Abstract
The determination of the past and the future of a physical system are complementary aims of measurements. An optimal determination of the past of a system can be achieved by an informationally complete set of physical quantities. Such a set is always strongly noncommutative. An optimal determination of the future of a physical system can be obtained by a Boolean complete set of quantities. The two aims can be reconciled to a reasonable degree with using unsharp measurements.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Beltrametti, G. Cassinelli, and P. Lahti, “Unitary measurements of discrete quantities in quantum mechanics,”Rep. Ser.-Turku-FTL-R139 (1988).
E. Beltrametti and G. Cassinelli,The Logic of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1981).
J. Jauch,Foundations of Quantum Mechanics (Addison-Wesley, New York, 1968).
N. Hadjisavvas, “Properties of mixtures of non-orthogonal states,”Lett. Math. Phys. 5, 327–332 (1981).
S. T. Ali and H. D. Doebner, “On the equivalence of nonrelativistic quantum mechanics based upon sharp and fuzzy measurements,”J. Math. Phys. 17, 1105–1111 (1976).
E. Prugovecki, “Information-theoretical aspects of quantum measurement,”Int. J. Theor. Phys. 16, 321–331 (1977).
P. Busch, “Unsharp reality and the question of quantum systems,” inSymposium on the Foundations of Modern Physics 1987, Lahti, P., and Mittelstaedt, P., eds. (World Scientific, Singapore, 1987), pp. 105–126.
E. B. Davies,Quantum Theory of Open Systems (Academic Press, London, 1976).
V. S. Varadarajan,Geometry of Quantum Theory (Springer, New York, 1985).
S. Pulmannova and A. Dvurecenskij, “Uncertainty principle and joint distribution of observables,”Ann. Inst. H. Poincaré A 42, 253–265 (1985).
B. van Fraassen, “Foundations of probability: a modal frequency interpretation,” inProblems in the Foundations of Physics, G. Toraldo di Francia, ed. (North-Holland, Amsterdam, 1979), pp. 344–394.
W. Salmon,The Foundations of Scientific Inference (University of Pittsburgh Press, Pittsburgh, 1966).
M. Ozawa, “Quantum measuring processes of continuous observables,”J. Math. Phys. 25, 79–87 (1984).
G. Cassinelli and P. Lahti, “On the measurement statistics interpretation of quantum mechanics: possible values and possible measurement results of physical quantities,”Found. Phys. 19 (1989), to appear.
W. Pauli,General Principles of Wave Mechanics (Springer, Berlin, 1980); original German text in 1933.
B. Williams,Compton Scattering (McGraw-Hill, New York, 1977).
P. Lahti and K. Ylinen, “On total noncommutativity in quantum mechanics,”J. Math. Phys. 28, 2614–2617 (1987).
H. Reichenbach,Philosophic Foundations of Quantum Mechanics (University of California Press, Berkeley, 1944).
J. Corbett and C. Hurst, “Are wave functions uniquely determined by their position and momentum distributions?”J. Austral. Math. Soc. 20, 181–201 (1978).
C. F. von Weizsäcker, “Heisenberg's philosophy,” inSymposium on the Foundations of Modern Physics 1987, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1987), pp. 277–296.
J. Klauder and B.-S. Skagerstam,Coherent States (World Scientific, Singapore, 1985).
W. Thirring,Quantum Mechanics of Atoms and Molecules (Springer, New York, 1981).
M. Grabowski, “A-entropy for generalized observables,”Int. J. Theor. Phys. 17, 635–642 (1978).
M. Pavicic, “Complex Gaussians and the Pauli non-uniqueness,”Phys. Lett. A 122, 280–282 (1987).
P. Busch and P. Lahti, “On various joint measurements of position and momentum observables in quantum theory,”Phys. Rev. D 29, 1634–1646 (1984).
P. Busch, “Indeterminacy relations and simultaneous measurements in quantum theory,”Int. J. Theor. Phys. 24, 63–92 (1985).
J. Klauder and J. McKenna, “Continuous-representation theory.” IV,J. Math. Phys. 5, 878–896 (1964).
R. Werner, “Quantum harmonic analysis in phase space,”J. Math. Phys. 25, 1404–1411 (1984).
P. Busch, T. Schonbek, and F. Schroeck, “Quantum observables: compatibility versus commutativity and maximal information,”J. Math. Phys. 28, 2866–2872 (1987).
P. Busch, “Unsharp reality and joint measurements of spin observables,”Phys. Rev. D 33, 2253–2261 (1986).
C. F. von Weizsäcker, “Komplementarität und Logik,” inZum Weltbild der Physik, C. F. von Weizsäcker, ed. (S. Hirzel-Verlag, Stuttgart, 1976), 12th edn., pp. 281–331 (1955).
P. Busch, “Some realizable joint measurements of complementary quantities,”Found. Phys. 17, 905–937 (1987).
C. Piron,Foundations of Quantum Theory (Benjamin, New York, 1976).
P. Mittelstaedt,Quantum Logic (Reidel, Dordrecht, Holland, 1978).
P. Mittelstaedt,Sprache und Realität in der Modernen Physik (B.I.-Wissenschaftsverlag, Mannheim, 1986).
P. Halmos,A Hilbert Space Problem Book (Springer, Berlin, 1982).
N. Dunford and J. Schwartz,Linear Operators, Parts 1 and 2 (Interscience, New York, 1958, 1963).
T. Kato,Perturbation Theory of Linear Operators (Springer, Berlin, 1966).
A. Lenard, “The numerical range of a pair of projections,”J. Funct. Anal. 10, 410–423 (1972).
P. Busch and P. Lahti, “To what extent do position and momentum commute?,”Phys. Lett. A 115, 259–264 (1986).
P. Lahti, “States of minimal uncertainty and maximal information for quantum position and momentum observables in quantum theory,”Rep. Math. Phys. 23, 289–297 (1986).
P. Busch and P. Lahti, “Minimal uncertainty and maximal information for quantum position and momentum,”J. Phys. A: Math. Gen. 20, 899–906 (1987).
J. Maczynski, “A compactness theorem in generalized probability theory,”Bull. Polon. Acad. Sci., Ser. Sci. Math. (in print).
K. Ylinen, “Position and momentum as spectral measures on locally compact abelian groups,” inSymposium on the Foundations of Modern Physics 1987, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1987), pp. 505–513.
P. Busch, “On joint lower bounds of position and momentum observables in quantum mechanics”,J. Math. Phys. 25, 1794–1797 (1984).
P. Busch and P. Lahti, “A note on quantum theory, complementarity, and uncertainty,”Phil. Sci. 52, 64–77 (1985).
P. Lahti, “Complementarity and uncertainty: some measurement-theoretical and information-theoretical aspects,” inSymposium on the Foundations of Modern Physics 1987, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1987), pp. 181–208.
N. Bohr, “The quantum postulate and the recent development of atomic theory,”Nature 121, 580–590 (1928).
N. Bohr, “Quantum physics and philosophy—causality and complementarity,” inEssays 1958–1962 on Atomic Physics and Human Knowledge, Richard Clay, ed. (Bungay, Suffolk, 1963).
P. Mittelstaedt, “Language and reality in quantum physics,” inSymposium on the Foundations of Modern Physics 1987, P. Lahti and P. Mittelstaedt, eds. (World Scientific, Singapore, 1987), pp. 229–250.
P. Busch and P. Lahti, “Some remarks on unsharp quantum measurements, quantum non-demolition, and all that,”Ann. Physik (in print).
P. Busch and F. Schroeck, “On the reality of spin and helicity,”Found. Phys. 19 (1989), to appear.
A. Barchielli, L. Lanz, and G. Prosperi, “Statistics of continuous trajectories in quantum mechanics: operation-valued stochastic processes,”Found. Phys. 13, 779–812 (1983).
Author information
Authors and Affiliations
Additional information
This work was partly supported by the Bundesministerium für Forschung und Technologie, Bonn, the Research Institute for Theoretical Physics, Helsinki, and the University of Turku Foundation, Turku.
Rights and permissions
About this article
Cite this article
Busch, P., Lahti, P.J. The determination of the past and the future of a physical system in quantum mechanics. Found Phys 19, 633–678 (1989). https://doi.org/10.1007/BF00731904
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00731904