Feynman Path Integral Formulation

18 1 Continuum Formulation
+ c 1 {R − 1 2 hα αR + h α β Rβ α − 1 8 Rhα αh β β + 1 4 Rhα β hβ α
− h ν β hβ αR α ν + 1 2 hα αh ν β Rβ ν − 1 4 ∇ νh α β ∇ν h β α
+ ∇ ν h α α∇ ν h β β − 1 2 ∇ β h α α∇ μ h β μ + 1 2 ∇α h ν β ∇ νh β α}] ,
(1.94)
up to total derivatives. Here ∇ μ denotes a covariant derivative with respect to the
metric g μν .Forg μν = η μν the above expression coincides with the weak field
Lagrangian contained in Eqs. (1.7) and (1.67), with a cosmological constant term
added, as given in Eq. (1.55).
To this expression one needs to add the gauge fixing and ghost contributions, as
was done in Eq. (1.67). The background gauge fixing term used is
− 1 2 C2 μ = − 1 2√ g(∇α h α μ − 1 2 ∇ μh α α)(∇ β h βμ − 1 2 ∇μ h β β ) , (1.95)
with a corresponding ghost Lagrangian
L ghost = √ g ¯η μ (∂ α ∂ α η μ − R μ αη α ) . (1.96)
The integration over the h μν field can then be performed with the aid of the standard
Gaussian integral formula
∫
ln [dh μν ] exp{− 1 2h · M(g) · h − N(g) · h}
= 1 2 N(g) · M−1 (g) · N(g) − 1 2
trlnM(g)+const. , (1.97)
leading to an effective action for the g μν field. In practice one is only interested in
the divergent part, which can be shown to be local. Specific details of the functional
measure over metrics [dg μν ] are not deemed to be essential at this stage, as in perturbation
theory one is only doing Gaussian integrals, with h μν ranging from −∞ to
+∞. In particular when using dimensional regularization one uses the formal rule
∫
d d k =(2π) d δ (d) (0)=0 , (1.98)
which leads to some technical simplifications but obscures the role of the measure.
In the flat background field case g μν = η μν , the functional integration over the
h μν fields would have been particularly simple, since then one would be using
h μν (x)h αβ (x ′ ) → = G μναβ (x,x ′ ) , (1.99)
with the graviton propagator G(k) given in Eq. (1.77). In practice, one can use the
expected generally covariant structure of the one-loop divergent part
ΔL g ∝ √ g ( α R 2 + β R μν R μν) , (1.100)
with α and β some real parameters, as well as its weak field form, obtained from