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Journal of Statistical Physics

, Volume 53, Issue 1–2, pp 323–344 | Cite as

Equivalence relations between deterministic and quantum mechanical systems

  • Gerard't Hooft
Articles

Abstract

Several quantum mechanical models are shown to be equivalent to certain deterministic systems because a basis can be found in terms of which the wave function does not “spread.” This suggests that apparently indeterministic behavior typical for a quantum mechanical world can be the result of locally deterministic laws of physics. We show how certain deterministic systems allow the construction of a Hilbert space and a Hamiltonian so that at long distance scales they may appear to behave as quantum field theories, including interactions but as yet no mass term. These observations are suggested to be useful for building theories at the Planck scale.

Key words

Hilbert space massless fermions extensive Hamiltonian primordial observables reversible cellular automaton Baker-Campbell-Hausdorff expansion discrete time 

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References

  1. 1.
    N. G. van Kampen, Ten theorems about quantummechanical measurements, Utrecht preprint THU-88-3.Google Scholar
  2. 2.
    M. Jammer,The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966).Google Scholar
  3. 3.
    J. S. Bell, Talk given at the Schwinger Symposium, UCLA, March 12, 1988.Google Scholar
  4. 4.
    J. M. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, 1968).Google Scholar
  5. 5.
    A. Einstein, B. Podolsky, and N. Rosen,Phys. Rev. 47:777 (1935).Google Scholar
  6. 6.
    J. S. Bell,Physics 1:195 (1964).Google Scholar
  7. 7.
    G. 't Hooft, On the quantization of space and time, inProceedings of the Moscow Seminar on Quantum Gravity (Academy of Sciences of the USSR, 1987), Utrecht preprint 1987.Google Scholar
  8. 8.
    G. 't Hooft, inRecent Developments in Gravitation, M. Lévy and S. Deser, eds. (Plenum, New York); C. J. L. Mannion and J. G. Taylor,Phys. Lett. 100B:261 (1981); T. D. Lee, Physics in terms of difference equations, Talk given at the Niels Bohr Centenary Symposium, Copenhagen, October 1985; L. Bombelliet al., Spacetime as a causal set, Syracuse, preprint, March 1987; P. Menotti and A. Pelissetto,Phys. Rev. D 35:1194 (1987).Google Scholar
  9. 9.
    E. Fredkin and T. Toffoli,Int. J. Theor. Phys. 21:219 (1982); T. Toffoli,Int. J. Theor. Phys. 21:165 (1982); R. P. Feynman,Int. J. Theor. Phys. 21:467 (1982).Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • Gerard't Hooft
    • 1
  1. 1.Institute for Theoretical PhysicsTA UtrechtThe Netherlands

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