Volume 95B, number 1 PHYSICS LETTERS 8 September 1980

ON THE DIRICHLET PROBLEM OF CHROMODYNAMICS ~

H. LEUTWYLER Institute of Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland

Received 14 July 1980

We show that the solutions to the euclidean field equations of chromodynamics are not in general uniquely determined by their boundary values. In the two-dimensional case there is a denumerable infinity of solutions satisfying any prescribed set of Dirichlet boundary data.

As pointed out by Nielsen and Olesen [1 ] constant chromomagnetic fields are unstable against quantum fluctuations. The purpose of the present paper is to show that this instability is related to an ambiguity in the boundary value problem of classical chromody- namics: the Dirichlet problem in general admits more than one solution.

The Dirichlet problem is the natural boundary value problem associated with the action principle of euclid- ean chromodynamics: consider a euclidean region V, say a sphere Ix] ~<R; specify the potential A u on the boundary ~3 V of this region and look for a solution to the field equations that satisfies this boundary condi- tion. The variational principle guarantees that the in- terpolating field that minimizes the action is always a solution to the problem. In the familiar case of La- place's equation A~ = 0 the boundary values of ~ on

V specify the solution uniquely; in this case the in- terpolating field with least action is the only solution to the Dirichlet problem. We claim that in chromody- namics this is not the case: in general there are physi- cally inequivalent solutions of the field equations with the same boundary values. In fact, suppose that the boundary conditions on ~ V correspond to a constant abelian field:

Au = - ¼FuvXvr 3 . (1)

[For simplicity we restrict ourselves to the gauge theo- ry of SU(2). We work in the 2 × 2 matrix representa-

Work supported by the Schweizerischer Nationalfonds.

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tion of the potential Au : r l , r 2, r 3 are the correspond- ing Pauli matrices.] One easily checks that the field (1) satisfies the field equations for arbitrary constant field strength Fur. We have pointed out [2] that for ]FuviR2 >> 1 the field (1) has minimal action (i.e. is stable) if and only if Fuv is (anti-)self-dual. If the field is not (anti-)self-dual - e.g. if F,, v is purely magnetic then there are field configurations with the same boundary value but with less action. Since the config- uration with least action always obeys the field equa- tions, we conclude that there exists at least one other solution of the field equations with the same boundary values.

We have not been able to characterize the inequiva- lent solutions of the Dirichlet problem in the four-di- mensional case. In the following we discuss the much simpler case of two-dimensional chromodynamies for which it is possible to explicitly construct all solutions to the field equations that satisfy a given boundary condition.

(i) In two-dimensional chromodynamics every so- lution to the field equations is gauge equivalent to a constant abelian field

41 = -- ¼Fx2r3, 4 2 = ¼FXlr 3 . (2)

(ii) To determine all possible solutions of the field equations that coincide with a given boundary value A u on the circle x 2 + x 2 =R 2 we therefore only need to find all solutions U and F to the condition

Au = UAu U? + iOu UU? ,

at Ix[ = R.

Volume 95B, number 1 PHYSICS LETTERS 8 September 1980

The component o f A u normal to the boundary is of no relevance: it merely fixes the normal derivative of U at Ix[ = R. For the tangential component AhL = x 1A 2 - x 2 A 1 the boundary condition reads

Arl U = ¼FR2Ur3 + i0U/34 , (3)

where 4 is the azimuthal angle defined by x 1 = R X cos 4, x 2 = R sin 4. The solution to this differential equation is given by

0 d0 , 0>l) (4)

X VexP(¼iFR24r3 ) ,

where V is a unitary constant of integration and P denotes ordering with respect to 4. For U(4) to be acceptable we must require U(27r) = U(0):

2rr

P exP ( - i f d4Ai,)= Vexp (-½ifR27rr3) V?. (5) 0

The left-hand side in this condition is the loop integral P e x p ( - i ~ d x u A,) . Its trace

t r P e x p ( - i f d x u A u ) = 2 cos ~FR2rr (6)

may be viewed as an analogue of the winding number integral in four-dimensional chromodynamics. Accord- ing to eq. (6) the boundary value A u restricts the field strength F to a denumerable infinity of possibili- ties. If F 0 is one admissible value then the general solution is

F= e F 0 + 4N/R 2,

e = e l , N=0,+1,_+2 . . . . . (7)

One easily checks that for any of these values there exists a unitary matrix V such that eq. (5) is obeyed. This guarantees that there is an admissible transfor- mation U(4) satisfying eq. (3). Finally, since SU(2) is simply connected, there exists a continuous inter- polating matrix U(x 1, x 2) on the interior of the cir- cle with the required value U(4) at the boundary. Hence for any of the values (7) for F there is a suita- ble gauge transformation U(x 1, x2) that brings the constant solution in agreement with the prescribed boundary values. The action associated with these in-

equivalent solutions of the field equations may be cal- culated in the gauge (2):

fd 2x tr A 2 v = F2R 2n" (8)

Suppose e.g. that the boundary values are already of the form (2), with some value F 0 of the field strength. Among all possible interpolations Of these boundary values the constant field (2) has minimal action as long as F 0 <~ 2/R 2. I f F 0 is larger then the constant interpolation is unstable. It is interesting to note that the minimal value of the action consistent with a given boundary value is determined by the "winding num- ber" integral (6). Note also that the nonabelian char- acter of the gauge group is crucial: the Dirichlet prob- lem associated with the euclidean Maxwell equations does have a unique solution. The origin of the differ- ence between the two cases may be traced back to the step in the above discussion which required an exten- sion of the given boundary value U(4) to the interior of the circle. Such an extension does not exist for the U(1) group because this group is not simply con- nected.

Why should the Dirichlet problem of classical chro- modynamics be of interest for physics which presuma- bly involves quantum chromodynamics? As exempli- fied by the instanton solutions a semi-classical approx- imation scheme may be based on classical solutions different from the trivial field A u u = 0 characteristic of perturbation theory. A consistent semi-classical ex- pansion requires that the classical field around which one expands is stable, i.e. realizes a minimum of the action with respect to localized fluctuations of the field. The variational principle offers an evident class of such stable field configurations: those that mini- mize the action subject to a given boundary condition at ]xi --* oo. One may therefore use this boundary con- dition to label stable solutions of the classical field equations. To allow for instantons means to replace the perturbation theory boundary condition Au -* 0 by the weaker requirement A u v ~ O, A u --> i 0 u UU t . Actually, even this requirement seems to be an unrea- sonably strong boundary condition. The average size of the field strength B as measured by the expectation value ¼ (OiA~vAUVaiO)= B 2 is estimated [3] to be of order B ~ m 2. There is no reason to require the field strength to be smaller than this as ix[ ~ ~. One could

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Volume 95B, number 1 PtlYSICS LETTERS 8 September 1980

perhaps expect that the boundary value at ix [-+ o~ is irrelevant if one sticks to matrix elements involving only local fields in the vicinity of the origin. This ex- pectation may however well be false: the minimal action solutions associated with the boundary condi- tion A u v ~ 0 have zero average action density (iso- lated instantons on a space that is almost everywhere empty). I f it turns out that contributions to the func- tional integral from fields for which the average action density does not vanish are important then the instan- ton boundary condition A u v ~ 0 is the wrong boun- dary condition.

A timid step in the direction of nontrivial bounda- ry conditions at bxt-~ °°is to allow the field strength to tend to a constant, A u - ~F~vx 73. (i) We have shown in ref. [2] that if the constant Fur is self-dual then the constant f ie ldA u 1 v = - ~Fu~x r 3 is a stable interpolation for this boundary condition. There are, however, infinitely many inequivalent classical solu- tions that obey the same boundary condit ion and have precisely the same action; we refer to these inequiva- lent fields as chromon excitations. Minkowski [4] has constructed an exact chromon solution for which the equality of the action can be checked explicitly. (The limiting case [4] for which the action turns out to be less than for the constant field does not solve the same Dirichlet problem - one has to perform a singu- lar gauge transformation that modifies the asymptotic behaviour of the field.) (ii) If the constant Fu~ is not

(anti-)self-dual there still is an interpolation with mini- mal action, but this interpolation is not a constant abelian field. What is it? In particular, does the stable interpolating field for purely magnetic boundary con- ditions (the field into which a constant magnetic field decays) have nonvanishing action density? More generally, are there stable classical solutions with fi- nite action density other than (anti-)self-dual fields?

More information about the classical Dirichlet problem of chromodynamics might shed some light on the scope of semi-classical approximation methods to QCD.

I am indebted to Petr Hajicek, John Klauder and Peter Minkowski for stimulating discussions.

References

[1] N.K. Nielsen and P. Olesen, Nucl. Phys. B144 (1978) 376. [2] H. Leutwyler, Vacuum fluctuations surrounding soft

gluon fields, and Constant gauge fields and their quan- tum fluctuations, preprints Bern (1980), to be published.

[3] M.A. Shifman, A.J. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448; A.I. Vainshtein, M.B. Voloshin, V.I. Zakharov, V.A. Novikov, L.B. Okun and M.A. Shifman, Soy. J. Nucl. Phys. 27 (Feb. 1978).

[4] P. Minkowski, On the vacuum expectation value (~2L¼ × : V~u V uva :l ~2) = p > 0 and constant classical gauge fieldstrengths, preprint Bern (1980), to be published.

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