If the classical Maxwell theory describes electrodynamics (electromagnetism E&M) fairly well, how suitable would the classical Yang-Mills theory (say SU(3) color) describe the chromodynamics, and how precisely?

I suppose the heavy quarks (c,t,b), being more non-relativistic and we can still heavily use the classical Yang-Mills theory? For light quarks, being relativistic, we may need to consider the classical Yang-Mills + relativistic Dirac equations, but would classical versions of Yang-Mills + relativistic Dirac theories and their Equations of Motions (EOMs) would be suitable to certain levels? How?

p.s. This is a question concerning comparing the

classical v.s. quantum Maxwell/Yang-Mills theory.

and how applicable are them to describe the real world:

electrodynamics / chromodynamics.

  • 1
    $\begingroup$ I believe the problem is that you never see gluons in the wild, so you never have enough to make a classical field. I'm speculating at this point, but it might be interesting to see if a quark-gluon plasma might have some sort of classical description. $\endgroup$
    – Javier
    Commented Sep 29, 2017 at 23:21
  • $\begingroup$ I'd say that (according to the numerical lattice computations) classical Yang-Mills theory is not the classical limit of the (strongly coupled) quantum Yang-Mills theory. That's why it is a bad approximation. $\endgroup$ Commented Sep 30, 2017 at 8:34

1 Answer 1


This is nicely answered in Jaffe-Witten's "problem description" of the problem of quantization of Yang-Mills theory:

By the 1950s, when Yang–Mills theory was discovered, it was already known that the quantum version of Maxwell theory – known as Quantum Electrodynamics or QED – gives an extremely accurate account of electromagnetic fields and forces. In fact, QED improved the accuracy for certain earlier quantum theory predictions by several orders of magnitude, as well as predicting new splittings of energy levels.

So it was natural to inquire whether non-abelian gauge theory described other forces in nature, notably the weak force (responsible among other things for certain forms of radioactivity) and the strong or nuclear force (responsible among other things for the binding of protons and neutrons into nuclei). The massless nature of classical Yang–Mills waves was a serious obstacle to applying Yang–Mills theory to the other forces, for the weak and nuclear forces are short range and many of the particles are massive. Hence these phenomena did not appear to be associated with long-range fields describing massless particles.

In the 1960s and 1970s, physicists overcame these obstacles to the physical interpretation of non-abelian gauge theory. In the case of the weak force, this was accomplished by the Glashow–Salam–Weinberg electroweak theory with gauge group H=SU(2)×U(1). By elaborating the theory with an additional “Higgs field”, one avoided the massless nature of classical Yang–Mills waves. The Higgs field transforms in a two-dimensional representation of HH; its non-zero and approximately constant value in the vacuum state reduces the structure group from H to a U(1) subgroup (diagonally embedded in SU(2)×U(1). This theory describes both the electromagnetic and weak forces, in a more or less unified way; because of the reduction of the structure group to U(1), the long-range fields are those of electromagnetism only, in accord with what we see in nature.

The solution to the problem of massless Yang–Mills fields for the strong interactions has a completely different nature. That solution did not come from adding fields to Yang–Mills theory, but by discovering a remarkable property of the quantum Yang–Mills theory itself, that is, of the quantum theory whose classical Lagrangian is the Yang-Mills Lagrangian. This property is called “asymptotic freedom”. Roughly this means that at short distances the field displays quantum behavior very similar to its classical behavior; yet at long distances the classical theory is no longer a good guide to the quantum behavior of the field.

Asymptotic freedom, together with other experimental and theoretical discoveries made in the 1960s and 1970s, made it possible to describe the nuclear force by a non-abelian gauge theory in which the gauge group is G=SU(3). The additional fields describe, at the classical level, “quarks,” which are spin 1/2 objects somewhat analogous to the electron, but transforming in the fundamental representation of SU(3). The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD).

The use of QCD to describe the strong force was motivated by a whole series of experimental and theoretical discoveries made in the 1960s and 1970s, involving the symmetries and high-energy behavior of the strong interactions. But classical non-abelian gauge theory is very different from the observed world of strong interactions; for QCD to describe the strong force successfully, it must have at the quantum level the following three properties, each of which is dramatically different from the behavior of the classical theory:

(1) It must have a “mass gap;” namely there must be some constant Δ>0 such that every excitation of the vacuum has energy at least Δ.

(2) It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quark fields, that transform non-trivially under SU(3), the physical particle states – such as the proton, neutron, and pion -- are SU(3)-invariant.

(3) It must have “chiral symmetry breaking,” which means that the vacuum is potentially invariant (in the limit, that the quark-bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.

The first point is necessary to explain why the nuclear force is strong but short-ranged; the second is needed to explain why we never see individual quarks; and the third is needed to account for the “current algebra” theory of soft pions that was developed in the 1960s.

Both experiment – since QCD has numerous successes in confrontation with experiment – and computer simulations, carried out since the late 1970s, have given strong encouragement that QCD does have the properties cited above. These properties can be seen, to some extent, in theoretical calculations carried out in a variety of highly oversimplified models (like strongly coupled lattice gauge theory). But they are not fully understood theoretically; there does not exist a convincing, whether or not mathematically complete, theoretical computation demonstrating any of the three properties in QCD, as opposed to a severely simplified truncation of it.


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