Feynman Path Integral Formulation

8.5 Wilson Lines and Static Potential 281
8.5 Wilson Lines and Static Potential
In a gauge theory such as QED the static potential can be computed from the manifestly
gauge invariant Wilson loop. To this end one considers the process where
a particle-antiparticle pair are created at time zero, separated by a fixed distance
R, and re-annihilated at a later time T . In QED the amplitude for such a process
associated with the closed loop Γ is given by the Wilson loop
{ ∮
W(Γ )=< exp ie A μ (x)dx μ} >, (8.35)
Γ
which is a manifestly gauge invariant quantity. Performing the required Gaussian
average using the (Euclidean) free photon propagator one obtains
{ ∮
< exp ie A μ dx μ} ∮ ∮
}
> = exp
{− 1 2 e2 dx μ dy ν Δ μν (x − y) . (8.36)
Γ
Γ Γ
For a rectangular loop of sides R and T one has after a short calculation
{ ∮
< exp ie A μ dx μ} > ≃ exp
{− e2 e2 T
(T + R)+
Γ
4πε 4π R + e2
)+···}
2π 2 log(T ε
[ ]
(8.37)
∼ exp −V (R) T ) , (8.38)
T ≫R
where ε → 0 is an ultraviolet cutoff. In the last line use has been made of the fact
that for large imaginary times the exponent in the amplitude involves the energy for
the process multiplied by the time T . Thus for V (R) itself one obtains
1
{ ∮
V (R)=− lim
T→∞ T log < exp ie A μ dx μ} > ∼ cst. − e2
Γ
4πR , (8.39)
which is the correct Coulomb potential for two oppositely charged particles.
To obtain the static potential it is not necessary to consider closed loops. Alternatively,
in a periodic box of length T one can introduce two long oppositely oriented
parallel lines in the time direction, separated by a distance R and closed by the periodicity
of the lattice, and associated with oppositely charged particles,
{ ∫
< exp ie A μ dx μ} { ∫
exp ie A νdy ν} > (8.40)
Γ
Γ ′
In the large time limit one can then show that the result for the potential V (R) is the
same.
In the gravitational case there is no notion of “oppositely charged particles”, so
one cannot use the closed Wilson loop to extract the potential (Modanese, 1995).
One is therefore forced to consider a process in which one introduces two separate
world-lines for the two particles. It is well known that the equation for free fall can
be obtained by extremizing the space-time distance travelled. Thus the quantity