Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 009, 40 pages      arXiv:1401.6507      https://doi.org/10.3842/SIGMA.2014.009
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

The Heisenberg Relation - Mathematical Formulations

Richard V. Kadison a and Zhe Liu b
a) Department of Mathematics, University of Pennsylvania, USA
b) Department of Mathematics, University of Central Florida, USA

Received July 26, 2013, in final form January 18, 2014; Published online January 25, 2014

Abstract
We study some of the possibilities for formulating the Heisenberg relation of quantum mechanics in mathematical terms. In particular, we examine the framework discussed by Murray and von Neumann, the family (algebra) of operators affiliated with a finite factor (of infinite linear dimension).

Key words: Heisenberg relation; unbounded operator; finite von Neumann algebra; Type II1 factor.

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References

  1. Bernstein S., Démonstration du théoréme de Weierstrass fondeé sur le calcul des probabilités, Comm. Soc. Math. Kharkow 13 (1912), 1-2.
  2. Dirac P.A.M., The principles of quantum mechanics, Clarendon Press, Oxford, 1947.
  3. Dixmier J., Les algèbres d'opérateurs dans l’espace hilbertien (algèbres de von Neumann), Gauthier-Villars, Paris, 1969.
  4. Graves L.M., The theory of functions of real variables, McGraw-Hill Book Company, Inc., New York - Toronto - London, 1956.
  5. Heisenberg W., The physical principles of the quantum theory, University of Chicago Press, Chicago, 1930.
  6. Hille E., Phillips R.S., Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, Amer. Math. Soc., Providence, R.I., 1957.
  7. Kadison R.V., Transformations of states in operator theory and dynamics, Topology 3 (1965), suppl. 2, 177-198.
  8. Kadison R.V., Algebras of unbounded functions and operators, Exposition. Math. 4 (1986), 3-33.
  9. Kadison R.V., Operator algebras - an overview, in The Legacy of John von Neumann (Hempstead, NY, 1988), Proc. Sympos. Pure Math., Vol. 50, Amer. Math. Soc., Providence, RI, 1990, 61-89.
  10. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. I. Elementary theory, Pure and Applied Mathematics, Vol. 100, Academic Press Inc., New York, 1983.
  11. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. II. Advanced theory, Pure and Applied Mathematics, Vol. 100, Academic Press Inc., Orlando, FL, 1986.
  12. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. III. Elementary theory - an exercise approach, Birkhäuser Boston Inc., Boston, MA, 1991.
  13. Kadison R.V., Ringrose J.R., Fundamentals of the theory of operator algebras. Vol. IV. Advanced theory - an exercise approach, Birkhäuser Boston Inc., Boston, MA, 1992.
  14. Liu Z., On some mathematical aspects of the Heisenberg relation, Sci. China Math. 54 (2011), 2427-2452.
  15. Mackey G.W., Quantum mechanics and Hilbert space, Amer. Math. Monthly 64 (1957), 45-57.
  16. Murray F.J., Von Neumann J., On rings of operators, Ann. of Math. 37 (1936), 116-229.
  17. Murray F.J., von Neumann J., On rings of operators. II, Trans. Amer. Math. Soc. 41 (1937), 208-248.
  18. Murray F.J., von Neumann J., On rings of operators. IV, Ann. of Math. 44 (1943), 716-808.
  19. Stein E.M., Shakarchi R., Real analysis. Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, III, Princeton University Press, Princeton, NJ, 2005.
  20. Stone M.H., On one-parameter unitary groups in Hilbert space, Ann. of Math. 33 (1932), 643-648.
  21. von Neumann J., Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102 (1930), 370-427.
  22. von Neumann J., Die Eindeutigkeit der Schrödingerschen Operatoren, Math. Ann. 104 (1931), 570-578.
  23. von Neumann J., On rings of operators. III, Ann. of Math. 41 (1940), 94-161.
  24. von Neumann J., Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, 1955.
  25. Wielandt H., Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann. 121 (1949), 21-21.
  26. Wintner A., The unboundedness of quantum-mechanical matrices, Phys. Rev. 71 (1947), 738-739.

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