Quote by selfAdjoint
Mike2 on the other thread raised the question of whether the quantum gravity model assumed by AJL had been rigorously developed.

As far as I can tell, it's completely rigorous. And I'm a mathematician by training, so I'm more fussy about these things than most.
I know of rigorous (1,1)dimensional theories and maybe some (1,2)dimensional ones, but I don't know of any fully (1,3) relativistic quantized ones.

It's easier to make discrete models rigorous than models that assume spacetime is a continuum. That's the main reason I like discrete models.
In particular, all the 3+1dimensional spin foam models of quantum gravity I've worked on  various versions of the BarrettCrane model  are mathematically rigorous and backgroundfree.
The problem is, we haven't gotten good evidence that these spin foam models "work"  namely, that they reduce to general relativity in the limit of distance scales that are large compared to the Planck length.
See my Marseille talk for a taste of the problems:
http://math.ucr.edu/home/baez/spin_foam_calculations.ps
Since we don't have any experimental evidence concerning quantum gravity, mathematical rigor is one way to make sure we're not playing tennis with the net down. I will be very happy when we get
any rigorously welldefined backgroundfree quantum theory of gravity that
works in the sense defined above.
More precisely: I will be very happy if we get numerical evidence that it works, and
ecstatic if we can mathematically
prove that it works. But since such a model is likely to be nonperturbative, a mathematical proof of this sort might be very difficult. Nobody has even proved confinement in lattice QCD, even though numerical calculations have convinced everyone it's true.