Skip to main content
SpringerLink
Log in

Axiomatic Quantum Theory

  • Published:
Journal of Philosophical Logic Aims and scope Submit manuscript

Abstract

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra–ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  • Dirac, P. A. M. (1930): The Principles of Quantum Mechanics, Oxford University Press.

  • Fränkel, A. (1953): Abstract Set Theory, North-Holland, Amsterdam.

    Google Scholar 

  • Frege, G. (1879/1967): Begriffsschrift, in van Heijenoort (1967).

  • Hilbert, D. (1899): Grundlagen der Geometrie, Leipzig.

  • Jauch, J. M. (1968): Foundations of Quantum Mechanics, Addison-Wesley, Reading, MA.

    Google Scholar 

  • Kolmogorov, A. (1933/1950): Foundations of the Theory of Probability, Chelsea, New York.

    Google Scholar 

  • Mackey, G.W. (1963): Mathematical Foundations of Quantum Mechanics, Benjamin, New York.

    Google Scholar 

  • Peano, G. (1889/1967): The Principles of Arithmetic, in van Heijenoort (1967).

  • Suppes, P. (1960): Axiomatic Set Theory, van Nostrand, Princeton.

    Google Scholar 

  • van Heijenoort, J. (ed.) (1967): From Frege to Gödel, Harvard University Press, Cambridge, MA.

    Google Scholar 

  • von Neumann, J. (1932/1955): Mathematical Foundations of Quantum Mechanics, Princeton University Press.

  • Whitehead, A. N. and Russell, B. (1910–1913): Principia Mathematica, Cambridge University Press.

Download references

Author information

Authors and Affiliations

  1. Dept. of Philosophy, McGill University, Montreal, Canada

    Storrs McCall

Authors
  1. Storrs McCall
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

McCall, S. Axiomatic Quantum Theory. Journal of Philosophical Logic 30, 465–477 (2001). https://doi.org/10.1023/A:1012226116310

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012226116310

Search

Navigation