Subatomic Physics

12.3. Gauge Invariance for Non-Abelian Fields 389
How can we generalize the gauge invariance of the single vector field (Abelian
case) to theories of several non-commuting (non-Abelian) massless vector fields? An
example would be a vector field with internal degrees of freedom, such as charge;
suppose the photon had isospin unity and came in three charge states. In chapter 8
we saw that this generalization is possible for a global gauge transformation with
the introduction of isospin. However, there we used a constant phase rotation,
U =exp(−iω ˆα·I), Eq. (8.20). The extension to a space–time dependent phase was
formulated by Yang and Mills. (5) Their result lay dormant for many years because
the strong interactions were described by the exchange of massive bosons (e.g.,
π, ρ) only some of which are vector particles; the weak interaction also requires very
massive bosons, but no theory with such bosons was available.
Consider a vector field V , with three internal (not space) components, V (a) (e.g.,
isospin = 1, with a =1...3). In analogy to Eq. (8.20), we generalize Eq. (12.2)
by introducing a different function ξ (a) for each internal (isospin) component of the
vector field, V (a)
ψ ′ ≡ Uψ =exp[igI (a) ξ (a) (x,t)]ψ =exp[ig � I· � ξ(x,t)]ψ, (12.18)
where a sum over repeated indices is assumed and the quantities � I and � ξ are vectors
in the internal space. Thus there are now three separate space-and time-dependent
phase angles ξ (a) and three non-commuting isospin vectors I (a) . It is this noncommuting
property that makes the theory non-Abelian. The difference between
the local gauge invariance and the global one, described in chapter 8, can be stated
in terms of a neutron and proton. These particles represent two states of different
I3. The choice of phase, I3 =+1/2 for the protons, is a matter of convention,
but it is the same everywhere in space. Since we deal with local actions and forces
carried by fields, rather than actions at a distance, Yang and Mills (5) questioned
whether two nucleons separated by a large distance can communicate their phase
instantaneously. Stated another way, could the proton have I3 =1/2 in one place
and I3 = −1/2 in another one? They went on to investigate the consequences of a
local invariance as we are doing here.
In analogy to the electromagnetic case, where the interaction can be obtained
from the replacement of
∇µ =
� �
1 ∂
, −∇
c ∂t
we define a generalized operator Dµ by
by Dµ =(D0, D),
Dµ = ∇µ + ig� I· � Vµ
(12.19)
with Vµ =(V0, V ). Arrows are used for internal vectors. This equation is similar
to Eq. (12.5) with different dimensions and with q replaced by g, and is part of the
required generalization for a non-Abelian theory.
5C.N. Yang and R.L. Mills, Phys. Rev. 96, 191 (1954).