The U(n) ⊃ R(n) problem (I)

1.C:6.B I

Nuclear Physics 83 (1966) 632--638; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or m.~crofilm without written permission from the publisher

THE U(n) ~ R(n) PROBLEM (I) M. R E S N I K O F F

Department of Physics and Astronomy, University of Maryland, College Park, Maryland t Received 11 March 1966 A b s t r a c t : The n u m b e r of operators required, to label the rows of an irreducible representation of

U(n), in addition to those of the p r o p e r rotation group R(n), is calculated, b o t h for the case of n non-zero rows of a Y o u n g tableau of U(n), and m :< n non-zero rows. For the physical c a s e of m = 3, these additional operators are constructed.

1. Introduction The problem of imbedding the rotation group, or proper orthogonal group, of n dimensions R(n), into the unitary group of n dimensions U(n), arises from the following physical motivation. Say we wish to classify and determine states of n particles when these particles move in a three-dimensional harmonic oscillator central potential. The Hamiltonian H of this system is invariant under a unitary group of 3n dimensions, U(3n). This group may be decomposed into the product U(3n) U(3) x U(n), a separation of the configuration space from the particle indice groups, respectively; U(3) may be further decomposed by the group R(3), U(3) ~ R(3), which preserves the spherical symmetry of configuration space, and, of course, adds the angular momentum quantum number as a means of specifying the system. This U(3) ~ R(3) problem has been solved by Bargmann and Moshinsky 1). Next, states of definite permutational symmetry on the particle indices may be constructed by labelling states with operators of the chain: U(n) ~ U ( n - 1 ) D O ( n - 1 ) , where O(n) is the orthogonal group in n dimensions. Then finally, states associated with a particular representation D E"-t' ~1(S,), of the symmetric group S,, may be constructed from the states of O ( n - 1). This step and the step from U(n) to U ( n - 1), a removal of the centre-of-mass motion, have been accomplished by Kramer and Moshinsky 2). Further, using the U(3) ~ R(3) results, they were able to construct states for the entire chain for the special cases of the number of particles n = 2, 3, 4. For the case of n greater than four, the step of the chain U ( n - 1 ) ~ O ( n - 1 ) or U(n) ~ R(n), must be understood, and that is the purpose of this investigation. In sect. 1, the number of operators required to label the rows of an irreducible representation ~(s) of U(n), in addition to those of the rotation group R(n), is calt W o r k supported by the United States Air Force under Contract A F O S R 500-64. 632

U ( n ) ~ R ( n ) PROBLEM

. 633

culated, both for the case o f n non-zero rows of a Young tableau of U(n), and m _<_ n non-zero rows. These additional operators, for the physical case of m = 3, are constructed in sect. 2. The actual application of these operators to a base vector If, ~> of U(n) and the allied questions of whether these operators yield non-degenerate eigenvalues and states, and hence lead to a complete specification of the system, will be dealt with in a subsequent paper.

2. N u m b e r o f O p e r a t o r s

The vector space ~ : of base vectors If, c~) in U(n) may be constructed as a homogeneous polynomial of t h e n 2 complex variables z r = (z~ 1). . . . . zl")). Define the following operators on the space ~ : (ref. 1)) T,s = z, " O/6z s = #z~)/gJz~ ~), Cr j = ~ ~(i)/,~,(j) , /~, ,

c~ = 1 . . . . .

n,

fl = 1,. . . , m =< n,

r, s = 1 . . . . . i, j , = 1 . . . .

m < n,

(2.1a)

n.

(2.1b)

(Summation convention implied.) Moshinsky 4) has shown that spaces ~ : will be associated with the irreducible representations 2 (:) of U(n) if the conditions T,s]f, @ = O, T ~ ] f , ~> = f~[f, @ ,

r < s = 1. . . . .

(2.2a)

n

(no summation)

(2.2b)

are satisfied, where f = (f~ . . . . . f , ) and e represents the row labels. Eq. (2.2b) is Euler's condition - I f , @ must be homogeneous in z i of degreefv The correspondence of eqs. (2.2) with an n-rowed Young t a b l e a u f = ( f l . . . . . f , ) is established by consideringfr as the number of boxes in the ith row. Then, eq. (2.2a) represents the antisymmetry condition in the columns of the Young tableau. Since ~rr-('), z(: ~]J = 0,

~--,ra!'),d~ s)] = O,

~-.,ra!', z (:)]~

= ~j6r~,

~.,a!" -

~/~_(~)

"/~=r ,

the operators Cry satisfy the commutation relations [Crj, Cry,] = @ j C ~ j , - 6~j, C r j ,

(2.3)

and hence are the generators of U(n). Note that the operators T,~, Crj-, commute and therefore the vector space ~ : is invariant under the operation of C~j. Eqs. (2.2), labelling the invariant ~ : spaces, provide ½ n ( n - 1 ) + n = ½n(n+l) conditions on a base vector If, ~> of n 2 coordinates; n 2 - ½ n ( n + l ) = ½n(n-1) additional operators are required to label the rows of the representation ~ ( : ) , or, to completely specify If, @. As has been shown by Baird and Biedenharn s), these operators may be taken to be the Casimir operators in the subgroup chain: U ( n - 1 ) = . . . ~ U(2) = U(1). Alternatively, the rows of ~ ( : ) may be labelled by the chain R(n) ~ R ( n - 1 ) • . . ~ R(3) plus the z-component of the angular m o m e n t u m 6). Since the rank l of

634

R(k),

M. RESNIKOFF k = 3 . . . . . n, is l =

[ik]

ik

k even

½ ( k - l)

k odd,

-

(2.4)

the n u m b e r r of Casimir operators in the rotation group chain, for n non-zero rows o f a Y o u n g tableau, is r = ¼nz

= l ( n + 1 ) ( n - 1)

n even

(2.5a)

n odd.

(2.5b)

Thus, if the rows of a representation 9 (:) o f U(n) are labelled with the rotation g r o u p chain, q additional operators are required to completely label the states: q = kn(n-2)

= ¼ ( n - 1) z

n even

(2.6a)

n odd.

(2.6b)

F o r n = 3, U(3) D R(3), B a r g m a n n and Moshinsky 1) have determined the one additional operator 12. F o r n = 4, U(4) = R(4), Moshinsky and Nagel 7) constructed the two required operators, f2, 4. F o r n = 5, qe = 4 operators are needed. Consider a Y o u n g tableau f ' = (fl . . . . . f,,, 0 . . . . . 0) where ./'1 > f2 ->- - . . _->fm > 0 for m < n. Eq. (2.2b) requires that the function If, c~) be constructed of the mn variables ~i-(J), i = 1, . .,. m,. j . = . 1, , n. Eqs. (2.2) place i m ( m + 1) conditions on the mn variables, hence m n - i m ( m + 1) operators are required to label the rows o f the irreducible representation ~ ( : ' ) . I f m > [in], the rotation group chain provides r Casimir operators, eqs. (2.5), hence, q = m[n-½(m+ 1)]-¼n 2

n even

(2.7a)

n odd

(2.7b)

m > [in] = m [ n - ½ ( m + 1 ) ] - ¼ ( n z - 1)

additional operators are required. Consider m < [in] and let m = ix, x even. Then, the following chain m a y be considered R(n)

~

...

~

R(x)

(i)

~

R(x-1)

~

...

~

R(3).

(ii)

Since the rank of the rotation groups in (i) is m, the n u m b e r of Casimir operators in (i) is ro) = m ( n - x + 1). F o r x even, r(ii) = ¼ x ( x - 2 ) , hence r = r(i)+r(ii ) = m ( n - m ) . Thus, the n u m b e r o f operators, in addition to the operators in the rotation subgroup chain, required to label the rows o f 9 (:') is m =< [ln],

q = ½m(m--1).

(2.8)

U(n) ~ R(n) PROBLEM

635

For the number of particles n large, eq. (2.8), q=0,

m=l,

q=l,

m=2,

q=3,

m=3.

(2.9)

For the problem of physical interest, the Hamiltonian H, invariant under U(3n), has eigenvalues N, corresponding to energy levels of the oscillator. Since the Hamiltonian H of U(3n) is simply the first order Casimir operator, the Young tableau of U(3n) is (N, 0 . . . . . 0) with 3 n - 1 zeros. In general U(3) has a three-rowed tableau ( f l , f z , f 3 ) . In order for the product U(3) × U(n) to have the tableau (N, 0 . . . . . 0) Kramer and Moshinsky 2) have shown U(n) must necessarily have a tableau ( f l , f 2 , f 3 , 0 . . . . . 0) with n - 3 zeros. By eq. (2.9) then, three additional operators are required to completely label the rows of an irreducible representation ~(~') of U(n) for the case of physical interest. 3. Construction of Operators

Symmetric and anti-symmetric combinations of the generators Cu may be written 1) C~j = A~j+Q~j, Aij

-~ ½(Cij-

Q,j = l ( C , j +

(3.1)

Cji),

Cji),

(3.2)

~atisfying the commutation relations JArs, Atu] = l (cSst Aru-Jrt As,, + bru A s t - bsu Art) ,

(3.3a)

[Qrs, -/it,] = ½(C~stQr,, + 6rt Qsu-6ru Qst-CSsuQ,t) ,

(3.3b)

[Q,~, Qtu] = ½(6stAru'4-OrtAsu+C]ruAst+OsuArt).

(3.3c)

The operators Ars are the generators of the proper rotation group R(n) (ref. s)). The operators Qu are a set, corresponding to the set of unitary symmetric matrices of n dimensions. The algebra is not closed; the commutator [Qii, Qkt] is not expressible as a linear function of Q~j, but only A u. For general U(n), the operators T,s may label the irreducible representations ~tl), or, equivalently, the Casimir operators Cp = Tr (C) p = Cu,~,~ C u ~ . . . Cu~u~ may be used, Cp commutes with Cu, hence with A~j, Q~j, and any function of A~j, Qu" Further, if we define c(k~ = ~z~i)/~z~j), ij

i, j = 1,

" • "~

k,

fl = 1,

" • "~

min (m, k),

(3.4)

636

M. RESNIKOFF

as the generators o f U(k), then C~k), the pth order Casimir operator commutes with

A(k) i j , the generators of R(k) [defined as in eq. (3.2)]. The rotation group R(k), k = 3 , . . . , n, has the Casimir operators A (k)v (ref. 9)).

A (k) = Tr (A(k)) p

A //i//2 (k) A ,~//~ (k) . . . . .A ~,u, (k)

p even.

(3.5)

It may be shown that [A(pk), A~S (k') ] = 0 and hence [A(pk), A(pk')] = 0. Since [C(pk), A(p~)] = 0, and hence, [Cp, A(pk,)] = 0, k = 3 . . . . . n, the Casimir operators in the chain R(n) ~ R ( n - 1) D .... ~ R(3) plus the z - c o m p o n e n t of the angular m o m e n tum, may be used to label the rows of ~ ( I ' ) , while the operators T,~ or Cp label the invariant spaces. F o r the physical problem, it is necessary to find three additional operators which c o m m u t e with the above set of operators. This may be done in the following manner. Using eq. (3.1), Cp m a y be expanded in terms of Aij, Qiy: C-

C,, = Qa~,

Ca = T r ( C ) 2 = Tr(Q) 2 + Tr(A) z = Q2 + A2.

(3.6) (3.7)

The first and second order products yield no additional independent c o m m u t i n g operators. Further, Ca = Q3 + f 2 ' + f ( A 2 , C2),

(3.8a)

~2' = Qu~u2A,2//3 A//~//,.

(3.8b)

where

In the appendix, it is shown that

[Qp, A!k.)l = O. "-tJ A

(3.9)

Also, o f course, [Qp, Cp] = 0. Hence, either Q3 or f2' m a y be chosen as the first additional operator t. In a similar fashion,

C4 = Q,+O~'+g(A2, A,~, C2, C3, Q3),

(3.10a)

(b' = 2(0,,//2 A,2,,~ Q//3//, A//,//, + 20,1//2 Q//2//3 A,~//, Au,//,).

(3.10b)

where

Again, either Q4 or 4 ' m a y be chosen **. N o t e that the entire operator 4)' c o m m u t e s with Q3 and not each term separately. Finally, expansion o f C 5 yields Q5 and for rn > 3, there should be two additional fifth-order operators, one o f which m a y also be used to label the rows of a representation of U(n) since q = 4 for n = 5 [see eq. (2.6b)1. t Bargmann and Moshinsky 1) have constructed an operator .Q, similar to .Q' above, by picking off the coefficient of a third-order characteristic determinant. *t qS' is similar in form to the fourth-order operator constructed by Moshinsky and Nagel 7).

u(n) ~ R(,) VROBLEM

637

F o r the physically i m p o r t a n t case, three operators are required and these m a y be taken to be Qp, p = 3, 4, 5, since [Qp, Qp,] = O,

p,p'

= 3, 4, 5.

(3.11)

This m a y be proved using the c o m m u t a t i o n relations eqs. (3.3). It should be noted that eq. (3.11) is not an entirely expected result, for it is n o t true that Qp, Q i j commute, [Qp, Qij] # O. It remains to explicitly construct the states in terms of the 3n variables zl j~, j = 1, . . . . n, (i = 1, 2, 3 for the physically i m p o r t a n t case), by applying the operators A(pk), k = 3, 4 , . . . , n, p = 2, 4, 6, and Qp, p = 3, 4, 5. This is not an entirely trivial problem + The author thanks Professor M. Moshinsky for suggesting this p r o b l e m and for the hospitality extended to him at the Universidad de Mexico where this work was begun. The author also greatefully acknowledges helpful discussions with Professors E. Chacon, A. Dragt and A. J. Macfarlane.

Appendix P r o o f that [Qv, AI~)] = O.

[Qp, A J

+ QUl,2"-" Q u , - 2 , , - , [ Q u s - , U s , +Qu,u~ " " Qu,-,us[Q,,, "31-

•

•

.

.

....

[Q,,u,,

+(...

Qu,iQiu,+~ . . . ) + ( . .

pth row

(A.I)

. . Qus-~jQiu, . . .)-(.

. . Qu~-,IQj,.~.

.

Qu~_~jQI,~+, . . . ) + ( .

.

sth row ( s + 1)st row

Aij]Qus+,,,+2 . . i Q,,,~

Aij]"

+(...

-1-

Aij]Qusus+l'" Qua.,

•

+Q,,u2""

-[-

first row

= [Qulu2, Aij]Qu2~3 . . . Qupu,

. . Qu~_~iQj,, . . .)-(. . . Qus-,iQj,,+~

. . .)-(.

. . Qu~-~jQi,,+~ . . . ) - ( .

,...) . . Qu, iQiu~+ ,...)

.

+ (Qiu~""

Q.M) -f-('"

Q,,-,i

QJu~)- (""

Qu,-,.i Q,,i)-

(Qj,~...

Q,,i).

(a.2)

N o t e that the sum on the right side of eq. (A.2) collapses. First, two terms of the first row cancel with two terms of the p t h row. Assume that the second and third terms of the sth row cancel with ( s - 1)st row. Next, note that the first and fourth terms of + This work has already been initiated by E. Chacon of the Universidad de Mexico.

638

M. RESNIKOFF

the sth row cancel with the second a n d third terms o f the ( s + 1)st row. Since this h o l d s for all s, the sum collapses a n d the c o m m u t a t o r is zero. The o p e r a t o r _,jA! k), i,j = 1, ..., k, is defined by eq. (3.2), using -Ci j(k) o f eq. (3.4) where the sum runs f r o m fl = 1 to fl = min(m, k). W r i t e the g e n e r a t o r o f R ( k ) , AI~)[1, min

(m, k)]

(A.4)

exhibiting the limits o f the sum. T h e o p e r a t o r Qp contains p r o d u c t s o f the g e n e r a t o r s Cij, i , j = 1 , . . . , n, a n d fl = 1. . . . . m < n. C o n s i d e r the case m < k, [-Op(1,

m), a~'(1,

since this c o m m u t a t o r does n o t differ f r o m the case m > k. Then,

A(R)(1 k) ij k ~ F r o m eq. (A.5),

Qp(1, m)

[Qp, A~j]

= 0 f o r / j = 1. . . . .

A(*I(1 m)+A}~)(m+l,k). \ ,

=

"~ij

(a.5) k. C o n s i d e r

(a.6)

a n d _,jA(-k.)tl,,m) c o m m u t e . F u r t h e r , [Qp(1,

since [d~r), z~s/] =

m)] = 0,

61j3~ a n d

m), Alki)(m +

1, k)] = 0,

(A.71

i 4= j in eq. (A.7). Therefore,

[Qp(1 , m),

Aij(k)(1,

k)] -- 0,

m > k,

and

[Qv,

AI~)-] = 0, for all m.

References 1) V. Bargmann and M. Moshinsky, Nuclear Physics 23 (1961) 177 2) P. Kramer and M. Moshinsky, preprint (April, 19651 Instituto de Fisica, Universidad de Mexico, to be published in Nuclear Physics 3) M. Moshinsky, Nuclear Physics 31 (1962) 384 4) M. Moshinsky, J. Math. Phys. 4 (1963) 1128 5) G. E. Baird and L. C. Biedenharn, J. Math. Phys. 4 (1963) 1449 6) J. A. C. Alcaras and P. L. Ferreira, J. Math. Phys. 6 (19651 578 7) M. Moshinsky and J. G. Nagel, Phys. Lett. 5 (1963) 173 8) G. Raeah, Group theory and spectroscopy, Princeton Lecture Notes (19521; CERN preprint 61-8 9) J. Ginibre, J. Math. Phys. 4 (1963) 720