Commun. math. Phys. 10, 231--244 (1968)

On the Reduction of n-fold Tensor Product Representations of Noncompact Groups

W ~ n L t ~ H. KLts~ and GRA~VILL~ J. S ~ Department of Physics and Astronomy, The University of Iowa

Received3larch 29, 1968

Abstract. The reduction of n-fold tensor products of induced unitary represen- tations of noncompact groups into irreducible constituents is shown. CIebsch- Gordon coefficients are then calculated. The technique is applied to the n-fold tensor products of the positive mass representations of the Poincar6 group.

Introduction

Noncompact groups have been used in elementary particle physics in a number of ways. The classic use of a noncompact group was Wigner's anMysis of the irreducible representations of the Poincar6 group; more recently the Lorentz group has been used in connection with Regge poles and current algebras. Higher symmetry groups containing, for example, the Lorentz group and some internal symmetry group are also currently under investigation.

The problem which we wish to discuss in this paper is the reduction of n-fold tensor products of representations of a noncompaet group G into its irreducible constituents. Letting [Z] label an irreducible represen- tat ion of G and {x} denote a set of eigenvalues of a complete set of com- muting observables chosen from. G, we wish to find the coefficients which reduce the tensor product I[Z1] x1> I[z~] x~>... I[z~] x~> knto irreducible basis elements I[Z] x; ~> where U denotes the degenerary parameters which label irreducible subspaces of the tensor product space with the same [Z].

The reduction of n-fold tensor products is of course not a new problem. The most obvious way to carry out the reduction is to reduce the tensor product of two representations and then apply the two fold reduction n times in a stepwise fashion. For the Poinear6 group this reduction has been carried out by MAcFA~L~E [1 ] using the two fold reduction given by JAco]~ and WICK [2]. The big disadvantage of this stepwise reduction technique is its assymmetry. For compact groups such as SO(3) it necessitates introducing 3 J , 6 J , etc. symbols which connect different stepwise reduetions.

232 W.H. KLnVK and G. J. S~rr~:

What is actually needed is a reduction process which is symmetric in as many variables as possible. GOLDB~G [3] and W ~ L E [4] have begun such a program for the Poincar6 group while L~vr-LEBLO~D [5] and Lv~cA~ [6] have utilized the special properties of the Galilean group to effect a symmetric reduction.

ROrFM~X [7] has given a completely symmetric method for carrying out n-fold tensor product decompositions if the two fold tensor product decomposition is known. What we wish to show in this paper is tha t for groups having unitary irreducible representations which can be written as induced representations, a symmetric n-fold tensor product reduction can be carried out with no knowledge of the two fold tensor product reduction needed. In Section I a general discussion of the technique used to obtain Clebsch-Gordan coefficients will be given, while in Section I I the technique will be applied to the Poincar6 group. In the process of carrying out the reduction all of the invariants which parameterize the irreducible subspaces are automatically found and since the reduction is symmetric, a highly symmetric choice of degenerary labels can be made.

I. General Technique for Obtaining Clebseh-Gordan Coefficients

In this section Mackey's theory of induced representations [8] will be used to obtain Clebsch.Gordan coefficients resulting from the decomposi- tion of tensor product representations of a group G. An induced represen- tation of G is built from a representation 2F of a subgroup ~ of G and defined to be U(go) ](g) =/(ggo) [8] where go is an element of G and ](g) are functions which map g, an element of G, in.to the vector space V (~f) upon which 5¢F acts. The set of functions /(g) is restricted to those satisfying the condition/(hg) = ~ ( h ) 1(g) for all h in ~ and g in G. This restricted set of functions forms a vector space

V(gt~) = {/]/(g) E V ( ~ ) , / ( h g ) = ~¢t"(h) /(g) Y h C g) and g C G}. (1)

The induced representations act on the vector space ~?(~f). We shall consider only unitary huduced representations which implies tha t ~ ( ~ ) is actually a Hilbert space [8].

Now G can be decomposed into right cosets

S = U ~go (2)

where the elements g~ of G label right cosets. With this decomposition the functions /(g) may be thought of as functions over right cosets ] (g0), since the set of functions /(g) are restricted to Chose functions satisfying the condition ] (hg) = ~ (h) ] (g).

Consider now induced representations on n Hilbert spaces t?(gt°l), (~f~) , . . . , ~ (~f~) with norms

li/(g0~)ll ~ = j" dgo, [/(g0~)l ~ < o 0 . (3)

n-fold Tensor Products 233

The tensor product space I?(J(¢~) × l ~ ( ~ ) × • . - × I~(~f~) is the Hfl- bert space of functions F(g~I, go2, . . . . g~) with norm

IIF (go1, go~, • • -, g~)!t ~ = f dgo, g g o 0 , . . . , dg~, IF (g~1, - - -, g~) i ~ < ~ (4)

and the induced representation on this space is

U(go) F(gc~, g~, • • . , go,) = F(g~lgo, g~go, • • . , gc,~go) • (5)

The tensor product space is reducible, and in fact the problem is to reduce this space into a direct integral of hTeducible subspaees. The tensor product functions will be decomposed in two steps. Mackey has shown that a partial decomposition of the tensor product space can be obtained through a double cosec decomposition, since induced represen- tations defined on the tensor product spaces are equivalent to induced representations defined on subspaces labelled by double eosets.

Let ~ ' and ~ " be subgroups of G. By a double cosec is meant the subset of G consisting of all elements of the form O ' g 0 " for fixed g in G. A set of elements gD iu G can be found such that the union of a set of double cosets covers G:

G = U '2) 'g~)O". (6) D

In order to define the subgroup which induces representations on tensor product functions it is convenient to consider the outer product group of G, the set of ordered pairs of elements ((gl, g~ . . . . . g~)} = (G1, G~ . . . . . G~) with gl, g2 . . . . , gn arbitrary elements of G. Then the subgroup inducing representations on the tensor product space [Eq. (4)] is (~1, ~2 . . . . . ~)~) with ~1, ~2 . . . . , ~ subgroups of G [9]; a right cosec decomposition of (G1, G2, . . . , G~) with respect to (~1, ~2, • •., ~ ) is

( G , G~ . . . . , G~,) = u ( 0 . ~ . . . . , ~ ) (g~, g ~ , . . . , g ~ ) . (7)

If the outer product group is restricted to the diagonal subgroup (G, G , . . . , G), the ordered set of elements ((g, g , . . . , g)}, then the representations induced by the subgroup ( ~ , ~2 . . . . . ~ ) can be reduced into a direct integral of induced representations which ace on subspaces of the tensor product space (4) labelled by double cosets.

The double cose~ decomposition of (GI, G 2 . . . . , G~) is then

(G1, G2 . . . . . en) ~- jO1DIU..D, (~1, ~2 . . . . , On) (s)

• ( g ~ , g ~ , • - -, gD~) (G, G , . . . , G)

with (gD~, . . . . gD.) the elements labelling the double cosets; a direct integral decomposition of F ( g c ~ , . . . , go,,) using the double cosets is

H F ( g ¢ ~ , . . . , g~,)lI 2 = f d n x d D ~ . . , d n ~ i[/~ . . . . z), (G/0~)H ~ (9)

234 W.H. KL~-K and G. J. S~T~:

where/D .... 1)~ (G/~I)) are square integrable functions in the subspaces D1 . . . . . D~. OD is the subgroup which induces representations on these subspaces and is defined as [8]

8 ~ = (g~, . . . . g~) -1(~1 , ~ , . . . , ~,~,) (g~l, • . . , gD~) ~ ( ~ , G . . . G). (10)

The induced representations on these subspaces are generally reducible; the final reduction into irreducible subspaees cannot be further carried out because the method used is dependent on G.

Assume however that the decomposition of F ( g c l , . . . , gc,~) into irre- ducible subspaces can be carried out and that/tz,~l(gc) is an element of the irreducible subspaces with norm

I]f[x,,](g~)Ii2 = f dg~ i[[z,,l(g~)lg' < c¢. (11)

I tere Z denotes an irreducible representation of G while ~ is a set of degeneracy parameters which label mutual ly orthogonal subspaces carrying the same irreducible representation labels. The direct integral decomposition of F ( g c ~ , . . . , g J over [Z,V] with weight function

d [ z , V] is tlF(ge~, . . ., ge,~)H ~ = f d [ z , V] ll]-[z,n](g~)lt 2" (12)

Now ]Ix,,] is an element of the Hflbert space (11) so the inner product of/[z, ,l (9e) and/~xl is welt defined:

(l~xl (g~), /:z, ,: (go)) = f dgo /:z*~ (g~) /:z,,: (go) (13)

where S is the operator which carries F (go,, • • -, go,) to the irreducible subspaces from the tensor product space ? ( ~ 1 ) x . . . l?(~n).

In order to define Clebseh-Gordan coefficients we want to make a connection between the functions/[xl (g*) and basis elements [[Z] x), since Ctebsch-Gordan coefficients are the coefficients which reduce the tensor product of basis elements [[Z1] xa> ][Z~] x2>" • • l[Z,,] x.>; x, de- notes eigenvalues of a complete set of commuting observables chosen from G. Now the non-square integrable functions over cosets

Ix] Dz,~(gc) -- <Ix] x't U (g~) i[Z] x> (14)

can be considered as a specific realization of the basis element [[Z], x>, since u (g ) D~,~(eo) 2) ~ " (15) = ~'ztg~g) • Hence, we choose l~ (go) ~ D~,~ (go) (16) and

F (gc~, . . . . gc,~) ~ L,~:~ ~cl) • • " ~ x ~ tge,) • (17)

Eq. (13) can then be written as

<[Z] x; ~ i [~6] x~; - • • ; [Z-] x~> = N f dgo ~'~)fx]* (g~) r,~ ~[z,] [x~] (18)

• L ' ~ ' 4 , , (g , ,1) • • • D d , ~ (g~)]

n-fold Tensor Products 235

with N a normalization factor. Eq. (18) may be used to calculate the Clebsch-Gordan coefficients provided S is known and Ix] Dx.x(gc) can be calculated.

IL Clebseh-Gordan Coefficients for the Positive Mass Representation of the Poineare Group

In this section we calculate the Clebseh-Gordan coefficients for the positive mass representations of the Poincar6 group P. The covering group of P will be used throughout this section; however we will not discuss its properties, since they have been discussed in detail else- where [10].

One can easily check that the matrix form of the covering group of P can be written as

p = ( 0 A H(a) A-l+] A_I~ / (19)

where A is an element of SL(2, C) and

H (a) = (at + a~ az -- iav t (20) \a~ ~ ia v at a~ / "

A right eoset decomposition of P with respect to the inducing subgroup

S U(2) / (21) can be written:

p U (0SU(2) H(a)SU(2)~ A;~+) (22)

where A = S U (2) A~ and Ac is any choice of right cosets of A with re- spect to S U(2).

Then, square integrable functions over right cosets, denoted by ~j(A~), have norm

+s tI~j(Ao)II ~-- Z f d A ~ I~j(A~)[ 2 < ~ . (23)

The summation over ] comes from the fact that the inducing subgroup has 28 A- 1 dimensional representations.

The induced unitary irreducible positive mass representations of P are

U(Ao, ao) ~ j (Ac )~ (~i [(0 A¢ O~1+)(0 A° H(ao)

= e i~'Aca° ~ f)~] (S~(2)) ~)i,(Ac,) y~ -8

with S U (2) and A¢ defined by A~Ao = S U (2) A~,, and/3 = .

236 W.H. KLI~K and G. J. SMITtt:

A nonnormalizable basis element of P with mass M, spin s, mo- mentum p , and spin component a is denoted ][Ms]p@. The trans- formation properties of l[Ms] p@ under an arbitrary Poinear6 trans- formation are

+ s

U(Aoao) ][Ms]p@= d "top'ao X D~'~ (p, Ao)][Ms]Aop, a') (25) a p ~ - -8

where the SU(2) rotation (p, Ao) is defined as the rotation O-t(Aop) • AoO(p). O(p) is the Lorentz transformation from the rest frame of the particle with momentum ~ to the frame where its momentum is i0 = (E, p). In the coset decomposition given in Eq. (22) O(p) can be writ ten as

O (p) ~ A~ (p) (26) with

H(p)=( E+P~ p~-ip~l=AcH(~)A+ (27) \p~+ip~, E p~ /

Thus, the rotation (p, Ao) by our definition is

(p, Ao) =- A71 (Aop) Ao At (p) • (2S)

The D functions can now be calculated using Eq. (14) so tha t

DIMs] ( A t ) ~ <[Ms]p',jl U(A~) [[Ms]p@ ~ p ' j , p a (29)

= E ' d a ( p ' - A o p ) ~ s . .

Further it is not hard to cheek [10] [M,] tha t D~j,p~(A~) can be thought of as a non square integrable function with the same transformation pro-

perties as the function ~(A~) in Eq. (24). Thus, ~ , p a (A~) can be thought oI as a concrete realization of I[M, s] pa> and simultaneously as a function ~j (A¢) on which the Mackey induced representation theory can be applied.

Now we consider the tensor product space of positive mass represen- tations of P which consists of square integrable functions

qo . . . . . . . (A~., . . . . A~) with norm

81 8n

I1~% ........ ~ ( A ~ , - - - , A ~ . ) t [ 2 = X " " Z ~,=-'~ ~ = - ~ (30)

f dA¢ . . , f dA~ l~v~,~,~, ...... o,~ (A~ . . . . . A~,)I 2 < oo.

In order to decompose the tensor product space it is necessary to consider the double coset decomposition of the outer product group (P1, P~ . . . . , P~) with respect to the subgroup (~1, ~ . , - - . , t)~) and (P, .P,..., P). Now ~ , ~2 . . . . , On are all the same subgroup, since we are considering only tensor products of positive mass representations.

n-fold Tensor Products 237

The double coset decomposition of the outer product, group is

(PI, P2 , . . P . ) = U (~1,. . ~ ) ( A ~ , , - AD~ ) (P, P)

, • . . , A ~ I +

D .... D~ S U(2)1 / . . . . . SU(2)~ 1/

[(Ao~ 0 0 . . . . . .

.[(o ~ ~_,+"~o,~-'+~, (o ~ "~o,~-'+~1 . . . . , A - I + ] j

A convenient choice of double eosets [11] is

[(0~o~ 0 . . . . . 0 AT,~ +) ( A'-s" A z j _ l ) ( I 7)] where

Ao, (g: ° ) = 1/D~ i = 1 , 2 , . . . , n - - 3

A~_~ = tlQ~-~l

Ao.='---(lo 7). The parameters D1, D ~ , . . . , D~_ 1 are real positive numbers and the parameters Q1 . . . . , Qn-~ are complex numbers. Tim choice of double cosets is fixed in the sense that for each element in ~he outer product group a unique element of the double coset must be defined. A unique association will exist if the inducing subgroup elements are divided out of the diagonal subgroup. Eq. (10) defines the inducing subgroup which induces representations in the subspaees of the tensor product space labelled by the double cosets. Using this definition and our choice of double cosets ~D can be calculated:

OD = (AD,, . . . . A ~ ) - I ( 0 1 , O~ . . . . . On) (AD,, . • . , A~ , ) ~ (P, . . . , P)

~-~ [(0/ /H(a>), . . . . (0 / H(a')] . (33)

Eq. (33) is valid for n ~ 3. For the case, n = 2,

~0 = [(0U(1) H(a) H(a) U(1) U(I)) ' (U(1) U(1)U(1))], (34)

where U(1) is the one dimensional unitary group [12]. Dividing the inducing subgroup 32) out of the diagonal subgroup, the double coset

238 W.H. KLII~K and G. J. S~IIT~:

decomposition of ( P1, P~ . . . . , P~.~) is:

[(A. A~'+'I<"t A~'+~I ' " " "' (0 ~" A:'+'<:'<°°> A~'+'tllj

(35) - ' . . . . s u ( 2 ) ~ /1

0 • [(Ao, . . . . , (Ao o ) ] [(o o_,+),..., (A 0A-'+)] A right coset decomposition of P with respect to ~D is:

(A H(a)A-I+~ A -I+ ] = AI'A j (/0 H(a))(0 A 01_:t+ ) (36)

so that square integrable functions in the subspaces of the tensor pro- duct space labelled by [~DiQi] =-- {o~1 . . . . . o~, D 1 . . . . . D, , Q1 . . . . . Qn} are defined as

F[~D~Q,I(A) = ~ . . . . . . (A~, Ae~, . . . , A~,,). (37)

From Eq. (37) it follows that under an arbitrary element of P the functions F[~D,Q,I (A) transform as

U (Aoao) F[~D~] (A) = e ~ " A~o F[~D~Q~] (AAo) (38)

with PD --= A ~ 1 P~ + A ~ 1 P~ + "" " + A~ 1 P~, and thus the subscripts [~ziDiQi] label the degenerate subspaces since they remain invariant un- der the transformation.

Hence, the first step of the tensor product decomposition is com- pleted. The direct integral decomposition of ~ . . . . . . . ( A , , , . . . , Ac~) into /~[~D~Qa (A) is:

1[9~ .... ~(A,~, . . . . A~)II ~= X f d Q ~ d D x . . . (39)

f dQ._~ dDn_ 1 IIF[~<,.,,<~,] (A)II ~ • Here the measure dQi dDi has not been calculated because it is not needed [8].

In order to complete the direct integral decomposition of the func- tions ~v ..... ~(A~ . . .Ae , ) into irreducible subspaces, the functions F [ ~ a (A) must be expanded in a series of irreducible functions of P. Since the irreducible lunctions of P are functions only over A~ and A = S U ( 2 ) A ~ , it is clear that ~[~Q~](A) must be expanded in the orthonormal Wigner functions D}~(S V (el).

However a difficulty appears since PD in Eq. (38) is not of the form of ~fi in Eq. (24). If we wish to induce with the representation e ~ " a, then we must find a set of transformations, denoted by S U(2)D, such that

n-fold Tensor Products 239

where

so that

S U (2)2) H (p~) S U (2) + = H (p;). Now H (p~)) = A~ (PD) H (~) A + (p~)

~ = 0 and M ~ = p ~ and H(~)=SU(2)H(,~)SU(2) +

0

A j 1 (pg) S U (2)9 Ao (PD) = S U (2). (40)

Hence the expansion in terms of the Wigner functions is actually written as D~,j(Aj 1 (PD) S U (2)D A~ (PD)) where A j 1 (Pl)) S U (2) D A0 (pj)) is an S U (2) element as seen from Eq. (40).

A right coset decomposition of P with respect to the inducing

snbgroup 0 = H(a) S u(2)51+~ SU(2)5~+ / can be written as

SU(2)5.+ ] (OA° 0;_1+) (41)

with A = S U(2) D A c so that we can proceed with the expansion of ~[~lDiQt] (A) ~ .F[a~DlQl] (S U (2)D Ac). Let

2 s ÷ l = ~ _~__ --~-~-D[~Ij(A[-I(pD) SU(2)DAc(pg)) (42)

s=0 i = - - s K s

with

i / ~ ~ 1 f d S U (2)v D~]~ ( A j 1 (p~) S U (2)~ A~ (PD))

• F ~ , ~ , ~ , ~ ( S u (2)~ A~). (43)

In order to prove that the function q~¢[~d)~Q~sx] (A~) is an element of an irreducible function space of P, we must check its transformation properties under an arbitrary Poinear6 transformation:

U(Ao'a°) qSJ[~'D'Q*sK] (Az) - - [ 4z f dSU(2)D

• D[al*(AjI(p~) SU(2)z)A~(p~) ) e iv~'Aa° F[~,D, Qd(AAo) (44) A-S ,.~

We ha~e used the fact that ~ l * ( R ) = Z D~*(R') ~! R" D~t.] ( ) where

JR = R 'R ''-~ and have defined A¢ and SU(2) by A~Ao = SU(2)A¢. Now Eq. (44) agrees with Eq. (24) which means q~i[~,2),q, sx](A~) is an element of an irreducible function space of P ; the subscripts [aiDi Q~s k] label the degenerate subspaees since they have not been transformed.

240 W.H. tfm~K and G. J. S~T~:

Now that the complete direct integral (and sum) decomposition of the tensor product space into irreducible subspaces is known, we can use Eq. (18) and (29) to calculate the Clebsch-Gordan coefficients. We choose ~5~ (A~) 1) IMSl (A = --~i,p~--~l = ~S(Pl) -- AcP) D~(p , Ac) and

yl[M1sl] / A A \ TI [Mas~] / A A h , .

I [ ~(P~ A1)~Ap,:) D ~] (- AD, A) i = 1

so that the Clebsch-Gordan coefficients are:

([Ms] pa; ~tDiQiK] [Mlsl] P l a l ; . . - ; [Mns~] p~an)

=NV ±I ÷" [ 4z z~ f d A f l S U ( 2 ) ~ D ~ * ( p , A ~ )

• d3 (p~) _ A p ) D~l* (Ag -~ (p~) S U (2)D A~ (PD))

. Ho~(p~ - A~, SU(2)~)A~p~) 1) [~d r , . A ~ SU(2)D A~ )

The delta function normalization factors have been absorbed in the normalization constant N.

The Clebseh-Gordan coefficients tell us how n single particle basis states I[Misi] piai} form a single particle basis state l[Ms] p a ; @ where

is a set of 4n - 6 degeneracy parameters;

[[Mis~] pic@ . . . I[M~%] p~a~}

"" ' E (46) ~ , K , a

• ([Ms] pa; ~,D~ Q~K I [Mls~] p ~ a l ; . . . ; [Mnsn] p~a~}

• t[Ms]pa; ~iDiQ~K}.

The 3n ÷ 3 integrations in Eq. (46) arc performed using the n ÷ 1 three dimensional delta functions. An investigation of the delta func- tions leads to the ~ollowing conclusions:

(1) da(p~ -- A~p ) i s equivalent to d 3 ( .~ P i - p ) , i.e. conservation 2

of momentum. (2) p~ = M ~ with p defined as the total fore" momentum of the n

particle system. (3) The 3 n - 6 continuous degeneracy parameters D~, i = l, 2 . . . n - 1,

Q~,]= 1 , 2 , . . . n - 3 , and IQ*-e] are fixed by scalars constructed from the momenta p~, bu~ the set of scalars which fix the parameters D~, Q~. is not unique.

n-fold Tensor Products 241

Proo] (1) . The n delta functions imply

Ao p~ = S U (2)~ 1 A ~ 1 p~ so tha t

Ac Pl = S U (2)~ 1 ~ A~, 1 Pi = S U (2)~ 1 p~ = P p . i i

Therefore,

which implies conservation of momenta.

P r o o / ( 2 ) . M ~ = I H ( p ) I = H (~ . A ~ , l ~ )

M1 M~ + . M~-I D--~ + ~)~ "" + ~ + ~ ,

QI* M1 - Q_~ M~ . . . . Q*~-2 M,~_~

: D1 1)2 D,~_~

- - -~i M1 . . . . Q"-~ M=_= , J~n--2

MID~ + M2D'~ + • .. + Mn-~D~_, M~

+ ÷ (47)

- [ - Q* ~ . . . . Q*-~ M~_ ~] [ D1 1 D._~

D ~ _ 2 •

Also n

p~ = (f~ + p~ + " + p~)~ = 2 : M~ + 2 Z P~ "P, . (4S) i=z i,j

i<i

The scalars Pi " P~ can be calculated using the delta functions.

2pi . p , = ]H(p, + p,)] - [H(pi) I • - ]H(pj)]

= IA~,~H('p~)A~, ~+ + A ~ H ( p , ) A ~ ~+] - M ~ - M2 (49) [ D~ D~ ~_ ]Q~t ~ tQ~f ~ (Q*Q~ + Q~Q~) ]

= M~M~ [ D~ + D~ -- D~ 4 D~ D~D~ . "

With the substitution of Eq. (49) into Eq. (48) we see that Eq. (48) and Eq. (47) are equal.

n(n -- 1) P r o o / ( 3 ) . Eq. (49) gives us 2 p~. p~ scalars in terms of the

degeneracy parameters. A set of n - 3 pseudoscalars denoted by e~r~ p~ pn~_l pvn_ ~ p] can also be written in terms of the degeneracy parameters as

~ r ~ P~ P~n-1 Pvn-~. l°~ = M~ M~_x M~_, M, 1 _ Dn~_ 1 4 ~ D~_~ D~ i = L 2 . . . . . n - 3 . (50)

242 W . H . K L ~ and G. J. S ) ~ H :

Now the quest ion arises whe ther the 3 n - 6 pa r ame te r s Di, Q~ can be solved for in t e rms of t he scalars a n d pseudoseatars given in Eq. (49)

and (50). F r o m the set of n(n - 1) 2 Pt " Pj scalar invarian~s (49) consider

the 2 n - 3 scalars Pi " Pn, i = l , 2, . . . , n - 1, and P~ " Pn-1, J = 1, 2 . . . . , n - 2 .

[ J),~ ~_~ IQ~I ~ ] 2p~- P~- I = M i M~_I [D~_I t ~ D , + D2--~-_~ J " (53)

Eq. (51) to (53) can be used to c~lculate 1)1, D 2 , . . . , Dn-1, IQI[ 2 . . . . . IQ,~-3i ~, IQ~-~I, in te rms o~ the 2 n - 3 soalars p~ "p~ and p~ " W - l - The imag ina ry p a r t ol Qi, i = 1, 2 , . . . , n - 3 can be found using the n - 3 pseudosealars (50), a n d the real p a r t of Qt, i = 1, 2 . . . . , n - 3 can be found using the n - 3 scalars Pi" Pn-~, i = 1, 2, . . . , n - 3 where

÷ ,_: 1Q~[ ~ ~ tQ~_~I 2 2/Q~_~[ReQ~ 2 Pi " P n - ~ = M i M ~ _ 2 D~ ÷ t .-~ ~ D~_~ D~ D,D~_~ J"

(54)

Thus~ we have un ique ly de t e rmined the 3n - 6 pa r ame te r s D 1, D ~ , . . . , D~-I , Qi, Q~, • •. , Q~-3, and [Qn-2[ in t e rms of 4 n - 9 invar ian ts . Obvi- ously, there exis t o ther sets of i nva r i an t s which could jus t as well be used to specify the 3 n - 6 pa ramete rs . (The ex t r a n - 3 invar ian t s , Eq. (50) serve to fix the sign of I raQi . )

Then per forming the in tegra t ions in Eq. (46) gives

l[M~sl]pla~} ] [Mns , ]p~a~} = N X - - ~ 12s + 1

" '" 4~ ~,8, a

D~, , , (p , A~, S U (2) 9 A~) (55) i = 1

wi th # denot ing the set of cont inuous scalar invar ian ts . I t is n o t difficult to show t h a t the ro t a t ion A~ -1 (Pl)) S U (2)D Ac Ac (p)

in the center of mass of the n-par t ic le sys tem ro ta tes all of the m o m e n t a t h rough the same angles, so t h a t if one visualizes the m o m e n t a as being fixed in a r igid body (i.e. f ixed wi th respect to each other) , t hen the ro t a t i on ro ta t e s the r ig id body .

I n o rder to prove this s t a t e m e n t we define a m o m e n t u m vec to r £~ to be

P~ ~- A~ -~ (Pz)) Pz), (56)

= A~)~ Pt so t h a t wi th p ~ defined as P.D~ ~ AD~ ~ Pi" Recal l t h a t p ~ ~ -~

n-fold Tensor Products 243

PD, is just one of the n momenta which combine to give Pl). Ag -1 (PD) is

the unique transformation which carries p~ to p = ~ so that Ag -1 (p~)

0 carries Pl)~ to a well defined momentum vector in the center of mass system. Now the delta function ( ia(pi-A1) ,SU(2)~)A~p~) implies Pi = AD~ S U (2)D At Pi, and the delta function ~a (P J9 - A~ p) implies PD = At p or At = Ao(pD) A j 1 (p), since p = A j 1 (Pl)) PD = A J 1 (P) P; therefore

P~ =-- A J 1 (Pz)) Pz~,

= A~ -1 (P.D) A ~ Pt (57)

= Ag-l(pD) S U(2)D Acp~

= A J ~ (PD) S U (2) D At A~ (p) p~o,,

where Pio,, = A~ -1 (P) Pi is a vector in the center of mass system. Thus, A J 1 (PD) S U (2)~ A~ At (p) is the rotation which carries the vector Pi~ into the vector p~, as was first discussed by W]~L~ [4].

Finally the remaining n rotations (Pi, AI)~ S U(2)DA~) are defined by Eq. (28) and are uniquely specified by the n momenta Pi.

Conclusion

I t has been shown that, for induced representations, Clebsch-Gordan coefficients can be obtained if it is possible to decompose an n-fold tensor product space and if it is possible to calculate the D-functions [x~

= <[x] ~oI u(g0) IEx] x>. The decomposition of the n-fold tensor product representation con-

sists of two steps, the decomposition into invariant subspaees labelled by double cosets and then the further decomposition of these subspaces into irreducible subspaces. This last decomposition depends on the sub- group inducing representations on the subspaces labelled by double cosets. This subgroup must be at least the identity subgroup which would then generate the regular representation. Thus, we have shown quite generally tha t if the decomposition of the regular representation of the non- compact group is known, then it is possible to decompose an n-fold tensor product representation into irreducible representations.

The method applied to the Poinear6 group leads naturally to a set of equations relating Poincar6 scalars to invariants parameters labelling the irreducible subspaces of the n-fold tensor product space. However, the number of equations relating the Poincar6 scMars to the invariant parameters is much larger than the number of invariant parameters. Thus it is possible to solve for different sets of Poinear6 scalars which 17 Commun. math. Phys., Vol. I0

244 W.H. KLL-~ and G. J. S~H~: n-fold Tensor Products

un ique ly determine the i nva r i an t parameters . I f two of the part icles ' momen ta are restr icted to a z-axis, t hen the n u m b e r of invar ian t s necessary to completely specify the reduct ion is 3 n - 10 in agreement with ROHRLICH [13].

I n a subsequent publ ica t ion we will t rea t the problem of a generalized part ial wave analysis for "2 in, n ou t" reactions.

References

1. MAcFAlcLANE, A. J.: Rev. Mod. Phys. 84, 41 (1962). 2. JAcoB, M., and G. C. WICk: Ann. of Phys. 7, 494 (1959). 3. GOLDBERG, H.: ~uovo Cimento 47, 495 (1967). 4. WERL]~, J.: Rclatistic Theory of Reactions. (London: John Wiley and Son

1966), other references arc listed on page 288. 5. LEvY-LEBLAND, J . M. : J . Math. Phys. 7, 2217 (1966). 6. LVRCAT, F.: Phys. 1, 95 (1964). 7. ROFF~AN, E. : J. Math. Phys. 9, 62 (1968). 8. Our method rests mainly on what Mackey calls the subgroup theorem. We have

not used the most general definition of induced representation. Further, measures defining Hilbert spaces and double cosets can be found in this reference. See MACKAY, G. W. : Theory of Group Representations. Dept of Mathematics, The University of Chicago. Chicago, Illinois 1955.

9. G1, G2 . . . . . G~, refer to the same group G where as 01, ~ , • - -, 0~ can be different subgroups of G.

10. See, for example, KLImG W. H.: Preprint 67--59, University of Iowa (1967). 11. R ID~u : Ann. Insti. Henri Poiucar6 3, 339 (1965). 12. The fact that the inducing subgroups differ for n = 2 and n ~ 3 implies that

the calculation of Clebsch-Gordan coefficients must be carried through separately for each case. We will consider only n ~ 3. For the case, n -~ 2, see reference 10.

13. RO~RLIC~, F. : Nuovo Cimento 38, 673 (1965).

W. H. KLI~K Department of Physics and Astronomy The University of Iowa Iowa City, Iowa 52240, U.S.A.