QCD | Too Oversimplified?

In physics, the term "Abelian" refers to a specific kind of symmetry that is associated with particles that do not interact with each other via a strong force. The term is named after the Norwegian mathematician Niels Henrik Abel.

In an Abelian theory, the force-carrying particles (called gauge bosons) do not interact with each other, which means that the theory is much simpler than non-Abelian theories, where the gauge bosons do interact. The simplest example of an Abelian theory is quantum electrodynamics (QED), which describes the interaction between electrically charged particles (such as electrons and photons). Abelian gauge theories are important in many areas of physics, including particle physics, condensed matter physics, and cosmology. They are used to describe a wide range of phenomena, including the behavior of superconductors, the properties of magnetic materials, and the behavior of particles in the early universe.

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  k>0 such that every excitation of the vacuum has energy at least k. 

(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.

-Kaylin Thornton