Feynman Path Integral Formulation

7.2 Lattice Weak Field Expansion and Transverse-Traceless Modes 233
h 13 = − 1
2k 1 k 3
(h 11 k 2 1 − h 22 k 2 2 − h 11 k 2 3 − h 22 k 2 3)
h 23 = − 1
2k 2 k 3
(−h 11 k 2 1 + h 22 k 2 2 − h 11 k 2 3 − h 22 k 2 3)
h 33 = −h 11 − h 22 , (7.32)
and show that the second (trace) part vanishes.
The above manipulations underscore the fact that the lattice action, in the weak
field limit and for small momenta, only propagates transverse-traceless modes, as
for linearized gravity in the continuum. They can be used to derive an expression
for the lattice analog of the result given in (Kuchař, 1970) and (Hartle, 1984) for
the vacuum wave functional of linearized gravity, which gives therefore a suitable
starting point for a lattice candidate for the same functional.
A cosmological constant term can be analyzed in the lattice weak field expansion
along similar lines. According to Eqs. (6.41) or (6.42) it is given on the lattice by
the total space-time volume, so that the action contribution is given by
I V = λ 0 ∑ V h , (7.33)
edges h
where V h is defined to be the volume associated with an edge h. The latter is obtained
by subdividing the volume of each four-simplex into contributions associated with
each hinge (here via a barycentric subdivision), and then adding up the contributions
from each four-simplex touched by the given hinge. Expanding out in the small edge
fluctuations one has
I V
∼ ∑ (ε (n)
1
+ ε (n)
2
+ ε (n)
4
+ ε (n)
8 )+ 1 2 ∑ ε (m) T
i M (m,n)
i, j ε (n)
j . (7.34)
n
mn,ij
One needs to be careful since the expansion of ε i in terms of h μν contains terms
quadratic in h μν , so that there are additional diagonal contributions to the small
fluctuation matrix L ω ,
ε 1 +ε 2 +ε 4 +ε 8 = 1 2 (h 11+h 22 +h 33 +h 44 )− 1 8 (h2 11+h 2 22+h 2 33+h 2 44)+··· . (7.35)
These additional contributions are required for the volume term to reduce to the
continuum form of Eq. (1.55) for small momenta and to quadratic order in the weak
field expansion.
Next the same set of rotations needs to be performed as for the Einstein term,
in order to go from the lattice variables ε i to the continuum variables ¯h μν .After
the combined S ω - and T ω -matrix rotations of Eqs. (7.10) and (7.16) one obtains for
the small fluctuation matrix L ω arising from the gauge-fixed lattice Einstein-Regge
term [see Eq. (7.12)]
L ω = − 1 2 k2 V, (7.36)