The quantisation of gauge theories in the path integral formalism requires more discussion, since the gauge condition (like the Lorentz gauge∂µAµ=0) seems to remove only one degree of freedom of the two that are eliminated in the Hamiltonian formulation (see Chapter 16). From a simple example it is easily demonstrated what the effect of gauge fixing on a (path) integral is. For this we take f (�x) to be a function on IR3, which is invariant under rotations around the origin, such that it is a function f (r) of the radius r = |�x|only. The symmetry group is hence SO(3), and we can attempt to compute the integral d3�x f (�x) by introducing a ‘gauge’ fixing condition like x2=x3=0. But it is clear that
d3x f (� x)� �=
d3x�δ(x2)δ(x3) f (x)� =
d x1 f (x1). (20.1) We know very well that we need a Jacobian factor for the radial integral
d3�x f (�x)=4π ∞
0
r2dr f (r). (20.2)
This Jacobian, arising in the change of variable to the invariant radial coor-dinates and the angular coorcoor-dinates, can be properly incorporated following the method introduced by Faddeev and Popov. The starting point is a straight-forward generalisation of the identity
d x |f�(x)|δ[ f (x)]=1, assuming the equation f (x) =0 to have precisely one solution (in a sense the right-hand side of the equation counts the number of zeros). It reads
1=
Dg|det M(gA)
|δ F(gA)
, (20.3)
where F( A) ∈ LG is the gauge-fixing function [with the gauge condition F( A)=0, e.g.,F( A) =∂µAµ=0]. The gauge transformation g of the gauge field Ais indicated bygA; see Equation (18.34). Furthermore, M( A) : LG →LG
plays the role of the Jacobian, M(gA)= ∂F(gA)
∂g ≡∂F(eXgA)
∂X at X=0. (20.4)
137
138 A Course in Field Theory Equivalently, with respect to the Lie algebra basis, whereF( A) ≡Fa( A)Ta, one has
Ma b( A)= dFa(exp(tTb)A)
dt at t=0. (20.5)
To relate this to the previous equation, one makes use of the fact that
h(gA)=(hg)A, (20.6)
which states that two successive gauge transformations, g and h, give the same result as a single gauge transformation with hg.
As an example we consider the Lorentz gauge, with F( A) = ∂µAµ, for which
F(exp( X)A)−F( A)= −q−1∂µDadµ( A)( X)+O( X2), (20.7) where Dµad( A) is the covariant derivative in the adjoint representation
Dadµ( A)( X)≡∂µX+q [Aµ, X]. (20.8) With respect to the Lie algebra basis, this gives
q Ma b( A)= −δa b∂µ∂µ+q fa bc(∂µAcµ+Acµ∂µ). (20.9) For an Abelian gauge theory, the structure constants fa bc vanish and M( A) becomes independent of the gauge field. This means that det[M( A)] can be absorbed in an overall normalisation of the path integral. For non-Abelian gauge theories this is no longer possible. Before describing how the A depen-dence of det[M( A)] is incorporated, it is important to note that we assumed the gauge conditionF(gA)=0 to have precisely one solution, which can be arranged with the help of Equation (20.6) to occur at g=1, in which case A is said to satisfy the gauge condition. This is, in general, not correct, as was discovered by Gribov. Even in our simple problem on IR3, the gauge condition x2 = x3 =0 does not uniquely specify the gauge, because we can go from (r, 0, 0) to (−r, 0, 0) through a rotation over 180 degrees. We have to introduce a further restriction to get the identity
d3x f (x)=4π
d3x x12δ(x2)δ(x3)θ(x1) f (x), (20.10) whereθ(x) = 0 for x < 0 andθ(x) = 1 for x ≥ 0. In perturbation theory only the gauge fields near the origin in field space are relevant, and gauge conditions are chosen so as to avoid this problem in a small neighbourhood of the origin. The Lorentz gauge is such a gauge condition, and the gauge fixing or Gribov ambiguity is not an issue for computing quantities in perturbation theory in q .
We still need to define what we mean byDg in Equation (20.3). It stands for the integration measure$
xdg(x), with dg(x) for every x defined as the
Gauge Fixing and Ghosts 139 so-called Haar measure on the group. It is best described in the example of SU(2), which as a space is isomorphic with S3. When S3 is embedded in IR4 as a unit sphere, n2µ = 1, it is not too difficult to see that g = n4+ iσknkgives an element of SU(2), whereas exp(iχskσk)=cos(χ)+i sin(χ)skσk, with s2k =1 shows that any element of SU(2) can be written in terms of nµ. The Haar measure coincides with the standard integration measure on S3, d4nδ(n2µ−1). The Haar measure is, in general, invariant under the change of variables g → hg and g → gh, for h some fixed group element. We can insert Equation (20.3) in the path integral to obtain
Z= We now use that the action S( A) is invariant under gauge transformations.
We leave it as an exercise to verify that likewiseDAµis invariant under the change of variables A→gA, which trivially implies that
Z=
The dependence of the integrand on g has disappeared, and the integration over g gives an overall (infinite) normalisation factor, which is irrelevant. We next note that Z has to be independent of the gauge-fixing functionF, in particularF( A)−Y is just as good for the gauge fixing [provided, of course, we show thatF(gA) = Y has a solution]. This modification does not affect the so-called Faddeev–Popov operator M( A) and we find
Z=
independent of Y. Suitably normalisingDY we can define
which combined with the previous equation gives Z=
For U(1) gauge theories withF( A) = ∂µAµ, this precisely reproduces the action of Equation (4.22) in the Lorentz gauge, and in that case det[M( A)] is a constant.
For non-Abelian gauge theories we are left with the task of computing det[M( A)] for each A, which is no longer constant. But here the path integral over Grassmann variables comes to the rescue. In Problem 25 we have seen
140 A Course in Field Theory up to an overall normalisation. This implies that the path integral can also be written in the Lorentz gauge as
Z=
Since ¯ηandηare auxiliary fields, they should never appear as external lines.
They are therefore called ghosts. Ghosts can only appear in loops and because of the fermionic nature of the ghost variables, every such loop gives a minus sign. The Feynman rules for the Lorentz gauge are given in Table 20.1.
Because one can easily derive that for a complex scalar field (up to an overall constant) we can view a ghost as the elimination of a complex degree of freedom. It is in this way that in the path integral the two unphysical degrees of freedom of a Lorentz vector are eliminated. For QED both the ghost and the unphysical degrees of freedom have no interactions and cannot appear as external lines either, which is why in QED the introduction of ghosts was never necessary for a consistent description of the theory. For non-Abelian gauge theories, because of the interaction of the ghost with the gauge field, ghosts can no longer be ignored. To have the ghosts eliminate the unphysical degrees of freedom, one should have the ‘masses’ (poles) of the ghosts coincide with the
‘masses’ of the unphysical degrees of freedom. Furthermore the couplings of TABLE 20.1
Feynman rules for ghosts.
no external ghost lines
-k
a b ≡ (q M)−1( A=0)ab= δa b
k2+iε ghost propagator (Lorentz gauge)
Gauge Fixing and Ghosts 141 the ghost and unphysical fields to the physical fields should be related. This is verified explicitly for the Georgi–Glashow model in Problem 35. In general it is guaranteed by the existence of an extra symmetry, discovered by Becchi, Rouet, and Stora, called the BRS symmetry s, which, for example, acts on the gauge field as follows:
s Aµ=Dµadη. (20.19)
This is precisely an infinitesimal gauge transformation. For more details, see Itzykson and Zuber, Section 12-4-1.
DOI: 10.1201/b15364-21