# To which extent is a semiclassical picture of QCD valid?

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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.

retagged Aug 20, 2014

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

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.

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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.

answered Aug 20, 2014 by (7,535 points)

It's not true that the lattice is only for numbers, you just haven't read Wilson's 1974 strong coupling paper. If you read it, you know exactly why QCD is confining, you get a complete picture of the full far-ultraviolet to far-infrared running of the pure gauge theory which you can verify is accurate using even the smallest simulations (4^4, 8^4) that you can run on a laptop and equilibrate quickly. You can see that the running stays one-dimensional, and the lattice action smoothly interpolates between g=0 at short distances and g=infinity at distances larger than the confinement radius with nothing special going on inbetween (no phase transition) except more and more randomness in the gauge field on every link-hop, according to logarithmic growth in the size of fluctuations at small distances (in 4d simulations) or linear running at small distances (in 3d simulation). You don't get numbers from Wilson, you get understanding. Once lattice QCD became a programming project, the focus shifted from understanding to numbers, but the insight from the recent Landau gauge simulations about the Landau gauge propagator is also qualitative and interesting, and gives the precise (easy) lattice counterpart to Gribov's copy-regions (which I only completely understood after seeing how it is done on a lattice just now).

The things that are known about QCD almost all come from lattice intuitions that are then converted to classical solutions before publication. If you aren't talking about global classical solutions, then I agree, you can learn something simple from these, but they neither describe the far infrared (which is completely random) or the far ultraviolet (which is free), and Wilson's picture describes the whole enchilada.

It's because of Wilson's picture that I was mystified that the QCD mass-gap was made a Clay problem, because Wilson solved it except for crossing the t's and dotting the i's. It's not so easy to do this, but it's not so hard either, and such straightforward work I didn't think compares to the insight required by Hamilton and Perelman.

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.

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

''I was mystified that the QCD mass-gap was made a Clay problem, because Wilson solved it except for crossing the t's and dotting the i's.''

The difficulty is precisely in the latter that you think is trivial. For the lattice is ordinary QM (finitely many degrees of freedom only), which poses no longer any deep matrhematical problems But the hard part is to prove that  for all interesting observables the continuum limits exist with full mathematical rigor. Like many problems in mathematical physics, they are physically of little interest but require considerable mathematical ingenuity. In this case since many people tried in vain to construct YM, so that it is clear that completely new conceptual ideas are needed to make it work.

With respect to mathematical technique, QFT is still at the level of Euler's use of power series - he could do marvellous tricks with them, but without any logical basis. Bringing it to the level of Weierstrass requires complex analysis to be developed, with its many new tools that Euler didn't dispose of. In a similar way, QFT must be enhanced from Feynman's intuitive formal level to one where rigorous tools enable deductions that Feynman (the Euler of QFT) could not dream of.

Note that the millennium problems are problems in mathematics, not in physics. The global existence of solution of the Navier-Stokes equations is physically also obvious, but another one of the millennium problems. Its solution also requires completely new conceptual ideas, but of a very different kind. Problems are highly prices if their solution is likely to add significant new tools to the mathematical enterprise.

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.

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.

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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