IC/66/23

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

ON THE REPRESENTATIONSAND COUPLING OF THE U(6,6) TOWER

D. A. AKYEAMPONG

1966

PIAZZA OBERDAN

TRIESTE

IC/66/23

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE REPRESENTATIONS AND COUPLING

OF THE U(6, 6) TOWER+

D. A. AKYEAMPONG*

TRIESTE

21 March 1966

' Submitted to Nuovo Cimento

* Permanent address Imperial College, London.

ABSTRACT

Some representations of U(i>,i/) 13 V(v, v) are constructed and the eigen-

value problem involved in the approach is solved and shown to lead, under

suitable boundary conditions, to a discrete spectrum. The coupling of the

different XJ(vtv) rungs of the tower is also discussed.

ON THE REPRESENTATIONS AND COUPLING OF THE U(6, 6) TOWER

I. INTRODUCTION

1-4Several authors have, in a series of papers, recently emphasized

the hope that infinite-dimensional representations of some non-compact

groups might be used for classifying elementary particles. In one of

these series in particular, it was shown by SALAM, DELBOURGO and2

STRATHDEE , that by relativistically boosting the unitary representation

of the rest-symmetry U(6, 6) group, one is led to the infinite-dimensional

representations of the U(6, 6) & U(6, 6) group, with U(6, 6)p as the little

group and a chain of subgroups GL(6, C), for 2-momentum processes, U(3, 3)

for 3-momentum processes and GL(3, C) for 4-momentum processes.

Apart from the little group U(6, 6)_ , (appropriate to a particle with 4-mo-

mentum p^), there is another U{6, 6) subgroup, called the U(6, 6)D in re-

ference 2, which contains the Lorentz transformations. As is well known,

the Bargman-Wigner equations will ensure a unitary norm even though the

representations of U(6, 6) we start with are non-unitary. It therefore

suffices to take the finite-dimensional representations for this subgroup in

any ladder representations and note that it is here acting as the maximal

"compact" subgroup in analogy to the GL(6, C) reduction relative to U(6),

say.

In this note, we study the representations of U(6, 6) Ef U(6, 6) C U(12,12)

and carry out the reduction relative to the subgroup U(6, 6)D . This group

has been chosen since it contains the compact rest-symmetry U(6) 3S U(6)

which has been shown to be a good strong interaction symmetry, as a sub-

group. In carrying out the reduction we shall follow the procedure pro-

posed by DOTHAN et_al., and used extensively by DELBOURGO et al. The

idea of the formalism is to construct an infinite tower which creates an in-

variant space in which the irreducible representation of the parent group

acts, and on each rung of the tower, to put the irreducible finite-diinension-

al representations of the maximal compact subgroup.

- 1 -

In Section 2, we write down the necessary commutation relations and

from these construct the representations of the general group V(v,v) ES

U(i/, v). Section 3 deals with the important question of the discreteness

or otherwise of the spectrum of the eigenvalues, and;following a prescript-

ion given in reference 3, we work out in Section 4 some recoupling co-

efficients of U(i/, v) S U(v,u). It is found that a discrete spectrum of

eigenvalues is obtained for this U(i/, v) S3 J3\v,v) Feynman tower.

2. THE U(v, v) B U(v, v) REPRESENTATIONS

Although we are mainly interested in the U(6, 6) & U(6, 6) group, owing

to the simplicity involved in working with the general group, we shall con-

sider below some of the representations of U(v, v) & X5{v, v). Following

reference 1, we define creation and annihilation operators S a and a"

with a - \,2,. ., ,2v for fictitious boson quarks and another set of creation

and annihilation operators ba and b a for equally fictitious boson anti-

quarks. These obey the commutation relations

where Q *'- (^)* ( *.)£ , V - 1*$* (*.) * ««>

Y - /' °YyC . The generators of U(v,i/) Ei \J(v, v) are now given by

\0 -l/

| [M^±Nj] , where

and these satisfy the commutation relations

- 2 -

s

M + » * « * * * * , M + «It is clear from the above relations that Mfi are the generators of U(6, 6)D

To construct the irreducible representations of XJ{v,v) 3S V{v,v), it is

feasible to fix on a lowest or "ground" level and apply the generators M^

and N** to it repeatedly. Whilst N^ will lead to new levels, M£ will

produce a mixing of states at each U(i/,v)D rung. The representation

then appears as a tower of finite-dimensional non-unitary representations

of V(v,v)D. With the definition that

it is seen that the operator N£ is projected out of the vacuum state into

(aa B01) [ 0 >. Hence we choose the lowest state as

•ii_ ^ n l / i - •f1 /T: - \ 0 I n ->. ; n i l . (2l> + 1 1 - 1 ) ' .

with < 0 1 (b. aj (b.a) | 0> = (-1) n1. — ^ ^ '^ .

The constructions of the representations of U(i/, v) and GL(i/, v) have been

discussed elsewhere and the generalization to X5(v,v) J3 U(i/, v) follows

closely on those lines.

We construct an irreducible representation from an infinite sequence

of symmetrized, traceless tensors

- 3 -

, rpu). },where

(3)

i = 0, l t 2,.. . , 2v and r is the difference between the number of lower and

upper indices. M^ , N£ are then defined by the relations

X

..a (4)

^ A

'r+t,

- 4 -

where the notation P 1 . . . (j)... pj indicates that j3j is missing from the

sequence ^ 2 . . . Pe and

at (6)

and X is a real number defined below in Section 3. We demand also that

Af , Bc be non-zero. The Casimir operators are (with appropriate3

modifications) similar to those already obtained for U(v,v), Thus we

have obtained a two-parameter set of representations of U{v,v) Ei U(i/,v).

3. THE EIGENVALUE PROBLEM

The reduction under V(v, v) of each level in the U(i/, v) S U(i/, v) tower

is unique. In the V(v, v) !S T5{v,v) tower, however, there exists an infinite

number of distinct irreducible representations of particle states, repeated

throughout the tower, the only way of distinguishing among the same part-

icle state being in the assignment of different values of the X , the Casimir

operator. Thus the U(6, 6) rungs of, say, the meson tower will be made

up of the series

1 © 143 © 5940 © . . .

-5-

Under U(6) S U(6), for example, the following decompositions occur:

1= (l.D-

143 = (1,1) + (6,6)+ (6,6) +(35,1) +(1,35).

5940 = (1,1) + (6,6) + (6,6) + (35,1) + (1,35) + 21, 21) + (21,21)

+ (35,35) + (120,6) + (6,120) + (120,6) + (6,120) + (405,1)

+ (1,405).

The various (1,1) singlet states, for instance, coming from the U(6, 6)D

irreducible representations 1,143 and 5940 are distinct particle states

and are only distinguished from each other by the different values of X

they take. We now determine the spectrum of X , where for convenience

we consider the meson tower. Let us construct the U(i/,i/)D irreducible

particle state of the U(v, v) E! U(v, v) by demanding that M^ , N£ satisfy the

equations

(7)

where X is real. Note that for the special case where we consider the

singlet of U(i/, v) , |3. = or. . We are especially interested in knowing

under what conditions, if any, the spectrum of X is discrete. Applying

Eq. (3) to Eq. (7), we obtain the recurrence relation

Li) - Xf

By means of the function

Eq. (8) is replaced by the equivalent differential equation

-6 -

0)

To solve his we make the substitution

Then Eq. (9) reduces to the confluent hyper geometric equation

which has the solution

and hence

(11)

Note that only the above solution is allowed since v is necessarily an

integer and we are imposing the boundary condition that the solution be

finite at x = 0 . It now remains to see under what conditions the ortho-

gonality relation

- Oholds.

From the properties of ,F( {\(v -X); v ; 2x} , we see that this function

increases with x for positive i(v-X). However, if k(v-X) is zero or

negative, then it reduces to a finite polynomial which tends to zero for

large x . Hence for (12) to be fulfilled, it is sufficient for \{v -X) to be

- 7 -

zero or negative, i. e . ,

i ( v - X ) = - n for n = 0 , 1 , 2 , . . . (13)

Thus the discrete spectrum of X is given by the relation X = v + 2n ,

Now for \{v -X) = -n (n = 0 , 1 , . . . )

Z = ,F, (-n; v ; 2x)

and then- ,F ( (-n ; v ; 2x) is related to the generalized Laguerre poly-

nomial by the equation

The orthogonality condition now becomes

d -x •=. M x _f^ (15)

where

(15a)

and n' = |(X' - i /) .

Hence the required solution is

- 8 -

Thus with the condition 0 ^ x ^ w , we obtain a complete' set of solutions

by taking X - v + 2n , implying that the spectrum of X is determined.

The coefficients fm can now, in principle, be evaluated. These are,

however, complicated as can be seen above and so for purposes of il-

lustration, we put n = .0 (corresponding to X = v) in Eq. (16). In this

very special c.ase, we have

ml

and hence from Eq. (2) the lowest state is now defined as

which clearly betrays the fact that we are not, as is well known, working

in a Hilbert space. The usual practice in this case is to apply the

Bargmann-Wigner equations to project out the positive-definite subspace.

Under this projection procedure the above state becomes normalizable

with a discrete spectrum X , a pleasant result physically.

Eq. (7) can now be used for the construction of higher meson states.

For example, with this as the lowest state <£(X) , the relation

will take us to the next state, etc.

- 9 -

4. THE TRILINEAR COUPLING

The ^eneral prescription for coupling of towers has been discussed in

reference 4 and we shall refer the reader to it for the necessary details.

Here it is our intention to employ this technique to determine the first

few coupling coefficients of a U(i/,i/) IS \J{v,v) tower, assuming that coup-

ling does exist in general. We shall also, for simplicity, consider the

coupling of two baryon towers to a meson tower. The general form of

such a trilinear invariant will be

where

) Y(i2) *(i3)) denotes a U(i/,v)D invariant, g labels the

distinct invariants which may be obtained with the same three tensors and

the coefficients [^1i2^3 ]g must be determined so as to make the sum a

U(vtv) BJ U(v,i/) invariant. To achieve this is to impose the condition

Njl = 0. (19)

and from this we obtain a number of recursion relations for the various

coupling coefficients. The meson-baryon invariant vertex function for

any determination of the recoupling coefficients follows from Eq. (18).

Applying Eq. (19) to Eq. (18) and equating appropriate coefficients leads

to a number of equations. Since in actual practice, only the first few

terms are of interest, we characteristically equate the coefficients of

J- Xlf "8 ~t T respectively to zero and obtain the following

equations:

-10-

(20)

= 4 fooa]

(22)

Following BISIACCHI and FRONSDAL , we choose [001] arbitrarily and

normalize it to unity. We then have

(23)rooo1 = ^±v /21KL0 0°J 3fx,+ )̂ ( ^vt.;

- 1 1 -

(24)

A similar method can be used to derive the recoupling coefficients for the

meson-meson vertex tower.

ACKNOWLEDGMENTS

The author is indebted to Professor Abdus Salam for his guidance and

advice and to Prof. C. Fronsdal, Drs. J. Strathdee and R. White for help-

ful comments and fruitful discussions. He would also like to express his

gratitude to the University of Ghana for financial assistance. Finally to

the IAEA the.author's thanks are due for the hospitality extended to him at

the International Centre for Theoretical Physics, Trieste.

-12-

*

REFERENCES

1 Y. DOTHAN, M. GELL-MANN and Y. NE'EMAN, Phys. Letters 1/7,

148 (1965).

A.O. BARUT, Proceedings of the Seminar on High-Energy Physics

and Elementary Part icles , Trieste , 1965 (IAEA, Vienna) p. 679.

C. FRONSDAL, Proceedings of the Seminar on High-Energy Physics

and Elementary Part icles , Trieste , 1965 (IAEA, Vienna) p. 665.

2 R. DELBOURGO, ABDUS SALAM and J. STRATHDEE, Proc . Roy.

Soc. 289A, 177 (1966).

3 ABDUS SALAM and J. STRATHDEE, Proc. Roy. Soc. (to be published).

4 ABDUS SALAM and J. STRATHDEE, Phys. Rev. (to be published).

C. FRONSDAL, J. Math. Phys. (to be published).

5 A.. ERDELYI et al. Higher Transcendental Functions, McGraw-Hill,

1, 248 (1954).

6. See,for example, S. WEINBERG, Phys. Rev. 139, B597 (1965).

7. G. BISIACCHI and C. FRONSDAL, Nuovo Cimento 41., 35(1966).

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