previous work (10) that this representation is identical with the irreducible representation (n - 1, 1). We may remark that the choice of relative coordinates (2.4) leads in this way to the conventional Young-Yamanouchi representation for (n - 1, 1) of s, . Since the grand radius R (or grand relative momentum P) is invariant under rotation and permutation, as is seenfrom (2.10)) the normalized relative coordinates pi (or momenta 7er)behave just in the same way as ri (or pi) with respect to rotation and permutation. As for the solution (2.11) of the n-pion Schrodinger equation, we need not consider the grand radial function Z(R), as long as we are concerned with its behavior under rotation and permutation. The grand angular part is expressed in terms of the harmonic polynomials, and when the latter have degree A, they are homogeneous polynomials of degree A in el , e2, . . . , en-1 . Therefore any member of Hifn-‘) goes over under any permutation to a linear combination of Hynw3”s with the same A. In other words, the harmonic polynomials with a specified degree make up an invariant subspace of S, . Moreover, it is evident that the total angular momentum, J and J3 , remain invariant under any permutation of particles. Consequently, our classification of the grand angular functions by A, J, and J3 , discussed in Section II. D.4, leads actually to a construction of invariant subspacesof rotation and permutation spanned by solutions of the Schrodinger equation. III.
NUMBER
OF CONFIGURATIONAL W, A, AND J, AND
WITH
WAVE DEFINITE
FUNCTIONS SYMMETRY
WITH
GIVEN
As is obvious from (2.11) and the invariance of R under rotation and permutation, the number of configurational wave functions with specified values of W, A, and J, and with specified symmetry property is determined by the number of grand angular functions with given A and J and with given symmetry. In this section we shall be concerned with a general method of working out the latter number. A. SYMMETRY
PROPERTIES hI0ikfEN~A'~
OF PRODUCTS
OF THE RELATIVE
COORDINATES
OR
1. We have already mentioned in last subsection that the relative coordinates ri or pi (i = 1, 2, . . . , n - 1) transform under permutation according to the irreducible representation (n - 1, l} of S, . In order to extend this result we shall now investigate the symmetry property of the tensors formed by products of various components of these relative coordinates. they
I9 All the arguments in this subsection are apply equally well to relative momenta.
given
in terms
of relative
coordinates,
but
432
GRYNBERG
Let
US
denote the three spherical
AND
KOBA
components
ri + = - ri= ~+ i?“i” ,
riz ~z - h-i” ,
ri- =
where Tiz, ri’, and rrE are the Cartesian
of ri by
components;
ri’ = ri
(3.1)
and correspondingly,
pi+ = -
pio =
piz
(3.2)
which satisfy n-1
z
i (Pa2
-
2Pi+Pi-J
=
(3.3)
1
The grand radius is expressed in a similar way, n-1
R2 = sz { (Y:)~ - 2ri+ ri-j
(3.4)
Now we introduce a linear space L6,^,’ spanned by the product of A elements taken from (3.1), out of which a are + components, b are - components, and c are 0 components. Thus A = a + 3 + c, and L% has the dimension (n+;-2).?+;-2).(n+:-2)
(3.5)
Obviously L$ , L% , and L$ represent the linear spaces spanned by (n - 1) elements of +, -, and 0 components, respectively. Examples, (n = 4). L (1) _ 100
L
(2)110
L
(2)200
-
-
IT1 +, r2+, IT1 {rl
++
r1 rl
, rl +
, n
r3+}
+r2 +
r2
, rl +
, rl
+r3 +
r3
, r2 +
++
f-1 , r2
f-
r2
, r2
+-
r3
, r3
+-
7-1 , r3
+r2 -, r3+r3-)
(3.6)
+
, r2 r2 , r2 + r3+, rs+r;.3+j
As is seen from our argument in Section II. E, the linear space LLt: is an invariant subspace of the symmetry group S, . Then our problem can be formulated as follows: Knowing that each of Lii& , L% , and Liti makes up basesof the irreducible representation (n - 1, l} of S, , to find out how many times and which irreducible representations of S, are included in the representation acting on LLad,’. 2. Each monomial in the Lb&)-space is symmetric in its a factors with indices +, b factors with indices - , c factors with indices 0. (See the last example of (3.6) : in L$i we do not discriminate between rl+ r2+ and r2’ rl+.) In other words, we form ath, bth, and cth order symmetrized product of L%, Li$ and L$Ai, respectively, which we shall denote by [L$]“, [L$lb, and [%I”; then we make their direct product and in this way construct the linear spaceLit: .
SYMMETRY
OF
433
?&-PIONS
We shall show how to find out all the irreducible represetations of S, acting on [L$]“. Then we can of course apply the same method to [Lh:&]b and [L$]“. For this purpose we have to decompose the character [xl” of the representationin general, reducible-acting on L$i into a sum of the characters of the irreducible representations of S, . The character [xl” can be found with the help of the formula” (1%‘) (3.7? where g represents any element of the symmetry group S, , and x (g) is the character of the irreducible representation (n - 1, 11 acting on LiAi . The summation is taken over all the possible ways of dividing a in the form a = a1 w1 + ffz w2 + . . . + LY”w,
(3.8)
with 0 < Wl < wz < . * * < w, LY,w
positive
integers
For example, when a = 2, the possible ways of division
(3.8) are
2 = 1.2
(a1 = 1, Wl = 2)
2 = 2.1
(01 = 2,wr
= 1)
= I,@ = l,wz
= 2)
and when a = 3 (cry1= l,w,
3 = 1.1 + 1.2 3 = 1.3
(a1 = 1, WI = 3)
3 = 3.1
(a1 = 3, w1 = 1)
in this way we obtain the following
formulas.
[xl” (9) = ?4x (8) + %x2 (9) [xl” (9) = %x (g3) + Mx (2) x (9) + %x3 (9) 1x1”(9) = xx (g4) + 36x b3) x (9) + %x2 (8) + xx (g2!x2 (9) + %*x4 (9)
(3.9a) (3.9b) (3.9c)
3. When the element g of S, represents a permutation of n objects which includes k, unchanged objects (one-term cycles), k2 pairs of exchanged objects (two-term cycles), k3 trios of objects cyclically exchanged (three-term cycles), *O We are indebted products.
to K. Olbrychski
for
helpful
information
concerning
the symmetrized
434
GRYNBERG
AND
KOBA
and so on, we can show that, for the irreducible
representation
{n - 1, I}, (3.10a)
Xln
- 1,ll
($7)
=
h
-
1,
X(n
- 1, II (g2)
=
kl
+
2k2
X{n
- 1.11
(g3)
=
kl
+
3k3 - 1,
X(n
- 1. 11 (g4)
=
h
+
2k2
-
+
(3.10b)
1,
4k4
(3.1Oc) -
(3.10d)
1.
The formula (3.10a) was proved previously (10) ; the other formulas can be obtained from this by noticing that the number of one-term cycles (unchanged objects) in g2, g3, and g4 are given respectively by k, + 2kt , k, + 3ka , and kl + 2kz + 4k4 . Example. Let g represent the permutation A A
BCDEF c B E
p
D
= (A)(BC)(DEF)
>
of Se . ( kl = 1, k2 = 1, k, = 1, k4 = 0). Then by direct calculation we find g2 = (A) (B) (Cl (DW, g3 = (A) (BC) CD) C-W (F) and g4 = (A) (B) (0 (DEF) . Thus we get X16,
11 (9)
=
X(6,11
0,
(g2)
=
2,
X(6,11 (g3) = 3,
X(5,
11 (g4)
=
2
Now we put (3.10) into the right hand side of (3.9)) and, with the help of the general expressions of the characters of the irreducible representations of S, , listed in Appendix 1, we can perform a decomposition of the character [xl” for a $ 4 into a sum of the characters of the irreducible representations. (3.11a)
[xl” = Xlnl + XIn-l,ll + Xk-79.~ , [xl3
=
Xlnl
+
[xl” = 2xbd+ +
X(n-3.31
2Xlr+l,ll
+
3x1,-1,~
+
Xb-2.1.11 +
xi+3,2,11
xh-2,1,1)
+
+
X1+-2,2) +
Xln-4,41
+
Xin--3,3)
3x1+2,21
,
(3.11b) (3.11c)
.
The above general formulas are exact when n 2 8. For smaller values of n some modifications are necessary. 4. The characters of [L$]* and [I$:Jc can be obtained by the same formulas as (3.11). The final step is to construct the direct product of these three spaces in order to find out how the character xCabc)of the representation acting on LL,^) is decomposed into a sum of the characters of the irreducible representations. X
(labc) (9) = [Xi?+I,J (9) *txb41b (9) ~[XLn--l,I~IC (9)
(3.12)
With the help of (3.11) the right hand side of (3.12) for a, b, c 4 4 is expressed as a sum of products of the characters of the irreducible representations of S, .
SYMMETRY
OF
n-PIONS
Each of these products can be further transformed into a sum representations by repeated application of the “multiplication irreducible representations, which are listed in Appendix 2. Thus we are now in a position to write down how many irreducible representations are included in the representation Lb,^,’ up to h $ 4. 5. Examples. ( 1) Monomials ri+ ~j+, j-i- ).j-, rLO t.,” (i 2 j) the spaces
In this case it is sufficient representation acting on representations (n) , {n (2) Monomials Ti+ rj-,
The representation (n - 1, 11 X {n (n - 1,l)
43 5
of the irreducible formulas” of the times and which acting on any form respectively
to use the formula (3.11a) and it is found that in the each of the spaces L$, L%, and LA%, the irreducible 1, 1) and (n - 2, 2) are included, each once. ri+ rjO, vi- pjo form, respectively, the spaces
acting on each of them is the product representation 1, 1). From the formula given in Appendix 2, we have
X (n - l,l]
= {n] + (n - 1,l) + (n - 2,2) + (n - 2, 12)
And we find that in the reducible representation acting on Lifi, I,$ or L% the irreducible representations (n], (n - 1, l}, (n - 2, 2} and {n - 2, 1’) are included once. For spaces of monomials of higher order the results are more complicated because we have to apply the reduction formulas more than once. B.
RELATION
BETWEEN
L$
AND
Hy"-3'
In the foregoing subsection we have examined the symmetry of the linear product space of ri’s. But this space L6:? is intimately related to the linear space spanned by the harmonic polynomials Hy n--3).In fact we can show that by using gi% instead of ri’s, that is to say, taking explicitly the condition (3.3) into account, the space Lib”,’ is reduced to the space Hyne3’. In order to see this we notice first that all the harmonic polynomials (2.36) can be expressed as sum of products of pi+, pi-, and pi’, which belong to Lit? with a - R = c;Z: 112~ . By the definition and orthogonality of (@” + . . . + p;+J’jGyj
(
. . ., . . -,
&+I e1” + . . . + &+I )
the identity (3.3) is taken into account here. On the other hand, the number of independent harmonic polynomials of degree
436
GRYNBERG AND KOBA
A is equal to the number of independent polynomials inxhxa+bti L6b^c’when the identity (3.3) is applied. The latter statement can be proved as follows. Denote the number of independent polynomials in C.L~+~+~ L$j (A fixed, but without any further condition imposed) by u(A), and the number of independent polynomials in &aSb+c L2, aft er CZ: ((pi’)* - 2 pi+ pi-) is put equal to one, by ~p(A). Then we have U(A) =(p(A)
+q(A-2)
+p(A-4)
+ ...
-I-p(lorO)
(3.13)
where ( 1 or 0) means0 if A is even and 1 if A is odd. From this we get immediately p(A) = u(A) - cr(h-- 2)
(3.14)
But it is known (8) that a(A) =
A+3n-4 A
>
(3.15)
From (3.14) and (3.15) p(A) = (2A + 3n - 5) (’ + 3n - 6)! A! (3n - 5)!
(3.16)
which is identical with (2.25)) the number of independent harmonic polynomials of degree A. C. NUMBER OF INDEPENDENT J, AND SYMMETRY
GRAND ANGULAR
FUNCTIONS
WITH DEFINITE
A,
On the basis of the results obtained in the foregoing two subsections, we shall now work out the number of independent grand angular functions (see (2.11) (2.13)) which have specified values of A, J, and Ja (see (2.35)-(2.39)), and which behave under permutation according to a specified irreducible representation of X, . This number will be denoted by % (A, J, Ja ; sym.), A L J, -J 5 Ja 6 J. Actually VI is independent of Ja , since neither A nor the symmetry depends on the spatial direction of the total angular momentum. Thus (32(A, J, JB , sym.) = % (A, J; sym.). 1. We have to rearrange the linear spaces L6i? into those corresponding to definite values of J and JB . For this purpose define first L(b).r3 = &
L2
(3.17)
LA = ,,&
L(“)=Q
(3.18)
and further
The assembly of the functions with the same A provides a set of basesfor a
SYMMETRY
OF
437
TGPIONS
representation-in general, reducible-which is equivalent to the one acting on the space L”. This equivalence follows from the fact, which we have mentioned in Section III. B, that there exists a linear transformation which expresses each HA by monomials and that this transformation is nonsingular because HA’s form a complete set and each monomials can be inversaly expressed by HA’s. Thus we can divide L” into subspaces with definite J and Ja and from these we can find % (A, J; sym) by a procedure described in the following. 2. We begin with the “most stretched” space LcA”. This of course corresponds to the largest possible value of JS for the fixed A, and the only value of J compatible with it is J = A. Thus by finding the number X of a certain irreducible representation included in the representation acting on I,‘A’A with the help of the conclusions of the subsection 1, we get the answer for J = A.
%( A, J = A; sym)
= 3z(L’*‘“;
sym)
= X( Lk%; sym)
(3.19)
h’ext we consider LcA)‘-‘, which will include the functions belonging either to J = A, Ja = A - 1, or to J = A - 1, Ja = A - 1. But the first of them is known by our above result. Thus, %( A, J = A - 1; sym)
= % (L’*‘“-‘;
sym)
- 31 (L’“‘*;
sym)
= 32.(L~,o,I;
sym)
- fn(LE0;
sym)
(3.20)
Sirnt’n&rry, we would expect to get % (A, J = A - 2; sym) by subtracting ; sym) from X( L’“‘A-2; sym). Due to the presence of the identity(3.3), WL however, not all the monomials included are independent when 1JBl 5 A - 2. The number of redundant monomials is equal to the number of independent monomials in LcAp2jJ3, LcnmoJ3, . . . , L’“-29’ J3, where Q is the largest integer which satisfies J, 5 A - 2q. The elimination of redundant numbers can be carried out along the same line as in (3.13) and (3.14), and we obtain
5X( A, J; sym)
= x(L’*‘~;
sym)
- X(L’“-2’J;
- x(L(*)‘+~;
sym)
sym)
+ X( L”‘-2’ ‘+l; sym)
(3.21)
Here we make a convention X(L (*jJ3; sym)
= 0
if
lJsl > A
(3.22)
and then (3.21) is quite general and includes (3.19) and (3.20). The results worked out in this way up to A 5 4 are listed in Table I. 3. Examples. (1) The monomial of order A = 0 is constant, which is of course invariant under rotation and permutation. So in this case we have
X(0,0; For all the other symmetries
In)) = X(L’“‘o;
x (0, 0; sym)
In])
vanish.
= 1
438
GRYNBERG
AND
TABLE J
0
0
@I
1
1
(72 -
1,l)
2
2
inI,
(1 -
1
{n -
2, lP}
4
I
Irreducible
A
3
KOBA
&2)
Sn2) (n -
in -
representations
2, 21
1, 11, tn -
0
(1 -
3
(1 - CM {n), (2 - an2 - h) (n - 2, 12}, (n - 3, 3)
2
(1 -
1
(1 - S,z) in), (2 - 2 &a - h3) (n - 1, 11, (2 - W (2 - h3) tn - 2, 121, 1% - 3, 31, b - 3, 2, 11
0
(n -
4
(2 - Se2 - &3) (n), (3 - 3 6,,2 - L3 - 6,4) {n - 1, 11, (3 - srr4 - 6n5) X In - 2, 21, (1 - L3) In - 2, 121, (1 - 8,) (n - 3, 31, {n - 3, 2, 11, {n - 4, 41
3
(2 - 6n3 - 2 6,2) (n - 1, 11, (2 (3 - 2 k3 - hd X in - 2, la], (n - 3, la), (n - 4, 3, 11
ann2) (72 -
1, I},
(n -
1, l),
(n -
2, 2) (n 2, 2),
1, l), (1 -
6,~)
(1 (n -
k)
(n 2, 12),
2, 2), (n -
1% -
2, 21,
3, 13)
2 -
h) (n - 2, 21, (2 - h5) ln - 3, 2, Ii,
in -
2
(2 (6 (3 (n
1
(2 - 2 6,2 - 6,3) ln - 1, 11, (2 - 6,4) (n - 2, 21, (3 - 3 L3 - 6d X (n - 2, l*), (n - 3, 31, (3 - 6,~) (n (2 - Bn4) (n - 3, 13), (n - 4, 3, 11, In - 4, 2, l”1
3, 31,
,sn2 - bn3) In), (5 - 5 Bn2 - 3 8,~ - 64 In - 1, 11, 3 bn4 - 6,J (n - 2, 21, (4 - 3 &S - &d in - 2, I”\, 6,d X (n - 3, 31, (4 - &J (n - 3, 2, 11, {n - 3, l”1, 4, 22), (n - 4, 3, l), (n - 4, 4)
3,2,
11,
(2 - 2 fin2 - sn3) (n] , (4 - 4 6,~ - 3 bn3 - 2 I,4 - 66 - W In - 1, 11, (5 - 3 &A - 2 6,s - 6,~) In - 2,21, 2 tn - 3, 2, 11, (1 - an3) (n - 2, 12) (2 - h) {n - 3, 31, (n - 4, 27, (n - 4, 41
0
(2) Case of
3, 2, 1)
A
= 2. We start with
Ja = 2 space:
L(Z)2 =; L(2) 200 and from Example 3-L(L’2’2; {nf)
1 of Section III. A.5 we have = 3z(L’2’2; (n - 1, lj)
= 3qL’2’2; (n - 2,2))
= 1
SYMMETRY
and for all the other symmetries tractions J therefore X(2,2;
OF
x(L’~‘~; sym)
439
‘GPIONY
sym)
vanish.
Here we need no suh-
= X(LC2’*; symj
In order to find ~(2.1; sym) we consider the space LC2” = L:i:. According Example 2 of Section III. A.5, 3x( LC2” ; {n) j = x(L’*“;
(n -
to
1, 1)) = 31(L’2”; (n - 2, 2) )
= 37@2”; (n - 2, 1’1 Subtracting from these numbers the above obtained x(L”‘~; X(2.1; (n - 2, l”]j
symj’s we obtain
= 1
and for all the other symmetries X( 2.1; sym) = 0. Finally we consider L’*” - space L cao= L:;i + Lg: From Examples 1 and 2 of Section III.
A.5 we have
51(L’2’o; (n}) = 3Z(L’2’o; (n - 1, 1)) = X(L’2’o; {n - 2,2)) = 2 3Z(Lc2’O;{n - 2, 12)) = 1 Subtracting X( L(““; sym)‘s and also, according to (3.21), X( L’““; from the above numbers, we get X(2,0;
(n - 1,l))
= X(2,0;
{n) j = 1
(n - 2,2) ) = 1
while for all the other symmetries X( 2.0; sym) vanish. APPENDIX
I
The character X(X) for an irreducible representation, corresponding to partition{X1,Xq, ... , X,), of an element of the permutation group S, characterized by k, , k, , . . . , k, (numbers of 1 term, 2 term, . . . , n term cycles included in it), is given by the coefficient of r:’ $ . . . r”,” in the expression (11) :
fj (Xi- 5) Ii1 (2 XiP)“” i=l
where k = A, + (n - q), Following this prescription we find: X{n1 = 1, Xjn-l,ll
=
k,
-
1,
(1 =
I,?,
...
.!I
440
GRYNBERG
AND
- h(kl
-
1) + k3,
+ kdkl
-
1) +
KOBA
(kz- 1) + 0~4 - I), XP-3~1) Xl~4,2~)
XIn-4,3,1)
X(n-4,2,12)
= %kl(kl
- 2)(kl
= Mzh(kl =
h 0 2
= kl(kl
-
- 4) -k3,
l)h
- 4)(k1
- 5) + kdk2 - 2) - k3(kl -
. (3 - h + kd + 3 (2)~+(;)+k,(l-iii), - 2) -
3 (;)+3(2)-(;)+ii,+kd&(;). APPENDIX
2
Using the expressions for the characters of irreducible representations Appendix 1, we find the following formulas for the direct products: {n - 1, 1) x (n -
1, 11= (n) + (1 +
{n -
I),
1, lj x {n - 2,2)=
(n
-
(n - 1,l)
h&2>
(n
-
1,
1)
+
(n
-
given in
2,2}
2,1’} + (1 -
6,4)
(n - 2,2j + {n - 2,l’j
+ (n - 3,3} + in - 3,2,1} (n - 1,l)
x (n - 2, 12)
= (n - 1,l)
+ {n - 2,2) + (1 -
+ (n - 3,2,11
S,,)
{n - 2, 12}
+ In - 3,1”1
{n - 1, 1) x {n - 3,3} = (n - 2,2) + {n - 3,2, 11 + {n - 4,3, lj + (1 - &A) In - 3,31 + tn - 4,4j 1% - 1, lj X {n - 3,2, 1) = (2 - S,,) (n - 3,2, 1) + (n - 4,2’} + (n - 3,3} + (n - 3, l”] + {n - 4,2,1’]
SYMMETRY
OF
441
WPIONS
+ {n - 2,2) + In - 2, l”] + {n - 4,3, 11 jn - 2,2) x (n - 2; 2) = {n) + (1 - S,,) {n - 1, l} + (1 - s,,)jn
- 2, l’}
+ (2 - &4 - 6,5) jn - 2, 2) + (n - 3, l”} + jn - 4,27 + (2 - 6,s) (n - 3, 2, 1) + (n - 4,3, 11 + (1 - &It?) (n - 3,3! + {n - 4,4) If, for given n, any partition
is impossible,
one has to substitute
zero for it.
ACKNOWLEDGMENT One of us (Z. K.) expresses his gratitude to the Polish L. Infeld, and Professor M. Danysz for their hospitality RECEIVED:
Academy of Sciences, Professor during his stay in Warsaw.
June 18, 1963 REFERENCES
1. Z. 2. M. 3. M. 4. A. 5. M. 6. G. 7. F. 8. A. 9. 10. 11. 22.
L. M. H. G.
KOBA, Phys. Letters 1, 34 (1962); Acta Phys. Polon. 99, 103 (1962). GRYNBERG AND 2. KOBA, Phys. Letters 1,130 (1962); Acta Phys. Polon. 23,503 (1963). GRYNBERC AND Z. KOBA, Proc. 1962. High Energy Conj. CERh’, Geneva, 1962, p. 178. PAIS, 9nn. Phys. (A;. Y.) 9,548 (1960); Ann. Phys. (‘$7. Y.) in press. GRYNBERG AND 2. KOBA, Bull. Acad. Polon. Sci., in press. C. WICK, Bnn. Phys. (N. Y.) l&65 (1962). T. SMITH, Phys. Rev. l!XJ. 1058 (1960). ERDELYI, W. MAGNUS, W. OBERHETTINGER, AND F. G. TRICOMI,“ Higher Transcendental Functions,” Vol. 2, Chap. XI. McGraw-Hill, New York, 1953. M. DELVES, Nucl. Phys. 9,391 (1958-59). GRYNBERG AND Z. KOBA, Nucl. Phys. 42, 313 (1963). WEYL, “Gruppentheorie und Quantenmechanik.” Hirael, Leipzig, 1931, J. LYUBARSKI, “Group Theory and its Application in Physics.” Moscow, 1957 (in Russian).
OF PHYSICS:
Symmetry
26,
418441
(1964)
Properties
of n-Pion
Wave
Functions
M. GRYNBERG AND Z. KOBA* Institute
for
Nuclear
Research,
Warsaw,
Poland
Symmetry properties of the configurational wave functions of an arbitrary number of pions are examined in the nonrelativistic approximation. In Section II general behavior of solutions of the n-pion SchrBdinger equation is described in the case where the interaction Hamiltonian depends only on the over-all relative distance R, and a scheme of generalized partial wave classification is introduced by means of the grand angular momentum A and the total angular momentum J. Each of these subsets of the configurational wave functions forms an invariant subspace of the symmetry group 8, . Section III presents a systematic method of working out how many times and what kinds of irreducible representations of 8, are included in the representation acting on this space of wave functions specified by A and J. The results thus obtained for A I 4 are tabulated. I. INTRODUCTION
AND
SUMMARY
A number of pion-pion resonances have been found and it seems quite possible that in the near future other resonances consisting of a larger number of pions may be detected and a new region of “pion spectroscopy”’ may become of vital interest. In this connection it will be desirable to have a general kinematic scheme of describing the system of an arbitrary number of pions, in order that we can separate intrinsically dynamic properties of the latter from those which are independent of the detailed mechanism of the interaction. In the present work we should like to discuss symmetry properties of the n-pion wave functions. Since pions obey the Bose statistics, an n-pion wave function must remain unchanged under exchange of any pair of the pions. Such a totally symmetric wave function in the centre of mass system with definite values of the total energy, W, the total angular momentum J, its third component, Js , the parity, II, the total isospin, T, and the total charge, T3 , can be constructed by putting
@(Jf’, J, Jo , II; TTs) = C a&(W, id
Here fi( W, J, JB , II; sym.)
J, Ja , II; sym)Tj( T, Ts ; sym.)
denotes a configurational
Present address: University Institute for Theoretical 1 This name has been suggested by Dr. B. C. Magli6. 418
Physics,
wave function Copenhagen.
(1.1)
(function
SYMMETRY
OF
‘Il-PIONS
419
of coordinates or momenta) of n pions with definite values of W, J, J3 , H, as well as a definite symmetry property. By a “definite symmetry property” we mean here that this function should transform under any permutation of pions according to an irreducible representation of the symmetry group S, . When this representation has a dimension Q I 2, then the Q functions making up the bases of the representation are labelled by the index i. The second factor qj( T, TX ; sym.) represents, on the other hand, an isospin wave function of n pions with definite values of T and TZ , and with the same symmetry property as .$ . The index j has the same meaning for 7 as i has for & It is known that we can then find such a set of coefficients a;? , depending on explicit forms of the representation matrices, that the sum of the Q” products on the right hand side of (1.1) is invariant under any permutation of coordinates (or momenta) and charge of n pions simultaneously. Thus we obtain an over-all wave function which satisfies the Bose statistics. An arbitrary n-pion wave function with definite W, J, Ja , II, T, and T, , can be expressed as a superposition of the functions of the type ( 1.1)) if we have worked out a complete orthonormal set of wave functions for ti’s and ~,~‘s separately.* In this way we shall have obtained a complete orthonormal set of over-all wave functions. From ( 1.1 j we realize that the Bose statistics implies certain correlations between the charge properties and the space-time properties of the system. For instance, the charge branching ratios and the angular and momentum distributions of the decay products-when the n-pion system decays through strong interactions-are related to each other depending on the isospin, the total angular momentum (spin), and the parity of the system3 (3). The symmetry properties of the n-pion isospin wave functions4 were investigated in detail by Pais (4). In this note, therefore, we shall be concerned with the symmetry properties of the configurational functions. The relativistic effects are, in general quite essential in the n-pion problem.’ 2 In our previous works (1, 2) explicit expressions of 5%‘~ and vi’s for 3- and 4.pion systems are given in the nonrelativistic approximation. As is also seen from these examples, we need of course other indices, too; in order to specify them uniquely. 3 One of t,he simplest example of the correlation is the case of a neutral system of two pions. Depending on T = 2, 1, and 0, the branching ration ~0 G/T+ X- of the decay products is 2, 0, and f/i, respectively. For 2’ = 2 and 0, the relative angular momentum of the two pions is always even; for T = 1, the relative angular momentum is always odd. 4 Explicit forms of the 3-, 4-, and 5.pion isospin wave functions with definite 7 and definite symmetry are given in (I, 2, 6). 5 In a fully relativistic treatment of the problem, we can proceed, for instance, in t,he following way. First we construct a wave function of n pions, $ (W, J, Jc , T; kr , kz , . . . , k,) with definite values of W, J, JB , and T, but without any definite symmetry property, in t,erms of t,he momenta of n pions. $ will be found, for instance, by repeated application
420
GRYNBERG
AND
KOBA
Here we restrict ourselves, however, to certain particularly simplifying aspects of the n-pion problem which appear in the nonrelativistic approximation. In this limit we can namely define a nonnegative integer A as the eigenvalue’ of the magnitude of “grand angular momentum”’ introduced ’ by Smith (7). This value has a direct physical meaning as a generalized angular momentum. (See the examples given in Section 1I.C.) Moreover, it can be shown that the set of independent wave functions with a given value of A forms an invariant subspace of the symmetry group S, . Since the energy and the total angular momentum in the center-of-mass system are also invariant under any permutation of n pions, we can classify the wave functions by W, A, J, and JB (where A 2 J 2 1Ja 1 ), and each subset thus obtained makes again an invariant subspaceof S, . The main purpose of the present work is to find formulas for an arbitrary n, which show how many times and what kind of irreducible representations are included in the representation of S, acting on the subset of wave functions specified by W, A, J, and Ja . Actually they depend only on A and J. Generally speaking, not all the symmetries are included in a subset specified by A and J. This is the caseespecially with smaller value of A and A - J. On the other hand, not all the kinds of symmetry can appear, in general, in the functions v( T, T, ; sym) with specified T. (This does not depend on T3.) These circumstances lead, in view of the form of the over-all wave function (1) to certain selection rules, simpler examples of which are known individually and have been utilized. Although these selection rules are strictly valid only in the nonrelativistic limit, we expect that they will also help us in semiquantitative analysis at a not too high energy. Further, a similar mathematical procedure may be applied also in a fully relativistic treatment, although the physical meaning of the parameter is no longer so clear-cut as in the nonrelativistic case. In Section II we discussthe solution of n-pion Schriidinger equation. This part is not essentially new, but summarizes and generalizes known results (8, 9, 7, 1, 2, 3)) formulating them in a way adapted for our purpose and adding some of Wick’s procedure, which has been given explicitly in the case of three pions (6). Then we can make use of the known procedure of successive symmetrieation and antisymmetrization in order to obtain a function of a given definite symmetry. 6 To be precise, the square of the grand angular momentum tensor has eigenvalues A (A + 3n - 5). 7 In our previous work (1,s) we called il “effective angular momentum.” We have noticed, however, that this quantity had been systematically discussed by Smith (7) and others (except its relations to the symmetry properties, which are our main interest in (1, 2) and in the present work) and called by him “grand angular momentum.” So we shall follow his nomenclature.
SYMMETRY
421
OF 'WPIONS
examples. Section III presents a method of enumerating the possible kinds of symmetry appearing in the configurational wave functions when the latter are classified by the grand angular momentum A and the total angular momentum J. Explicit forms of wave functions ti in (l.l), with definite values of A, J, Ja , and lI, and with definite symmetry, will be given in a forthcoming work by one of US (;\I. G.j. II.
SOLUTION
A. ELIMINATIOX
OF
THE
n-PION
OF THE CENTER-•
SCHRODINGER
EQUATION
F-~/IASS ;\IOTION
Throughout this work we restrict ourselves to the nonrelativistic approximation and describe the system of n pions by a configurational wave function 9” . satisfying the SchrGdinger equation
& $ kj2 + V(x, , xz, . . . , x,,) ‘ko = wo\ko
(2.1)
with
In order to eliminate the center-of-mass motion we introduce a unitary transformation IT and go over to a new set of canonical variables, ri’s and pj’s.
ro ‘I
\
Xl
r1 I r2 = u “.” I ,
1:
!h-1 I
1&9
PO fkll Pl f/ pz = U I+ 1. li, 1 IPn-1,
(2.2)
Here we identify r. and POwith the center-of-mass coordinate and momentum, respectively.’ rO = L
(x1 + x2 + . . . + xn),
po =
+
d/n
n
(kl + k, + . . . + k,)
(2.3)
The rest of the variables, the relative coordinates, rl , rz , . . . , rnWl, and the relative momenta, pl , p2 , . . . , pn-l , can be defined in a variety of ways, each set being related to other ones through unitary transformations for the (n - 1) variables. The choice of a particular set depends of course on the physical prob8 These definition ventional formulation
of center of mass coordinates ones by factors &and l/d- 72, respectively. with respect to ri’s and pi’s.
and
momentum In this way
deviate from the conwe get a more symmetric
422
GRYNBERG
lem in which we are interested. definitions :
AND
KOBA
For example, one can make the following
simple
xi = s2 (XI - x21 r2 = &
1x1 + x2 - 2x3)
.. . (2.4) 1 {Xl
+
X2
+
’ ’ * +
Xj
-
jXj+l]
rj = dJj . . .
r,-l
=
dnci
_
1)
and the same transformation Another choice is: 11 +
{x1
x2 + - -. + xnwl -
+
(n -
1)x%]
from ki’s to pj’s.’
x2
+
* * *
+
Xj
-
jXj+l)
j = 1,2, * . ’ , h -
1
(2.5)
- h(Xh+l + .*- + xn>l
rh+i = dzc:+ 1) I(xh+l + -. . + Jh+z)- kb+z+1] 1 = 1,2, * . * , n - h -
1
and the same transformation for the momenta. This set of relative coordinates and momenta corresponds to a division of the whole system into two subsystems consisting of particles 1, 2, . . . , h, and particles h + 1, h + 2, . . . , n, respectively. The center-of-mass motion is eliminated by putting \k0 = exp (ipo.ro)Wrl where po is a constant
vector
representing
9 This corresponds to the following procedure. relative motion, then add to this two-particle motion of the latter with respect to the center forth.
, r2, . . . , r,-d the center-of-mass
(2.6) momentum,
and
First take two particles and specify their system a third one and specify the relative of mass of the two-particle system, and so
SYMMETRY
423
OF 'CPIONS
we get (2.7) with ps2=-v+-
.E+A!L+L (
w = wo-
&-$, )
22
21
(po%d
Here we have assumedthat V does not depend on r. , that is to say, no external field is present. B. SEPARATION ANGULAR"
OF THE PARTS
EQUATION
INTO
“GRAND
RADIAL'
AND
“GRAND
Let us define “grand radius,” R, and “grand relative momentum,” P, by n-1 iz
n-1 v-22 =
R2,
2
P:
=
(2.8)
P2.
and normalized relative coordinates and momenta by pi = ri/‘R,
ni = p/P
(2.9)
R and P represent, so to speak, an over-all relative distance and an over-all relative momentum of n particles, respectively, as is seen from the following relations.
(2.10)
In the following we confine ourselves to specifically simple caseswhere the interaction potent,ial V depends only on R. (Two examples will be given in the next subsection.) Then the Schrijdinger equation can be separated into a “grand radial” part, which depends only on R, and a “grand angular” part, which depends only on ~1, p2, . . . , en-1 (8). We put Wrl,r2,...,
r,-,)
= Z(R)Hy”-3’(~,
m, ... 9 en-1)
Here Hy n-3) is a harmonic polynomial of degree A in 3 (n it is homogeneousof degree A, so that Hii3”3’(aF;1 , a&, . . . , a&-~) = u*H~+~‘(&
(2.11)
1) variables (8) ;
, f2, . . . , i&-1)
(2.12)
424
GRYNBERG AND KOBA
and satisfies the (3n-3) -dimensional Laplacian equation,
(2.13) (RAHi3n-3)(~l,e2,
*** .en-dl
= 0
Then the Schrodinger equation (2.7) is reduced to an equation for the grand radial function Z(R) . d2Z im+
3JJ7 - &JT(R)
-
*(*
+i2n - 5)
Z = 0
(2.14)
>
The parameter A which appears in (2.11)-(2.14) is called (the eigenvalue of) the “grand angular momentum” (7).‘” This is a natural generalization of the relative orbital angular momentum of the two-body problem. C. GRAND RADIAL
FUNCTIONS:
Two EXAMPLES
In certain simple casesthe equation (2.14) can be explicitly solved. Two such examples are given in the following. It is seenthat they show features remarkably analogous to the well-known two-body cases. 1. System qf n Free Pions In this case V = 0, and the solution is given by ~+(3,-6,/2(PR)
+
bNA+C3n-&P@1
with P2 = 2/.lw,
a, b = con&.,
(2.16)
where J, and N, are the v-th order Bessel functions of the first and the second kind, respectively. If we require that no singularity is present at the origin, the second term must disappear and we get ZA(R> = R&
Jn+~3n-dPR)
(2.17‘)
which gives, when combined with normalized harmonic polynomials to be discussed below, a complete orthogonal set of n free particle wave functions normalized for continuous values of P. The physical meaning of the parameter A is now obvious: it represents the degree of penetration of n particles through the over-all centrifugal barrier. A = 0, 1, 2, *** ) correspond to generalized S, P, D, * . . , waves, respectively. lo See footnote
7.
SYMMETRY
OF
‘ai5
‘GPIONS
A quantitative measure of concentration of particles can be obtained by the relative probability of finding all the particles effectively within the over-all distance,” R 5 Ra . lrn / Z*(R)
1’ exp (-g)
lF4dR
2. System o;f n Pions Bound through Oscillator Potentials between Each Pair We assume,
or, with
the help of (2.10), v zz I/‘(R)
The eigensolution z dk
of (2.14) with .A)
= n?$R2
(2.19)
(2.19) is given by 2/&wR2)
(2.20)
The normalization
factor
= N. R” exp ( - $&“&R2)L~+C3’-s,12(
where Lka is the (generalized) N is given by
Laguerre
polynomial.”
l/Z
[r(k The corresponding W, = &i
+ A :!+(n
-
l))]”
>
(2.21)
eigenvalue of the energy is o(2k + A + j?$(n
= v% w{q •k x(n
- l)}
- l)] 4 = 2k + A = 0, 1, 2, . . .
(2’22)
Thus, for a specified energy level Q,the grand angular momentum A can take the values A = q, q - 2, q - 4, . . . , 1 or 0
(2.23)
eleI* In deriving (2.18) we have used IIrzi Sri = R3~ dR d& where & is the ‘kurface ment” of the 3(n - I)-dimensional hypersphere R = 1. The harmonic polynomials are so normalized that the integration over this surface gives 1. The function I, on the right hand side is the Bessel function of imaginary argument. I2 This function satisfies the following differential equation.
426
GRYNBERG AND KOBA
Equations (2.20)-( 2.23) represent a straightforward generalization of the three-dimensional oscillator (bound two-body system). D. GRAND ANGULAR FUNCTION; PROPERTIES OF THE HARMONIC POLYNOMIALS The harmonic polynomial Hk3%-‘) of degree A in 3(n - 1) variables, el , @2, . * . ) @n-l(c pi2 = l), has been defined by (2.12) and (2.13). In the simplest caseof two-body problem (n = 2)) it is reduced to a superposition of well-known surface harmonics,
Hi3’M
= ,=g* A(m) YA”h?l),
1el 1 = 1 (2.24)
where A(m)% are arbitrary coefficients with c 1A 1’ = 1, and we see that there are (2A + 1) linearly independent polynomials, specified by the value of m and here they are so chosen as to be orthogonal to each other and normalized on the unit sphere. Now we shall generalize these relations to arbitrary n. 1. Number of Independent Harmonic Polynomials There are (‘*
+ 3n - 5,
(A+3n-6)! A! (3n - 5)l
linearly independent harmonic polynomiaIs of degree A in 3 (n - 1) variables (8). In choosing them orthogonal to each other and normalized on the unit hypersphere, we find it convenient for our later purpose to specify them by means of the following set of indices: 11,
12,
..-
, L-1
;ml,
m2,
..a
,m,-I
; VI,
VZ,
*a-
, h-2
(2.26)
lj and mi represent respectively the magnitude and the third component of the angular momentum related to the relative coordinate ri and relative momentum pi . The physical meaning of vi’s will be clarified later. li and vI can take the nonnegative integral values which satisfy n-1 h
=
gl
n-2 li
+
2 gl vj
(2.27)
and rn( can take all the integral values within the limits -1i 6 mi s li
(2.28)
Enumeration of possible values of (2.26) with (2.27) and (2.28) leads to (2.25). 2. Decomposition of a Harmonic Polynomial In order to work out explicit forms of the above-mentioned orthonormal set of harmonic polynomials, we have to make use of a decomposition formula,
SYMMETRY
OF
H y3)(@,
@) . . . ) &-1)
X HYj’(el
7 ~2, . . . , ~j)HYT-33’-3)(ej+l,
=v*E”Ab,
427
‘WPIONS
A’, A”)N(j,
n - j -
1; A’, A”, v>
@jj+t, . *. 3 en-1 ) (2.29)
e3+1 + n-1 + ... + &-1 g @: ) &+2
where A’s are normalized satisfies
arbitrary A’
coefficients, +
A”
2v
+
v is a nonnegative
=
integer that (2.30)
A
and G, is the Jacobi polynomial13 of degree v. The formula (29) is written, for the purpose of later application, in a general form, so that it remains valid in a case CYL: pi2 # 1. The normalization factor N is given by [N(j,
n - j -
1; A’
A”
v)12
x
A’
A”
+
I’[v
f
+
=
(4v
+
2A’
(3% - 5)/2]
+
2A”
-I-
3n - 5)
I’[v + A” + (3n - 3j - 3)/2]
v!(I’[AN+(3n-3j-3)/2])2T
(
(2.31)
v+A’+$)
The fifth factor in (2.29), ( ~:ZI’ eie)“Gy, can be expressed as a homogeneous polynomial of degree v in (&+I + &+2 + e-e + &-I)
and
(et + ez” + ...
+ et),
thatistosay,ofdegree2vin]~I/,]p2j,~~~,I@,-11. A particular case of (2.29) is a recurrence formula suitable for application to the set of relative coordinates (2.4), Hi3n-3? el ,
e2
, . . . , en-d =
CA&-2,
L-1,
vn-2)N(n
-
2,
1;
L-2,
L-1,
X Hk3nn_-;@(el, e2, - * * , en-dH~i!,(en-l) X I3 The
Jacobi
Gv,-,(L-2
polynomial
+ L-1 + (3n - 5)/2, L-1 + 24; &-1) is defined
by
vn-2)
(2.32)
428
GRYNBERG
AND
ICOBA
with A,,
-I- L-1 + 2v,-n = An-1 = A
(2.33)
Another corollary to (2.29)) which is convenient for application to the set of relative coordinates (2.5)) where the whole system of n particles is decomposed first into two subsystems of h and (n - h) particles respectively, is obtained by using the formula twice.14 Hyn-3)(pl
, @2) . * * ) @n-l> = c
A(lh ) A’, A”, v, v’>
X N(n
-
2, 1; A’ + A” + 2v’,k,
x N(h
-
1, n - h -
x
(&(A’
+
A”
)( (1 - pt)“G,r(A
+
l,, +
v)
1; h’, A”, v’)
2v’ +
(h
-
5)/2,
+ A” + (3n - 8)/2,
h
+
%;
eh2>
A” -I- (3n - 3h-
<&+1+ *** +
3)/2;
@“n-1)/1 -
&>
(2.34)
Here v and v’ specify the way of dividing the grand radius, R (or the grand relative momentum, P) , into three parts: the relative distance (or the relative motion) of the two subsystems, and the internal distances (or internal motions) of the two subsystems themselves. lt, , A’, and A” are respectively the magnitudes of the grand angular momenta associated with these three motions.‘5 3. Orthonormal Set of Grand Angular Functions Specijied by A, li , mi , vi We can apply the recurrence relation (2.32) and (2.33) in succession, and finally arrive at a complete decomposition:
= zA(b,L, x
fly
n-3)(
e.. ,Zn--l,ml, Zl , - - - , L1
, ml
.e. ,m,-1, , . - . , m,-1
VI, *+* , vn-I)
, VI , * * f , vn-2
e1 ,
;
e2 3
* . . , en-d
(2.35)
I4 Equation (2.34) was used in the description of the 4-pion system, n = 4, h = 2 (9). 16 For the relative motion of the two subsystems, the grand angular momentum is of course reduced to the usual angular momentum.
SYMMETRY
OF
429
‘WPIONS
where16 H!p'(Zl
) . . . ,L
,rnl
) *-.
,m,-1
,Vl
, . . . ,vn-2
;@1,@2,
***
) en-1)
n-2
(2.36 XGvj
(
Aj+lj+l+(3j+1)/2,Zj+l+%;,
x ffi=l 1@iIZi YF
l2
+p2J’+1 2
. . . +
$+1
)
(fi)
with” Al Ai+1
=
11,
=
Ai
-I-
l&l
+ 215,
n-1
An-1 E A = glL
j = 1, 2, . . . , n -
+ 2Fl vi.
In this way we have worked out harmonic (3n - 4) indices,” Ii , mi , and vj ; they satisfy
polynomials orthonormal
s
. . ..vL2.p1,@2,
H
..-,Zk-l,ml’,
~!3n-3)(Z1',Zi,
2 (2.37)
n-2
--.,mL,s',
(2.36) specified by relations, .**,pn-l
4. Classification by A, J, and J3 We can apply further a well-known unitary transformation to the set (2.36) with the help of products 01 the Clebsch-Gordan coefficients, and obtain a new complete set of orthonormal harmonic polynomials which are specified by the total angular momentum and its third component, J and J3 , together with (n - 3) intermediate angular momenta, instead of m(‘s. rfi We have
used
an evident
I7 As is well known, Zi operators associated with construct operators whose these operators are given I* h is uniquely determined
relation,
and rni can be defined as the relative coordinates eigenvalues are just the in Smith’s work. when Z;‘s and Y~‘S are
eigenvalues of the angular momentum and momenta ri and pi We ran also parameters AC and Yj . Explicit forms of given.
430
GRYNBERG AND KOBA
(J,
Hk3"-3'
J3
; h
, 13 , 113 , 13 , 1123 , * * * , 623
1..
n-2
, k-1
, VI , . . . , h-2
;
el,ez,~~~,en-l)
= $a m,
C(J,
J3
; h , 12 , 112 , 13,
1123 , ”
’ , 1123 . . . n-2
, k-1
; li
, mi)
(2.39)
3
x H~“-3’(Z1,Z2,
...
,
1+I
, ml
, m2
, . . . , m,-1
, VI , . . . , h-2 @l,
...
;
, en-d
Notice that this transformation is relevant only to the last factor IIIyT(e
J1
20 J32
(2.40) -J
Each subset includes only a finite number of linearly independent polynomials, which can be made orthonormal by specifying by Zi’s, Zi, ...‘s. and vj’s. E. SYMMETRY OPERATION ON THE WAVE FUNCTIONS The representation matrices of the symmetry group S, , when acting on the original set of individual coordinates, z1 , x2 , . . . , xn (or momenta, kl , k2 , . . . , k,) , are obviously given, for example, by unitary matrices,
$02)
=
‘0 1 0 0
1 0 0 0
0 0 1 0
0 0 0 1
‘*a** *** e-0
0’ 0 0 0 ’
,o
0 0 0
-**
1,
‘Q(123)
=
(0 10 0 0 0
0
1 0 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
*** *** -*a --. -**
0’ 0 0 0 0
,o
d 0 0 0
*-*
1,
(2.41)
When acting on our new set of coordinates (or m_omenta,(2.2))) the symmetry operations are represented by unitary matrices p !g = @jju-’ (2.42) Since the first member of the new set of variables (2.2)) r. or _po, is completely symmetric with respect to all the particles, all the matrices g must have the following form: 1 f 0 0 -a- 0 ___,_-----------0 I g= (2.43) 0 I 2-J
SYMMETRY
431
OF 'TL-PIONS
where T stands for an (n - 1)-dimensional representation of X, , which acts on the relative coordinates (or momenta). It has been shown in a previous work (10) that this representation is identical with the irreducible representation (n - 1, 1). We may remark that the choice of relative coordinates (2.4) leads in this way to the conventional Young-Yamanouchi representation for (n - 1, 1) of s, . Since the grand radius R (or grand relative momentum P) is invariant under rotation and permutation, as is seenfrom (2.10)) the normalized relative coordinates pi (or momenta 7er)behave just in the same way as ri (or pi) with respect to rotation and permutation. As for the solution (2.11) of the n-pion Schrodinger equation, we need not consider the grand radial function Z(R), as long as we are concerned with its behavior under rotation and permutation. The grand angular part is expressed in terms of the harmonic polynomials, and when the latter have degree A, they are homogeneous polynomials of degree A in el , e2, . . . , en-1 . Therefore any member of Hifn-‘) goes over under any permutation to a linear combination of Hynw3”s with the same A. In other words, the harmonic polynomials with a specified degree make up an invariant subspace of S, . Moreover, it is evident that the total angular momentum, J and J3 , remain invariant under any permutation of particles. Consequently, our classification of the grand angular functions by A, J, and J3 , discussed in Section II. D.4, leads actually to a construction of invariant subspacesof rotation and permutation spanned by solutions of the Schrodinger equation. III.
NUMBER
OF CONFIGURATIONAL W, A, AND J, AND
WITH
WAVE DEFINITE
FUNCTIONS SYMMETRY
WITH
GIVEN
As is obvious from (2.11) and the invariance of R under rotation and permutation, the number of configurational wave functions with specified values of W, A, and J, and with specified symmetry property is determined by the number of grand angular functions with given A and J and with given symmetry. In this section we shall be concerned with a general method of working out the latter number. A. SYMMETRY
PROPERTIES hI0ikfEN~A'~
OF PRODUCTS
OF THE RELATIVE
COORDINATES
OR
1. We have already mentioned in last subsection that the relative coordinates ri or pi (i = 1, 2, . . . , n - 1) transform under permutation according to the irreducible representation (n - 1, l} of S, . In order to extend this result we shall now investigate the symmetry property of the tensors formed by products of various components of these relative coordinates. they
I9 All the arguments in this subsection are apply equally well to relative momenta.
given
in terms
of relative
coordinates,
but
432
GRYNBERG
Let
US
denote the three spherical
AND
KOBA
components
ri + = - ri= ~+ i?“i” ,
riz ~z - h-i” ,
ri- =
where Tiz, ri’, and rrE are the Cartesian
of ri by
components;
ri’ = ri
(3.1)
and correspondingly,
pi+ = -
pio =
piz
(3.2)
which satisfy n-1
z
i (Pa2
-
2Pi+Pi-J
=
(3.3)
1
The grand radius is expressed in a similar way, n-1
R2 = sz { (Y:)~ - 2ri+ ri-j
(3.4)
Now we introduce a linear space L6,^,’ spanned by the product of A elements taken from (3.1), out of which a are + components, b are - components, and c are 0 components. Thus A = a + 3 + c, and L% has the dimension (n+;-2).?+;-2).(n+:-2)
(3.5)
Obviously L$ , L% , and L$ represent the linear spaces spanned by (n - 1) elements of +, -, and 0 components, respectively. Examples, (n = 4). L (1) _ 100
L
(2)110
L
(2)200
-
-
IT1 +, r2+, IT1 {rl
++
r1 rl
, rl +
, n
r3+}
+r2 +
r2
, rl +
, rl
+r3 +
r3
, r2 +
++
f-1 , r2
f-
r2
, r2
+-
r3
, r3
+-
7-1 , r3
+r2 -, r3+r3-)
(3.6)
+
, r2 r2 , r2 + r3+, rs+r;.3+j
As is seen from our argument in Section II. E, the linear space LLt: is an invariant subspace of the symmetry group S, . Then our problem can be formulated as follows: Knowing that each of Lii& , L% , and Liti makes up basesof the irreducible representation (n - 1, l} of S, , to find out how many times and which irreducible representations of S, are included in the representation acting on LLad,’. 2. Each monomial in the Lb&)-space is symmetric in its a factors with indices +, b factors with indices - , c factors with indices 0. (See the last example of (3.6) : in L$i we do not discriminate between rl+ r2+ and r2’ rl+.) In other words, we form ath, bth, and cth order symmetrized product of L%, Li$ and L$Ai, respectively, which we shall denote by [L$]“, [L$lb, and [%I”; then we make their direct product and in this way construct the linear spaceLit: .
SYMMETRY
OF
433
?&-PIONS
We shall show how to find out all the irreducible represetations of S, acting on [L$]“. Then we can of course apply the same method to [Lh:&]b and [L$]“. For this purpose we have to decompose the character [xl” of the representationin general, reducible-acting on L$i into a sum of the characters of the irreducible representations of S, . The character [xl” can be found with the help of the formula” (1%‘) (3.7? where g represents any element of the symmetry group S, , and x (g) is the character of the irreducible representation (n - 1, 11 acting on LiAi . The summation is taken over all the possible ways of dividing a in the form a = a1 w1 + ffz w2 + . . . + LY”w,
(3.8)
with 0 < Wl < wz < . * * < w, LY,w
positive
integers
For example, when a = 2, the possible ways of division
(3.8) are
2 = 1.2
(a1 = 1, Wl = 2)
2 = 2.1
(01 = 2,wr
= 1)
= I,@ = l,wz
= 2)
and when a = 3 (cry1= l,w,
3 = 1.1 + 1.2 3 = 1.3
(a1 = 1, WI = 3)
3 = 3.1
(a1 = 3, w1 = 1)
in this way we obtain the following
formulas.
[xl” (9) = ?4x (8) + %x2 (9) [xl” (9) = %x (g3) + Mx (2) x (9) + %x3 (9) 1x1”(9) = xx (g4) + 36x b3) x (9) + %x2 (8) + xx (g2!x2 (9) + %*x4 (9)
(3.9a) (3.9b) (3.9c)
3. When the element g of S, represents a permutation of n objects which includes k, unchanged objects (one-term cycles), k2 pairs of exchanged objects (two-term cycles), k3 trios of objects cyclically exchanged (three-term cycles), *O We are indebted products.
to K. Olbrychski
for
helpful
information
concerning
the symmetrized
434
GRYNBERG
AND
KOBA
and so on, we can show that, for the irreducible
representation
{n - 1, I}, (3.10a)
Xln
- 1,ll
($7)
=
h
-
1,
X(n
- 1, II (g2)
=
kl
+
2k2
X{n
- 1.11
(g3)
=
kl
+
3k3 - 1,
X(n
- 1. 11 (g4)
=
h
+
2k2
-
+
(3.10b)
1,
4k4
(3.1Oc) -
(3.10d)
1.
The formula (3.10a) was proved previously (10) ; the other formulas can be obtained from this by noticing that the number of one-term cycles (unchanged objects) in g2, g3, and g4 are given respectively by k, + 2kt , k, + 3ka , and kl + 2kz + 4k4 . Example. Let g represent the permutation A A
BCDEF c B E
p
D
= (A)(BC)(DEF)
>
of Se . ( kl = 1, k2 = 1, k, = 1, k4 = 0). Then by direct calculation we find g2 = (A) (B) (Cl (DW, g3 = (A) (BC) CD) C-W (F) and g4 = (A) (B) (0 (DEF) . Thus we get X16,
11 (9)
=
X(6,11
0,
(g2)
=
2,
X(6,11 (g3) = 3,
X(5,
11 (g4)
=
2
Now we put (3.10) into the right hand side of (3.9)) and, with the help of the general expressions of the characters of the irreducible representations of S, , listed in Appendix 1, we can perform a decomposition of the character [xl” for a $ 4 into a sum of the characters of the irreducible representations. (3.11a)
[xl” = Xlnl + XIn-l,ll + Xk-79.~ , [xl3
=
Xlnl
+
[xl” = 2xbd+ +
X(n-3.31
2Xlr+l,ll
+
3x1,-1,~
+
Xb-2.1.11 +
xi+3,2,11
xh-2,1,1)
+
+
X1+-2,2) +
Xln-4,41
+
Xin--3,3)
3x1+2,21
,
(3.11b) (3.11c)
.
The above general formulas are exact when n 2 8. For smaller values of n some modifications are necessary. 4. The characters of [L$]* and [I$:Jc can be obtained by the same formulas as (3.11). The final step is to construct the direct product of these three spaces in order to find out how the character xCabc)of the representation acting on LL,^) is decomposed into a sum of the characters of the irreducible representations. X
(labc) (9) = [Xi?+I,J (9) *txb41b (9) ~[XLn--l,I~IC (9)
(3.12)
With the help of (3.11) the right hand side of (3.12) for a, b, c 4 4 is expressed as a sum of products of the characters of the irreducible representations of S, .
SYMMETRY
OF
n-PIONS
Each of these products can be further transformed into a sum representations by repeated application of the “multiplication irreducible representations, which are listed in Appendix 2. Thus we are now in a position to write down how many irreducible representations are included in the representation Lb,^,’ up to h $ 4. 5. Examples. ( 1) Monomials ri+ ~j+, j-i- ).j-, rLO t.,” (i 2 j) the spaces
In this case it is sufficient representation acting on representations (n) , {n (2) Monomials Ti+ rj-,
The representation (n - 1, 11 X {n (n - 1,l)
43 5
of the irreducible formulas” of the times and which acting on any form respectively
to use the formula (3.11a) and it is found that in the each of the spaces L$, L%, and LA%, the irreducible 1, 1) and (n - 2, 2) are included, each once. ri+ rjO, vi- pjo form, respectively, the spaces
acting on each of them is the product representation 1, 1). From the formula given in Appendix 2, we have
X (n - l,l]
= {n] + (n - 1,l) + (n - 2,2) + (n - 2, 12)
And we find that in the reducible representation acting on Lifi, I,$ or L% the irreducible representations (n], (n - 1, l}, (n - 2, 2} and {n - 2, 1’) are included once. For spaces of monomials of higher order the results are more complicated because we have to apply the reduction formulas more than once. B.
RELATION
BETWEEN
L$
AND
Hy"-3'
In the foregoing subsection we have examined the symmetry of the linear product space of ri’s. But this space L6:? is intimately related to the linear space spanned by the harmonic polynomials Hy n--3).In fact we can show that by using gi% instead of ri’s, that is to say, taking explicitly the condition (3.3) into account, the space Lib”,’ is reduced to the space Hyne3’. In order to see this we notice first that all the harmonic polynomials (2.36) can be expressed as sum of products of pi+, pi-, and pi’, which belong to Lit? with a - R = c;Z: 112~ . By the definition and orthogonality of (@” + . . . + p;+J’jGyj
(
. . ., . . -,
&+I e1” + . . . + &+I )
the identity (3.3) is taken into account here. On the other hand, the number of independent harmonic polynomials of degree
436
GRYNBERG AND KOBA
A is equal to the number of independent polynomials inxhxa+bti L6b^c’when the identity (3.3) is applied. The latter statement can be proved as follows. Denote the number of independent polynomials in C.L~+~+~ L$j (A fixed, but without any further condition imposed) by u(A), and the number of independent polynomials in &aSb+c L2, aft er CZ: ((pi’)* - 2 pi+ pi-) is put equal to one, by ~p(A). Then we have U(A) =(p(A)
+q(A-2)
+p(A-4)
+ ...
-I-p(lorO)
(3.13)
where ( 1 or 0) means0 if A is even and 1 if A is odd. From this we get immediately p(A) = u(A) - cr(h-- 2)
(3.14)
But it is known (8) that a(A) =
A+3n-4 A
>
(3.15)
From (3.14) and (3.15) p(A) = (2A + 3n - 5) (’ + 3n - 6)! A! (3n - 5)!
(3.16)
which is identical with (2.25)) the number of independent harmonic polynomials of degree A. C. NUMBER OF INDEPENDENT J, AND SYMMETRY
GRAND ANGULAR
FUNCTIONS
WITH DEFINITE
A,
On the basis of the results obtained in the foregoing two subsections, we shall now work out the number of independent grand angular functions (see (2.11) (2.13)) which have specified values of A, J, and Ja (see (2.35)-(2.39)), and which behave under permutation according to a specified irreducible representation of X, . This number will be denoted by % (A, J, Ja ; sym.), A L J, -J 5 Ja 6 J. Actually VI is independent of Ja , since neither A nor the symmetry depends on the spatial direction of the total angular momentum. Thus (32(A, J, JB , sym.) = % (A, J; sym.). 1. We have to rearrange the linear spaces L6i? into those corresponding to definite values of J and JB . For this purpose define first L(b).r3 = &
L2
(3.17)
LA = ,,&
L(“)=Q
(3.18)
and further
The assembly of the functions with the same A provides a set of basesfor a
SYMMETRY
OF
437
TGPIONS
representation-in general, reducible-which is equivalent to the one acting on the space L”. This equivalence follows from the fact, which we have mentioned in Section III. B, that there exists a linear transformation which expresses each HA by monomials and that this transformation is nonsingular because HA’s form a complete set and each monomials can be inversaly expressed by HA’s. Thus we can divide L” into subspaces with definite J and Ja and from these we can find % (A, J; sym) by a procedure described in the following. 2. We begin with the “most stretched” space LcA”. This of course corresponds to the largest possible value of JS for the fixed A, and the only value of J compatible with it is J = A. Thus by finding the number X of a certain irreducible representation included in the representation acting on I,‘A’A with the help of the conclusions of the subsection 1, we get the answer for J = A.
%( A, J = A; sym)
= 3z(L’*‘“;
sym)
= X( Lk%; sym)
(3.19)
h’ext we consider LcA)‘-‘, which will include the functions belonging either to J = A, Ja = A - 1, or to J = A - 1, Ja = A - 1. But the first of them is known by our above result. Thus, %( A, J = A - 1; sym)
= % (L’*‘“-‘;
sym)
- 31 (L’“‘*;
sym)
= 32.(L~,o,I;
sym)
- fn(LE0;
sym)
(3.20)
Sirnt’n&rry, we would expect to get % (A, J = A - 2; sym) by subtracting ; sym) from X( L’“‘A-2; sym). Due to the presence of the identity(3.3), WL however, not all the monomials included are independent when 1JBl 5 A - 2. The number of redundant monomials is equal to the number of independent monomials in LcAp2jJ3, LcnmoJ3, . . . , L’“-29’ J3, where Q is the largest integer which satisfies J, 5 A - 2q. The elimination of redundant numbers can be carried out along the same line as in (3.13) and (3.14), and we obtain
5X( A, J; sym)
= x(L’*‘~;
sym)
- X(L’“-2’J;
- x(L(*)‘+~;
sym)
sym)
+ X( L”‘-2’ ‘+l; sym)
(3.21)
Here we make a convention X(L (*jJ3; sym)
= 0
if
lJsl > A
(3.22)
and then (3.21) is quite general and includes (3.19) and (3.20). The results worked out in this way up to A 5 4 are listed in Table I. 3. Examples. (1) The monomial of order A = 0 is constant, which is of course invariant under rotation and permutation. So in this case we have
X(0,0; For all the other symmetries
In)) = X(L’“‘o;
x (0, 0; sym)
In])
vanish.
= 1
438
GRYNBERG
AND
TABLE J
0
0
@I
1
1
(72 -
1,l)
2
2
inI,
(1 -
1
{n -
2, lP}
4
I
Irreducible
A
3
KOBA
&2)
Sn2) (n -
in -
representations
2, 21
1, 11, tn -
0
(1 -
3
(1 - CM {n), (2 - an2 - h) (n - 2, 12}, (n - 3, 3)
2
(1 -
1
(1 - S,z) in), (2 - 2 &a - h3) (n - 1, 11, (2 - W (2 - h3) tn - 2, 121, 1% - 3, 31, b - 3, 2, 11
0
(n -
4
(2 - Se2 - &3) (n), (3 - 3 6,,2 - L3 - 6,4) {n - 1, 11, (3 - srr4 - 6n5) X In - 2, 21, (1 - L3) In - 2, 121, (1 - 8,) (n - 3, 31, {n - 3, 2, 11, {n - 4, 41
3
(2 - 6n3 - 2 6,2) (n - 1, 11, (2 (3 - 2 k3 - hd X in - 2, la], (n - 3, la), (n - 4, 3, 11
ann2) (72 -
1, I},
(n -
1, l),
(n -
2, 2) (n 2, 2),
1, l), (1 -
6,~)
(1 (n -
k)
(n 2, 12),
2, 2), (n -
1% -
2, 21,
3, 13)
2 -
h) (n - 2, 21, (2 - h5) ln - 3, 2, Ii,
in -
2
(2 (6 (3 (n
1
(2 - 2 6,2 - 6,3) ln - 1, 11, (2 - 6,4) (n - 2, 21, (3 - 3 L3 - 6d X (n - 2, l*), (n - 3, 31, (3 - 6,~) (n (2 - Bn4) (n - 3, 13), (n - 4, 3, 11, In - 4, 2, l”1
3, 31,
,sn2 - bn3) In), (5 - 5 Bn2 - 3 8,~ - 64 In - 1, 11, 3 bn4 - 6,J (n - 2, 21, (4 - 3 &S - &d in - 2, I”\, 6,d X (n - 3, 31, (4 - &J (n - 3, 2, 11, {n - 3, l”1, 4, 22), (n - 4, 3, l), (n - 4, 4)
3,2,
11,
(2 - 2 fin2 - sn3) (n] , (4 - 4 6,~ - 3 bn3 - 2 I,4 - 66 - W In - 1, 11, (5 - 3 &A - 2 6,s - 6,~) In - 2,21, 2 tn - 3, 2, 11, (1 - an3) (n - 2, 12) (2 - h) {n - 3, 31, (n - 4, 27, (n - 4, 41
0
(2) Case of
3, 2, 1)
A
= 2. We start with
Ja = 2 space:
L(Z)2 =; L(2) 200 and from Example 3-L(L’2’2; {nf)
1 of Section III. A.5 we have = 3z(L’2’2; (n - 1, lj)
= 3qL’2’2; (n - 2,2))
= 1
SYMMETRY
and for all the other symmetries tractions J therefore X(2,2;
OF
x(L’~‘~; sym)
439
‘GPIONY
sym)
vanish.
Here we need no suh-
= X(LC2’*; symj
In order to find ~(2.1; sym) we consider the space LC2” = L:i:. According Example 2 of Section III. A.5, 3x( LC2” ; {n) j = x(L’*“;
(n -
to
1, 1)) = 31(L’2”; (n - 2, 2) )
= 37@2”; (n - 2, 1’1 Subtracting from these numbers the above obtained x(L”‘~; X(2.1; (n - 2, l”]j
symj’s we obtain
= 1
and for all the other symmetries X( 2.1; sym) = 0. Finally we consider L’*” - space L cao= L:;i + Lg: From Examples 1 and 2 of Section III.
A.5 we have
51(L’2’o; (n}) = 3Z(L’2’o; (n - 1, 1)) = X(L’2’o; {n - 2,2)) = 2 3Z(Lc2’O;{n - 2, 12)) = 1 Subtracting X( L(““; sym)‘s and also, according to (3.21), X( L’““; from the above numbers, we get X(2,0;
(n - 1,l))
= X(2,0;
{n) j = 1
(n - 2,2) ) = 1
while for all the other symmetries X( 2.0; sym) vanish. APPENDIX
I
The character X(X) for an irreducible representation, corresponding to partition{X1,Xq, ... , X,), of an element of the permutation group S, characterized by k, , k, , . . . , k, (numbers of 1 term, 2 term, . . . , n term cycles included in it), is given by the coefficient of r:’ $ . . . r”,” in the expression (11) :
fj (Xi- 5) Ii1 (2 XiP)“” i=l
where k = A, + (n - q), Following this prescription we find: X{n1 = 1, Xjn-l,ll
=
k,
-
1,
(1 =
I,?,
...
.!I
440
GRYNBERG
AND
- h(kl
-
1) + k3,
+ kdkl
-
1) +
KOBA
(kz- 1) + 0~4 - I), XP-3~1) Xl~4,2~)
XIn-4,3,1)
X(n-4,2,12)
= %kl(kl
- 2)(kl
= Mzh(kl =
h 0 2
= kl(kl
-
- 4) -k3,
l)h
- 4)(k1
- 5) + kdk2 - 2) - k3(kl -
. (3 - h + kd + 3 (2)~+(;)+k,(l-iii), - 2) -
3 (;)+3(2)-(;)+ii,+kd&(;). APPENDIX
2
Using the expressions for the characters of irreducible representations Appendix 1, we find the following formulas for the direct products: {n - 1, 1) x (n -
1, 11= (n) + (1 +
{n -
I),
1, lj x {n - 2,2)=
(n
-
(n - 1,l)
h&2>
(n
-
1,
1)
+
(n
-
given in
2,2}
2,1’} + (1 -
6,4)
(n - 2,2j + {n - 2,l’j
+ (n - 3,3} + in - 3,2,1} (n - 1,l)
x (n - 2, 12)
= (n - 1,l)
+ {n - 2,2) + (1 -
+ (n - 3,2,11
S,,)
{n - 2, 12}
+ In - 3,1”1
{n - 1, 1) x {n - 3,3} = (n - 2,2) + {n - 3,2, 11 + {n - 4,3, lj + (1 - &A) In - 3,31 + tn - 4,4j 1% - 1, lj X {n - 3,2, 1) = (2 - S,,) (n - 3,2, 1) + (n - 4,2’} + (n - 3,3} + (n - 3, l”] + {n - 4,2,1’]
SYMMETRY
OF
441
WPIONS
+ {n - 2,2) + In - 2, l”] + {n - 4,3, 11 jn - 2,2) x (n - 2; 2) = {n) + (1 - S,,) {n - 1, l} + (1 - s,,)jn
- 2, l’}
+ (2 - &4 - 6,5) jn - 2, 2) + (n - 3, l”} + jn - 4,27 + (2 - 6,s) (n - 3, 2, 1) + (n - 4,3, 11 + (1 - &It?) (n - 3,3! + {n - 4,4) If, for given n, any partition
is impossible,
one has to substitute
zero for it.
ACKNOWLEDGMENT One of us (Z. K.) expresses his gratitude to the Polish L. Infeld, and Professor M. Danysz for their hospitality RECEIVED:
Academy of Sciences, Professor during his stay in Warsaw.
June 18, 1963 REFERENCES
1. Z. 2. M. 3. M. 4. A. 5. M. 6. G. 7. F. 8. A. 9. 10. 11. 22.
L. M. H. G.
KOBA, Phys. Letters 1, 34 (1962); Acta Phys. Polon. 99, 103 (1962). GRYNBERG AND 2. KOBA, Phys. Letters 1,130 (1962); Acta Phys. Polon. 23,503 (1963). GRYNBERC AND Z. KOBA, Proc. 1962. High Energy Conj. CERh’, Geneva, 1962, p. 178. PAIS, 9nn. Phys. (A;. Y.) 9,548 (1960); Ann. Phys. (‘$7. Y.) in press. GRYNBERG AND 2. KOBA, Bull. Acad. Polon. Sci., in press. C. WICK, Bnn. Phys. (N. Y.) l&65 (1962). T. SMITH, Phys. Rev. l!XJ. 1058 (1960). ERDELYI, W. MAGNUS, W. OBERHETTINGER, AND F. G. TRICOMI,“ Higher Transcendental Functions,” Vol. 2, Chap. XI. McGraw-Hill, New York, 1953. M. DELVES, Nucl. Phys. 9,391 (1958-59). GRYNBERG AND Z. KOBA, Nucl. Phys. 42, 313 (1963). WEYL, “Gruppentheorie und Quantenmechanik.” Hirael, Leipzig, 1931, J. LYUBARSKI, “Group Theory and its Application in Physics.” Moscow, 1957 (in Russian).