This sort of covariant quantization method is often referred to as the “Gupta-
Bleuler” method, and is described in more detail in many quantum field theory
textbooks.
In the Yang-Mills case, each of the methods that we have discussed for
dealing with the gauge symmetry runs into problems:
- In theA 0 = 0 gauge, there again is a symmetry under the group of time-
independent gauge transformations, and a moment mapμ, withμ= 0
the Yang-Mills version of Gauss’s law. The symplectic reduction however
is now a non-linear space, so the quantization method we have developed
does not apply. Gauss’s law can instead be imposed on the states, but
it is difficult to explicitly characterize the physical state space that this
gives. - A Yang-Mills analog of the Coulomb gauge condition can be defined, but
then the analog of equation 46.13 will be a non-linear equation without a
unique solution (this problem is known as the “Gribov ambiguity”). - The combination of fieldsχ(A) now no longer satisfies a linear wave equa-
tion, and one cannot consistently restrict to a positive energy subspace
and use the Gupta-Bleuler covariant quantization method.
Digression.There is a much more sophisticated Lorentz covariant quantiza-
tion method (called the “BRST method”) for dealing with gauge symmetry that
uses quite different techniques. The theory is first extended by the addition of
non-physical (“ghost”) fermionic fields, giving a theory of coupled bosonic and
fermionic oscillators of the sort we studied in section 33.1. This includes an op-
erator analogous toQ 1 , with the property thatQ^21 = 0. One can arrange things
in the electromagnetic field case such that
Hphys={|ψ〉:Q 1 |ψ〉= 0}
{|ψ〉:|ψ〉=Q 1 |ψ′〉}In the BRST method one works with unconstrained Lorentz covariant fields, but
non-unitary state spaces, with unitarity only achieved on a quotient such as
Hphys. This construction is related to the Gupta-Bleuler method in the case of
electromagnetic fields, but unlike that method, generalizes to the Yang-Mills case.
It can also be motivated by considerations of what happens when one imposes a
gauge condition in a path integral (this is called the “Faddeev-Popov method”).
46.7 For further reading
The topic of this chapter is treated in some version in every quantum field
textbook. Some examples for Coulomb gauge quantization are chapter 14 of
[10] or chapter 9 of [16], for covariant Lorenz gauge quantization see chapter
7 of [35] or chapter 9 of [78]. [17] has a mathematically careful discussion of
both the Coulomb gauge quantization and covariant quantization in Lorenz