Feynman Path Integral Formulation

208 6 Lattice Regularized Quantum Gravity
Δ(g)= 1 √ g
∂ μ
√ gg μν ∂ ν , (6.144)
for a given background metric. This then allows one to define the massless lattice
scalar propagator as the inverse of the above matrix, G ij (l 2 )=Δij
−1 (l 2 ).The
continuum scalar propagator for a finite scalar mass m and in a given background
geometry, evaluated for large separations d(x,y) ≫ m −1 ,
1
G(x,y|g) =
∼
d(x,y)→∞ d−(d−1)/2 (x,y) exp { −md(x,y) } , (6.145)
involves the geodesic distance d(x,y) between points x and y,
∫ √
τ(y)
d(x,y|g)= dτ g μν (τ) dxμ dx ν
τ(x)
dτ dτ . (6.146)
Analogously, one can define the discrete massive lattice scalar propagator
[
G ij (l 2 1
)=
−Δ(l 2 )+m
]ij
2
∼
d(i, j)→∞ d−(d−1)/2 (i, j) exp { −md(i, j) } , (6.147)
where d(i, j) is the lattice geodesic distance between vertex i and vertex j. The
inverse can be computed, for example, via the recursive expansion (valid for m 2 > 0
to avoid the zero eigenvalue of the Laplacian)
1
( ) 1 n
m 2 Δ(l2 ) . (6.148)
−Δ(l 2 )+m 2 = 1 m 2 ∞
∑
n=0
The large distance behavior of the Euclidean (flat space) massive free field propagator
in d dimensions is known in the statistical mechanics literature as the Ornstein-
Zernike result.
As a consequence, the lattice propagator G ij (l 2 ) can be used to estimate the lattice
geodesic distance d(i, j|l 2 ) between any two lattice points i and j in a fixed
background lattice geometry (provided again that their mutual separation is such
that d(i, j) ≫ m −1 ).
d(i, j) ∼ − 1
d(i, j)→∞ m lnG ij(l 2 ) . (6.149)