100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Consistent Discrete Space-Time 439for the black hole. We found out 19 that|ρ 12 (T max )| ∼ |ρ 12 (0)|( ) 2MPlanck3M BH(62)where ρ 12 is an off-diagonal element of the density matrix of the state ofinterest at the time the black hole would have evaporated. For a Solar sizedblack hole the magnitude of the off diagonal element is 10 −28 , that is, itwould have de facto become a mixed state even before invoking the Hawkingeffect. The information paradox is therefore unobservable in practice. Itshould be noted that this estimate is an optimistic one. In reality clocksfare much worse than the estimate we worked out and the fundamentaldecoherence will operate even faster than what we consider here.6. Connections with Continuum Loop Quantum GravityIn spite of the possibilities raised by the discrete approach, some readersmay feel that it forces us to give up too much from the outset. This wasbest perhaps captured by Thiemann 20 , who said “While being a fascinatingpossibility, such a procedure would be a rather drastic step in the sensethat it would render most results of LQG obtained so far obsolete”. Indeed,the kinematical structure built in loop quantum gravity, with a rigorousintegration measure and the natural basis of spin foam states appear asvery attractive tools to build theories of quantum gravity. We would like todiscuss how to recover these structures in our discrete approach.To make contact with the traditional kinematics of loop quantum gravity,we consider general relativity and discretize time but keep space continuous,and we proceed as in the consistent discretization approach, that is,discretizing the action and working out the equations of motion. We startby considering the action written in terms of Ashtekar variables 21 ,∫ ( )S = dtd 3 x ˜Pai F0a i − N a C a − NC(63)where N, N a aare Lagrange multipliers, ˜P i are densitized triads, and thediffeomorphism and Hamiltonian constraints are given by, C a = ˜P i aFab i ,C = ˜P a ˜P ()bi j√detqɛ ijk Fab i − (1 + β2 )K[a i Kj b]where βKa i ≡ Γ i a − A i a and Γ i a is thespin connection compatible with the triad, and q is the three metric. Wenow proceed to discretize time. The action now reads,∫ [ (S = dtd 3 x Tr ˜Pa ( A a (x) − V (x)A n+1,a (x)V −1 (x) + ∂ a (V (x))V −1 (x) ))−N a C a − NC + µ √ detqTr ( V (x)V † (x) − 1 )] (64)