Volume 212, number 2
PHYSICS LETTERSB
22 September 1988
P O L Y G O N DISCRETIZATION OF THE LOOP-SPACE EQUATION
Yu. M. MAKEENKO
Institute for TheoreticalPhysics. Universityof California, Santa Barbara, CA 93106, USA
and Institute for Theoreticaland Experimental Physics, 117 259 Moscow, USSR
Received 10 June 1988
Methods for studyingequations on the loop space are developedusing interpretation of the loop equation of many-colorQCD
as the functional (inhomogeneous) Laplace equation. We discretize the loop space by M-vertex polygonsto approximate the
functional laplacian by a (finite-dimensional) second order partial differential operator. A path-integral representation for the
Green function of the functional laplacian is obtained as the M--,~ limit of the Green function of the approximatingoperator.
Finally, we speculateon the possibilityof extending this approach to relativistic string.
The loop-space equation (for a review, see refs. [ i ,2], and references therein) had been proposed originally
to describe the many-color limit of QCD. Recent interest in this subject is due both to attempts at numerical
solution [ 3-6 ] and to studies of general properties of the loop space in the context of string field theory.
The goal of the present work is to develop systematic methods for studying equations on the loop space. Our
approach is based on the fact that the loop equation can be interpreted as the functional (inhomogeneous)
Laplace equation. A discretization of the loop space by M-vertex polygons is used to approximate the functional
laplacian by a (finite-dimensional) second order partial differentia! operator. We obtain the Green function of
this operator whose limit as M - , m determines the Green function of the functional laplacian and present a
path-integral representation for the latter Green function. Finally, we speculate on the possibility of applying
this approach to the case of relativistic string.
We deaI with the loop space whose elements are continuous closed loops which can be described in a parametric form by functions, xz (a), where the parameter a0 ~<6~< 6f. These functions have to satisfy three conditions: (i) the values o'=60 and a=6~ are identified - the loops are closed, (ii) the functions x,,(6) and
A,,,x~ ( a ) + ot~,, with Au, and oe~ being independent of a, represent an equivalent loop - rotational and translational invariance, (iii) x u ( 6 ) and xz ( a ' ) with a' = f ( 6 ) represent the same loop - reparametrization invariance.
Two famous examples of the loop space are ( 1 ) lines of force in Yang-Mills theory and (2) trajectories of the
end of a string.
The standard form [1,2] of the loop equation involves derivatives with respect to path, 0~, and area,
8/Scru~ (x), which are well-defined for so-called functionals of the Stokes type. These functionals satisfy the backtracking condition: they are not changed when an appendix passing back and forth is added to the loop at some
point. This condition represents the Bianchi identity of Yang-Mills theory. The path and area derivatives can
be expressed via the variational derivative, 8/8xu (~,), by the formula
8
8
where 2~ (6) = do:,,( a ) / d o and the sum on the RHS is present for the case of functionals having marked (irregular) points x~=x(tr,).
221
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22 September ; 988
The original form of the loop equation of many-color QCD, written for the Wilson loop expectation vaiue,
W(C), reads
C
w ( c ) =2. f dx; ~ ( x - x ' ) W(C~,) W(C~,,4,
c
(2)
where 2 =gZN remains finite in the many-color limit. To find W(C), eq. (2) should be solved for the class of
Stokes functionals with the initial condition W(0) = I for loops which are shrunk to poims.
However, the original eq. (2) is not formulated entirely on the loop space. The (d-vector) operator on the
LHS depends on the point x and involves non-trivial (non-commuting) path and area derivatives. These derivatives are closely related to Yang-Miiis perturbation theory but it is difficult to apply them to a general ansatz
for W(C) or to implement them in numerical studies.
A much more convenient form of the loop equation can easily be obtained by integrating both sides ofeq. (2)
over dx, along the same contour, C, which yields
f dx~ ~-~
,, ~-----~-~3e~.,(x) v / ( c ) = ~ f dx~ 3:~ d ~ , c ' ~ ( x - x ' ) W ( C ~ , ) W ( C ~ , ~ ) "
c
c
c
(3)
Now, both sides of eq. (3) are weii-defined on the loop space. The operator on the LHS ca~ be interpreted as
some (infinitesimal) variation of the eiements of the loop space. The proof of the fact that scalar eqo (3) is
equivalent to d-vector eq. (2) is based on the important property of eq. (2) both sides of which are annihilated
identically by the operator &.~ [ 7 ].
When applied to reguIar z~nctionals which do not have marked points, the operator on the LHS of eq. (3)
can be represented using eq. ( i ) in an equivalent form
8
8
Such an operator was discussed by Gervais and Neveu [ 8 ] who first pointed out that the (classical) loop equation is nothing but a (homogeneous) functional Laplace equation [ 9]. Thus, eq. (3) can be represented as an
(inhomogeneous) functional Laplace equation
Aw(c) =J(c),
(5)
where J(C) stands for the RHS ofeqo (3).
We develop operator calculus on the functional loop space by making finite-dimensional approximations° The
simplest approximations by finite-dimensionai Laplace operators [ 9 ] cannot be applied to our equation with
] ( C ) that corresponds to many-color QCD. For this reason, we shall consider the polygon approximation.
Any continuous loop can be approximated by the M - ~ limit of M-vegex polygons whose vertices, x i ( i = 1,
.... M), coincide with M points, x(ai), chosen arbitrarily on the loop. The functionAs W(C) can be approximated by functions W(x,. .... , x ~ ) of d.M variables provided they satisfy some relations that guarantee the
desired functional limit. For our case, any solution ofeq. (5) wilt satisfy these relations when ](x~, ..., xM) - an
approximation of J ( C ) - does. The explicit form of ](x~ .... , XM) given below possesses this propeny~
We suggest the following finite-dimensionA approximation of the functional taplacian by a second order partiaI differentia operator
a~')=
L ~(2iJ
i,j= !
~xj
•
(6)
Due to the cyclic symmetry, the MxM-matrix Qo sb.ould depend only on i i - j ! :
Q . . . . . =qx
222
(K=0, 1..... M - 1),
qx=q ..... *
(7)
Volume 212, number 2
PHYSICS LETTERSB
22 September 1988
Here and below all indices are defined mod 3/.
The simplest choice of the numbers qK which guarantees that A (M~ tends to the functional laplacian as M ~ c o
reads
qx=(q) x, K = 0 ..... [M/2],
(8)
where the number q < 1 satisfies
q=*l,
M(1-q)--,c~
asM-,~.
(9)
The proof of the fact that our choice ofq~: gives A(M)~A as M--,m is given in the appendix.
An explicit form of iX(M~ reads as follows:
~(g
zX
where the sum over K contains/M/2J+ 1 terms. However, only K ~ ( 1 - q) - ~<< M terms are non-vanishing as
M-+ ~ for our choice of qx given by eqs. (8) and (9). In this sense our discretization of the functional laplacian
is local. It is worthwhile mentioning that Z~(M~, which is a second order differential operator at finite M, becomes
a first order operator (satisfies Leibnitz rule) as M-+ ae.
The operator a (M) is used for discretizing the LHS of eq. (5). The RHS can be approximated by
or(.......... ~=a 2 ~ (x,-x,_l),(x~-xj_~),
Xa(½(xi+x,_,)--½(xj+xa_~)) W(xi, x,+~ ..... xj_2, x2_~)W(xj, xj+l ..... x,_z,&_~).
(11)
Other approximations of J ( C ) can be made with the differences vanishing as M+oa. A convenient approximation is when the region I i - j ] ~<( 1 - q) - ~is eliminated from the sum overj in eq. ( 11 ). That corresponds to
the principal value integral over d_v; of the RHS of eq. (3).
Thus we propose the following discretization of the loop-space eq. (5):
2~{M)W(x, ..... XM) =J(x~ ..... XM)
(12)
with k (M) defined by eqs. ( 6 ) - (9) and J(x,, ..., XM) given by eq. ( 11 ). Below we discuss some properties of
this equation and transform it to an integral form which is more convenient for practical purposes (at least for
iterative solution).
The operator A(v~ can easily be inverted using the standard Green function to give
W(x ...... xM) = 1- ½i d A (J(x~ ~
0
~...... xM + vTA ~M) ) e,
(13)
where
( F ( ~ ...... {.u) )e d~=rJ E ~
d~ exp( - ~ Y¢,J{,Q~7~{J) F({ ......
f~',~ 1 d~, exp( - ½2,j~Q,7 ~ )
~,,u)
(14)
and
i.~=l ,ia~l,j~-.. i~l ( ll~q ,2i ...~ l@q2 (,,-,i_l )2) .
(15)
This formula requires some comments. The Green function describes evolution from the initial M-angle, xe,
to the final one, x,+ x/CA{j, during the proper time A. It is represented in a form that admits the M-+oo limit. For
Volume 212, number 2
PHYSICS LETTERSB
22 September ! 988
our choice of the matrix Q~/(i ) the exponent in eq. ( 14 ) is positive definite (ali eigenvaiues of Q~j are positive ),
(ii) the inverse matrix Q [ : is !ocal (only diagonal and next to diagonal elements are non-vanishing), (iii)
typical loops that contribute to the integral are continuous [ ( ~ - ~ _ ~)2 ~ ( 1 - q ) ] but not smooth.
The property (iii) can be reformulated in a slightly different form. Let us consider the corre!ator of{~ and 6,
which equais
<~,Y~'~.~j> =6U~Q~j =6.""q I,-Jl ,,~,odM)
( 16 )
When q-~ 1, ~¢and ~ are correlated for i i - i f ~ ( 1 - q) - ". Since I << ( i - q) - ~<< ld, this implies that continuum
limit exists° For q=0~ which corresponds to approximating the functional iaptacian by means of the ordinao ~
finite-dimensional laplacian [ 9 ], ~ and ~ are not correlated so that there is no continuum limit.
Eqs. ( 13 ) - ( ! 5 ) allow a path-integral representation for the M - , m limit of the finite-dimensional Green function. Introducing .:he parameter <r whose discretization is defined by ~ = ~ o + ( o ~ - ~ o ) i / M ( i = I ..... M ) , one
gets the following representation for the Green function of the functional iaplacian:
~/(x(.))=~-½
7
1 d~ < ~ ( x ( o ) + , / ~ ( . ) ) > : ,
o
(17)
where
~-o
fe(o):e{~/,)~e -s
(!8)
and
S = ½ j~ d~ ( ~ ~2(~)+ E~2(~;)).'
~u
(19)
To obtain this representation we first fix !/~ = M ( 1 - q ) as M ~ c ~ and then let e-~0. This expression looks iike
a partition function for a harmonic oscillator at temperature e while the condition ~(0) = ~( 1/e) represents the
fact that the toop is closed. The second term on the RHS ofeqo (19) cannot be omitted, however, when ~ 0 .
While reparametrization invaria~ce is lost in eq. ( 19 ) for e # 0, it can be shown that non-invariant terms are
of order ~ as e~0, i f F ( ~ ( . ) ) is invariant. Therefore reparametrization invariance is recovered when e-~0.
The above formulas allow us to iterate eq. ( 17 ) in 2 starting from the solution W(C) = 1 at 2 = 0. It is easy to
show how the simplest diagram with the gluon propagator is reproduced to order 2. I have verified as we11 how
the diagram with the ,:hree-gluon vertex appears to order 22 In this case~ the calcu!afions are more tedious. I will
present those in another publication [ i 01.
Let us now discuss whether it is possible to extend our approach to other theories including the theory of
relativistic string. An important observation is that the functional iapIacian is welLdefined for a wider cIass of
fnnctionals than the Stokes class. It seems to yield a finite answer acting on any reparametrization4nvariant
functional The functional Lapiacian is in some sense a uniclue reparametrizationdnvariant operator in the loop
space and might be tried for describing other theories.
Let us consider the case of the scalar field p (x). [ have verified that the reparametrization4nvariant equation
Aw(c)=~ f d. ~ d~' .j~./,~(~.)~(~(.)-~(.,)) w(c),
where ~ is a constant of dimensionality of [mass ] 2-~u2 does describe the free-field case. I have shown that
(2o)
Volume 212, number 2
PHYSICSLETTERSB
22 September 1988
where the average goes over the quantum fluctuations of the free scalar field, satisfies eq. (20).
An interesting issue of our approach might be application to the string theory,. To compare the functional
laplacian A with the standard operator g2/8xu(a) 2, it is convenient to diagonalize Q~j introducing the normal
coordinates
o~ = 1 ~ gj~exp(2rcijn/M), o~5_~ = ( ~ ) * .
(22)
:vl 2= 1
For smooth loops, the eel's coincide as M + o e with the standard normal modes of the string. They differ, however, for n ~ M f o r the case of non-smooth loops. The diagonalized operator A <v) reads
~M/2;
(l_q2)
1 0 ~
A(M)= ,=~, M [ I + q ~ - - 2 q - 7 ~ ( 2 n n / M ) ] . ~
'
(23)
For smooth loops, this operator becomes proportional to the standard one while they differ by the n ~ M terms.
An advantage of the operator (23) compared to the standard one is that contribution of the n ~ M terms is
suppressed.
It would be very interesting to check whether it is possible to construct a reparametrization-invariant term
associated with the string tension as well as that describing the string interaction.
I am indebted to A.A. Migdal for many useful discussions. It is a pleasure to thank the organizers of the "QCD
and its applications" program at ITP and especially A1 Mueller for the invitation and for their warm hospitality
at Santa Barbara. This research was supported in part by the National Science Foundation under Grant No.
PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration, at the University of California at Santa Barbara.
Appendix
Let us prove that the second order partial differential operator A ('vt) defined by eqs. ( 6 ) - ( 9 ) tends to the
functional lapiacian A as M--,oo. For this purpose, let us apply A ('a) to the non-abelian ansatz
~( . . . . . . . . . ) = l t r ( ~
'
N
\,=~
L~(. . . . .
~'
))
(A1)
with
L~(x . . . . x~) = 1 + A + ½A.2+ 6A3.+ ~ c~0.2A.- ~2c2[ 0A., A. ] + O (3/1-4)~
(A2)
Here A -=A, (½ (&+x~_ ~) ) (xi-&_ ~)u is antihermitian NXN matrix which is of order O(M-~ ) for smooth
loops when i x i - x, _ ~[ ~ M - ~. For c~ = c2 = 1, Ui coincides with the non-abelian phase factor up ~0 terms of order
O ( M - 3 ) . While these terms are not essential for 0 itself, they can become of order O ( 1 ) after A(M) is applied.
Our criterion, which determines the matrix Qo, is that A(~t)~ should not depend on the choice of c~ and c2 as is
required for continuum limit.
A straightforward differentiation of 0 defined by eqs. (A1) and (A2) yields
Volume 212, number 2
PHYSICS LETTERS B
,(
+ ~ (x,-x,-:.)~Ntr
22 September t988
P 1~ & ) { ~ ( 2 q o + q : ) V v L , ( ½ ( x , + x , - : ) ) + 6 ( c ~ - 1 ) ( q o - q . , ) @ 2 A ~
i=!
+ ½(c - c : ) (go - q : ) [~vAv, A~] +-~ (c2 - 1 ) (q; - q o ) [ (@~A,,+ 0~A~), Av] } ) .
(A3)
The first term having the double sum over i a n d j is o f order [ M ( ! - q ) I - " for our choice of qK given by eqs.
( 8 ), ( 9 ) and vanishes when M ( I - q) -~ Go. The second term having the single sum reproduces A~ (x (~) ) if and
only ifq~ ~qo-+ I as M - + m . Otherwise there is no continuum limit.
Thus we have proven the fact that ~he finite-dimensionaI eq. ( i I ) reproduces the functional eq. (5) as M-~ceo
References
[ 11 Yu.M. Makeenko, Lecture Notes in Physics Vol. 18 l (Springer, Berlin, 1983) p. 67.
[2i A.A. Migdai, Phys. Rep. I02 (!983) 199.
[ 3 ] G. Marchesini, Nucl. Phys. B 239 (1984) 135.
[4] G. Marchesini and E. Onofri, Nut!. Pi~ys.B 249 (I985) 225.
[51A.A. Migdal, Phys. Left. B i47 (1984) 347
[6~ A.A. Migdal, NucL Phys. B265 [FS15] (!986) 594.
[7 ] Yu.M. Makeenko and A.A. MigdaI, Nucl. Phys. B i88 ( 198t ) 269.
[ 8 ] J.L. Ge~'ais and A. Neveu, N~acLPhys. B ! 53 ( t 979) 445.
[9] P. Ldvy, Probl~mes concrets d'analyse fonctionnelie (Paris, 1951 );
for a recent review see M,N. FelIer, Sow J. Usp. Matem. Nauk 4! ( 1986 ) 97.
[ I0] Yu.M. Makeenko, to be published.
226