Open Systems & Information Dynamics

, Volume 7, Issue 2, pp 101–130 | Cite as

Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century

  • V. P. Belavkin


A history and drama of the development of quantum theory is outlined starting from the discovery of the Plank's constant exactly 100 years ago. It is shown that before the rise of quantum mechanics 75 years ago, the quantum theory had appeared first in the form of the statistics of quantum thermal noise and quantum spontaneous jumps which have never been explained by quantum mechanics. Moreover, the only reasonable probabilistic interpretation of quantum theory put forward by Max Born was in fact in irreconcilable contradiction with traditional mechanical reality and causality. This led to numerous quantum paradoxes; some of them, related to the great inventors of quantum theory such as Einstein and Schrödinger, are reconsidered in the paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for the resolution of this interpretational crisis by a divorce of the algebra of dynamical generators and a subalgebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule together with self-compatibility of measurements ensuring the consitency of histories is called the nondemolition principle. The application of these rules in the form of dynamical commutation relations leads to the derivation of the von Neumann projection postulate, as well as to more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution in the form of dynamical filtering equations to the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting from Schrödinger and Bohr. The simplest Markovian quantum stochastic model for time-continuous measurements involves a boundary-value problem in second quantization for input "offer" waves in one extra dimension, and a reduction of the algebra of "actual" observables to an Abelian subalgebra for the output waves.


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  1. 1.
    M. Planck, Scientific Autobiography, and Other Papers, Williams & Norgate LTD, London, 1949.Google Scholar
  2. 2.
    H. Kangro, Planck's Original Papers in Quantum Physics, Taylor & Francis, 1972.Google Scholar
  3. 3.
    W. Heisenberg, Z. Phys. 33, 879 (1925).Google Scholar
  4. 4.
    M. Born, W. Heisenberg, and P. Jordan, Z. Phys. 36, 557 (1926).Google Scholar
  5. 5.
    E. Schrödinger, Quantization as an Eigenvalue Problem, Ann. Phys. 79, 361 (1926).Google Scholar
  6. 6.
    E. Schrödinger, Abhandlundgen zur Wellenmechanik, J.A. Barth, Leipzig, 1926.Google Scholar
  7. 7.
    W. Moore, Schrödinger life and thought, Cambridge University Press, 1989.Google Scholar
  8. 8.
    W. Heisenberg, On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics, Z. Phys. 43, 172 (1925). English translation in: J.A. Wheeler and W. fi Zurek, eds., Quantum Theory and Measurement, Princeton University Press, 1983, pp. 6284.Google Scholar
  9. 9.
    J. von Neumann, Mathematische Grundlagen der Quantummechanik, Springer, Berlin, 1932.Google Scholar
  10. 10.
    A. Einstein, B. Podolski, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 47, 777 (1935).Google Scholar
  11. 11.
    N. Bohr, Phys. Rev. 48, 696 (1935).Google Scholar
  12. 12.
    E. Schrödinger, Naturwis. 23, 807, 823, 844 (1935).Google Scholar
  13. 13.
    G. Birkgofi and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics 37, 823 (1936).Google Scholar
  14. 14.
    G. C. Wick, A. S. Wightman, and E. P. Wigner, The Intrinsic Parity of Elementary Particles, Phys. Rev. 88, 101 (1952).Google Scholar
  15. 15.
    J. P. Jauch and C. Piron, Can Hidden Variables be Excluded in Quantum Mechanics?, Helv. Phys. Acta 36, 827 (1963).Google Scholar
  16. 16.
    V. P. Belavkin, Nondemolition Principle of Quantum Measurement Theory, Foundations of Physics 24, No. 5, 685 (1994).Google Scholar
  17. 17.
    R. L. Stratonovich and V. P. Belavkin, Dynamical Interpretation for the Quantum Measurement Projection Postulate, Int. J. of Theor. Phys. 35, No. 11, 2215 (1996).Google Scholar
  18. 18.
    G. Ludwig, J. Math. Phys. 4, 331 (1967), 9 1 (1968).Google Scholar
  19. 19.
    E. B. Davies and J. Lewis, Commun. Math. Phys. 17 239 (1970).Google Scholar
  20. 20.
    E. B. Ozawa, J. Math. Phys. 25, 79 (1984).Google Scholar
  21. 21.
    V. P. Belavkin, Quantum Stochastic Calculus and Quantum Nonlinear Filtering Journal of Multivariate Analysis 42, No. 2, 171 (1992).Google Scholar
  22. 22.
    G. Accardi, A. Frigerio, and J. T. Lewis, Publ. RIMS Kyoto Univ. 18, 97 (1982).Google Scholar
  23. 23.
    K. Itô, On a Formula Concerning Stochastic Difierentials, Nagoya Math. J. 3, 55 (1951).Google Scholar
  24. 24.
    V. P. Belavkin, On Quantum ItôAlgebras and Their Decompositions, Lett. Math. Phys. 45, 131 (1998).Google Scholar
  25. 25.
    R. L. Hudson and K. R. Parthasarathy, Quantum Itô's Formula and Stochastic Evolution, Comm. Math. Phys. 93, 301 (1984).Google Scholar
  26. 26.
    R. L. Stratonovich, Conditional Markov Processes and Their Applications to Optimal Control, Moscow State University, 1966.Google Scholar
  27. 27.
    V. P. Belavkin, Quantum Filtering of Markov Signals with White Quantum Noise, Radiotechnika and Electronika 25, 1445 (1980).Google Scholar
  28. 28.
    V. P. Belavkin, in: Modelling and Control of Systems, ed A. Blaquifiere, Lecture Notes in Control and Information Sciences 121 245, Springer, 1988.Google Scholar
  29. 29.
    V. P. Belavkin and P. Staszewski, Nondemolition Observation of a Free Quantum Particle, Phys. Rev. 45, 1347 (1992).Google Scholar
  30. 30.
    V. P. Belavkin, A New Wave Equation for a Continuous Nondemolition Measurement, Phys. Lett. A 140, 355 (1989).Google Scholar
  31. 31.
    V. P. Belavkin, A Continuous Counting Observation and Posterior Quantum Dynamics, J. Phys. A: Math. Gen. 22, L1109 (1989).Google Scholar
  32. 32.
    E. Schrödinger, Sitzberg Preus Akad. Wiss. Phys.-Math. Kl. 144 (1931).Google Scholar
  33. 33.
    J. G. Cramer, Rev. Mod. Phys. 58, 647 (1986).Google Scholar
  34. 34.
    V. P. Belavkin, On the Equivalence of Quantum Stochastics and a Dirac Boundary Value Problem, and an Inductive Stochastic Limit, Submitted for publication, January 2000.Google Scholar
  35. 35.
    M. Jammer, The Conceptual Development of Quantum Mechanics, McGraw-Hill, 1966.Google Scholar
  36. 36.
    A. Pais, Niels Bohr's Times, Clarendon Press-Oxford, 1991.Google Scholar
  37. 37.
    D. C. Cassidy, Uncertainty. Werner Heisenberg, W. H. Freeman, New-York, 1992.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • V. P. Belavkin
    • 1
  1. 1.Department of MathematicsThe University of NottinghamNottinghamUK

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