Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century
 V. P. Belavkin
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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 selfcompatibility 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 timecontinuous measurements involves a boundaryvalue 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.
 M. Planck, Scientific Autobiography, and Other Papers, Williams & Norgate LTD, London, 1949.
 H. Kangro, Planck's Original Papers in Quantum Physics, Taylor & Francis, 1972.
 W. Heisenberg, Z. Phys. 33, 879 (1925).
 M. Born, W. Heisenberg, and P. Jordan, Z. Phys. 36, 557 (1926).
 E. Schrödinger, Quantization as an Eigenvalue Problem, Ann. Phys. 79, 361 (1926).
 E. Schrödinger, Abhandlundgen zur Wellenmechanik, J.A. Barth, Leipzig, 1926.
 W. Moore, Schrödinger life and thought, Cambridge University Press, 1989.
 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.
 J. von Neumann, Mathematische Grundlagen der Quantummechanik, Springer, Berlin, 1932.
 A. Einstein, B. Podolski, and N. Rosen, Can QuantumMechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 47, 777 (1935).
 N. Bohr, Phys. Rev. 48, 696 (1935).
 E. Schrödinger, Naturwis. 23, 807, 823, 844 (1935).
 G. Birkgofi and J. von Neumann, The Logic of Quantum Mechanics, Annals of Mathematics 37, 823 (1936).
 G. C. Wick, A. S. Wightman, and E. P. Wigner, The Intrinsic Parity of Elementary Particles, Phys. Rev. 88, 101 (1952).
 J. P. Jauch and C. Piron, Can Hidden Variables be Excluded in Quantum Mechanics?, Helv. Phys. Acta 36, 827 (1963).
 V. P. Belavkin, Nondemolition Principle of Quantum Measurement Theory, Foundations of Physics 24, No. 5, 685 (1994).
 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).
 G. Ludwig, J. Math. Phys. 4, 331 (1967), 9 1 (1968).
 E. B. Davies and J. Lewis, Commun. Math. Phys. 17 239 (1970).
 E. B. Ozawa, J. Math. Phys. 25, 79 (1984).
 V. P. Belavkin, Quantum Stochastic Calculus and Quantum Nonlinear Filtering Journal of Multivariate Analysis 42, No. 2, 171 (1992).
 G. Accardi, A. Frigerio, and J. T. Lewis, Publ. RIMS Kyoto Univ. 18, 97 (1982).
 K. Itô, On a Formula Concerning Stochastic Difierentials, Nagoya Math. J. 3, 55 (1951).
 V. P. Belavkin, On Quantum ItôAlgebras and Their Decompositions, Lett. Math. Phys. 45, 131 (1998).
 R. L. Hudson and K. R. Parthasarathy, Quantum Itô's Formula and Stochastic Evolution, Comm. Math. Phys. 93, 301 (1984).
 R. L. Stratonovich, Conditional Markov Processes and Their Applications to Optimal Control, Moscow State University, 1966.
 V. P. Belavkin, Quantum Filtering of Markov Signals with White Quantum Noise, Radiotechnika and Electronika 25, 1445 (1980).
 V. P. Belavkin, in: Modelling and Control of Systems, ed A. Blaquifiere, Lecture Notes in Control and Information Sciences 121 245, Springer, 1988.
 V. P. Belavkin and P. Staszewski, Nondemolition Observation of a Free Quantum Particle, Phys. Rev. 45, 1347 (1992).
 V. P. Belavkin, A New Wave Equation for a Continuous Nondemolition Measurement, Phys. Lett. A 140, 355 (1989).
 V. P. Belavkin, A Continuous Counting Observation and Posterior Quantum Dynamics, J. Phys. A: Math. Gen. 22, L1109 (1989).
 E. Schrödinger, Sitzberg Preus Akad. Wiss. Phys.Math. Kl. 144 (1931).
 J. G. Cramer, Rev. Mod. Phys. 58, 647 (1986).
 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.
 M. Jammer, The Conceptual Development of Quantum Mechanics, McGrawHill, 1966.
 A. Pais, Niels Bohr's Times, Clarendon PressOxford, 1991.
 D. C. Cassidy, Uncertainty. Werner Heisenberg, W. H. Freeman, NewYork, 1992.
 Title
 Dynamical Solution to the Quantum Measurement Problem, Causality, and Paradoxes of the Quantum Century
 Journal

Open Systems & Information Dynamics
Volume 7, Issue 2 , pp 101130
 Cover Date
 20000601
 DOI
 10.1023/A:1009663822827
 Print ISSN
 12301612
 Online ISSN
 15731324
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 V. P. Belavkin ^{(1)}
 Author Affiliations

 1. Department of Mathematics, The University of Nottingham, NG7 2RD, Nottingham, UK