Feynman Path Integral Formulation

306 9 Scale Dependent Gravitational Couplings
The interaction in real space is often obtained by Fourier transform, and the above
expression is singular as k 2 → 0. The infrared divergence needs to be regulated,
which can be achieved by utilizing as the lower limit of momentum integration m =
1/ξ . Alternatively, as a properly infrared regulated version of the above expression
one can use
⎡
⎤
( ) 1
G(k 2 ) ≃ G c
⎣ m
2 2ν
1 + a 0 + ... ⎦
k 2 + m 2 . (9.2)
The last form for G(k 2 ) will only be necessary in the regime where k is small, so
that one can avoid unphysical results. From Eq. (9.2) the gravitational coupling then
approaches at very large distances r ≫ ξ the finite value G ∞ =(1 + a 0 + ...)G c .
Note though that in Eqs. (9.1) or (9.2) the cutoff no longer appear explicitly, it is
absorbed into the definition of G c . In the following we will be mostly interested in
the regime l P ≪ r ≪ ξ , for which Eq. (9.1) is completely adequate.
The first step in analyzing the consequences of a running of G is to re-write the
expression for G(k 2 ) in a coordinate-independent way. The following methods are
not new, and have found over the years their fruitful application in gauge theories
and gravity, for example in the discussion of non-local effective actions (Vilkovisky,
1984; Barvinsky and Vilkovisky, 1985). Since in going from momentum to position
space one usually employs k 2 →−✷, to obtain a quantum-mechanical running of
the gravitational coupling one should make the replacement
and therefore from Eq. (9.1)
G(✷) =G c
[
G → G(✷) , (9.3)
( ) 1
]
1 2ν
1 + a 0 + ...
ξ 2 . (9.4)
✷
In general the form of the covariant d’Alembertian operator ✷ depends on the specific
tensor nature of the object it is acting on,
( )
✷ T αβ...
γδ...
= g μν ∇ μ ∇ ν T αβ...
γδ...
. (9.5)
Only on scalar functions one has the fairly simple result
✷S(x) = 1 √ g
∂ μ g μν√ g∂ ν S(x) , (9.6)
whereas on second rank tensors one has the already significantly more complicated
expression ✷T αβ ≡ g μν ∇ μ (∇ ν T αβ ).
The running of G is expected to lead to a non-local gravitational action, for example
of the form