To which extent is a semiclassical picture of QCD valid?

Question

To which extent is the semiclassical picture of QCD painted in

CG Callan Jr, R Dashen, DJ Gross,
Toward a theory of the strong interactions,
Physical Review D 17 (1978), 2717-2763

still valid today?

I ask this partially in view of the discussion starting at  http://www.physicsoverflow.org/21786/energy-mass-spectrum-yang-mills-bosons-infinite-and-discrete?show=21843#c21843, where Ron Maimon claims that the classical limit is completely irrelevant for quantum Yang-Mills theory (and hence QCD), while the authors of the above paper use essentially semiclassical arguments to unravel (on a heuristic level) the main properties of QCD.

Comments

Note that I'll be travelling for a few days, and might respond not before Tuesday.

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This paper should be imported in reviews, I think. In this case, it would be nice to write a summary of the arguments, since it is not freely available.

Answer 1

Classical global solutions are irrelevant to quantum Yang-Mills, you are confusing "classical solutions" (meaning a point in the classical field theory phase space) with "local classical solutions" (action minimizing local bumps, like instantons).

You can partially describe the random configuration of the Euclidean gauge field by sprinkling classical local solutions at random, and patching these local solutions together into a global approximate action minimum which serves as a basis for a near-minimum quadratic expansion. The key word is "sprinkled at random". The instanton gas is such a picture, so is Polyakov 2+1 monopole confinement, and the Argyres version in compactified 4d gauge theory, or any other soup-of-classical-solutions picture. The soup is what makes it a description of a gauge vacuum.

An instanton gas configuration is not a classical solution, a single instanton is. A configuration of an instanton gas is only an approximate solution, the approximation getting better when the centers are far apart, and it is made by superposing instantons, which don't superpose exactly (this isn't a linear theory).

These approximate classical configurations are not in any way related to precise classical points in field theory phase space, these are patched together local solutions used to approximate the vacuum by summing random configurations which are locally action minimizing. These methods can perhaps work to describe the gauge field at long distances, but they have no relation to phase space methods in 0+1 d quantum systems.

I am skeptical of classical methods, because the lattice picture is much easier and much more numerically accessible strong coupling lattice gauge theory, so I don't care about this. There's not much point to my mind in even considering the classical equations for quantum Yang-Mills, every useful classical-solution concept has a much easier analog on the lattice, but people like it, and it never hurts to learn about special solutions. They just like it too much in math, because they have all sorts of fancy specialist theorems about classical solutions, and not so many theorems about lattice configuraitons.

Comments

I am not confusing anything (wasn't talking about ''global classical solutions'' or ''precise classical points in phase space''), just pointing out that almost everything known about QCD can be derived qualitatively by the semiclassical analysis in the paper cited. (In contrast, in lattice studies one gets numbers but very little insight.)

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First, the lattice is not a finite number of degrees of freedom, as the lattice is infinite in volume. This is not a trivial point, as finite volume QCD does not have a confining behavior at small coupling, while infinite volume lattice QCD is qualitatively the same at all couplings. Talking about the measure on infinite volume lattice gauge theory is not significantly easier than talking about the measure in continuum gauge theory, the continuum theory is simply the limit of small lattices with a coupling that vanishes as the log of the lattice spacing.

Field theory is not at the level of Euler, it hasn't been since Wilson. Wilson played the role of Weierstrauss in field theory, just he wasn't a mathematician, and the mathematicians don't understand it. No difficult new ideas are needed beyond Wilson. The Wilson paper is this one: http://journals.aps.org/prd/abstract/10.1103/PhysRevD.10.2445 . The point of this paper is that he goes from an obviously confining model (lattice theory at big g) and shows you how it continuously is linked to the continuum theory (lattice theory at small g), and there is nothing special going on in the middle, something which you can see intuitively, verify by explicitly simulating, and also see how to prove rigorously (although it's tedious).

The mathematical subtleties here are all of the stupid sort, defining what convergence means (easier), establishing that the running is one-dimensional (a little harder). There is nothing that is conceptually difficult after Wilson, everything is pedestrian stuff that mathematicians don't understand only because they don't deal with probability constructions very well.

Hairer's construction was so important for me because it gives a framework for easily defining limits of statistically chosen distributions. He does things in continuous space, using differentiable functions that approximate distributions as his regulator is relaxed, but the framework is just to define what the distributional limits mean, and the renormalization of their product. It is what physicists know as the OPE, except not as completely rigorous mathematics.

Since this Clay problem pisses me off, I think it should never have been posed, I think about it once in a while.

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OK, Wilson's lattice is (unlike the numerical lattices) infinite. But his functional integral is Euclidean, there is no Hilbert space, hence no rigorous quantum mechanics. Constructing the Hilbert space is highly nontrivial, and Wilson says nothing about it, hence doesn't contribute to making the theory well-defined.  The Euclidean path integral on the lattice is presumably constructible, though here we have a case that is not superrenormalizable, and Hairer's techniques are probably too weak.

But even when this has been established, a proof of the continuum limit requires bounds on the Euclidean path integral that are extremely difficult to establish. Not to speak of the requirements for analytic continuation to Minkowski space, needed to construct the Hilbert space and to get a well-defined dynamics. Your handwaving is completely inadequate to address these issues.

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Wilson discusses in Section III.B of his paper the quantization. He says after (3.15):

A theory defined only for discrete imaginary values of the time leaves much to be desired. Fortunately, one can generalize the theory to define a Hamiltonian for a quantized theory. 

He then takes a number of steps each of which are highly nontrivial to establish rigorously: He defines in (3.18) an object $V$ in terms of an infinite product of measures representing a functional integral (whose existence is taken for granted), reinterprets it as a Hermitian operator (though it is only given as a bilinear form), assumes it has a full system of eigenvalues (which requires proving $V$ to be self-adjoint and the Hilbert space to contain a nuclear subspace on which $V$ is defined as an operator), defines the Hamiltonian in (3.23) by taking logarithms of the eigenvalues (which might result in imaginary energies), and states:

A problem arises with this definition if $V$ has any negative eigenvalue [...] whether it happens here the author does not know; this question must be studied further.

Thus he gets stuck even on the informal physical level. What he does (on the mathematical level) is not more than outlining an ambitious research program for how one can possibly get a Hilbert space with an appropriate Hamiltonian acting on it. And even if this were achieved, one still has to verify the properties that make it a relativistic quantum field theory (Wightman axioms) satisfying QCD field equations.

This is very far from your claim that he presents ''something which you can see intuitively, verify by explicitly simulating, and also see how to prove rigorously (although it's tedious)''! Only the intuition for the physical part of the picture is there, no intuition at all for the mathematical part; instead an admission of a glaring gap - that $V$ is not proved positive definite, an absolute necessity for the construction to work.

Witten, when he posed the YM millennium problem, was well aware that Wilson only scratched the surface, as far as the mathematics was concerned. You didn't present more insight than Witten surely had, hence you shouldn't claim something to be trivial that Witten found worthy of a big prize.

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Yes! You have found the main difficulty in the second comment--- establishing rigorously that there is exactly one dimensional flow on the space of lattice theories. This is true.

The difficulties in the first problem are not really difficult. Once you know the rigorous 1d flow, you can establish the continuum limit relatively simply, because the continuum field sample distributions are easy to define from the lattice links by a limiting procedure. This picking process used to be annoying to make rigorous, not so much anymore.

Regarding the reconstruction of the quantum theory, the correlations obeying reflection positivity are sufficient, and this is not so difficult, and is covered well in Streater and Whitman. This was relatively difficult and thankless work,  but the reconstruction theorem for Minkowski theories linked to statistical Euclidean theories was sorted out in the 1960s.

The main point of the Wilson paper is to produce the strong coupling expansion and to show it's properties. Please review this, it's a beautiful result. The main conjecture is the absence of a phase transition from strong to weak coupling, and the pure 1-d flow from weak to strong coupling, and this is amply verified by the smallest lattice experiments you can do at home.

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This is a comment that has been converted to an answer. I made it a comment for a reason, it doesn't answer the question! I haven't even read the relevant paper yet, I don't know what parts of what they say is right or not.

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Please give a link to Wilson's paper, or import it into the reviews section.