1. Gerard ’t Hooft Santiago de Compostela December 15 2008 Utrecht University Can be at the Planck scale?

2. The Quantum Discussion The probability interpretation Pilot waves The ontological interpretation EPR paradox Bell inequality Conway – Kochen “Free Will theorem” Why all these results are important Why all these results might be wrong Local hidden variables A model that seems to work

3. in his robā‘īyāt : “And the first Morning of creation wrote / What the Last Dawn of Reckoning shall read.” Determinism Omar Khayyam (1048-1131)

4. 1. Any live cell with fewer than two neighbours dies, as if by loneliness. 2. Any live cell with more than three neighbours dies, as if by overcrowding. 3. Any live cell with two or three neighbours lives, unchanged, to the next generation. 4. Any dead cell with exactly three neighbours comes to life.

5. Paul Dirac’s state vectors E.Schrödinger’s wave equation M.Born’s probability interpretation

6. D. Bohm’s Pilot waves

7. EPR paradox

8. J.S. Bell Particles (1) and (2) are “entangled”. Let’s talk about spin rather than position and momentum A1 A2 B1 B2 if you measure spin A, you know spin B, and they commute z x

9. A1 A2 B1 B2

10. A1 A2 B1 B2 However, all σ only take values ±1, and you can’t have all four of these entries contribute +1, so this number should be between ―2 and 2. Bell’s inequality: contradicts QM

11. 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.

12. A new variety of the same idea: the Conway – Kochen Free Will Theorem Consider two entangled massive spin 1 particles, with total spin S = 0 :

13. In case of spin 2 : 1 1 0

14. The 4 cubes of Conway & Kochen It is impossible to attach 0’s and 1’s to all axes at the positions of the dots, such that all orthogonal triples of axes have exactly the (1,1,0) combination.

15. Source 1 2 Conclude: Free Will Theorem: If observers on the two different sites have the free will to choose which axes to pick, the spin values of the two particles cannot be pre-determined. No “hidden variables ”

16. But is there “Free Will” ??? What is Free Will ??

17. Starting points Suppose one assumed a ToE that literally determines all events in the universe: determinism “Theory of Everything” Our present models of Nature are quantum mechanical. Does that prove that Nature itself is quantum mechanical?

18. We imagine three scales in physics: Pico: The Planck scale: 10 -33 cm Micro: The microscopic, or atomic, scale: 10 -8 cm Macro: The macroscopic scale (people, planets): > 1 cm At the Planck scale (pico) there are strictly deterministic laws. At the atomic scale (micro) everything seems chaotic. What we call atoms and molecules, are just minimal deviations from the statistical average (deviations of only a few bits of information on many billions! At the macroscopic scale, these deviations seem to adopt classical behavior (“classic limit”)

19. TOP DOWN BOTTOM UP 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: “Beable” “Changeable”

20. Cauchy line How could a local, deterministic dynamical system obtain a quantum field theory at large distance scales? Example: a C ellular A utomaton (CA) x t

21. x t Cauchy line The evolution rule is of the type: This is time reversible:

22. On the Cauchy surface at time t : At even time t : At odd time t :

23. At even time t : Then the next step : only if Now write :

24. Thus we find the Hamiltonian: H from Campbell-Baker-Hausdorff :

25. Note that is a finite-dimensional hermitean matrix, hence has a lower bound. Therefore, has a lowest energy eigenvalue: the “vacuum”. All terms obey: If CBH would converge, then we would have

26. The above may seem to be a beautiful approach towards obtaining a local quantum field theory from a deterministic, local CA. However, CBA does not converge, especially not when sandwiched between states with energy difference One could conjecture that our theory needs to be accurate only at energy scales << Planck length.

27. What happened to Bell’s inequalities ??? “vacuum” B “vacuum” A Rotating B is not possible without affecting vacuum B ; Rotating A is not possible without affecting vacuum A ; vacuum A and vacuum B will affect the entangled particles α and β. The “Quantum non-locality” may merely be a property of what we define to be the vacuum !

28. It is essential to distinguish: The use of quantum statistics (Hilbert space) while assuming deterministic underlying laws. This is not a contradiction !

29. If there is info-loss, our formalism will not change much, provided that we introduce

30. Two (weakly) coupled degrees of freedom

31. The (perturbed) oscillator has discretized stable orbits. This is what causes quantization.

32. The equivalence classes have to be very large these info - equivalence classes are very reminiscent of local gauge equivalence classes. It could be that that’s what gauge equivalence classes are Two states could be gauge-equivalent if the information distinguishing them gets lost. This might also be true for the coordinate transformations Emergent general relativity

33. Consider a periodic system: E = 0 1 2 3 ― 1― 1 ― 2― 2 ― 3― 3 a harmonic oscillator !! kets bras

34. A simple model generating the following quantum theory for an N dimensional vector space of states: 2 (continuous) degrees of freedom, φ and ω :

35. stable fixed points In this model, the energy ω is a beable.

36. Quite generally, contradictions between QM and determinism arise when it is assumed that an observer may choose between non-commuting operators, to measure whatever (s)he wishes to measure, without affecting the wave functions, in particular their phases. But the wave functions are man-made utensils that are not ontological, just as probability distributions. A “classical” measuring device cannot be rotated without affecting the wave functions of the objects measured.

37. One of the questionable elements in the usual discussions concerning Bell’s inequalities, is the assumption of Propose to replace it with

38. Free Will : “Any observer can freely choose which feature of a system he/she wishes to measure or observe.” Is that so, in a deterministic theory ? In a deterministic theory, one cannot change the present without also changing the past. Changing the past might well affect the correlation functions of the physical degrees of freedom in the present – the phases of the wave functions, may well be modified by the observer’s “change of mind”.

39. R. Tumulka: we have to abandon one of [Conway’s] four incompatible premises. It seems to me that any theory violating the freedom assumption invokes a conspiracy and should be regarded as unsatisfactory... We should require a physical theory to be non-conspirational, which means here that it can cope with arbitrary choices of the experimenters, as if they had free will (no matter whether or not there exists ``genuine" free will). A theory seems unsatisfactory if somehow the initial conditions of the universe are so contrived that EPR pairs always know in advance which magnetic fields the experimenters will choose. Do we have a FREE WILL, that does not even affect the phases? Citations: Using this concept, physicists “prove” that deterministic theories for QM are impossible. The existence of this “free will” seems to be indisputable. Bassi, Ghirardi: Needless to say, the [the free-will assumption] must be true, thus B is free to measure along any triple of directions.... Conway, Kochen: free will is just that the experimenter can freely choose to make any one of a small number of observations... this failure [of QM] to predict is a merit rather than a defect, since these results involve free decisions that the universe has not yet made.

40. General conclusions At the Planck scale, Quantum Mechanics is not wrong, but its interpretation may have to be revised, not only for philosophical reasons, but also to enable us to construct more concise theories, recovering e.g. locality (which appears to have been lost in string theory). The “random numbers”, inherent in the usual statistical interpretation of the wave functions, may well find their origins at the Planck scale, so that, there, we have an ontological (deterministic) mechanics For this to work, this deterministic system must feature information loss at a vast scale Holography: any isolated system, with fixed boundary, if left by itself for long enough time, will go into a limit cycle, with a very short period. Energy is defined to be the inverse of that period: E = hν

44. What about rotations and translations? One easy way to use quantum operators to enhance classical symmetries: The displacement operator: Eigenstates: Fractional displacement operator: This is an extension of translation symmetry

45. Other continuous Symmetries such as: rotation, translation, Lorentz, local gauge inv., coordinate reparametrization invariance, may emerge together with QM... They may be exact locally, but not a property of the underlying ToE, and not be a property of the boundary conditions of the universe

46. Rotation symmetry momentum space

47. Renormalization Group: how does one derive large distance correlation features knowing the small distance behavior? K. Wilson

48. momentum space Unsolved problems: Flatness problem, Hierarchy problem

49. One might imagine that there are equations of Nature that can only be solved in a statistical sense. Quantum Mechanics appears to be a magnificent mathematical scheme to do such calculations. Example of such a system: the ISING MODEL L. Onsager, B. Kaufman 1949 In short: QM appears to be the solution of a mathematical problem. As if: We know the solution, but what EXACTLY was the problem ?

50. Maar hoe kan dit klassieke gedrag nu afwijken van de Bell-ongelijkheden ? De Bell-ongelijkheden zijn afgeleid uit de veronderstelling dat onze huidige beschrijving een “werkelijkheid” betreft. Maar de lege ruimte, het “vacuüm”, correspondeert niet met één werkelijkheid. Het vacuüm is en lineaire superpositie van werkelijkheden, die wij in onze beschrijving ervan hebben ingevoerd. Wat doet dat met de Bell-ongelijkheden ?

51. Dit is hoe de Bell-ongelijkheden omzeild kunnen worden Echter, we hebben hier nog geen harde wiskundige formulering van Een grote moeilijkheid is de precieze wiskundige definitie van het vacuüm als toestand met exact omschreven correlatiefuncties. In de QM is het vacuüm de toestand met de laagste energie mogelijk. Dat is een verstrengelde toestand. Maar juist het energiebegrip is in deze modellen lastig te behandelen.