The answer to this question, as posed, is "yes", but not in the way implied by other answers. In particular, the long-distance properties of the Yang-Mills vacuum are not described by considering small fluctuations of global classical solutions of the classical Yang-Mills equations. The* local *properties of Yang-Mills fields (near any one point, in a radius much smaller than the confinement radius) are described well by fluctuations around classical field theory. But the path-integral paste together these local descriptions into something monstrously quantum at long distances.

To understand quantum Yang-Mills, you should first examine the lattice version in detail, because this can be used to define the theory algorithmically, and compute all correlation functions. The perturbative formalism for Yang-Mills is secondary, because it is only justified at weak coupling, it is a weak coupling expansion.

In the lattice formulation, you have a G valued matrix field on each link, and an action which is $1/g^2$ times the sum over each plaquette of the trace of the holonomy on the plaquette. You can do something on the lattice which are interesting and different from what you do in perturbative field theory, you can ask what the perturbation theory looks like at *strong* coupling, when g is effectively close to infinite.

In this case, you start with 0 action as your leading approximation, and every matrix on every link is chosen uniformly randomly according to Haar measure. With a small 1/g^2, you have some correlations between the matrices on neighboring links, but the main result of the strong coupling expansion, described in classic lattice papers of Wilson, is that the correlations die off at long-distances, so that there is an exponential decay of all correlation functions in the strong coupling expansion to any order.

The qualitative reason for this behavior is simple to understand--- the field is nearly random on every link, except for a little bit of correlation between neighboring links. When you look at the product of the G's on two links going in the same direction, the product is closer to random than each factor, simply because multiplying together large random elements of a gauge group fills the gauge group ergodically.

This observation is intuitively obvious (and also correct), the result of looking at the gauge field at long distances is no different from shuffling a deck of cards, you are doing a random walk with large steps in a gauge group as you take each step on the lattice. This intuitive explanation is not Wilson's argument, which was different--- he showed that if you place colored sources far away, the correlations in the strong coupling expansion must follow a string between the sources, and the fluctuations of the string are small at large coupling. He also showed how to compute the correlation functions using these correlation strings, which dominate the long-distance correlations at large distances.

What this means is that the correlation functions in gauge theory are exponentially decaying at strong coupling, and the long-distance theory is entirely random, corresonding to a Euclidean action which is *zero*. Zero action means completely uncorrelated ultra-local fields, in other words, at scales far below the confinement scale, you don't see any gauge field, the dynamics is confined, and the exponential decay rate is (by Euclidean definition) the mass gap.

This is completely the opposite behavior than at short distances, where the coupling is weak, you have a good perturbation theory, and the fluctuations are described by a nearly classical loop expansion. The place where classical solutions describe the dynamics is in this short-distance limit.

In this limit, you have 4d instantons, and 3d instantons which are similar to monopoles. These objects, when their classical radius is small, are appropriate for describing effects in the gauge theory. Instantons in 4d are least Euclidean action configurations which (in Lorentzian space) describe tunneling between different classical vacua, the 2+1 instantons, which, when you line-extend, become weird monopole-like things, can be superposed statistically in an instanton gas, and then they have the effect of randomizing the 2+1 dimensional gauge field, as shown by Polyakov. This allows you to understand the interpolation between the short-distance physics, which is described by the classical theory and perturbative small fluctuations, and the long-distance physics, with mass gap and no correlations and zero action. The classical configurations, when placed randomly at a given density, can randomize the gauge field in 3d.

When instantons were discovered in 1976, David Gross and others hoped that a sea of instantons would randomize the gauge field at long distances to make the long-distance limit emerge from an instanton gas calculation and prove that there is a mass gap. This is not what happens, the instantons by themselves don't randomize the gauge field. However, in recent years, Argyres has considered the case of compactified 4d gauge theory, and in the limit that the dimension of the box is shorter than the confinement scale, you have to reproduce Polyakov's mechanism for mass-gap in 3d gauge theory. You do reproduce it, and the configurations which are involved are line-extended versions of Polyakov's solutions.

These line extended things can close into loops when you open the dimension fully, and in this case, you expect that these monopole loops will produce the randomization somehow, and these classical solutions can inform the long-distance limit. But the main ingredient here is the random placement of these solutions at long distances, that random scattering of instantons, or monopole loops, is what randomizes the gauge field at enormously long distance in the continuum picture, when you look at the description at weak coupling.

The randomization at long distances is the central mechanism for the mass gap, and this means that the classical descriptions are literally confined to a bag, beyond which you need to consider configurations which randomly fluctuate between one classical region and another. The gauge vacuum can be described by classical configurations only locally, the gluing between local descriptions has to reproduce the totally random fluctuations at long distances.

The mathematics work which starts with classical field theory starts with* global *classical solutions to Yang Mills theory, which, is simply mentally defective. If the procedure looked at local solutions, not global solutions, maybe there would be some way to paste together the close-to-classical behavior at each patch into some sort of description, but that's not what they do. They just consider global solutions to the equations and then try to produce a quantum theory from this object by deforming these classical global solutions. It is a procedure which is hopeless, because the solutions to classical Yang-Mills are rigidly deterministic, while the Euclidean gauge theory at longer scales is entirely randomized, so that the qualitative properties are entirely different at distances longer than the confinement radius.

It is possible that you can produce a rigorous construction of quantum field theory starting from the solutions to classical field theory at short distances, and patching them together at long distances like the definition of a manifold. Hairer in mathematics has done exactly this for solutions of stochastic PDE's whose stationary distributions are Euclidean versions of quantum theories, i.e. stochastically quantized field theories. His approach uses a "regularity structure" which is used to define the patching of the different regions together, and it reproduces the correct operator product relations in these theories.

His approach reproduces the renormalization of low-dimensional theories, it works, and can be used to define 4d Yang-Mills (apart from technical details about how to control the operator dimensions at short distances, which he doesn't know how to do in the case of logarithmic running). Approaches which simply try to take global solutions of Yang Mills and globally reconstruct the quantum theory from these are totally wrongheaded, they cannot produce the quantum theory, because they don't consider the randomized configurations at long distances.