2.3 Faddeev–Popov Gauge Fixing
39
the kinetic term) for a general curved background. For example, the inner product
for the scalar fields X μ is
(δX
μ , δX
μ ) g =
d
2 σ
√ g G μν (X)δX
μ δX
ν .
(2.42)
It is not possible to normalize anymore the measure to set det G(X) = 1 like in
(2.32) since it depends on the fields. On the other hand, this factor is not important
for the manipulations performed in this chapter. Any ambiguity in the measure will
again correspond to a renormalization of the cosmological constant [11, p. 923].
Moreover, as explained above, it is not necessary to write explicitly the matter
partition function as long as it describes a CFT.
2.3
Faddeev–Popov Gauge Fixing
The naive integration over the space Met(( g ) of all metrics of g (note that the
genus is fixed) leads to a divergence of the functional integral since equivalent
configurations
(f
∗ g, f
∗ ) ∼ (g, ,),
(e
2ω g, ,) ∼ (g, ,)
(2.43)
give the same contribution to the integral. This infinite redundancy causes the
integral to diverge, and since the multiple counting is generated by the gauge group,
the infinite contribution corresponds to the volume of the latter. The Faddeev–Popov
procedure is a means to extract this volume by separating the integration over the
gauge and physical degrees of freedom
d(fields) = Jacobian × d(gauge) × d(physical).
(2.44)
The space of fields (g, ,) is divided into equivalence classes and one integrates
over only one representative of each class (gauge slice), see Fig. 2.1. This change
of variables introduces a Jacobian which can be represented by a partition function
with ghost fields (fields with a wrong statistics). This program encounters some
complications since G is a semi-direct product and is non-connected.
Example 2.1: Gauge Redundancy
A finite-dimensional integral which mimics the problem is
Z =
R 2
dx dy e
−(x−y) 2 .
(2.45)
39
the kinetic term) for a general curved background. For example, the inner product
for the scalar fields X μ is
(δX
μ , δX
μ ) g =
d
2 σ
√ g G μν (X)δX
μ δX
ν .
(2.42)
It is not possible to normalize anymore the measure to set det G(X) = 1 like in
(2.32) since it depends on the fields. On the other hand, this factor is not important
for the manipulations performed in this chapter. Any ambiguity in the measure will
again correspond to a renormalization of the cosmological constant [11, p. 923].
Moreover, as explained above, it is not necessary to write explicitly the matter
partition function as long as it describes a CFT.
2.3
Faddeev–Popov Gauge Fixing
The naive integration over the space Met(( g ) of all metrics of g (note that the
genus is fixed) leads to a divergence of the functional integral since equivalent
configurations
(f
∗ g, f
∗ ) ∼ (g, ,),
(e
2ω g, ,) ∼ (g, ,)
(2.43)
give the same contribution to the integral. This infinite redundancy causes the
integral to diverge, and since the multiple counting is generated by the gauge group,
the infinite contribution corresponds to the volume of the latter. The Faddeev–Popov
procedure is a means to extract this volume by separating the integration over the
gauge and physical degrees of freedom
d(fields) = Jacobian × d(gauge) × d(physical).
(2.44)
The space of fields (g, ,) is divided into equivalence classes and one integrates
over only one representative of each class (gauge slice), see Fig. 2.1. This change
of variables introduces a Jacobian which can be represented by a partition function
with ghost fields (fields with a wrong statistics). This program encounters some
complications since G is a semi-direct product and is non-connected.
Example 2.1: Gauge Redundancy
A finite-dimensional integral which mimics the problem is
Z =
R 2
dx dy e
−(x−y) 2 .
(2.45)