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Deterministic Quantum Mechanics and Information Loss in Quantum Mechanics

This article discusses the concept of deterministic quantum mechanics and its connection to information loss in quantum mechanics. It explores the rules for physical calculations, the role of basis choices, locality, gauge and ontological equivalence classes, and the use of Hilbert Space techniques. The article also examines the concept of beables and their identification in various systems, such as atoms, neutrinos, and photons. Finally, it delves into the role of information loss in introducing constraints and quantization in systems like the harmonic oscillator and black holes.

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Deterministic Quantum Mechanics and Information Loss in Quantum Mechanics

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  1. Utrecht University Information Loss Determinism and QUANTUM MECHANICS Gerard ’t Hooft Isaac Newton Institute, December 15, 2004

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

  3. 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: Diagonalize:

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

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

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

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

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

  11. Neutrinos aren’t sheets ... 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. They are equivalence classes Note:

  12. Two coupled degrees of freedom

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

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

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

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

  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

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