Abstract
We consider the following statistical problem: suppose we have a light beam and a collection of semi-transparent windows which can be placed in the way of the beam. Assume that we are colour blind and we do not possess any colour sensitive detector. The question is, whether by only measurements of the decrease in the beam intensity in various sequences of windows we can recognize which among our windows are light beam filters absorbing photons according to certain definite rules?
To answer this question a definition of physical systems is formulated independent of “quantum logic” and lattice theory, and a new idea of quantization is proposed. An operational definition of filters is given: in the framework of this definition certain nonorthodox classes of filters are admissible with a geometry incompatible to that assumed in orthodox quantum mechanics. This leads to an extension of the existing quantum mechanical structure generalizing the schemes proposed by Ludwig [10] and the present author [13]. In the resulting theory, the quantum world of orthodox quantum mechanics is not the only possible but is a special member of a vast family of “quantum worlds” mathematically admissible. An approximate classification of these worlds is given, and their possible relation to the quantization of non-linear fields is discussed. It turns out to be obvious that the convex set theory has a similar significance for quantum physics as the Riemannian geometry for space-time physics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bergman, P. G.: The general theory of relativity — case study in the unfolding of of new physical concepts. Problems in the Philosophy of Science, p. 249. Amsterdam: North Holland Publ. Co. 1968.
Birkhoff, G., and J. von Neumann: Ann. Math.37, 823 (1936).
Dähn, G.: Attempt of an axiomatic foundation of quantum mechanics and more general theories (IV). Commun. Math. Phys.9, 192 (1968).
Finkelstein, D.: The logic of quantum physics, N. Y. Acad. Sci.1963, 621;
-- The physics of logic. Preprint IC/68/35 (Triest, 1968).
Finkelstein, D., J. M. Jauch, S. Schimonovich, and D. Speiser: Foundations of quaternionic quantum mechanics. J. Math. Phys.3, 207 (1962); — Some physical consequences of generalQ-covariance. J. Math. Phys.4, 788 (1963).
Gunson, J.: Structure of quantum mechanics. Commun. Math. Phys.6, 262 (1967).
Haag, R.: Lectures on axiomatic quantum mechanics. 1960 (unpublished).
Haag, R., and D. Kastler: An algebraic approach to quantum field theory. J. Math. Phys.5, 848 (1964).
Jordan, P.: Über das Verhältnis der Theorie der Elementarlänge zur Quantentheorie. Commun. Math. Phys.9, 279 (1967).
Ludwig, G.: Versuch einer axiomatischen Grundlegung der Quantenmechanik und allgemeinerer physikalischer Theorien. Z. Physik181, 233 (1964);
—— Attempt of an axiomatic foundation of quantum mechanics and more general theories (II). Commun. Math. Phys.4, 331 (1967); (III) Commun. Math. Phys.9, 1 (1968).
Landsberg, P. T.: A proof of Temple's laws of transition. Ann. Phys.6, 14 (1954).
Mackey, G. W.: The mathematics foundations of quantum mechanics. New York: W. A. Benjamin, Inc. 1963.
Mielnik, B.: Geometry of quantum states. Commun. Math. Phys.9, 55 (1968).
Piron, C.: Axiomatique quantique. Helv. Phys. Acta37, 439 (1964).
Plymen, R. J.: A modification of Piron's axioms. Helv. Phys. Acta41, 69 (1968).
Pool, J. C. T.: Baer *-semigroup and the logic of quantum mechanics. Commun. Math. Phys.9, 118 (1968);
—— Semimodularity and the logic of quantum mechanics. Commun. Math. Phys.9, 212 (1968).
Śniatycki, J.: On the geometric structure of classical field theory in Lagrangian formulation. Preprint, Dept. of Appl. Math. and Comp. Sc. Sheffield University (1969).
Trautman, A.: Noether equations and conservation laws. Commun. Math. Phys.6, 248 (1967).
Temple, G.: The general principles of quantum theory. London: Methuen 1934.
Tulczyjew, W.: Seminar on phase space theory. Warsaw, 1968 (unpublished).
Woronowicz, St.: Lectures on the axiomatic quantum field theory. Warsaw, 1968 (to be published).
Zierler, N., and M. Schlessinger: Boolean embeddings of orthomodular sets and quantum logic. Duke Math. J.32, 251 (1965).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mielnik, B. Theory of filters. Commun.Math. Phys. 15, 1–46 (1969). https://doi.org/10.1007/BF01645423
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01645423