We formulate a general method of collective fields in quantum theory, which represents a direct generalization of the Bo

*521*
*110*
*863KB*

*English*
*Pages 17*
*Year 1980*

Nuclear Physics @ North-Holland

B165 (1980) 511-527 Publishing Company

THE QUANTUM APPLICATION

COLLECTIVE

FIELD

TO THE PLANAR

METHOD

AND

ITS

LIMIT*

A. JEVICKI Department of Physics, Brown University, Providence, RI 02912, USA

B. SAKITA Department of Physics, City College of the City University of New York, New York, NY 10031, USA Received

7 November

1979

We formulate a general method of collective fields in quantum theory, which represents a direct generalization of the Bohm-Pines treatment of plasma oscillations. The present method provides a complete procedure for reformulating a given quantum system in terms of a most general (overcomplete) set of commuting operators. We point out and exemplify how this formalism offers a new powerful method for studying the large-N limit. For illustration we discuss the collective motions of N identical harmonic oscillators. As a much more important application, we show how, based on the present formalism, one solves the planar limit of a non-trivial SU(N) symmetric quantum theory.

1. Introduction

The study of collective motions in many-body systems was one of the active subjects [l, 21 of the 1950’s. We consider this old subject in the present paper in order to explore and motivate applications of similar ideas and methods to modern problems of quantum field theory. Here we especially have in mind non-abelian Yang-Mills gauge theories whose ultimate solution obviously requires a development of new non-perturbative methods. Although in this work we confine our discussions to many-particle quantum mechanical systems, we shall formulate the coIlective field method in a sufficiently general and abstract fashion so that one can apply the formalism to quantum field theories. The basic ideas of the method are best described in the following example. Consider the quantum mechanics of N Bose particles. One way to solve this manybody problem is, of course, to solve the Schrodinger equation and then select the totally symmetric wave functions. Another approach would be that one regards the * Supported

in part by the National

Science

Foundation

BHE-13085. 511

Grant

No. PHY-78-24888

and PSC-

512

A. Jeoicki,

B. Sakita

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Quantum

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field method

wave function as a function of all possible symmetric combinations of the coordinates (these are: I;“=, x8,IL, x?,, ...) and then try to solve the problem. This latter approach leads to the collective field theory. The essence of the quantum collective field method is to consider a most general (overcomplete) set of commuting operators and explicitly perform a change of variables to this new set (the collective field). For instance, in the above Bose particle example this set can be given by the density variables. In classical N-particle statistical mechanics probIems the usefulness of reformulating the partition function in terms of this collective field is quite well known (for a recent application the reader is referred to ref. [16]). In the same fashion in the present quantum mechanical case it is simple to rewrite the wave functions as functionals of the collective field and also to determine the jacobian. The additional more difficult problem now is to reformulate the hamiltonian. This can be done in general but with constraints present which signify the fact that .the collective variables are not all independent. Now, it is at this point that the relevance of the large-Nlimit becomes obvious. Namely, in this limit the new variables become almost independent and one can define a genuine field theory which approaches the original quantum mechanical system. This is the collective field theory. To recapitulate, what has been done in this procedure is nothing but a non-trivial “change” of variables, for example, from N original degrees of freedom one goes to a new set of infinitely many variables. The general method described in the present paper is very useful and applies to the treatment of Yang-Mills gauge theories. Namely, in that case, the general set of invariant commuting operators is given by all possible gauge-invariant phase factors. Following the method of quantum collective fields, the formulation of the YM hamiltonian in terms of these new variables is established in a separate paper by Sakita [3]*. As the main application in this paper, we would like to show how the present formalism provides a new effective method for studying the large-N limit in quantum theory. A simple illustrative example is given by considering the system of N-identical harmonic oscillators. We find how the collective motions (i.e., the largeN behavior) of this system are, in fact, given by the classical solution of the collective field theory, and the formalism provides a systematic framework for the l/N expansion. As a more remarkable example, we consider an interacting SU(N) symmetric quantum system. This is an analog problem to the Yang-Mills theory in the sense that in the limit N+co, one has all the planar diagrams. We show how after rewriting the hamiltonian in terms of a collective field (which is defined employing an * We mention that the present method is in a sense the precise opposite to coordinate method [4] used for soliton quantization. Namely, in that case from a field theory a single quantum mechanical variable (which describes quantum particle), while in the present case we in fact grossly enlarge the of degrees of freedom.

the collective we have extracted the soliton as a original number

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overcomplete set of SU(N) invariants) the planar limit can be solved in a rather straightforward way. Namely, as in the previous case, it is again simply given by a classical solution of the collective field theory. To summarize, the content of this paper is as follows. First, in sect. 2, we describe the collective field formalism in the example of the quantum mechanical system of N Bose particles. We comment on and clarify some of the earlier works on this subject. Then, in sect. 3, we apply the above formalism to study the collective motions of a system of N identical harmonic oscillators. In sect. 4 we present the general method as a simple generalization of the formalism described in sect. 2. The important application of this general method to an SU(N) symmetric quantum system is given in sect. 5. There, the planar limit of this theory is solved.

2. The collective field theory of N Bose particles To illustrate in detail the main ideas, let us consider a quantum mechanical system of N Bose particles in one dimension. The hamiltonian representing the system is taken to be

H=$

c”g+i f i=l

c(&,i,)+

i#j

F VG,). r=l

(2.1)

The second term is the two-particle interaction energy, while the third is the energy due to a common potential V(x). The Schrodinger wave function is a totally symmetric function of xi(i = 1 . N). Since the density operators p*(X)= F 6(x_x^i), ,=l

*L>xa-$L,

(2.2)

are the most general commuting symmetric operators, one may regard, in general, the wave function as a functional of the density function p(x); $(x1, x2, *.., xPJ) =@b(.)l*

(2.3)

Notice that eq. (2.3) restricts to totally symmetric wave functions so that the Bose statistics is automatically imposed. Since p(x) should satisfy the constraint

J

dxp(x)=

N,

(2.4)

it would be better to use its Fourier components:

(2.5)

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where k takes discrete values ,+,

n = *l, *2, . .. .

Expression

(2.6)

(2.3) is then given by

4G1,

x2,

.*.,

XN)

=

“;,;

k ;k .I

Pk#kz

. . . Pk,@‘n(kl

...

n

k,) .

(2.7)

The Schrodinger equation of the system is given by

= E*(xl,

x2,

...

(2.8)

XN) .

We insert (2.7) into (2.8) and convert Cia2/dxf the following chain rule of differentiation:

into the functional

derivatives using

emikx accordingly,

+ii,

Wk k’; bl)(

-i&)( -i&) ,

(2.9)

where w(k; [PI) = -k’Pk, R(k, k’; [p]) = Fpk-k’ Then, Schrodinger

+i

(2.10)

.

equation,

(2.8), becomes H@[p] = E@[p]:

I I dyp(xb(x, yMy)+ I dxp(x)(V(x)-4x,x)). dx

Now let us consider a field theory which is defined by H[r,

+i

41=$x dk;

[db-k

+i

2,

W,

k’;

[dh-kTk’

k

I dx I dy4(xMx,yM(y)+ I

dxdx)(V(x)-4x,x)).

(2.12)

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This field theory, which we call collective field theory, is obtained from (2.11) replacing P& and -id/@& by C#!Q and -id/d& = r-k, respectively. The only difference between (2.11) and (2.12) is that in (2.11) the P&‘s are defined by (2.5) so that all the P&‘s are not independent variables, while in (2.12) all the C#J~‘S are considered to be independent. The relevance of the collective field theory to the original N-body problem is almost obvious, because using solutions of (2.12) one obtains solutions of (2.11), accordingly the original Schrodinger equation (2.8) is obtained through (2.3) or (2.7). Note, however, that it is impossible to determine the Fock wave function [5] Q,(k* . .. k,)(n = 0, 1, ... CO)from the N-body wave function I++(x,.. . xN). The mapping between the solutions of these two theories is many (collective field theory) to one (N-body problem). In order to obtain the energy eigenvalues of the system it is sufficient to consider the collective field theory. However, many degenerate states of the collective field theory may correspond to few states in the N-body problem. Although we must be careful in counting the number of states, we would expect no mistakes involved in the low-lying states. The new representation for the hamiltonian (2.12) agrees with the form derived originally by Bogoliubov and Zubarev [2]. It is not hermitian under the operation 4’k =4-k, (2.13)

a:==-,,

and, in fact, a direct perturbative expansion based on this hamiltonian led to incorrect results [8]. These difficulties are resolved by understanding that the scalar product in the new functional representation involves a non-trivial jacobian coming from the change of variables. Namely, the hamiltonian (2.11) is really hermitian but in the original Hilbert space whose inner product is given by ($1,

$2)

=

j- dxl

dx2

. . . dxr-,llrT

(xl,

xz,

. . . x~Mz(xl,

x2 . . . XN)

.

(2.14)

In order to deduce from (2.14) the corresponding inner product in the functional space of 4, we insert the following identity into (2.17): 1 ...I

~od9k6(~k-~i~1e~ik~‘)=l.

(2.15)

We then obtain (2.16)

(2.17) is the jacobian induced by the change of variables.

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Now, perturbative calculations which take into account the contributions coming from the non-trivial scalar product (2.16) were performed by Jevicki in ref. [6] and, not surprisingly, the correct results are obtained. The jacobian (2.17) can be explicitly evaluated at large density (l/N expansion) to arbitrary high order. Here, we point out that there is an alternative, much more efficient, way to determine and take into account the jacobian. Namely, consider the similarity transformation

(2.18) such that the new scalar product becomes trivial. Obviously then, after this transformation the kinetic energy term in (2.12) should become manifestly hermitian. Now the point is that J[#I] is a real functional of C$and it can be uniquely determined from the requirement of hermiticity of the hamiltonian (2.12) after the substitution (2.18). This requirement , after some calculation, leads to the following relatively simple condition which has to be satisfied:

This equation can be solved formally and we obtain &In

J[4] = -1 K’(k,

k’; [4])k’2&.

(2.20)

k’

Inserting (2.20) into (2.18) and (2.12), one obtains finally the hermitian effective hamiltonian of the collective field theory: Hea=+&

+;

rkfi(k,

k’;

[dlhk

‘“2,

UC-k;

W(k)-idk, k)kf-k,

[4lW’(k k’;4Mk’; [d~l)

(2.21)

where v(k, k’) =

II

dx dx’ e

pikx+ik’x’V(X,

xrj

,

(2.22)

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The third term of (2.21) is given by -$ Ck k*, which should be cancelled by the infinite zero-point energy of the density field. The expression of the effective hamiltonian has a much neater form if one uses the coordinate representation of the field: (2.23)

dxb(x)[V(x)--(x,x)].

(2.24)

In this expression we included a part of the contribution from the kinetic energy term in the potential term. The r(x) used in (2.23) is defined by r(x)=-

1

1 eikXnk,

(2.25)

LkZO

so that [v(x), 4(x’)] = -ii

,:,

e’k(x-x”

(2.26) Concerning the final hermitian hamiltonian (2.23), we mention that it could be alternatively deduced using the second quantized formulation of Bose systems and changing directly to current and density variables as was done by Sharp and collaborators [7]. However, for more general cases and the applications we have in mind there will not exist such a direct procedure. We close this section by describing for completeness the physics of the BohmPines electron plasma. In this case, V(x) = 0 and V(X,x’) is the Coulomb interaction: (2.27) Although in the above discussions we obtained the expression of the collective field theory in one space dimension, we convert it for 3-space dimensions simply by changing x and k into 3-dimensional vectors. The Fourier transform of (2.27) is given by 4ne* v(k, k’) = k2

v6k.k’

t

(2.28)

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where V is the total space volume. For large N one may approximate

Inserting

Pk by

(2.28) and (2.29) into (2.21) one obtains in the quadratic approximation: (2.30)

Thus, the plasma frequency is given by 47W2N

“‘, = -+k4,

which is the well-known

(2.31) result.

3. Collective motions of N-identical harmonic oscillators As the main application of the above formalism, we would like to point out and exemplify that it offers a new powerful method for studying the large-N limit in quantum theory. A simple illustrative example is given by the system of N identical harmonic oscillators. Obviously this problem allows for an exact solution; however, the physics of the large-N limit, namely the collective motions of these oscillators, look quite different. To apply the general representation (2.23) we have in this case V(x) = $d2X2, Therefore,

u(x,y)=O

the effective potential

(3.1) is now given by

2 V[c,b]=Q\

dx%+fu2

x24(x) dx,

(3.2)

and one must satisfy the costraint equation

I

dxd(x)=N,

(3.3)

which is the basis for relevance of the l/N expansion. We expand the potential around the minimum do, which is a solution of

fm41

---+A

@J(x)

=o

and the condition (3.3). Here A is a Lagrange multiplier. the classical field

(3.4) The solution is given by

(3.5)

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Now, since 4” is proportional to NW the perturbation expansion about this classical field is a l/N expansion. Keeping up to quadratic terms in ~[c#J= &+ ~1 one obtains H -:

dx &(x)(V+))2

The fluctuation

I

+$

(3.6)

q(x) is subject to the following condition:

dxv(x)=O.

(3.7)

In order to have a form of normal modes, we first make a unitary transformation r, 77+ 6,$ such that )

7r(x) = ~01’2(x)7j(x) v(x) = ~~“~.4i(~,

,

(3.8)

and use the explicit form of &J&) (3.5). After a short calculation we obtain H(2)

= t

t$/ The condition

I

dx +(x)[-V2, t co2x2-o]+(x)

dx&x)[-VZ,+,2x2-~]&x)]

(3.9)

(3.7) becomes (3.10)

dx eC’X2’2&(x) = 0.

We then expand 6 and 7i in terms of the eigenfunctions

of harmonic oscillators:

(-~v’:t~w2x2,~n(X)=(n+~)O*n.

(3.11)

Namely, i(x) = Lo 42wMx)b,, 1 G(x) = X, J2wnX”(X)7i”.

(3.12)

The condition (3.10) implies that in this expansion the n = 0 mode should be removed. One obtains finally

f&2,=+ ; [7T:+n2w2&].

(3.13)

n#O

Since this system is a system of non-interacting harmonic oscillators, the results we have obtained would be explained by the classical considerations. There are two

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A. Jevicki, B. Sakita / Quantum collective field method

b)

n = 2 mode

Fig. 1. Collective

I

modes

of non-interacting

harmonic

oscillators.

collective modes, which can easily be conceived from classical motions. Namely, (i) all the particles are at x = 0 for t = 0 with positive initial velocities (see fig. la); (ii) all the particles are at x = 0 for f = 0 with zero averaged initial velocity (see fig. lb). These correspond to n = 1 and II = 2, respectively. The n = 2 mode is precisely the collective mode considered by Goshen and Lipkin [9] previously. Based on this example we emphasize the following most important point about the present approach to the l/N expansion. Namely, as seen above, the large-N limit is determined by a single classical field q&(x) which is obtained as a static solution of our collective field theory. This, we argue, will also be the case in more complicated problems’. l

Moreover, in a work with Papanicolaou [lo] we argue and demonstrate in the framework of several N-component models that, in fact, in that case the large-N limit can be understood and given directly in terms of a special real time classical solution to the original equations of motion.

A. Jevicki, B. Sakita / Quantum collective field method

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4. The general formalism We now develop the general method of collective fields as a direct extension of the method described in the previous sections. We derive general expressions which can then easily be applied to specific problems. Let us consider an operator hamiltonian

(4.1) where (qr, .. .. qM) are the basic degrees of freedom and p^;= (l/i)?~/~?q, are the conjugate variables. Suppose that the interaction potential V is of such a form that it can be expressed in terms of an infinite number of combinations given, for definiteness, by

d(x) =f(x; 41, ...>&I).

(4.2)

We consider these as new variables and term this set a collective field. We stress here that this set can be, and almost always is, overcomplete. The basic idea then is to re-express the theory in terms of the collective field variables 4(x). Now the important point is that it must be such that the wave function of the problem can also be expressed as a functional of Q%(X).This can come about as, for example, a consequence of a demand that one is interested in only the invariant subspace of the full Hilbert space. In any case we assume that

*bh, ...,q‘w)= w-(x; aI= @[4(.)I

(4.3)

holds. Then to deduce the scalar product in terms of 4 we use the fact that

x

I n” II W(x)-Rx; dqi

i=l

4)).

(4.4)

x

The second integral in this expression defines the jacobian J[$J] and then the resealed wave functionals

have a simple scalar product

(4.6)

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Next, the kinetic energy term with the use of a chain rule takes the form K=i

-:

I

I

dxw(x:[mli&

dx

I

dy 0(x, y;

,41164&.dcyJ 7

(4.7)

where we have denoted

-CV?fkd=wb;MlL C vif(X;

qlV,f(Y

i

2)= 0(x, Y; [dl)

(4.8)

.

Here, the fact that these two quantities are expressed as functionals of 4 is an important non-trivial statement. Now in order to make the hamiltonian expressed in terms of d(x) and n(x) = (l/i)a/a4(x) hermitian, consider the similarity transformation (4.5). The effect is

3x)@[41=v~41)-“*b+(x)-iC(x; [~I)~~[~],

(4.9)

where (4.10) Then the kinetic term (4.7) can be written in the form K-fij

+f\

[w+ij

,c-+i/j

(AI-21

CQ]rr+i//

(7rRr-CfIC)

Q?rC).

(4.11)

The important point which we emphasize now is that as an alternative to a direct evaluation of the jacobian based on (4.4) it is possible to determine it much more efficiently by demanding hermiticity of the hamiltonian after similarity transformation. Namely, inspection shows that in (4.11) it is only the first term which is non-hermitian and then the demand that it vanishes produces an equation for C:

@(xiMl)+] dy S’(x’y’~)-21dyn(x,y:[rbl)C(y;C)=0. s4cyj

(4.12)

The solution is formally given by C = &‘(w

+ i(4.n)) ,

where RP’(x, y) is the inverse of 0.

(4.13)

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With the above information we can now explicitly write down the final hermitian hamiltonian. Some calculation leads to

(4.14) This is the general representation field d(x) and its conjugate r(x).

of the hamiltonian

in terms of the collective

5. Planar limit in an SU(N) quantum system ‘t Hooft has shown [13] that in the large-N limit the Yang-Mills theory reduces to planar diagrams and it is indeed a challenging task to sum up this large set of graphs [14]. In solvable cases like the O(N) cT-models or the Gross-Neveu model, only bubble diagrams appear and one can use, for example, the Lagrange multiplier method to generate the l/N expansion. But in the SU(N) case, which involves general planar diagrams, there exist no similar methods and the problems simply look untractable. In what follows we shall point out and demonstrate that the collective field formalism offers a powerful new approach to this problem. The only non-trivial example of summing planar diagrams is the work of Brezin, Itzykson, Parisi and Zuber [ll] who considered an SU(N) symmetric quantum mechanical system: 2’=iTr(fi*)-Tr

(

$?*+;““),

(5.1)

where 6f is a hermitian N x N matrix. The above authors have in a remarkable fashion succeeded in restating the problem as that of the one-dimensional Fermi gas and then solving the large-N limit. Applying the present method to this problem we shall show that the planar limit is solved in a rather straightforward fashion. In our approach no special ingenuity is needed and, in fact, the SU(N) case is solved in the same direct way as the Ncomponent oscillator problem considered in sect. 3. The hamiltonian corresponding to (5.1) reads . Here we have denoted fi = 1, t”M,, with ta representing are generators of U(N).

(5.2) N x N matrices which

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We restrict our attention to the singlet subspace in what follows. Then, in accordance with our general strategy consider the general set of invariants C#Q= Tr (eCikM) ,

(5.3)

where k is arbitrary real number. Now we propose to change variables from the original set iVf, to this new set (& k E R). Obviously this new set of variables is overcomplete, but that is precisely the essence of our approach. Using the Fourier transformed field

we see that the potential

term now reads (5.4)

and there is also the important

I

constraint

(5.5)

dx d(x) = Tr (U)= N.

In order to rewrite the kinetic term we simply apply the general formulas derived in sect. 4. First of all for the functionals w and fI [defined by (4.8)] we now have the explicit forms

fW, k’; [cbl)=;

(&dk)(

$&k’) a

a

In deriving these we used the differentiation 1 d ;a!+&(-ikQ) -ik&l dcu ezge

I

=.

= kk’4k-k, .

(5.6)

formula -i(l-n)k,,?

Fe

(5.7)

and the fact that c (%3(~a)&3’ a In the coordinate

= &&3,,

representation

.

(5.8) eqs. (5.6) read

(5.9) where the integral is a principal value integral.

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Next, substituting everything into the general expression (4.14) for the hamiltonian, we obtain, after some calculation, the new representation H=+

dx dy~(x)R(x,y;[~])~(y)+~[~]+AV. I I Here the effective potential is

(5.10)

Y[cS]=(dx{tGz(x;&+:x2+;x4]eS(x),

(5.11)

with (5.12) The additional

term AV denotes a singular form coming from

dx MdllW4x). Now that we have completed the reformulation of the hamiltonian in terms of new variables we are in a position to directly generate the l/N expansion. Since the potential Y’[c#J]contains a tadpole term and in view of the fact that we have the constraint (5.5) present we obviously need to minimize the functional: -:I

&#~)=~Ml+++j-

dxd(xi).

Here the constant e denotes a Lagrange multiplier. d(x) and e we get the system of equations

(5.13) Varying E(e, 4) with respect to

:(fdyE)*-fdy$fdy$$

(5.14a)

I

dx4(x)=N.

(5.14b)

The solution to the integral equation (5.14a) is obtained following the standard procedure of continuing into the complex z plane which is cut along the real interval (-A, A). The solution C&(X)vanishes outside this interval and inside one finds that it is given by (5.15) A is determined

from the equation

2e-,12-2gp=o N

(5.16)

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and the Lagrange multiplier constant, e, is fixed by the constraint explicitly reads

(5.14b) which

l/2 =N.

(5.17)

Now, the ground-state energy in the leading order in N is simply given as the classical energy of the field &(x) l Eb” = V[&J].

(5.18)

This is explicitly evaluated using the fact that from the classical equation follows that j- dx9u(x)G(x;[9a])‘=:I

dx&,(x)(2e-x2-$x4).

(5.14a) it

(5.19)

Consequently, ?‘“[&]=ej-d&(x)-:j-dx&,(x)(?e-x2-$x4), and then after substituting E;” = eN-

(5.20)

the explicit form (5.15) for &J”(X)we obtain

(5.21)

This final expression for the ground-state energy coincides with the result obtained by BIPZ in ref. [ll]. In connection with the above derivation of the planar limit we mention a recent investigation of related problems by Itzykson and Zuber [15]. Namely, exhibiting the more complex nature of matrix models in respect to simple N-component vector models, these authors arrived at a pessimistic conclusion that in the SU(N) case it is not possible to describe the planar limit in terms of a classical configuration. Now, in the present formalism we have in fact succeeded in working out and exhibiting the planar limit in terms of a classical stationary point &,(x). To summarize, the two basic steps in our collective field theory approach are the following: first, we change variables to an overcomplete set of invariants; second, we find the classical stationary point which then directly determines the large-N behavior. * The additional

singular tadpole term AV which is present in (5.10) does not give a contribution in the leading order in N. Namely the classical contribution (5.18) is of order N2 while AV gives contributions of order N and consequently should be considered only in the next correction in the l/N expansion.

A. Jeuicki, B. Sakita 1 Quantum collective field method

527

6. Conclusions and outlook In conclusion let us reflect on the most important problem to be solved, namely the Yang-Mills theory. The first step towards the solution, which is the reformulation of the hamiltonian in terms of the collective field, is already done [3]. The new variables in this case are given by the general gauge-invariant phase factors: W[T]=Tr[Pexp(i~A(x).dl)], Iwhere I are arbitrary space contours. Now, in order to solve the large-N limit, one should proceed in parallel to our examples considered in sects. 3, 5. Namely, in this framework the problem reduces to that of solving the functional equation which gives the classical field IV,(I). Now, in view of the rather non-standard form of this equation this is not yet an easy task; however, we are hopeful that it can be done. One of us (A. J.) is happy to acknowledge the hospitality of the Aspen Center for Physics where part of this work was done.

[l]

[2]

[3] [4] [5] [6] [7]

[8] [9] [lo] [ll] [12] [13] [14] [15] [16]

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