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SU(2) x SU(2) x U(1) basis for symmetric SO(6)
representations : matrix elements of the generators
R. Piepenbring, B. Silvestre-Brac, Z. Szymanski
To cite this version:
SU(2)
SU(2)
xU(1)
basis for
symmetric SO(6) representations :
matrix elements of the
generators
R.
Piepenbring,
B. Silvestre-Brac and Z.Szymanski
(+ )
Institut des Sciences Nucleaires, 53 avenue des Martyrs, 38026 Grenoble Cedex, France
(Reçu
le 30septembre
1986, accepté le 28 novembre1986)
Résumé. 2014 Nous calculons de
façon explicite et
analytique,
les éléments de matrice des générateurs du groupeSO(6)
pour la représentation irréductiblesymétrique ;
nous employons la chaîne dedécomposition
SO(6) ~
SU(2)
xSU(2)
xU(1)
(qui
est différente du schéma très connu dessupermultiplets
deWigner).
Nousindiquons
la relation avec la méthode de Gel’fand et Tsetlinqui
utilise la chaîneSO(6) ~ SO(5) ~
... ~SO(2).
Nous présentons aussi un
exemple d’application physique.
Abstract. 2014 Matrix elements
of the group generators for the symmetric irreducible
representations
ofSO(6)
are
explicitly
calculated in a closed formemploying
thedecomposition
chainSO(6) ~ SU(2)
xSU(2)
xU(1)
(which
is different from the well knownWigner supermultiplet scheme).
The relation to the Gel’fand Tsetlin methodusing SO(6) ~
SO(5) ~
... ~SO(2)
is indicated. Anexample
of aphysical application
is given.Classification
Physics
Abstracts 02.20 - 21.60F1. Introduction.
The
algebra
of the rotation group in 6 dimensionsSO(6), equivalent
to that ofSU(4), plays
animport-ant role in nuclear and
particle physics.
Thedecom-position
chainSO(6) => SU(2)
xSU(2)
has beenwidely
known as theWigner supermultiplet
scheme(see
e.g.Wigner
[1]
or Hecht andPang
[2]).
Another well-knowndecomposition
of the irreducible rep-resentations ofSO(n) (and
inparticular
ofSO(6))
has beenprovided
by
the work of Gel’fand and Tsetlin[3]).
However,
in some cases ofphysical
interest(see
e.g. Krumlinde andSzymanski [4]
orPiepenbring
et al.
[5])
adecomposition
different from the abovementioned
Wigner
supermultiplet
scheme is needed. It is based on theembedding
SO(6) > SO(4)
xU(1) ~ SU(2)
xSU(2)
xU(1).
It is aparticular
caseof
decompositions
of Lie groups intoproducts
involv-ing
theU(I) subgroup
as discussed in references[6,
7].
Relevantdecompositions
have also been revie-wed in references[8-10].
In the present paper we intend to
give
acomplete
andelementary
derivation of theexplicit
formulae for the matrix elements of theSO(6)
group(*)
Work partlysupported
by the Polish-US Maria Sklodowska Curie foundation, grant number PF 7037 P.(+ )
On leave of absence from University of Warsaw,Poland.
generators in the
symmetric
irreducible represen-tation. In section 2 we establish the tensorialcharac-ter of the 15 group generators with respect to
SU(2)
x SU(2)
xU(1).
Sections 3 to 5 are devotedto a discussion of the
symmetric
irreduciblerepresen-tations of
SO(6).
In section 6 weexplicitly
calculate in a closed form the matrix elements of the groupgenerators for the
symmetric
representations
ofSO(6) employing
thedecomposition
chainSO(6) :D
SU(2)
xSU(2)
xU(1).
In section 7 we indicate the relations of the obtained basis with that of Gel’fand Tsetlin.Finally,
section 8brings
anexample
of aphysical application
and section 9 the conclusions. 2.Group
generators and their tensorial clas-sification.The 15 generators of the
SO(6)
Liealgebra
can be defined as theoperators
ofgeneralised
momentaJag
with a, b= 1, 2, ... 6
(cf.
forexample
[2, 4]).
In the case of theWigner
supermultiplet
decomposition
the twoSU(2)
subgroups
aresimply
formedby
Jab
with a, b= 1, 2,
3 andJa6
witha, b
=4, 5, 6,
respectively.
The Cartansubalgebra
may be chosenas defined
by
the three operatorsJ12,
J34
andJs6.
Thisscheme, however,
has thedisadvantage
that none of these three operators commutesimul-taneously
with all the generatorsforming
the twosubgroups
SU(2).
For our purpose itis, therefore,
578
more convenient to choose first the four-dimension-al
subspace
(say, corresponding
toJab
with a,b
= 1, 2, 3, 4).
It is well-known that theresulting
SO(4)
subgroup
isequivalent
to theproduct
SU(2)
xSU(2)
of the twocommuting ordinary
angular
momenta. We shall call themquasispins
asto maintain the
analogy
with thetypical
application
of section 8(see
also[5]).
Now,
the operatorJ56
commutes with all the sixgenerators of the
SO(4)
subgroup
and can thus define aU(I)
subgroup
completing
our chainSO(6) > SU(2)
xSU(2)
xU(1).
In
general
(see
Racah[11])
the statesforming
anirreducible
representation
basis of the groupSO(b)
can becompletely
characterizedby
specifying
1/2 -
r
= 6 quantum numbers in addition to the rquantum numbers
specifying
therepresentation
itself(by
itshighest weight).
Here p = 15 denotes the number ofgenerators
ofSO(b)
while r = 3 is therank of the group
(see
e.g.[2, 11]).
However,
we shall showthat,
in the case of asymmetric
represen-tation,
fourquantum
numbers will be sufficient todetermine the
representation
basiscompletely.
We shall now defineexplicitly
all the 15 generators ofSO(6)
in terms of thegeneralised
angular
momen-ta
Ja6
(a,
b =1, 2
...6).
The twoquasispins,
say, Kand L can be introduced as
and
It is easy to see that the two
quasispins
commute, i. e.It is also obvious that the operator
J56
commuteswith the 6
components
of the twoquasispins
K and L.The
threemutually commuting
operatorsJ12
=Ko
+Lo, J34
= -(Ko - Lo )
andJ56
are chosen toform the Cartan
subalgebra.
Theremaining
8generators of
SO(6)
can bearranged
as tensors of theorder 1 2
withrespect
to both thequasispins
K and L.Simultaneously,
these tensors carry + 1 or -1 of thequantity d
which is defined as aneigenvalue
of(- J56 ).
In other wordsand
The choice of the
quantum
number d as aneigenvalue
of(- J56 ) (instead
ofJ56)
has been madeas to remain consistent with the results of
[5]
where d has asimple physical meaning
(1).
Closed-form
expressions
of theeight
componentsT.,2112 8
in terms of thegeneralised angular
momentaJab
can begiven explicitly.
In order tosimplify
the notationTf ’k 1 8
we shall omit thesuperscripts k
and 1 that arealways
equal
to 1/2 andreplace
quantum
numbers K(=
±1/2), A (= ± 1/2)
and 8(= ± 1)
by
their
signs.
Thus forexample
T 112 112
willbe
replaced by
T + ,
etc.and
Tensors
Tf21f2 8
obey
thefollowing hermiticity
conditionThe
eight
generatorsTk 1 8
with k = 1=1
and2
6 = ± 1 exhibit the difference of the present classifi-cation as
compared
with theWigner super-multiplet
scheme[2]
where the nineanalogous
generatorsTK a
are characterizedby k
= I = 1.(1)
Throughout
this paper, we use thesymbol d
todenote the
eigenvalue
of -J56,
which can take any integervalue
satisfying
relation(25).
Thesymbol
6, which is anindex of the tensors we introduce, takes only the two
3.
Symmetric
representations.
The irreducible
representations
are characterisedby
their
highest weights (see
e.g.[11, 12]).
The rep-resentations may be labelledby
the vectorI Jí2, J34, J56) of
the threeeigenvalues
of thegenerators
J12,
J34
andJ56 forming
the Cartansub-algebra. Alternatively,
one canemploy
partitions
[A]
=[A A2 A3]
of aninteger
number 1(1 = A 1 + A 2 + A 3 )
specifying
thecorresponding
Young
diagram
(see
e.g.[12, 13]).
Thesymmetric
irreducible
representations correspond
to theYoung
diagrams [1 j
with one rowonly,
i. e.Al
= 12 =l ,
A2 = A3
= 0(with
thehighest weight corresponding
to the vector with components
d2, 0,
0 in the Cartanspace).
Thepositive integer f2
may have a certainphysical meaning
as it will become clear below. Tospecify
therepresentation
basis,
it seems natural to use the twocommunting quasispins
K and Ltogether
with theeigenvalue
d of(- Js6 ).
Thus weintroduce vectors
with
MK denoting
theeigenvalue
ofKo
definedby
equation
(1)
andK,
the totalquasispin K
quantum number(i.e. K(K
+1)
is theeigenvalue
ofK2).
Themeanings
ofML and
L areanalogous
toMK
and Kexcept that
they
refer to thequasispin
L(Eq. (2)).
Ingeneral,
one motequantum
number is needed in order tospecify
the basis vectors of the irreduciblerepresentations
ofSO(6)
asalready
mentioned in thepreceding
section.However,
in the case of thesymmetric
irreduciblerepresentation
even the basis(7)
is toogeneral
since thesymmetry
of the represen-tationsimplies
thatIn order to demonstrate the existence of this con-straint we can
employ
thebranching
rulecorresponding
to theembedding
given
by King [6].
Here(s. t ) corresponds
to theouter
product
of two Schur functions related to thetwo one-row
Young
diagrams,
while[À I (s . t)]
denotes the outer division[A ]/ [s . t ]
of two Schur functions(e.g. [12,
14,
15]). Finally
fs - t}
denotes therepresentations
ofU(I) specified by
theinteger
s - t. For thesymmetric
irreduciblerepresentation
[I]
ofSO(6)
thebranching
rule(9)
issimply
reducedto
implying
that the irreduciblerepresentations
[1 - s - t ]
of thesubgroup
SO(2 k - 2 )
are alsosymmetric.
In the case of k =3,
onegets
SO(4)
which isisomorphic
toSU(2)
xSU(2).
The symmet-ry of theSO(4) representation requires
thevanishing
of its secondlabel,
i.e. K - L = 0 and henceequation
(8).
It follows from
equation
(8)
that we may charac-terise the basis states of thesymmetric
representa-tions(7) by
only
fourindependent
quantum numbersin addition to the
highest-weight
labels(f2, 0, 0)
characterising
therepresentation
itself. In the case of ageneral
irreduciblerepresentation
ofSO(6)
we have of courseL # K
and there remains one morequantum number to
specify
therepresentation
basis(cf.
ananalogous
problem
in theWigner
supermul-tiplet
scheme is discussedby
Hecht andPang
[2]).
4.Step
downoperators.
We shall now demonstrate an
explicit
construction of all the statesforming
arepresentation
basis.Acting
with operatorsK+
andL,
andstarting
withan
arbitrary
state K, K ;
MK, ML ;
d)
we may al-ways reach theMK
=ML
= K state which we callhereafter the maximum
aligned
(MA)
state for agiven
K :where the
right
hand side is a convenient short-handnotation for the MA states. Thus in order to obtain any state
belonging
to asymmetric
representation
basis it is sufficient to construct all
possible
MAstate-vectors. One can show that the two operators
with 8 = + 1 or 8 = -1 are the
step-down
operators
thatby acting
on the MA states lower the fourquantum numbers K,
L =K,
MK
= K andML
= Kby
one half. Thus we havewhere the constant c
appearing
in thisequation
will be discussed below(Sect. 6).
We can see from
equation
(15)
that therepresenta-tion basis contains both
integer
andhalf-integer
580
and
5.
Representation
basis.Now,
we are in aposition
to constructexplicitly
all the MA statesbelonging
to therepresentation
basisand,
consequently (employing
in addition theK-and L_
operators),
all therepresentation
basis. It issimply
sufficient to act several times with theoperators
U:t l’ on the statecorresponding
to thehighest weight
I Ji2,
J34, J56 ) =
In, 0, 0).
We obtainwith
and where C is a constant which will be calculated
later.
It follows from
equations (16-20)
that thequantum
numbersappearing
on theright-hand
side ofequation (22)
do notdepend
on the order of theoperators
cr+iand
a_ i in thisequation.
Moreover,
one can show that the constant C
appearing
inequâtion (22)
does notdepend
on the order ofu + 1
and o,- 1,
either. It caneasily
be shown that[or+i, u-1]1 {K} ; d)
= 0.Explicitly
onegets
and
which lead to the same result since T =
T_ ±
commute.
Equations
(23)
and(24) imply
that for agiven
value of K we obtainOn the other
hand,
equation
(20) together
withequations (23)
and(24) imply
that d is aninteger
changing
instep
of 2 between the limits indicated inequation
(25).
The total number of states obtained
by
acting
with various powers of thestep-down
operators(18)
onthe
highest weight
state can now be calculated. Thequantum
number K varies between 0and f2 .
2
For each
given
value of K there exists fi - 2 K + 1 differentvalues
of d(cf. inequalities
(25)),
and,
finally,
there is a(2 K + 1 )2
multiplicity
for each
pair (K, d )
connected with the variations in thequantum
numbersMK
andML.
Thus the total number of states results asIt is easy to see that this is
exactly
equal
the dimension of thesymmetric
irreducible representa-tion of theSO(6)
groupcorresponding
to thehighest
weight
In,
0,
0).
This can be shown eitherby
the methods ofWeyl
[16] (see
also Ref.[12],
Chap-ter
14.2)
or elseby employing
theequivalence
of theSO(6) algebra
toSU(4).
In fact(see [2])
thesymmetric
representation
ofSO(6)
withhighest
weight
,fl ,
0,
0)
can be shown tocorrespond
to the irreduciblerepresentation
ofSU(4)
labelledby
aYoung
diagram
{n, n}
i.e. which hasobviously
the dimension
given
by
(27) (see
e.g. Close[17]).
In this way, we haveproved
that the states(12)
indeed span the totalrepresentation
space.6. Matrix elements of the group
generators.
We shall now calculateexplicitly
the matrix elements of all the 15generators
of the groupSO(6)
in the basis of thesymmetric
irreduciblerepresentation
discussed in aprevious
section. Matrix elements ofthe
quasispin
operators
K and L are well known. Foretc. The three operators
Ko, Lo
andJ56 (or J12,
J34
andJ56)
arediagonal
witheigenvalues Ko
=MKI
Lo
=ML
andJ56
= - d,
respectively.
We arefinally
left with the 8 tensor components
(5a-h).
In order tocalculate their matrix elements we need first the
norm of an
arbitrary
MA state(cf. Eq. (22)).
Let usdenote the norm of the state
(22) by JY’(m, n)
i.e.As has been
already
mentioned in theprevious
section,
one can show that the order of theoperators
u +1 and
u -1 inside
the matrix element(28)
does notchange
the value ofN(m, n ).
It can also be shown(see Appendix)
thatN(m, n )
satisfies two recursion relations which areconsiderably simplified
and canbe reduced to one recursion
equation
if oneintro-duces
thequantity
Then the recursion relations
together
with somelimiting
values becomeIt is not difficult to find the solution of the above
set of
equations
Having
the aboveresult,
the rest of the calculation becomesstandard.
Owing
to theSU(2)
symmetry
of the twoquasispins
we may use twice theWigner-Eckart theorem
with d’ = d + 6 and where the
symbol
(Kmkm [ K’ M’ )
inequation (32)
denotes aClebsch-Gordan coefficient. The
doubly
reduced matrix elementappearing
in thisequation
can becalculated when the MA states are taken in
equation (32) together
with K = A =+ 1
or K =A
= -1.
Then the matrix element can be either 2expressed directly by
the normsN(m, n )
orequiva-lently by
thequantity p (m,
n ).
In this case the lefthand side of
equation
(32)
can be calculateddirectly.
Generally,
tensoroperators
Tk1/2 f2 8 can
only
change
K
quantum
numberby
± 2 1
andquantum
number dby - 1.
In order to illustrate theprocedure
let uscalculate a
typical
matrix elementIt is useful to note that the tensor
operator
T - when sandwiched between the two MA statescan be
replaced by
astep
-downoperator
u -1(up
toa
factor).
Indeed,
it follows fromequation (14)
thatonly
the first term of o, - 1 contributes in the matrixelement
given
above.Now,
the factor(2 Ko + 1 )
(2 Lo
+1 ) acting
to theright gives
a c-numberequal
to
(2 K + 1
)2.
Thus we have582
On the other
hand,
we can useequation
(32) directly
for
K = A = -1,16 = -1,
MK=ML=K
and 2M’ K
=M’ L
= K’ =K -1.
Thus,
we can determine2
the- reduced matrix element
for K’ =
K -1
1
and d’ = d -1. We can
repeat
the for K’2
and d’ == d - 1. We canrepeat
the calculation for allpossible non-vanishing
matrix elementsK’ = K ± 1, d’ = d ± 1
2
and substitute them intoequation (32).
The results for thenon-vanishing
matrix elements of all the tensorsT.1121/2+1
1are
given
belowand
where sgn K and
sgn A
denote thesigns
of K and A,respectively.
Matrix elements of the tensorTK ’12 A 1/2 - 1
can be obtained from the above formulae
by
Hermi-tianconjugation (cf.
Eq.
(6)).
7. Relation to the Gel’fand Tsetlin basis.
Another
possible decomposition
of theSO(6)
ir-reduciblerepresentations
is offeredby
the well-known Gel’fand Tsetlin method[3].
It is related tothe chain
S0(6) :D
SO (5 ) z) SO (4 ) > SO (3 ) > SO(2) .
(37)
The
symmetric
representations
ofSO(6)
can belabelled
[4] by a single
column of 5 numbers the first of which(n)
characterizes therepresentation
while theremaining
4 numbers(A, Sm, S
andSo)
determine the states of the basis. It seems convenient to define theSO(3)
subgroup
of the chain(37)
as a totalquasispin
where K and L have been
already
defined in thepreceding
sections. Then the irreduciblesymmetric
representations
of theSO(4)
may be definedby
the maximumpossible
value of S(i.e.
Sm
=(K
+L )L
= K = 2K). Finally,
A may characterizethe irreducible
symmetric representations
of thesubgroup SO(5) (where 8m
A --D).
The matrix elements of all theSd(6)
groupgenerators
can becalculated
relatively easily
with the method used in the case ofsymmetric representation (see
Ref.[4]).
However,
asalready
mentioned in theintroduc-tion,
there exist somephysical problems
where it is more convenient to haveJ56
as agood
quantum
number. This is not the case in the Gel’fand Tsetlin
scheme
[3].
Thus in thistype
ofproblem
it is necessary todiagonalize
the operatorJ56 explicitly
in the latter basis. We thus arrive at the transformationwhere columns with
parentheses
denote theoriginal
Gel’fand Tsetlinrepresentation
basis while those with square brackets denote the new basis in which theoperator
(-
J56 )
isdiagonal
witheigenvalue
d. It follows from the Gel’fand Tsetlin rules that theoperator
J56
does notchange
thequantum
numbersSm, S
andSo.
Thus the summation inequation
(39)
extendsonly
over A. Theeigenvalues occurring
in thediagonalisation
are d =0,
±1, ±
2... It has beenalso our numerical
experience
thatthey
fulfilrela-tions
equivalent
toinequalities (25)
and that dchanges
insteps
of 2 forgiven K (i.e. Sm
and d are both even numbers or else both oddnumbers,
simultaneously).
The transformation coefficientsC J( n,
Sm )
have also to be determinednumerically.
All theseproperties
become clear when we notice that there exists asimple
relationused
throughout
this paper(cf.
Eqs.
(8)
and(9)).
Nevertheless,
the latter is much more convenient forpractical
calculations.8. An
example
ofphysical application.
The
decomposition
of thesymmetric
irreduciblerepresentation
discussed in this paper has been found[5]
in the course of the calculation devoted tothe
coupled
beta and gamma vibrations in deformed nuclei in the framework of anexactly
solvablemicroscopic
model. The model involves twodegener-ate
single-particle
levels with thedegeneracy
2 n for each level withpairing
andquadrupole
forces. The model Hamiltonian used in[5]
can beexpressed by
the generators of
SO(6)
in astraightforward
wayThe first term describes the
single-particle
splitting
between the twolevels,
the second term is apairing
force while the third and fourth terms
correspond
tothe
(À,JL)= (2, 0)
and(À,JL)= (2, ± 2)
compo-nents of the
quadrupole
force,
respectively.
It is easyto see that both
J56
andJ12
aregood
quantum
numbers i.e.
they
commute with the Hamiltonian.Thus,
their values can befixed, a
priori,
and the dimensions of submatrices to bediagonalized
areconsiderably
reduced ascompared
to the numbergiven by equation
(27).
In fact the dimensions of the submatrices become of the order ofn 2
instead of then 4 dependence
following
fromequation
(27).
Theexplicit
formulae for the dimensions aregiven
in[5].
We have
performed
thediagonalisation
of the Hamiltonian(41)
in two ways.First,
by employing
the Gel’fand Tsetlin method with the
preliminary
diagonalization
ofJ56
followedby
thediagonalisation
of H. The other method consisted inusing
therepresentation
discussed in this paper(cf.
Sect. 3 to5)
with theexplicit
expressions
for the matrix elementsgiven
inequations (35)
and(36).
The numerical results obtained for the energyeigenvalues
and matrix elements of
physical
interest(i.e.
quad-rupole
moments, transitionprobabilities etc.)
wereof course identical for both methods.
However,
thelatter has
proved
to bedefinitely superior
in the numericalcomputations.
9. Conclusions.
We have demonstrated a new
decomposition
chainfor the
symmetric
irreduciblerepresentations
of the groupSO(6)
which differsessentially
from that of theWigner
supermultiplet
scheme. It has turned out to bepossible
to constructexplicitly
therepresenta-tion basis and to calculate the matrix elements of the group
generators
in a closedand,
infact,
verysimple
form. The scheme can, for
example,
beapplied
tothe exact solution of the
microscopic
treatment of thepairing plus quadrupole
two-level model in thecase of the
coupled
beta and gamma vibrations indeformed nuclei
[5].
Thegeneralization
of the aboveprocedure
into the case ofarbitrary (i.e.
notneces-sarily symmetric)
irreduciblerepresentations
ofSO(6)
seems to be alsopossible.
Acknowledgments.
One of us
(Z. S.)
wishes to express his warm thanksto the Universite
Scientifique
et Mddicale de Greno-ble for the one-year invitation and to the staff of the Institut des Sciences Nucldaires in Grenoble for the wonderfulworking
conditions.Appendix.
Instead of
giving
a full derivation for the calculationof the norm
(28)
and theresulting
recursion relation(30)
we shallonly
outline the main idea of theproof.
We shall follow
essentially
ananalogous
calculationgiven by
Hecht[18]
for thesimpler
chainSO(5) =)
SU(2)
xSU(2).
We start withequation
(28)
for thearguments
(m
+1,
n )
Now the operator
ul
acts to theright (and
in fact tothe
left,
aswell)
on an MA state andconsequently
can be reduced toonly
one termT+ +
multiplied
from the left
by (2 Ko
+1) (2 Lo
+1)
which reducesto a number
(cf.
ananalogous
trick usedalready
in theexample
given
in Sect.6).
Thus :The last
part
of thisequation
is,
of course, anidentity.
Let us call the first and second terms of the lastpart
inequation (A.2) (I)
and(II), respectively.
Calculating
the commutatorappearing
in(II)
andneglecting
what vanishes when sandwiched in584
Casimir operator
C
of theSO(6)
group and someadditional terms that we discuss below. The Casimir
operator
C
is defined asIt has an
eigenvalue n (n + 4).
The additionalterms
following
from the calculation of thecom-mutator in
(II)
are eithersimple
functions ofJ12, J34
andJ56
and are thusdiagonal
or else form anexpression
that is to be sandwiched between the MA states as
indicated
by equation
(A.2).
We can now express(I)
as well as both contributionscoming
from(A.4) by
using
the same trick as in section 6 andby
observing
that
and
with
proportionality
coefficientsdepending
onN (m, n)
andN (m - 1, n ),
orN (m, n )
andN (m, n -1 ), respectively. Taking
all the termstogether
wefinally
arrive at a recursion relationUsing
relation(A.7)
and itsanalogous
forN (m, n
+1 )
one can calculate the first values of the norms.Further,
by
induction,
one mayeasily
showthat
Now,
using
the definition(29)
ofp (m, n ) (observe
that p(m, n )
# p(n, m ))
one caneasily
derive the recursion relations(30).
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