Abstract
We provide a mathematical description of quantum measurements with a finite exactness. The exactness of a quantum measurement is used as a new metric on the space of quantum states. This metric differs very much from the standard Euclidean metric. This is the so-called ultrametric. We show that a finite exactness of a quantum measurement cannot he described by real numbers. Therefore, we must change the basic number field. There exist nonequivalent ultrametric Hilbert space representations already in the finite-dimensional case (compare with ideas of L. de Broglie). Different preparation procedures could generate nonequivalent representations. The Heisenberg uncertainty principle is a consequence of properties of a preparation procedure. The uncertainty principle “time-energy” is a consequence of the Schrödinger dynamics.
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References
L. E. Ballentine,Rev. Mod. Phys. 42 (4), 358–381 (1970); L. E. Ballentine,Quantum Mechanics (Prentice Hall, Englewood Cliffs, 1989).
N. Bohr,Phys. Rev. 48, 696 (1935).
P. A. M. Dirac.The Principles of Quantum Mechanics (Clarendon, Oxford, 1932); J. von Neumann,Matematische Grundlagen der Quantenmechanik (Berlin, 1932).
L. de Broglie,The Current Interpretation of Wave Mechanics: A critical Study (Elsevier, Amsterdam, 1964); L. de Broglie.La physique quantique restera-t-elle indéterministe?, (Gauthier-Villars, Paris, 1953).
K. Mahler,Introduction to p-adie Numbers and their Functions (Cambridge University Press, Cambridge, 1973); W. Schikhof,Ultrametric Calculus (Cambridge University Press, Cambridge. 1984); Z. I. Borevich and I. R. Schafarevich,Number Theory Academic, New York, 1966).
V. S. Vladimirov, I. V. Volovich, E. I. Zelenov,P-adic Numbers in Mathematical Physics (World Scientific, Singapore, 1993).
A. Yu. Khrennikov,P-adic-Valued Distributions in Mathematical Physics (Kluwer Academic, Dordrecht, 1994).
I. V. Volovich, “Number theory as the ultimate physical theory.” preprint TH.4781/87(1987);Class. Quantum Gravit. 4, L83-L87 (1987).
P. G. O. Freund and M. Olson.Phys. Lett. B 199, 186 (1987); P. G. O. Freund, M. Olson, and E. Witten,Phys. Lett. B 199, 191 (1987); P. H. Frampton and Y. Okada,Phys. Rev. Lett. 60, 484 (1988).
A. Yu. Khrennikov,Russ. Math. Surr. 45, 87 (1990);J. Math. Phys. 32, 932. (1991);Theor. Math. Phys. 83, 124 (1990);Sov. Phys. Dokl. 35, 867 (1990).
R. Cianci and A. Yu. Khrennikov,Phys. Let. B 328, 109–112 (1994).
H. S. Snyder,Phys. Rev. 7, 38 (1947); A. Shild.Phys. Rev. 73, 414 (1948); E. J. Hellund and K Tanaka,Phys. Rev. 94, 192 (1954); P. Gibbs. “The small-scale structure of space-time: a bibliographical review.” preprint HEP-TH/9506171.
A. N. Kolmogoroff,Grundbegriffe der Wahrscheinlichkeitslehre. 1933. English translation by N. Morrison (Chelsea. New York, 1950).
A. Yu. Khrennikov,Phys. Lett. A 200, 219–223 (1995):Physica A 215, 557–587 (1995);Int. J. Theor. Phys. 34, 2423 2433 (1995);J. Math. Phys. 36, 6625–6632 (1995).
W. Muckenheim,Phys. Rep. 133, 338–401 (1986).
E. Prugovecki.Can. J. Phys. 45, 2173–2219 (1967);Found. Phys. 3, 3–18 (1973).
H. Margenau and J. L. Park,Found. Phys. 3, 19–29 (1973).
A. Yu. Khrennikov,Physics-Doklady. to be published.
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On leave from Moscow Institute of Electronic Engineering.
This research was supported by the Alexander von Humboldt-Stiftung.
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Khrennikov, A. The ultrametric Hilbert-space description of quantum measurements with a finite exactness. Found Phys 26, 1033–1054 (1996). https://doi.org/10.1007/BF02061402
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DOI: https://doi.org/10.1007/BF02061402