1. Utrecht University Gerard ’t Hooft and Isaac Newton Institute, December 15, 2004

2. Conventional Quantum Mechanics Deterministic Quantum Mechanics All choices of basis are equivalent There is a preferred basis The rules for physical calculations are identical Locality applies to commutators outside the light cone Locality can only be understood in this basis Gauge equivalence classes of states Ontological equivalence classes

3. The use of Hilbert Space Techniques as technical devices for the treatment of the statistics of chaos... A “state” of the universe: A simple model universe: Diagonalize:

4. (An atom in a magnetic field) An operator that is diagonal in the primordial basis, is a BEABLE. Other operators such as H, or: are CHANGEABLES In the original basis:

5. Deterministic evolution of continuous degrees of freedom: but, … this H is not bounded from below !

6. The harmonic oscillator Theorem: its Hilbert Space is that of a particle moving along a circle H ?

7. Our assignment: Find the true beables of our world! Beables can be identified for: An atom in a magnetic field Second quantized MASSLESS, NON-INTERACTING “neutrinos” Free scalar bosons Free Maxwell photons

8. Beables for the first quantized neutrino:

9. But, single “neutrinos” have Dirac’s second quantization: } empty } full But a strict discussion requires a cut-off for every orientation of : But, how do we introduce mass? How do we introduce interactions? How do the “flat membranes” behave in curved space-time ?

10. A key ingredient for an ontological theory: Information loss Introduce equivalence classes

11. Neutrinos aren’t sheets... They are equivalence classes There is an ontological position x, as well as an orientation for the momentum. The velocity in the direction is c, but there is “random”, or “Brownian” motion in the transverse direction. Note:

12. Two coupled degrees of freedom

13. Does dissipation help to produce a lower bound to the Hamiltonian ? Consider first the harmonic oscillator: The deterministic case: write

14. Important to note: The Hamiltonian nearly coincides with the Classical conserved quantity We now impose a constraint (caused by information loss?) Two independent QUANTUM harmonic oscillators!

15. This oscillator has two conserved quantities: Write Alternatively, one may simply remove the last part, and write Or, more generally: Then, the operator D is no longer needed.

16. Compare the Hamiltonian for a (static) black hole. We only “see” universe # I. Information to and from universe II is lost. We may indeed impose the constraint:

17. But, even in a harmonic oscillator, this lock-in is difficult to realize in a model. Projecting onto states with can only happen if there is information loss. The “classical quantization” of energy: Let H be the Hamiltonian and U be an ontological energy function.

18. This way, one can also get into grips with the anharmonic oscillator. Since H must obey where T is the period of the (classical) motion, we get that only special orbits are allowed. Here, information loss sets in. The special orbits are the stable limit cycles! If T is not independent of, then the allowed values of H are not equidistant, as in a genuine anharmonic oscillator.

19. The perturbed oscillator has discretized stable orbits. This is what causes quantization.

20. A deterministic “universe” may show POINCARÉ CYCLES: Equivalence classes form pure cycles: Gen. Relativity: time is a gauge parameter ! Dim(  ) = # different Poincaré cycles Heisenberg Picture: fixed.

21. For black holes, the equivalence classes are very large!

22. The black hole as an information processing machine These states are also equivalence classes. The ontological states are in the bulk !!

23. The cellular automaton

24. Suppose: ★ a theory of ubiquitous fluctuating variables ★ not resembling particles, or fields... Suppose: ★ that what we call particles and fields are actually complicated statistical features of said theory... One would expect ★ statistical features very much as in QM (although more probably resembling Brownian motion etc. ★ Attempts to explain the observations in ontological terms would also fail, unless we’d hit upon exactly the right theory...

25. dobbelgod