Feynman Path Integral Formulation

204 6 Lattice Regularized Quantum Gravity
Fig. 6.14 Labeling of edges
and fields for the construction
of the scalar field action.
φ 3
l 2
3
l 1
φ 1
1
2
l 3
φ 2
with the following definitions
(
l
g ij (Δ)=
3
2 1
2 (−l2 1 + l2 2 + l3)
2 )
1
2 (−l2 1 + l2 2 + l3) 2 l2
2
(6.126)
detg ij (Δ)= 1 [
4 2(l
2
1 l2 2 + l2l 2 3 2 + l3l 2 1) 2 − l1 4 − l2 4 − l3
4 ]
≡ 4A
2
Δ (6.127)
(
g ij 1
l
(Δ)=
2 1
2 2 (l2 1 − l2 2 − l 2 )
3)
1
detg(Δ)
2 (l2 1 − l2 2 − l3) 2 l3
2 . (6.128)
The scalar field derivatives get replaced as usual by finite differences
∂ μ φ −→ (Δ μ φ) i = φ i+μ − φ i , (6.129)
where the index μ labels the possible directions in which one can move away from
a vertex within a given triangle. Then
(
(φ
Δ i φΔ j φ = 2 − φ 1 ) 2 )
(φ 2 − φ 1 )(φ 3 − φ 1 )
(φ 2 − φ 1 )(φ 3 − φ 1 ) (φ 3 − φ 1 ) 2 . (6.130)
Then the discrete scalar field action takes the form
I =
16 1 1 [
∑ l
2
Δ
A 1 (φ 2 − φ 1 )(φ 3 − φ 1 )+l 2 2(φ 3 − φ 2 )(φ 1 − φ 2 )+l 2 3(φ 1 − φ 3 )(φ 2 − φ 3 ) ] ,
Δ
(6.131)
where the sum is over all triangles on the lattice. Using the identity
(φ i − φ j )(φ i − φ k )= 1 [
2 (φi − φ j ) 2 +(φ i − φ k ) 2 − (φ j − φ k ) 2] , (6.132)
one obtains after some re-arrangements the slightly more appealing expression for
the action of a massless scalar field (Itzykson and Bander, 1983)
I(l 2 ,φ) = 1 2
∑
A ij
( φi − φ j
l ij
) 2
. (6.133)