ON A SOL TION OF T U
f,o
STATE LAB L I ROBLEMFOR TI'IO-ROÌíED REPRESENTATIONS
by
P.D. Jarvís, B.Se. (Ilons.)
A thesis
submitted for the degree of Master of Science
Departmeot of Mathemat,ical Physics, Unívers ity of Adel aide.
March, L974
I
TABLE OF CONTENTS
ABSTRACT
PREFACE S
Ch. 1 INTRODUCTION
1.1 Irreducible Representations and Abstract ' Bases
L,2 Physical Applications
ch. 2 TtlE U(N)r O(N) REDUCTTON VrA TENSoR REPRE- SENTAT I ONS
Page No.
13
L7
18 21
1
2
2.I
2.2 2.3
2.4 2.5
25 35 39
Irreducíb1e Tensor Representations Symmetríes of Irreducíb1e Tensors
U(N) > Q(N) Branching Theorem for Two-Rowed
Representations
U(N). O(N) Reduction by Character Analysis Conparison for the Case U(4) [gr¿J
2.6 Explicít Tensor Reduction for Some Specíal
Cases
ch. 3 u(N) AND 0(N) TNVARTANTS
3.1 General Properties Independent Invariants
Invariants for Two-Rowed Representations 3,2
3.3
40
42 43 46 50
Ch. 4 EVALUATION OF INVARIANTS
4.2
Pre 1 iminar ie s
U(N) Invariants and Cubic Polynônial Identity
o(N) rnvarianÈ <t¿>
=
<â2)-
å<.,t).o(N) Invariant å(a¡a + ãaã).
Operational Def inition of /I 4.3
4 5
APPENDI CE S
REFERENCES 4-.4
Pace No.
52 53
56
6L 65 67
70
87
ABSTRACT
Considerations of the number of 1abe11ing oPeTators in a general state 1abel1ing problem inply that for two- rowed irreducible representations of U(N), 1abe1led
[n,e,o,..ro], jusE one additional commuting labelling
operaÈor, Â , í" required for the soluÈion of the U(N) = 0(N) state labe1ling problem. A detailed investigation of the U(N)) O(N) reduction via tensor rePresentaEion, f."a, to a proposal of an integral label x , irplying a non-orthogonal l_abelling s cheme. A simp 1e u (N) ) O (N) branching theorem for two-rohled representations is formulated in teïms of 1 .
The additional labe11ing operator A , with eigen- value tr, ís to be defined inplícitly by an equation of
ttre form f(A,iÞ,"') - o, where f i" a polynomial inA;
Ëhe additional O (N) invariants (D in the represenËatí on subduced by U(N) [pr9ro,..,o] , and other known labels and
invariants. The o(N) invariants, which are cubic and quartic in the generators of U(N) ' are discussed. Tech- níques developed by Green and nt""k"rr(1) for the U(3) ) SO (3) problem are used Èo evaluate the single independent cubic
ínvariant, and it is shown how this is used in obtaining the desired inplícit operational definition of A
A cubic characteristic polynomial identity, satisfied by the generaÈors of U (N) in two-rowed rePresentatíons, is also found using these teehniques.
Some possible physical applications of the U(N)> 0(N)
state labelling problem are briefly discussed.
t
This thesis conÈains no material which has been áccepted for the award of any other degree or diplorna in any university. To the best of the auEhorls knowledge and belief, the thesis contains no material prevíous1y published or \dritten by another Persoûr except when due reference ís made in the text.
London
ApríL 197 4
The author gratefully acknowledges the assistance of Professor H. S. Green, DeparÈment of MaÈhematical Physics, University of Adelaider urder whose guidance the work here presented was carried out.
