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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  [251Jump To The Next Citation Point], 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] .
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