1. Utrecht University Gerard ’t Hooft Gell-Mann Colloquium Singapore, February 24, 2010

3. prototype: (any number of space dimensions) prototype: (any number of space dimensions) The evolution law: variables: Margolus rule

4. (when x + t is odd) Alternatingly, the sites at even t and the ones at odd t are updated:

5. A and B are operators. Write them as is the is the permutation operator for the variable F for the variable F equal-time commutators:

6. Write: What is H ? Use Baker-Campbell-Hausdorff:

8. Faster convergence is reached if we limit ourselves to the conjugation class of H : Where F is chosen such that Write repeated commutators, for instance as:, to find as:, to find

9. appears to be a perfectly local, bounded appears to be a perfectly local, bounded quantum operator, similar to the Hamilton density operator of a QFT. similarly: stays outside the “light cone”: information does not spread faster than velocity v =1=c as an operator, is (practically) bounded as an operator, is (practically) bounded (from below and above), so H should have a lowest eigenstate. This is the vacuum state of the cellular automaton. only if one may terminate the BCH series But does the Baker-Campbell-Hausdorff expansion converge ?

10. One can argue that divergence occurs when two energy eigenvalues of H are considered that are apart. apart. “Planck energy” ? But does the Baker-Campbell-Hausdorff expansion converge ?

11. Qu: time translation invariance only strictly holds for time tranlations over integral multiples of Δt, the lattice time unit. Is conservation of energy violated by multiples of ? A1: yes, if you introduce a classical perturbation: allow the cellular automaton to be perturbed: Then, acts with the beat of the lattice clock. It only respects energy conservation modulo.

12. A2: no, if you expand the complete Hamiltonian H into a linearlized part and an interaction piece. The total energy, defined by is exactly conserved. is exactly conserved. Can one resum the BCH series ?

14. Converges only if at all t

15. This distinction may be of crucial importance for the following discussion:

17. t = 0 α and β are entangled. P cannot depend on B, and Q cannot depend on A → Bell’s inequality → contradiction! And yet no useful signal can be sent from B to P or A to Q.

18. It is essential to realize that Bell’s inequalities refer to the states a system is in, whereas our “hidden variables” are a theory for their dynamics. We can always assume our system to be in a state violating Bell’s inequalities, and evolve it backwards in time, to conclude that the initial state must have been a thoroughly entangled one. The Universe must have started out as a highly entangled state … must have started out as a highly entangled state … or rather, our understanding of it, or rather, our understanding of it, But so what ?

19. Our world is not quantum mechanical, but only our perception of it …

20. G. ‘t H, arXiv:0909.3426; P.Jizba, H. Kleinert, F.Scardigli, arXiv:012.2253, And others …

21. The Cellular Automaton Prototype Its evolution operator Hamilton formalism Convergence problem QM and GR Conclusion