It is shown that the symmetry Lie algebra of a quantum system with accidental degeneracy can be obtained by means of the Noether's theorem. The procedure is illustrated by considering a generalized anisotropic two dimensional harmonic oscillator, which can have an infinite set of states with the same energy characterized by an u(1,1) Lie algebra

Castanos, O.

Lopez-Pena, R.

English

NASA Technical Reports Server

N93-27348
FERMION REALIZATION OF EXCEPTIONAL LIE ALGEBRAS
FROM MAXIMAL UNITARY SUBALGEBRAS
A. SCIARRINO
Universitd di Napoli Federico H
Dipartimento di Seienze Fisiehe
Mostra d'Oltremare Pad.19
80125 NAPOLI ITALY
Abstract
From the decomposition of the exceptional Lie algebras (ELAs) under a maximal unitary
subalgebra a realization of the ELAs is obtained in terms of fermionic oscillators.
1 Introduction
Realizations of classical Lie algebras (LAs) in terms of bosonic and/or fermionic oscillators
are known long since and are very useful in several physical contexts. Via the embedding of
50(8) (_ SO(8) C E8 a realization of ELAs in terms of fermionic oscillators has been obtained
by the author [1]. However it is more convenient to dispose of several different realizations
of ELAs which allow to describe in a more appropriate way different subalgebras embedding
chains. Moreover, e.g., the embeddings G2 c 50(7) and F4 c Ee are not "deformable",
while the embeddin8_ SU(3) c G= and SO(9) c F4 are "deformable". The proposal of this
contribution is to present a realization of ELAs in terms of multilinears in fermionic oscillators
via the embedding of a maximal unitary subalgebra. It should be quoted that constructions
of ELAs as bilinears in fermionic fields in the basis SU(9) and SU(3)' has been obtained by
Koea [2]. While Koea's approach makes a more evident connection with physical applications
in a GUT framework, the multilinear approach keeps a closer connection with the algebraic
structure of LAs (roots, weights, etc.). Moreover this formalism allows to obtain multilinear
realizations for all the fundamental representations and for generators and vector spaces of all
maximal embedding$ of ELAs [3].
2 Composition law for fermionic multilinears
Let us introduce a set of N fermionic oscillators a,+, a, satisfying: (i,j = 1,2,...,N)
(i)
379
https://ntrs.nasa.gov/search.jsp?R=19930018158 2020-03-17T06:06:01+00:00Z
i
i
g
A fermionic multiline_ (f.m.) X is defined by the following formula:
(f, -- a+,f_, -- a_,i > O)
x=IIf, i_Icz" (2)
i
The number of f, willbe called the order of X.
We define the contraction of two bilinearsX and Y of,reap.,order N and N' as a operation
giving s f.m. (X-"Y) obtained from the m. XY by deleting the couples (ifany) (.f,,j'_,)with
]',"in" X and f__ "in" Y, multiplied by a factor (-1)", n being the number of transpositions
necessary to obtain allthe ]'inear to j'__in XY, and by a rational coe_cent C(N,N', Z), Z
being the number of contractions.
We definea composition law (X o Y) of two f.m. by the followingequation (ij=E I,j_E J)
We remark:
• XoY=-(YoX)
• X o Y = [X, Y] (N, N'E 1,2)
We put (N,N' = 1, 2, 3, 6; ArT = order of X"Y):
• C(N, N', 0) = 1
• C(N, N', I) -- _NT,N or 6NT,N ,
" • C(N, N, N:I) =
• C(N, 2N, N) =½ (N> 1)
=
• C(N, N, _) -1 (N even)
3 Realization of Es
We consider the embedding SU(9) c//78. The adjoint representation of Es decomposes as i
248 =_ 80 + 84 + 84
Introducing a set of 9 fermionic creation and annihilation operators and we cam write
(id = 1,2,..,9):
(4)
so _= { (._'.,} (i # j), _ .i.._ - _,+,.,+_+= h.- h.+,
1
+ + + ¢#ja,,,._.ala.,,a.apa, a,.}84 -- {a ia ia_ +_.
1 + + + + + +
84 =- {o_a¢a_ + "0.. _Pa_"P_'at a,,,a,, % aq a t }
In the following we call:
380
(,#o)} (s)
(o)
(v)
• a + "hermitian conjugate" (h.c.) of a,;
+ + + + + + "dual coniugate" (d.c.) of a_a_ak.
• _isk, lmnpqral area nap aq a r
Proposition I The above set of bilinears and trilinears in the fermionic oscillators closes and
satisfies the Jacobi identity under the composition law (o) defined in See. t_.
The generators corresponding to the simple roots axe:
o_, _ a'_aa, c_, -.-. a, a2as + d.c., ak _ aLla_ (3 <_ k _< 8) (8)
The generator corresponding to the highest root is a_'ao.
4 Realization of E7
In the embedding SU(8) c E7 the adjoint representation decomposes as:
133 ==_ 63 + 70 (9)
The SU(8) C E7 is not contained in the SU(9) C Es, Exploiting the property that the two
unitary algebras have a common maximal subalgebra SU(6), the following realization of E7 is
obtained (ij,k = 1,2,..6 ; r = 1,2,..,5):
5
63
- {aTai -+-+-+, "7"s_ +d'c', a +aT, a +a +a_+d.c., h.c.
2 1
h,-h,+l, _(h,+h.+h,)-_. _, 2h,-h.-h,} (10)
!
70 -= {a_a_a7 + d.c., a.ia._ah + d.c., h.c.}
Realization of ti70
In the embedding SU(6) (_ SU(2) c E6 the adjoint representation decomposes as:
78 ==_ (35,1) + (1,3) + (20,2)
We have (id,k -- 1,2,..6; r -- 1,2,..5):
(35,1) ___{a+a_,
(1, 3) =- {a + a; a_ + d.c., h.c.,
h, - h,+l}
2 1
g(h, + h, + _) - g _ h,}
(20, 2) = {a_a¢ak + d.c., h.c.}
(11)
(12)
(13)
(14)
(15)
381
6 Realization of F4
In the embedding SU(4) _ SU(2)' c F4 the adjoint representation decomposes as:
52 ==_ (15,1) + (1,3) + (4,2) + (4,2) + (6,3) (16)
The most convenient way to identifythe elements of F4 isthe following:
i)draw the Dynkin diagram of Ee;
ii)from i) draw, by folding,the Dynkin diagram of F4, identify the corresponding simple
roots and the highest root;
iii)draw the extended Dynkin diagram of F4 and then, by deleting a dot, identify
SU(4) _ SU(2)'.
We get for the 52 (ij,k= 1,2,.6):
7
7 "$ ""9 + d.c.,
a+ a_ + (-1)'+'-la_,'a,
a_a_ak+d.c. (i<j<k;i+j+k=M;M=6,7,9,10,..14), h.c.
a_ajat+a_a,a_+d.c. (t= l,3.4;i# j #k#l;i+J+k +l= 7),
2 1 __, hi, hs + h4
-_(h, + h, + hs) +
i
hi + ht - h2 - h6, h2 + h4 - hs - h5
(i + j = 7), h.¢. :_;::::
(i#i# k #1;i+j < k+t;i+i+k+l = 14),
h._.
Realization of G2
In the emdedding SU(3) C G2 the adjoint representation decomposes as
14=_8+3+3
where (i = 1,2,.9; j = 1,2...6):
h°c.
(17)
(18)
8 _- (axa2as + d.c., aTata9 + d.c., a+ as, h.c.
1 2 1
-_(hz + h, + h_)+ § _ h, - _(h, + h, + h,) + _ _. h,)
i>3 $
(19)
3 + S = {_,+_, _,_ + d._., .,_+_+_+-,-,d._., b.c.} (2O)
8 Conclusions
2 :: :z:_ :
One of the advantages of the oscillators construction of LAs is the knowledge of the Fock space
which becomes the carrier space of irrep, of the the LAs.
382
In the case of construction of LAs SU(N) by using fermionic oscillators it is well known that
the carrier space of antisymmetric irreps, can be realized on the Fock space. As the fundaznen-
tal irreps, of G2, Es, ET, of dimension, resp. 7, 27, 56, decompose under the maximal unitary
subalgebras as a sum of antisymrnetric irreps,as:
7 ==_ 3+3+1
27 ==_ (15,1) + C6,2)
56 ==# 28+28 (21)
one can think that on the Fock space of the fermionic oscillatorsitk possible build up the
fundamental representations,at least,of these ELAs.
Indeed for G2 this has already been obtained [4].
References
[1] A. Sciarrino, J.Math.Phys. 80, 1674 (1989)
[2] M. Koca, Phys. Rev. D 24, 2645 (1981)
[3] A. Sciarrino,"Fermionic multilinear realization of exceptional Lie algebras", in preparation
[41 L. Frappat and A. Sciarrino, J.Phys A 25, L383 (1992)
383



Symmetry algebra of a generalized anisotropic harmonic oscillator

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