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.
