Abstract
The aim of this paper is to reconcile the two modes of description of macrosystems, i.e., to remove certain inconsistencies between the classical phenomenological and the quantum-theoretical descriptions of a macrosystem. Starting from Ludwig's formulation of a general framework for classical theories and his ansatz for a compatibility condition between the quantum theoretical and the classical mode of description for a macrosystem, we try to make clear what the “classical content” of many-body quantum theory really is. It is shown that this classical content may be described by a certain “observable,” i.e., an operator-valued measure over the Borel sets of the classical trajectory space. There exists a reduced time evolution within this classical content which has the structure of a true semigroup in case of an irreversible classical time evolution.
Similar content being viewed by others
References
K. E. Hellwig and K. Kraus,Comm. Math. Phys. 11, 214 (1969);16, 142 (1970); K. Kraus,Ann. Phys. (N.Y.) 64, 311 (1971).
J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton, 1955).
V. M. Maksimov,J. Theor. Math. Phys. 20, 632 (1974).
G. Ludwig,Makroskopische Systeme und Quantenmechanik (Notes in Mathematical Physics, Nr. 5, Universität Marburg, 1972).
A. Hartkämper and H. Ncumann,Foundations of Quantum Mechanics and Ordered Linear Spaces (Lecture Notes in Physics 29, Springer-Verlag, Berlin, 1973).
R. Werner, Über eine statistische Rahmentheorie makroskopischer Systeme, Diploma-Thesis, Universität Marburg (1976).
G. Ludwig,Deutung des Begriffs “Physikalische Theorie” und axiomatische Grundlegung der Hilbert-Raumstruktur der Quantenmechanik durch Hauptsätze des Messens (Lecture Notes in Physics 4, Springer-Verlag, Berlin, 1970).
G. Ludwig,Einführung in die Grundlagen der Theoretischen Physik 3 (Vieweg-Verlag, Braunschweig, 1976).
G. Ludwig, A theoretical description of single microsystems, inThe Uncertainty Principle and Foundations of Quantum Mechanics, W. C. Price and S. S. Chissick, eds. (Wiley, New York, 1977).
H. Neumann,Comm. Math. Phys. 23, 100 (1971).
R. Haag and D. Kastler,J. Math. Phys. 5, 848 (1963).
R. V. Kadison,Topology 3, Suppl. 2 (1965).
R. Haag, R. V. Kadison, and D. Kastler,Comm. Math. Phys. 16, 81 (1970).
J. F. Arnes,J. Funct. Anal. 5, 14 (1970).
N. Dinculeanu,Vector Measures (Pergamon Press, 1967).
I. Batt and E. I. Berg,J. Funct. Anal. 4, 215 (1969).
I. Dixmier,Les Algebras d'Opérateurs dans l'Espace Hilbertien, (Gauthier-Villars, 1957).
S. Sakai,C*-Algebras and W*-Algebras (Springer-Verlag, Berlin, 1971).
N. Dunford and I. T. Schwartz,Linear Operators I (Interscience, New York, 1958).
E. B. Davies and U. T. Lewis,Comm. Math. Phys. 17, 239 (1970).
L. P. Horwitz, I. A. LaVita, and I. P. Marchand,J. Math. Phys. 12, 2537 (1971).
C. George, I. Prigogine, and L. Rosenfeld,Det Kgl. Danske Vidensk. Selskat Mat. Fys. Medd. 38, 12 (1972).
L. Lanz, L. A. Lugiato, and G. Ramella,Physica 54, 1 (1971).
Author information
Authors and Affiliations
Additional information
Dedicated to Prof. G. Ludwig on the occasion of his sixtieth birthday.
Rights and permissions
About this article
Cite this article
Melsheimer, O. On the classical content of many-body quantum mechanics. Found Phys 9, 193–215 (1979). https://doi.org/10.1007/BF00715179
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00715179