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The Algebraic Quantum Formalism

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The Formalisms of Quantum Mechanics

Part of the book series: Lecture Notes in Physics ((LNP,volume 893))

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Abstract

The physical observables of a quantum system are represented by the symmetric (self-adjoint) operators on the Hilbert space of pure states of the system (see Sects. 2.3.1 and in 2.3.3). They thus generate (by addition and multiplication) the set of all (not necessary symmetric) operators on the Hilbert space. This set forms an associative but non-commutative complex algebra of operators.

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Notes

  1. 1.

    The exact definition of the spectrum is slightly different for a general real Banach algebra.

  2. 2.

    See below for a more precise definition and discussion. For many authors the term of superselection sectors is reserved to infinite dimensional algebras which do have inequivalent representations.

  3. 3.

    in the mathematic sense: they are not defined with reference to a given representation such as operators in Hilbert space, path integrals, etc.

  4. 4.

    Unitary with respect to the real algebra structure, i.e. unitary or antiunitary w.r.t. the complex algebra structure.

Bibliography

  1. S. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)

    MATH  Google Scholar 

  2. Y. Aharonov, D. Bohm, Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. G. Auletta, Foundations and Interpretation of Quantum Mechanics (World Scientific, Singapore, 2001)

    Google Scholar 

  4. N.N. Bogoliubov, A.A. Logunov, A.I. Oksak, I.T. Todorov, General Principles of Quantum Field Theory. Number 10 in Mathematical Physics and Applied Mathematics (Kluwer Academic, Dordrecht, 1990)

    Google Scholar 

  5. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, vols. I and II (Springer, Berlin, 2002)

    Google Scholar 

  6. A. Connes, Noncommutative Geometry (Academic, New York, 1994)

    MATH  Google Scholar 

  7. A. Connes, M. Marcolli, Noncommutative Geometry, Quantum Fields and Motives, vol. 55 (AMS Colloquium Publications, AMS and Hindustan Book Agency, 2008)

    Google Scholar 

  8. P. de la Harpe, V. Jones, An introduction to C*-algebras: Chapters 1 to 9, vol. 1 (1995). Publications internes de la Section de mathématiques de l’Université de Genève

    Google Scholar 

  9. J. Dixmier, Les algèbres d’opérateurs dans l’espace Hilbertien: algèbres de Von Neumann, volume fasc. 25 of Cahiers scientifiques, 2e édition revue et augmenteé edition (Gauthier-Villars, Paris, 1969)

    Google Scholar 

  10. W. Ehrenberg, R.E. Siday, The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. B 62, 8–21 (1949)

    Article  ADS  MATH  Google Scholar 

  11. I.M. Gelfand, M.A. Naimark, On the embedding of normed rings into the ring of operators in Hilbert space. Mat. Sb. 12, 197–213 (1943)

    Google Scholar 

  12. K.R. Goodearl, Notes on Real and Complex C*-Algebras. Shiva Mathematics Series (Shiva Publishing Ltd., Nantwitch, UK, 1982)

    Google Scholar 

  13. R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1996)

    Book  MATH  Google Scholar 

  14. R. Haag, D. Kastler, An algebraic approach to quantum field theory. J. Math. Phys. 5, 848–861 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. L. Ingelstam, Real Banach algebras. Arkiv för Mat. 5, 239–270 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. S. Sakai, C*-Algebras and W*-Algebras (Springer, Berlin, 1971)

    Book  Google Scholar 

  17. I. Segal, Irreducible representations of operator algebras. Bull. Am. Math. Soc. 53, 73–88 (1947)

    Article  MATH  Google Scholar 

  18. I. Segal, Postulates for general quantum mechanics. Ann. Math. 48, 930–948 (1947)

    Article  MATH  Google Scholar 

  19. R.F. Streater, A.S.Wightman, PCT, Spin and Statistics and All That. Landmarks in Mathematics and Physics (Princeton University Press, Princeton, 2000)

    Google Scholar 

  20. E.C.G. Stueckelberg, Quantum theory in real Hilbert space. Helv. Phys. Acta 33, 727–752 (1960)

    MATH  MathSciNet  Google Scholar 

  21. S. Weinberg, The Quantum Theory of Fields, Volume 1: Foundations (Cambridge University Press, Cambridge, 2005)

    Google Scholar 

  22. W.K. Wooters, in Complexity, Entropy, and the Physics of Information, ed. by W.H. Zurek (Addison-Wesley, Reading, 1990), pp. 39–46

    Google Scholar 

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Authors and Affiliations

  1. Institut de Physique Théorique, CEA Saclay, Gif-sur-Yvette, France

    François David

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  1. François David
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David, F. (2015). The Algebraic Quantum Formalism. In: The Formalisms of Quantum Mechanics. Lecture Notes in Physics, vol 893. Springer, Cham. https://doi.org/10.1007/978-3-319-10539-0_3

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