Gerard ’t Hooft

1. Utrecht University the Determinististic QUANTUM Gerard ’t Hooft Tilburg, Keynote Address

2. QM vs determinism: opposing religions? Newton’s laws: not fully deterministic The heuristic theory: local determinism, info-loss and quantization (discretization) Quantum statistics: QM as a tool. Quantum states; The deterministic Hamiltonian The significance of the info-loss assumption: equivalence classes Energy and Poincaré cycles Free Will, Bell’s inequalities; the observability of non-commuting operators Symmetries, local gauge-invariance

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

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

5. Newton’s laws: not fully deterministic: CHAOS t=0: x = 1.23456789012345 prototype example of a ‘chaotic’ mapping: t=1: x = 2.41638507294163 t=2: x = 4.62810325476981 How would a fully deterministic theory look?

6. t=0: x = 1.23456789012345 t=1: x = 1.02345678901234 t=2: x = 1.00234567890123 There is information loss. Information loss can also explain QUANTIZATION

7. Information is not conserved This is a necessary assumption

8. Two (weakly) coupled degrees of freedom

9. 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 ?

10. The use of Hilbert Space Techniques as technicaldevices for the treatment of the statistics of chaos ... A “state” of the universe: A simple model universe: TOP DOWN BOTTOM UP “Beable” Diagonalize: “Changeable”

11. Quantum States With this Hamiltonian, the quantum system is identical to the classical system.

12. If there is info-loss, this formalism will not change much, provided that we introduce EQUIVALENCE CLASSES

13. Consider a periodic system: 3 ¹/₃T 2 ½T T 1 E = 0 a harmonic oscillator !! ― 1 ― 2 ― 3

14. For a system that is completely isolated from the rest of the universe, and which is in an energy eigenstate, is the periodicity of its Poincaré cycle ! The quantum phase appears to be the position of this state in its Poincaré cycle.

15. 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

16. 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

17. momentum space Rotation symmetry

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

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

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

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

22. Bell inequalities And what about the ? John S. Bell electron vacuum Measuring device

23. 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.

24. The most questionable element in the usual discussions concerning Bell’s inequalities, is the assumption of FREE WILL Propose to replace it with Unconstrained Initial State

25. 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”.

26. Do we have a FREE WILL , that does not even affect the phases? Using this concept, physicists “prove” that deterministic theories for QM are impossible. The existence of this “free will” seems to be indisputable. Citations: 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. 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. Bassi, Ghirardi: Needless to say, the [the free-will assumption] must be true, thus B is free to measure along any triple of directions. ...

27. 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 General conclusions

28. THE END

31. In search for a Lock-in mechanism

32. Lock-in mechanism

33. The vacuum state must be a Chaotic solution Just as in Conway’s Game of Life ... stationary at large distance scales

34. “Free will” is limited by laws of physics The ultimate religion: ( Moslem ?? ) “The will of God is absolute ...

35. Conway's Game of Life 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.

36. This allows us to introduce quantum symmetries Example of a quantum symmetry: A 1+1 dimensional space-time lattice with only even sites: x + t = even. ↑ t “Law of nature”: x →

37. even odd ↑ t Classically, this has a symmetry: x → an EMERGENT QUANTUM SYMMETRY But in quantum language, we have:

38. t x t x

39. 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

40. Consider twonon - interacting periodic systems:

41. The allowed states have “kets ” with and “bras ” with Now, and So we also have:

42. For every that are odd and relative primes, we have a series: 5 4 3 2 1 0 -1 -2 -3 -4 -5 The combined system is expected again to behave as a periodic unit, so, its energy spectrum must be some combination of series of integers:

43. The case This is the periodicity of the equivalence class:

44. We now turn this around. Assume that our deterministic system will eventually end in a limit cycle with period . Then is the energy (already long before the limit cycle was reached). If is indeed the period of the equivalence class, then this must also be the period of the limit cycle of the system ! In the energy eigenstates, the equivalence classes coincide with the points of constant phase of the wave function. Limit cycles The phase of the wave function tells us where in the limit cycle we will be.