Skip to main content
Log in

The mathematical foundations of quantum mechanics

  • Published:
Synthese Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

References

  1. Aronson, J. L. and Edwards, D. A., ‘Quantum Logics: I–IV’ (mimeographed notes).

  2. Bell, J. S., ‘On the Problem of Hidden Variables in Quantum Mechanics,’ Reviews of Modern Physics 38 (1966), 447.

    Google Scholar 

  3. Birkhoff, G. and MacLane, S., A Survey of Modern Algebra, Macmillan, New York, 1964.

    Google Scholar 

  4. Birkhoff, G. and von Neumann, J., ‘The Logic of Quantum Mechanics,’ Annals of Mathematics 37 (1936), 823.

    Google Scholar 

  5. Bohm, D., ‘A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden Variables I,’ Physical Review 85 (1972), 166.

    Google Scholar 

  6. Bohm, D., ‘A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden Variables II,’ Physical Review 85, (1952), 180.

    Google Scholar 

  7. Bohm, D., ‘Quantum Theory as the Indication of a New Order in Physics,’ Foundations of Physics.

  8. Bohr, N., ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’, Physical Review 48 (1935), 696.

    Google Scholar 

  9. Bohr, N., Kramers, H. A., and Slater, J. C., ‘The Quantum Theory of Radiation,’ Philosophical Magazine 47 (1924), 785.

    Google Scholar 

  10. Bohr, N. and Rosenfeld, L., ‘Field and Charge Measurement in Quantum Electrodynamics,’ Physical Review 78 (1950), 794.

    Google Scholar 

  11. Bopp, F., ‘The Principles of the Statistical Equations of Motion in Quantum Theory,’ in S. Körner (ed.), Observation and Interpretation in the Philosophy of Physics, Dover Publication, New York, 1962.

    Google Scholar 

  12. Bub, J., ‘On the Completeness of Quantum Mechanics,’ in C. A. Hooker (ed.), Contemporary Research in the Foundations and Philosophy of Quantum Theory, Reidel, Dordrecht, 1973.

    Google Scholar 

  13. Bub, J., The Interpretation of Quantum Mechanics, D. Reidel Pub. Co. Dordrecht, 1974.

    Google Scholar 

  14. Colodny, R. G., ‘The Philosophical Challenge of the Quantum Domain,’ in R. G. Colodny (ed.), Paradigms and Paradoxes, University of Pittsburgh Press, Pittsburgh, 1972.

    Google Scholar 

  15. Dirac, P. A. M., The Principles of Quantum Mechanics, Oxford University Press, London, 1958.

    Google Scholar 

  16. Domotor, Z., ‘The Probability Structure of Quantum Mechanical Systems,’ Synthese 29 (1974), 155.

    Google Scholar 

  17. Einstein, A., ‘Remarks to the Essays Appearing in This Collective Volume,’ in P. A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist, Harper and Row, New York, 1959.

    Google Scholar 

  18. Einstein, A., Podolsky, B., and Rosen, N., ‘Can Quantum-Mechanical Description of Physical Reality be Considered Complete?’, Physical Review 47 (1935), 777.

    Google Scholar 

  19. Emch, G. G., Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.

    Google Scholar 

  20. Federer, H., Geometrid Measure Theory, Springer-Verlag, New York, 1969.

    Google Scholar 

  21. Feynman, R. P., ‘Concept of Probability in Quantum Mechanics,’ Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1951, pp. 533–541.

  22. Feynman, R. P., ‘Space-time Approach to Non-Relativistic Quantum Mechanics, Reviews of Modern Physics 20 (1948), 367.

    Google Scholar 

  23. Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics, Addison-Wesley, 1963.

  24. Finkelstein, D., ‘The Physics of Logic’, in R. G. Colodny (ed.), Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, University of Pittsburgh Press, Pittsburgh, 1972.

    Google Scholar 

  25. Foulis, D. J. and Randall, C. H., ‘Empirical Logic and Quantum Mechanics,’ Synthese 29 (1974), 81.

    Google Scholar 

  26. Freistadt, G., ‘The Causal Formulation of Quantum Mechanics of Particles,’ Supplemento del Nouvo Cimento 5 (1957), 1.

    Google Scholar 

  27. Gleason, A. M., ‘Measures on the Closed Subspaces of a Hilbert Space,’ Journal of Mathematics and Mechanics 6 (1957), 567.

    Google Scholar 

  28. Greechie, R. J., ‘An Orthomodular PO set with a Full Set of States Not Embeddable in Hilbert Space,’ Caribbean Journal of Science and Mathematics 1 (1969), 15.

    Google Scholar 

  29. Greechie, R. J., ‘Some Results From the Combinatorial Approach to Quantum Logic,’ Synthese 29 (1974), 113.

    Google Scholar 

  30. Greechie, R. J. and Gudder, S., ‘Quantum Logics,’ in C. A. Hooker (ed.), Contemporary Research in the Foundations and Philosophy of Quantum Theory, Reidel, Dordrecht, 1973.

    Google Scholar 

  31. Grothendieck, A., ‘Technique de descente et théorèmes d'existence en géométrie algèbraique I–IV’, Seminar Bourbaki, Exp. 190, 195, 212, 221, 1959–60, 1960–61.

  32. Gudder, S., ‘Hidden Variables in Quantum Mechanics Reconsidered,’ Review of Modern Physics 40 (1958), 229.

    Google Scholar 

  33. Halmos, P.R., Measure Theory, D. van Nostrand Company, Princeton, 1950.

    Google Scholar 

  34. Heisenberg, W., Physics and Philosophy, Harper and Row, New York, 1966.

    Google Scholar 

  35. Hilbert, D., Foundations of Geometry, The Open Court Publishing Co., La Salle, 1965.

    Google Scholar 

  36. Jammer, M., The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York, 1966.

    Google Scholar 

  37. Jammer, M., The Philosophy of Quantum Mechanics, Wiley-Interscience, New York, 1974.

    Google Scholar 

  38. Jauch, J., Foundations of Quantum Theory, Addison-Wesley, New York, 1968.

    Google Scholar 

  39. Jordan, P., von Neumann, J., and Wigner, E. P., ‘On an Algebraic Generalization of the Quantum Mechanical Formalism,’ Annals of Mathematics 35 (1934), 29.

    Google Scholar 

  40. Kadison, R., ‘Transformations of States in Operator Theory and Dynamics,’ Topology 3 (1965), 177–198.

    Google Scholar 

  41. Kochen, S., and Specker, E., ‘The Problem of Hidden Variables in Quantum Mechanics, Journal of Mathematics and Mechanics 17 (1967), 59.

    Google Scholar 

  42. Kuhn, T., The Structure of Scientific Revolutions, University of Chicago Press, Chicago, 1962.

    Google Scholar 

  43. Lawyere, F., ‘The Category of Categories as a Foundation for Mathematics, in S. Eilenberg et al. (eds.), Proc. of the Conf. on Categorical Algebra, La Jolla, 1965, Springer-Verlag, New York, 1966.

    Google Scholar 

  44. Ludwig, H., ‘The Measuring Process and an Axiomatic Foundation of Quantum Mechanics,’ in Foundations of Quantum Mechanics, Il Corso, Academic Press Inc., New York, 1971.

    Google Scholar 

  45. Mackey, G. W., The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, Inc., New York, 1963.

    Google Scholar 

  46. Mackey, W. G., ‘Infinite Dimensional Group Representations,’ Bulletin of the American Mathematical Society 69 (1963), 628.

    Google Scholar 

  47. Mackey, G. W., Induced Representations of Groups and Quantum Mechanics, W. A. Benjamin, Inc., New York, 1963.

    Google Scholar 

  48. Mackey, G. W., ‘Ergodic Actions of the Euclidean Group and the Canonical Commutation Relations,’ in C. Hagen, G. Garalnik, and V. Mathur (eds.), Proceedings of the 1967 International Conference on Particles and Fields, Interscience Pub., New York, 1967.

    Google Scholar 

  49. Mackey, G. W., ‘Ergodic Theory and Its Significance for Statistical Mechanics and Probability Theory,’ Advances in Mathematics 12 (1974), 178–268.

    Google Scholar 

  50. MacLane, S., ‘Categorical Algebra’, Bull. Amer. Math. Soc. (1965).

  51. Maclane, S. and Birkhoff, G., Algebra, Macmillan, New York, 1970.

    Google Scholar 

  52. Mandelbrot, B., Fractals: Form, Chance and Dimension, W. H. Freeman and Co., San Francisco, 1977.

    Google Scholar 

  53. Matheron, G., Random Sets and Integral Geometry, John Wiley and Sons, Ltd., New York, 1975.

    Google Scholar 

  54. Messiah, A., Quantum Mechanics, North-Holland Publishing Co., Amsterdam, 1964.

    Google Scholar 

  55. Murray, F. J. and von Neumann, J., ‘On Rings of Operators,’ Annals of Mathematics 37 (1936), 116.

    Google Scholar 

  56. Nelson, E., ‘Derivation of the Schrödinger Equation from Newtonian Mechanics,’ Physical Review 150B (1966), 1079.

    Google Scholar 

  57. Newton, T. D. and Wigner, E. P., ‘Localized States for Elementary Systems,’ Reviews of Modern Physics 21 (1949), 400.

    Google Scholar 

  58. Putnam, H., ‘Is Logic Empirical?’ in R. S. Cohen (ed.), Proceedings of the Boston Colloquium in the Philosophy of Science, 1966–1968, Humanities Press, New York, 1969.

    Google Scholar 

  59. Reid, C., Hilbert, Springer-Verlag, Heidelberg, New York, 1972.

    Google Scholar 

  60. Sachs, M., ‘A New Approach to the Theory of Fundamental Processes,’ British Journal for the Philosophy of Science 15 (1964), 213.

    Google Scholar 

  61. Schilpp, P. A. (ed.), Albert Einstein: Philosopher-Scientist. 2 volumes. Harper and Bros., Harper Torchbooks, New York, 1959. Now published by The Open Court Publishing Co., La Salle, Ill.

    Google Scholar 

  62. Schweber, S., An Introduction to Relativistic Quantum Field Theory, Row, Peterson and Company, Evanston and New York, 1961.

    Google Scholar 

  63. Segal, I.E.,: Postulates for General Quantum Mechanics,’ Annals of Mathematics 48 (1947), 930.

    Google Scholar 

  64. Segal, I. E. Mathematical Problems of Relativistic Physics, Lectures in Applied Mathematics, American Mathematical Society, Providence, R. I., 1963.

  65. Segal, I. E., ‘Algebraic Integration Theory’, Bull. Amer. Math. Soc. (1965).

  66. Simon, B., The Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, 1974.

    Google Scholar 

  67. Sneed, J. D., ‘Quantum Mechanics and Classical Probability Theory’, Synthese 21 (1970), 34.

    Google Scholar 

  68. Stein, H., ‘On the Conceptual Structure of Quantum Mechanics,’ in R. G. Colodny (ed.), Paradigms and Paradoxes, University of Pittsburg Press, Pittsburgh, 1972.

    Google Scholar 

  69. Streater, R. F., Mathematics of Contemporary Physics, Academic Press, New York, 1972.

    Google Scholar 

  70. Streater, R. F. and Wightman, A. S., PCT, Spin and Statistics and All That, W. A. Benjamin, Inc., New York, 1964.

    Google Scholar 

  71. Varadarajan, V. S., ‘Probability in Physics and a Theorem on Simultaneous Observability,’ Communications in Pure and Applied Mathematics 15 (1962), 189.

    Google Scholar 

  72. Varadarajan, V. S., Geometry of Quantum Mechanics, D. van Nostrand Co., Princeton, 1968.

    Google Scholar 

  73. Velo, G. and Wightman, A., Constructive Quantum Field Theory, Springer Lectures Notes in Physics, Vol. 25, New York, 1973.

  74. von Neumann, J., Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, 1955.

    Google Scholar 

  75. Weyl, H., The Theory of Groups and Quantum Mechanics, Dover Publications, Inc., New York, 1931.

    Google Scholar 

  76. Weyl, H., The Classical Groups, Princeton University Press, Princeton, 1946.

    Google Scholar 

  77. Weyl, H., Space, Time, Matter, Dover Publications, Inc., New York, 1950.

    Google Scholar 

  78. Whittaker, E. A., A History of the Theories of Ether and Electricity, 2 volumes. Harper and Row, Harper Torchbooks, New York, 1960.

    Google Scholar 

  79. Wick, G. C., Wightman, A. S. and Wigner, E. P., ‘The Intrinsic Parity of Elementary Particles,’ The Physical Review 88 (1952), 101.

    Google Scholar 

  80. Wightman, A. S., ‘On the Localizability of Quantum Mechanical Systems,’ Reviews of Modern Physics 34 (1962), 845.

    Google Scholar 

  81. Wigner, E. P., ‘Unitary Representations of the Inhomogeneous Lorentz Group,’ Annals of Mathematics 40, (1939), 149.

    Google Scholar 

  82. Simms, D. and Woodhouse, N., Lectures on Geometric Quantization, Springer Lectures Notes in Physics, vol. 53, New York, 1976.

  83. Bleuler, K. and Teetz, A. Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Mathematics, vol. 570, New York, 1977.

  84. Guillemin, V. and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, American Math. Soc., Providence, R. I., 1977.

    Google Scholar 

  85. Brey, J. and Jones, B., Critical Phenomena, Springer Lecture Notes in Physics, vol. 54, New York, 1976.

  86. Nelson, E., Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, 1967.

    Google Scholar 

  87. Wightman, A. S., ‘Hilbert's Sixth Problem: Mathematical Treatment of Axioms of Physics’, in Mathematical Developments Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics, Vol. XXVIII, Part 1, American Mathematical Society, Providence, Rhode Island, 1976.

    Google Scholar 

  88. P. D. Finch, ‘On the Structure of Quantum Logic’, Journal of Symbolic Logic 34 (1969), 275–282.

    Google Scholar 

  89. V. Bargmann, Group Representations in Mathematics and Physics, Springer Lecture Notes in Physics, vol. 6, New York, 1970.

  90. E. Wigner, Group Theory, Academic Press, New York, 1959.

    Google Scholar 

  91. R. Jancel, Foundations of Classical and Quantum Statistical Mechanics, Pergamon Press, New York, 1969.

    Google Scholar 

  92. A. Volkel, Fields, Particles and Currents, Springer Lecture Notes in Physics, vol. 66, New York, 1977.

  93. H. Araki, ‘Local Quantum Theory I’ in R. Jost (ed.), Local Quantum Theory, Proceedings of the Int. School of Phys. Enrico Fermi, Course 45, Academic Press, New York, 1969, p. 65–96.

    Google Scholar 

  94. R. Haag and D. Kastler, ‘An Algebraic Approach to Quantum Field Theory’, Journal of Mathematical Physics 5 (1969), 848–61.

    Google Scholar 

  95. Ruelle, D., Statistical Mechanics Rigorous Results, W. A. Benjamin, New York, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Edwards, D.A. The mathematical foundations of quantum mechanics. Synthese 42, 1–70 (1979). https://doi.org/10.1007/BF00413704

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00413704

Keywords

Navigation