Feynman Path Integral Formulation

6.11 Lattice Higher Derivative Terms 201
∫
1
4
d d x √ (
gR μνλσ R μνλσ δh ) 2
→ ∑ V h . (6.112)
hinges h
A Ch
The above construction then leaves open the question of how to construct the remaining
curvature squared terms in four dimensions. If one takes the form given
previously in Eq. (6.36) for the Riemann tensor on a hinge and contracts one obtains
R(h)=2 δ h
A Ch
, (6.113)
which agrees with the form used in the Regge action for R. But one also finds readily
that with this choice the higher derivative terms are all proportional to each other
(Hamber and Williams, 1986),
(
1
4 R μνρσ(h)R μνρσ (h)= 1 2 R μν(h)R μν (h)= 1 δh
) 2
4 R(h)2 = . (6.114)
A Ch
Furthermore if one uses the above expression for the Riemann tensor to evaluate the
contribution to the Euler characteristic on each hinge one obtains zero, and becomes
clear that at least in this case one needs cross terms involving contributions from
different hinges.
The next step is therefore to embark on a slightly more sophisticated approach,
and construct the full Riemann tensor by considering more than one hinge. Define
the Riemann tensor for a simplex s as a weighted sum of hinge contributions
] [ δ
[R μνρσ = ∑ ω h U μν U ρσ
s
h⊂s
A C
]h , (6.115)
where the ω h are dimensionless weights, to be determined later. After squaring one
obtains
[
R μνρσ R μνρσ] [ δ
] [ δ = ∑ ω h ω h ′ U μν U ρσ U μν U ρσ] . (6.116)
s
h,h ′ A
⊂s
C h A C h ′
The question of the weights ω h introduced in Eq. (6.115) will now be addressed.
Consider the expression for the scalar curvature of a simplex defined as
[ ]
R
s = ∑
h⊂s
ω h
[
2 δ A C
]h . (6.117)
It is clear from the formulae given above for the lattice curvature invariants (constructed
in a simplex by summing over hinge contributions) that there is again a
natural volume associated with them : the sum of the volumes of the hinges in the
simplex
V s = ∑ V h , (6.118)
h⊂s