Feynman Path Integral Formulation

282 8 Numerical Studies
∫ τ(b) √
μ
τ(a)
dτ g μν (x) dxμ
dτ dxν
dτ
, (8.41)
can be taken as the Euclidean action contribution associated with the heavy spinless
particle of mass μ.
Next consider two particles of mass M 1 , M 2 , propagating along parallel lines in
the “time” direction and separated by a fixed distance R. Then the coordinates for the
two particles can be chosen to be x μ =(τ,r,0,0) with r either 0 or R. The amplitude
for this process is a product of two factors, one for each heavy particle. Each is of
the form
∫
L(0; M 1 )=exp
{−M 1
√
}
dτ g μν (x) dxμ
dτ dxν
dτ
, (8.42)
where the first argument indicates the spatial location of the Wilson line. For the
two particles separated by a distance R the amplitude is
Amp. ≡ W(0,R; M 1 ,M 2 )=L(0; M 1 ) L(R; M 2 ) . (8.43)
For weak fields one sets g μν = δ μν +h μν , with h μν ≪ 1, and therefore g μν (x) dxμ
dτ dxν
dτ
= 1 + h 00 (x). Then the amplitude reduces to
∫ T
W(M 1 ,M 2 )=exp
{−M 1 dτ √ }
∫ T
1 + h 00 (τ) exp
{−M 2 dτ ′√ }
1 + h 00 (τ ′ ) .
0
0
(8.44)
In perturbation theory the averaged amplitude can then be easily evaluated (Hamber
and Williams, 1995)
{
< W(0,R; M 1 ,M 2 ) > = exp −T (M 1 + M 2 − G M 1M )+···}
2
, (8.45)
R
and the static potential has indeed the expected form, V (R) =−G M 1 M 2 /R. The
contribution involving the sum of the two particle masses is R independent, and can
therefore be subtracted, if the Wilson line correlation is divided by the averages of
the individual single line contribution, as in
1
V (R)=− lim
T→∞ T log < W(0,R; M 1 ,M 2 ) >
< L(0; M 1 ) >< L(R; M 2 ) > ∼−G M 1M 2
R
. (8.46)
If one is only interested in the spatial dependence of the potential, one can simplify
things further and take M 1 = M 2 = M. To higher order in the weak field expansion
one has to take into account multiple graviton exchanges, contributions from
graviton loops and self-energy contributions due to other particles.
How does all this translate to the lattice theory At this point, the prescription for
computing the Newtonian potential for quantum gravity should be clear. For each
metric configuration (which is a given configuration of edge lengths on the lattice)
one chooses a geodesic that closes due to the lattice periodicity (and there might