Feynman Path Integral Formulation

4 1 Continuum Formulation
which, by comparison to the expected Newtonian potential energy −GMM ′ /r,gives
the desired identification κ = √ 16πG.
The pure gravity part of the action in Eq. (1.7) only propagates transverse traceless
modes (shear waves). These correspond quantum mechanically to a particle of
zero mass and spin two, with two helicity states h = ±2, as shown for example in
(Weinberg, 1972) by looking at the nature of plane wave solutions to the wave equation
in the harmonic gauge. Helicity 0 and ±1 appear initially, but can be made to
vanish by a suitable choice of coordinates.
It is relatively easy to see that the particle described by the field h μν has spin
two. Consider the wave equation for h μν in the absence of sources, and subject to
the harmonic gauge condition
✷h μν = 0 , ∂ μ h μ ν = 1 2 ∂ ν h λ λ . (1.19)
A plane wave
h μν (x) =e μν (k) e ik·x + e ∗ μν(k) e −ik·x , (1.20)
will satisfy the wave equation and the harmonic gauge condition, provided k 2 = 0
and
k μ e μ ν = 1 2 k ν e λ λ , (1.21)
respectively. Initially the polarization tensor e μν = e νμ has ten independent components,
which then get reduced to six by the imposition of the harmonic gauge
condition. These get further reduced to just two when one allows for local coordinate
changes x μ → x μ + ε μ (x), which here modify h μν according to
h ′ μν = h μν − ∂ μ ε ν − ∂ ν ε μ . (1.22)
To see the effects of this invariance, one first expands ε μ (x) as well in plane waves,
ε μ (x) =iε(k) e ik·x − iε(k) e −ik·x , (1.23)
so that under the gauge transformation h μν → h ′ μν one has
h ′ μν
= e ′ μν e ik·x + e ′ ∗
μν e −ik·x , (1.24)
with new polarization tensor
e ′ μν = e μν + k μ ε ν + k ν ε μ . (1.25)
One can explicitly verify that the new polarization tensor e ′ μν still obeys the harmonic
gauge condition, k μ e ′ μ
ν = 1 2 k νe ′ λλ , which shows that the residual gauge invariance
can indeed be used to eliminate four additional degrees of freedom.
To proceed further it will be useful to consider a wave propagating along the 3
direction, for which k 1 = k 2 = 0 and k 3 = k 0 = k > 0. The harmonic gauge condition
of Eq. (1.21) then gives the four equations