Feynman Path Integral Formulation

86 3 Gravity in 2 + ε Dimensions
and can be shown to vanish in d = 3. To show this one needs to compute the analog
of Eq. (1.17) in d dimensions, which is
− κ2
2
∫
[
d d x
T μν ✷ −1 T μν − (d − 2) −1 T μ
μ
→− d − 3 κ 2 ∫
d − 2 2
]
✷ −1 Tν
ν
d d−1 xT 00 G T 00 , (3.83)
where the Green’s function G is the static limit of 1/✷, and κ 2 = 16πG. The above
result implies that there are no Newtonian forces in d=2+1 dimensions (Deser,
Jackiw and ’t Hooft, 1984; Deser and Jackiw, 1984). The only fluctuations left in
3d are possibly associated with the scalar curvature (Deser, Jackiw and Templeton,
1982).
The 2 + ε expansion for pure gravity then proceeds as follows. First the gravitational
part of the action
L = −
με √ gR , (3.84)
16πG
with G dimensionless and μ an arbitrary momentum scale, is expanded by setting
g μν → ḡ μν = g μν + h μν , (3.85)
where g μν is the classical background field and h μν the small quantum fluctuation.
The quantity L in Eq. (3.84) is naturally identified with the bare Lagrangian, and
the scale μ with a microscopic ultraviolet cutoff Λ, the inverse lattice spacing in a
lattice formulation. Since the resulting perturbative expansion is generally reduced
to the evaluation of Gaussian integrals, the original constraint (in the Euclidean
theory)
detg μν (x) > 0 , (3.86)
is no longer enforced [the same is not true in the lattice regulated theory, where it
plays an important role, see the discussion following Eq. (6.70)].
A gauge fixing term needs to be added, in the form of a background harmonic
gauge condition,
L gf = 1 2 α√ (
gg νρ ∇μ h μν − 1 2 βgμν ∇ μ h )( )
∇ λ h λρ − 1 2 βgλρ ∇ λ h , (3.87)
with h μν = g μα g νβ h αβ , h = g μν h μν and ∇ μ the covariant derivative with respect
to the background metric g μν . The gauge fixing term also gives rise to a Faddeev-
Popov ghost contribution L ghost containing the ghost field ψ μ , so that the total Lagrangian
becomes L + L gf + L ghost .
In a flat background, g μν = δ μν , one obtains from the quadratic part of the Lagrangian
of Eqs. (3.84) and (3.87) the following expression for the graviton propagator
=