nLab quantization of Yang-Mills theory

Idea

Despite its fundamental role in the standard model of particle physics, various details of the non-perturbative quantization of the class of field theories known as Yang-Mills theory for non-abelian gauge group (such as QCD but not QED) are still open, such as derivations of the phenomena

• quark confinement

and notably

• the mass gap problem, the question of why the bound states of massless quarks (hadrons) have mass

From Jaffe-Witten:

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) $\times$ 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 $H$; 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 $SU(2) \times U(1)$. 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 has been given $[$above$]$. 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 $\Delta \gt 0$ such that every excitation of the vacuum has energy at least $\Delta$.

(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.

This is the problem of non-perturbative quantization of Yang-Mills theory.

Related concepts

• Nakanishi-Lautrup field, antighost field

References

Non-perturbative theory and mass gap

See at mass gap for more.

A survey and problem description of the non-perturbative quantization of Yang-Mills theory, in the context of the Clay Mathematics Institute’s list of “Millennium Problems” (“Yang–Mills and Mass Gap”) is in

• Arthur Jaffe, Edward Witten, Quantum Yang-Mills theory (pdf)

and a report on the progress (essentially none) is in

• Michael Douglas, Report on the Status of the Yang-Mills Millennium Prize Problem (2004) [pdf]

Notes reviewing more technical details of the problem are in

• Jay Yablon, The Origins of QCD Confinement in Yang-Mills Gauge Theory, January 2008 (pdf)

Useful discussion of details of the problem description is in

• Arnold Neumaier, PO comment 2014

Perturbative theory

The quantization of Yang-Mills theory in the guise of EW and QCD as perturbative quantum field theory is carried out rigorously via causal perturbation theory in

• Günter Scharf, Quantum Gauge Theories – A True Ghost Story, Wiley 2001

The locally covariant perturbative quantum field theory of Yang-Mill on curved spacetimes is constructed in

• Stefan Hollands, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys. 20:1033-1172, 2008 (arXiv:0705.3340)

Discussion of the adiabatic limit is in

• Pawel Duch, Massless fields and adiabatic limit in quantum field theory (arXiv:1709.09907)