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3.8
Other lattice approaches

In principle the discrete approaches described in Section
2.5
- in particular, the lattice descriptions of ’t Hooft and
Waelbroeck - should be straightforward to quantize. In practice,
there has been fairly little work in this area, and most of the
literature that does exist involves point particles rather than
closed universes. ’t Hooft has emphasized that the Hamiltonian in
his approach is an angle, and that time should therefore be
discrete
[251], in agreement with the Lorentzian spin foam analysis of
Section
3.6
. ’t Hooft has also found that for a particular representation of
the commutation relations for a point particle in
(2+1)-dimensional gravity, space may also be discrete
[254], although it remains unclear whether these results can be
generalized beyond this one special example. Criscuolo et
al. have examined Waelbroeck’s lattice Hamiltonian approach
for the quantized torus universe
[97], investigating the implication of the choice of an internal time
variable, and Waelbroeck has studied the role of the mapping
class group
[269]
.