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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
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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.
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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.
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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.
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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
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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
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, 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,
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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 © . . .
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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
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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
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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
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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.
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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:
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(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.;
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(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.
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*
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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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Available from the Office of the Scientific Informotion and Documentation Officer,
Internationol Centre for Theoretical Physics, Piazza Oberdan 6, TRIESTE, Italy
3634