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C lassic al C ellul ar A utoma ta

Utrecht University. C lassic al C ellul ar A utoma ta. and. Quantum Field Theory. Gerard ’t Hooft. Gell-Mann Colloquium Singapore, February 24, 2010. CELLULAR AUTOMATON. prototype: (any number of space dimensions). variables:. The evolution law:. Margolus rule.

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C lassic al C ellul ar A utoma ta

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  1. Utrecht University ClassicalCellularAutomata and Quantum Field Theory Gerard ’t Hooft Gell-Mann Colloquium Singapore, February 24, 2010

  2. CELLULAR AUTOMATON

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

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

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

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

  7. 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

  8. appears to be a perfectly local, bounded quantum operator, similar to the Hamilton density operator of a QFT. 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 ? similarly: stays outside the “light cone”: information does not spread faster than velocity v =1=c

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

  10. 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 . 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 ?

  11. 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. Can one resum the BCH series ?

  12. Converges only if at all t

  13. The EPR Paradox and BELL s inequalities , This distinction may be of crucial importance for the following discussion:

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

  15. 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 … or rather, our understanding of it, But so what ?

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

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

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

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