Comments on ‘The Formal Representation of Physical Quantities’
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In this paper, Prof, van Fraassen proposes a formal model for a class of physical theories which generalizes the formalisms used in classical and quantum mechanics. He shows how semantic analysis of his model, inspired by the work of Beth, leads to a study of the logic of the theory, here defined as the study of the relations of validity and entailment between the propositions, or elementary statements, of the theory and their logical compositions. He shows that this logic, or rather these logics, share many features with the quantum logics which have been the subject of much recent discussion. He is thus led to raise the question whether there is anything very useful about the quantum logic approach to quantum theory, since many of its features are not specific to that theory.
KeywordsHilbert Space Quantum Theory Physical Theory Quantum Logic Elementary Statement
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- 3.I should have added that one can get along with the traditional concepts of logic if one does not choose to apply the concept of elementary statement or proposition to such word groupings as “the position of the electron is x,” as Van Fraassen, following the quantum logicians, does. They may be regarded as incomplete phrases, in somewhat the same sense as “the viscosity of water is v” is incomplete without a specification of conditions of pressure, temperature, etc. of a definite sample of water. As Bohr always emphasized, the attribution of properties such as position to microsystems must always be in the context of a complete specification of the classically described arrangement in which these properties take on their meaning. For a full discussion of Bohr’s viewpoint, with a critique of the quantum logic approach, see the important paper by C. A. Hooker, ‘The Nature of Quantum Mechanical Reality: Einstein Versus Bohr’, in R. G. Colodny (ed.), Paradigms and Paradoxes, U. of Pittsburgh Press, 1972.Google Scholar
- 4.At last, one can recommend a textbook on this approach which has recently appeared: G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.Google Scholar
- 5.J. M. Cook in an article on ‘Banach Algebras and Asymptotic Mechanics’ outlines the algebraic approach to classical and quantum mechanics, as well as classical and quantum statistical mechanics. Application of Mathematics to Problems in Theoretical Physics (ed. by F. Lurcat), Gordon and Breach, New York, 1967.Google Scholar
- 6.Perhaps I exaggerate slightly; but Jauch in his Varenna lectures, says “When a theory is to be generalized or modified a postulational formulation is particularly useful since the empirical justification can be explicitly identified in a few well defined places and possible modifications can be more easily studied and surveyed” (p. 23). If only it were so! Later he says of the elementary propositions, interpreted as yes-no experiments, “These concepts are relatively easily interpreted in physical terms and therefore a good motivation for the axiomatics is available” (p. 24). J. M. Jauch, ‘Foundations of Quantum Theory’, in B. d’Espagnat (ed.), Foundations of Quantum Mechanics, Academic Press, New York, 1971.Google Scholar
- 7.Actually, this situation is a bit more complicated technically. For a review of this work, see Jauch’s book, J. M. Jauch, Foundations of Quantum Mechanics, Addison-Wesley.Google Scholar
- 8.For a survey of lattice theory, the article by Abbott is most useful: Trends in Lattice Theory (ed. by J. C. Abbott), Van Nostrand Reinhold, New York, 1970.Google Scholar
- 11.Stein has recently emphasized this point, following the work of Weyl, in the context of the interpretation of the quantum mechanical formalism: H. Stein, ‘On the Conceptual Structure of Quantum Mechanics’, in R. Colodny (ed.), Paradigms and Paradoxes Univ. of Pittsburgh Press , 1972.Google Scholar