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A NOTE ON SL(-) CLASSICAL YANG-HILLS THEORIES
E.G . Floratos s
Theoretical Physics Division, CERN, 1211 Geneva 23, Switzerland
In an interesting work, Hoppe has studied
the canonical quantization of a relativistic
spherical membrane of Dirac type , in the light
cone gauge .
He was able to show that this
theory is exactly the same as the quantum
mechanics of space constant SU(N) Yang-Hills
potentials4 in the limit N+ - .
The basic
ingredients of his proof are first, that the
SU(N) Lie algebra, when N + -, can be shown to
be isomorphic to the infinite dimensional Lie
algebra of area preserving (or symplectic)
di£feomorphisms of the sphere S2 which is the
symmetry of the relativistic membrane after
gauge fixing ; second, the particular form of
the membrane Hamiltonian in the light-cone gauge
is the same as that of SU(N) space constant
Yang-Hills potentials (in the gauge A0 = 0) for
N+- .
One would like to have an analogous geome-
trical interpretation for the full Yang-Mills
theory, that is for space-time dependent poten
tials . In this work, although wa do not find
any direct relation of the Yang-Mills theory to
the membrane theory or an extension of it, we do
find a simple geometrical picture of the SU(-)
Yang-Mills theory by making use of the equiv-
0920-5632/89/503 .50 © Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)
Nuclear Physics B (Proc. Suppl.) 11 (1989) 350-354i4orth-Holland, Amsterdam
alence between the SU(N), N ; -, Lie algebra and
that of the "symplectic" or area preserving co-
ordinate transformations of an "internal" sphere
S2 .
Indeed ure show that in the large N limit
the classical SU(N) fields (NxN matrices) :
AP(x) = Aat, ta
SU(N), a - 1, . . .,N2-1
are replaced by c-number functions of an
"internal" sphere S2 :
p = 0,1,2,3
(1)
- itA (X,O,e) = E
E
ARm(x)Y (O,e), (2)p
R=1 m=-.t p
~,m
where Ylm are the spherical harmonics on S2 .
The gauge transformations
and
SAp = ôNw + [Ap,w], w = wata
(3a)
8FpV = [FWV,w ],Fp.V
= ailAV - 8VA il +
(3b)
are replaced by
+ [A, t,AV ]
On leave of absence from Physics Department, Univ. of Crete, Iraklion, Crete and Research Centre ofof Crete .
Work done in collaboration with J. Iliopoulos ENS, Paris and G. Tiktopoulos, National Technical Univ .Athens .
according to
SN =
2 f d4x trFpVF"
gN
in the large N limit becomes
where
SAp(x,9,$) - 8 Iw(x,9,0) + JAp,w}
(44)
ÔFpV - IFhV'VI, FIIV=
ailAV - 8VAp +
FpV (Xse'0
and g is the limit of N-1-gN'
{f'SI=
af
aM -AL
asacose be
80 acose °
as we pass from Eq . (1) to (2) .
The action of SU(N) Yang-Mills theory
SN
=
12
JS2dQ jd4x F1WGx'e'O16ug
Fliv(x,e .e) - biAV(x,e'~) -
- aVAg(xse'm) + JApaAV I
E.G. Floratos/SU(oo) classical Yang-Mills theories
(4b)
where the Poisson bracket of two functions, f
and g on S2 is defined as
The commutators are replaced by Poisson brackets
lim N[Ag,AV] = JAil AV1
(6)N-b-
In this note we shall give a heuristic
proof of this construction . We begin by review-
ing the defining properties of the infinite
dimensional Lie algebra of symplectic or area
preserving diffeomorphisms of the sphere(SDiff(S2)) .
The generators of symplectictransformations of a surface, locally have theform
- af
a _ af
aIf
aQ2 aQlaQ i aQ2
and they satisfy the algebra
[Lf,L9]= L{f,gi
for f,g differentiable functions of S2 ,
f,g 60(S2) . Here{f,g} is the Poisson bracket
defined as
~f .g} =af$ - AL
a01 a02
a02 801 ' (12)
in local co-ordinates a,, a2 of the surface such
that the area element is
dA = daida2
(13)
that is, in canonical or symplectic co-ordi-
nates5 .
In the case of the sphere of = *, a2 -cose . For the sphere, we can form a basis of
generators choosing f = Yl'm(® .$) to be the
spherical harmonicsl '6.
L
BYIm
a - aYRm
a
(14)R,m = acose o
o acose
Relation (11) implies that
it . .m ..
(15)[LR'm,Lk,'m, ] = fR.m,R,m, , LR-m-
where the structure constantsflwm»
areIm,R ®m ®defined by expanding the Poisson brackets on the
basis YRm:
(16)
These structure constants have been calculated
explicitly in Ref . [1] .
To get a feeling of
352
the geo
note that for R = 1, m - 0, ±1,
usual angular momentum generators
lization), Lz9
so
trical meaning of these generators, we
these are the
(up to norma-
L+ . Also,
= -1Y1'mgyR®m°J
[Ll'+l,Lksmj = [R(j.+l)-M(m±l)1 Lt.mtl(18)
[L1,0,TJA.'mj= aLjt'm.
i"m"-fïm,R'meYR"m"
Equations (18) and (19) mean that the infiniteset of generators L.
is reduced to an infiniteA,msum of irreducible sets of operators of definiteangular momentum 1,
Lx'-v LA.,-R+1' . . ., L't R A. = 1,2, . . . (20)
with respect to the SU(2) Lie algebra, L1 0,L1+ 1 of solid rotations of the sphere . We nowchoose an appropriate basis for SU(N) 1 . Thespherical harmonics YRm(®,f) are harmonic homo-geneous polynomials of degree A in threeEuclidean co-ordinates xi, x2, x3
xl = cosesin®, x2 = sinesin®, x3 = COS®
x . . .xXI,
its
(17)
(19)
where a(m)
i Is a symmetric and tracelessis
. . ,s
1tensor . For fixed 1 there are 2141 linearlyindependent tensors a(m)
7 .fi' . .sii
m
Let Sl, S2, S3 be NxN Hermitian matriceswhich form an N-dim irreducible representationof the Lie algebra SU(2)
E.G. F'loratos%SU(oo) classical Yang-Mills theories
[Si psj ] = isijkSk
(23)
In Ref . 1, it was shown that the matrix polyno-mials
T(N)= E
a(m)
S
S
(24)A m
'k-1,2,3
11 . . .IR il
. .Lit
for A. = 1, . . .,N-1, m = k, . . .,R can be used to
construct a basis of N2-1 matrices for the fun-
damental representation of SU(N), with corres-
ponding structure constants f(N) :
T(N)
(N)
a
(N) Jt"m"
(N)
[ Rm'eTR'm°j
ifRM,R®m,, Tim (25)
There is a normalization of the generators TI N )such that the limit
N.f(N)I"m"
+
fit"m..
RM,A °m' N-
A,m,A °m° s (26)
exists and coincides with the structureconstants of SDiff(S2 ) as defined in Eq . (16)l .
After these preliminaries we proceed to esta-
blish the relation of the infinite dimensional
algebra (15), SDiff(S2 ), to the SU(N) algebra as
N + -, by an argument which avoids the explicit
computation of Ref. 1 for the structure
constants f(N) and f.
If we rescale the genera-
tors of SU(2) by 1/N,
Si-' Ti
= N S1(27)
they satisfy the algebra
[Ti9T~j s 1 N eijkTk
(28)
and the Casimir element
T2 =Ti+T2+ T3= 1+N
(29)
has ,a finite limit for N + co .
Under the norm
(21)
m(®9$) = E1 a(m)
kk=1, . . .,R
=18293 ,-
(22)
Ix/2 = Tr:K2 ,
(30)
for x
SU(2), the generators Ti , i = 1,2,3 have
definite limits as N + - which are three objects
x1, x2, x3 which commute [by (28)] and are
constrained [by (29)] according to
xi + x2 + x3 = 1 .
(31)
If we consider two polynomial functions of three
commuting variables f(xl,x2,x3) and g(xl,x2,x3),
the corresponding matrix polynomials
f(Tl,T2,T3), g(Tl,T2,T3) have commutation rela-
tions for large N which follow from (28) .
lim. i [f,g] = eijkxi Of j L9
I-
This is similar to the passage from quantum
mechanics to classical mechanics . This can be
generalized to all semi-simple Lie groups . Thus
the large representations of SU(2) give rise to
a symplectic structure on the sphere . Large N
limits of quantum models have been considered as
classical theories with particular symplectic
structures in Refs . 8 and 9. If we parametrize
xi by polar co-ordinates (21), we see that the
right-hand side of (32) is nothing else than the
Poisson brackets of Ref . 5 . Consider now the
basis Tkm) of SU(N) obtained by replacing in
(24) the matrices Si by the rescaled ones Ti .
Then according to (27) we obtain
lim i [TÂm) JR'm'] _ {Ykm'YR'm'1N-~
E.G. Floratos/SU(oo) classical Yang-Mills theories
(32)
(33)
If we replace the left-hand side of this
equation by Eq . (25) and the right-hand side
with Eq. (16) we obtain Eq . (26) .
To arrive at the new type of SU(-) gauge
invariance described by Eqs . (1)-(9), we expand
the classical Yang-Mills fields in the basis of
the matrices Tim) :
AL(x) = Alm(x) (N),R = 1, . . .,N-1,
(34)
Equation (33) implies that the commutator of the
gauge fields as N
m has as a limit the Poisson
bracket
N[AIl AV ]
+
JAlà,(x,0,40), V(x,0,4)1
(35)N-
The factor of N necessary for the existence of
this limit can be obtained by rescaling the
gauge field in the initial Yang-Mills theory by
the substitution
Ap + NAP.
(36)
353
It is easy to see now that the large N limit of
the classical Yang-Mills theory (1), (3a-b) and
(7) is the "Poisson gauge theory" (2),(4a-b) and
(8) .
The new gauge theory presented here may be
relevant for the study of the planar limit of
SU(N) Yang-Mills theory10-12
.
REFERENCES
1 . J . Hoppe, Ph .D thesis, MIT (1982) and Aachenpreprint PITHA86/24 .
2 . P.A.M. Dirac, Proc. Roy . Socl 268A (1962)57 .
3 . P.A . Collins and R.W . Tucker, Nucl . Phys .B112 (1976) 150.
4 . M. Löscher, Nucl. Phys . B219 (1983) 233 .
5 . V . Arnold, "Méthodes mathématiques de lamécanique classique", Edit . MIR, Moscow(1974) .
6. E.G . Floratos and J . Iliopoulos, Phys .Lett . B201 (1988) 237 .
7 . M. Na ermesh, "Group theory and itsapplication to physical problems", Reading,Massachusetts (1962) .
8. ®. Jevicki sad N. Papanicolaou, Ann. ofPhys. 120 (1979) 107 .
9. L .. Yaffe, Rev. Modern Physics 56 (1982)407 .
10. G. °t Hooft, Nucl. Phys . B17 (1974) 461 .
Il. G. Veneziano, Nucl. Phys. B117 (1976) 519 .
12 . R. Witten, Cargése Summer Institute, ed. byG.
't Hooft et al ., Plenum Press, (N.Y.),(1979) .
E.G. Floratos/SU(oo) classical Yang-Mills theories
