Feynman Path Integral Formulation

1.3 Wave Equation 9
with
s μν = 1 d η μν ∂ · ε
a μν = 1 2 (∂ με ν − ∂ ν ε μ )
t μν = 1 2 (∂ μ ε ν + ∂ ν ε μ − 2 d η μν ∂ · ε) . (1.53)
Then s μν (x) can be thought of describing local scale transformations, a μν (x) is
written in terms of an antisymmeric tensor and therefore describes local rotations,
while t μν (x) contains a traceless symmetric tensor and describes local shears.
Since both the scalar curvature R(x) and the volume element dx √ g(x) are separately
invariant under the general coordinate transformations of Eqs. (1.48) and
(1.50), both of the following action contributions are acceptable
∫
dx √ g(x)
∫
dx √ g(x) R(x) , (1.54)
the first being known as the cosmological constant contribution (as it represents the
total space-time volume). In the weak field limit, the first, cosmological constant
term involves
√ g = 1 +
1
2
hμ
μ + 1 8 h μ μ hν ν − 1 4 h μνh μν + O(h 3 ) , (1.55)
which is easily obtained from the matrix formula
√
detg = exp(
1
2
trlng)=exp[ 1 2
trln(η + h)] , (1.56)
after expanding out the exponential in powers of h μν . We have also reverted here to
the more traditional way of performing the weak field expansion (i.e. without factors
of κ),
g μν = η μν + h μν
g μν = η μν − h μν + h α
μ h αν + ... (1.57)
with η μν the flat metric. The reason why such a √ g cosmological constant term
was not originally included in the construction of the Lagrangian of Eq. (1.7) is that
it does not contain derivatives of the h μν field. It is in a sense analogous to a mass
term, but does have the very important property that it does not break the local gauge
invariance.
This is a good place to discuss another issue. There is an old question of whether
the graviton is exactly massless, or whether it has possibly a very small mass m.
It is clear that the mass has to be very small, otherwise it would cause observable
deviations from the Newtonian potential used to describe successfully large galactic
cluster scales. In principle there are a number of well known problems that arise
when a (Pauli-Fierz) spin-two mass term