Introduction to Representations of the Canonical Commutation and Anticommutation Relations

  • J. Dereziński
Part of the Lecture Notes in Physics book series (LNP, volume 695)


Since the early days of quantum mechanics it has been noted that the position operator x and the momentum operator D := −i∇ satisfy the following commutation relation:


Hilbert Space Real Hilbert Space Complex Hilbert Space Canonical Commutation Relation Fermionic Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of free Bose field, Journ. Math. Phys. 4 (1963) 1343–1362.zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Araki, H.: Type of von Neumann algebra associated with free field, Prog. Theor. Phys. 32 (1964) 956–854.zbMATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Araki, H.: Relative Hamiltonian for faithful normal states of a von Neumann algebra, Pub. R.I.M.S. Kyoto Univ. 9, 165 (1973).zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Araki, H.: On quasi-free states of CAR and Bogolubov automorphism, Publ. RIMS Kyoto Univ. 6 (1970) 385–442.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Araki, H.: On quasi-free states of canonical commutation relations II, Publ. RIMS Kyoto Univ. 7 (1971/72) 121–152.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Araki, H.: Canonical Anticommutation Relations, Contemp. Math. 62 (1987) 23.zbMATHMathSciNetGoogle Scholar
  7. 7.
    Araki, H., Shiraishi, M.: On quasi-free states of canonical commutation relations I, Publ. RIMS Kyoto Univ. 7 (1971/72) 105–120.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Araki, H.,Woods, E.J.: Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. Math. Phys. 4, 637 (1963).MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Araki, W., Wyss, W.: Representations of canonical anticommunication relations, Helv. Phys. Acta 37 (1964) 139–159.MathSciNetGoogle Scholar
  10. 10.
    Araki, H., Yamagami, S.: On quasi-equivalence of quasi-free states of canonical commutation relations, Publ. RIMS, Kyoto Univ. 18, 283–338 (1982).MathSciNetGoogle Scholar
  11. 11.
    Bach, V., Fröhlich, J., Sigal, I.: Return to equilibrium. J. Math. Phys. 41, 3985 (2000).zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Brauer, R., Weyl, H.: Spinors in n dimensions. Amer. Journ. Math. 57 (1935) 425–449.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to algebraic and constructive quantum field theory, Princeton NJ, Princeton University Press 1991.zbMATHGoogle Scholar
  14. 14.
    Berezin, F. A. The method of the Second Quantization, (Russian) 2nd ed. Nauka 1986.zbMATHGoogle Scholar
  15. 15.
    Bogolubov, N.N. Zh. Exp. Teckn. Fiz. 17 (1947) 23.MathSciNetGoogle Scholar
  16. 16.
    Brattelli, O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics, Volume 1, Springer-Verlag, Berlin, second edition 1987.Google Scholar
  17. 17.
    Brattelli, O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics, Volume 2, Springer-Verlag, Berlin, second edition 1996.Google Scholar
  18. 18.
    Cartan, E.: Théorie des spineurs, Actualités Scientifiques et Industrielles No 643 et 701 (1938), Paris, Herman.Google Scholar
  19. 19.
    Clifford, Applications of Grassmann' extensive algebra, Amer. Journ. Math. 1 (1878) 350–358.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Connes, A.: Characterization des espaces vectoriels ordonnces sous-jacentes aux algébres de von Neumann, Ann. Inst. Fourier, Grenoble 24, 121 (1974).zbMATHMathSciNetGoogle Scholar
  21. 21.
    Cook, J.: The mathematics of second quantization, Trans. Amer. Math. Soc. 74 (1953) 222–245.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    van Daele, A.: Quasi-equivalence of quasi-free states on the Weyl algebra, Comm. Math. Phys. 21 (1971) 171–191.zbMATHMathSciNetCrossRefADSGoogle Scholar
  23. 23.
    Davies, E. B.: Markovian master equations. Commun. Math. Phys. 39 (1974) 91.zbMATHCrossRefADSGoogle Scholar
  24. 24.
    De Biévre, S.: Local states of free Bose fields. Lect. Notes Phys. 695, 17-63 (2006).Google Scholar
  25. 25.
    Dereziński, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11, 383 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Dereziński, J., Jakšić, V.: Spectral theory of Pauli-Fierz operators, Journ. Func. Anal. 180, 241 (2001).Google Scholar
  27. 27.
    Dereziński, J., Jakšić, V.: Return to equilibrium for Pauli-Fierz systems, Ann. H. Poincare 4, 739 (2003).CrossRefzbMATHGoogle Scholar
  28. 28.
    Dixmiere, J.: Positions relative de deux varietés linneaires fermées dans un espace de Hilbert, Rev. Sci. 86 (1948) 387.Google Scholar
  29. 29.
    Dereziński, J., Jaksic, V., Pillet, C.A.: Perturbation theory of W *-dynamics, Liouvilleans and KMS-states, Rev. Math. Phys. 15 (2003) 447–489.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Dirac, P.A.M.: The quantum theory of the emission and absorption of radiation. Proc. Royal Soc. London, Series A 114, 243 (1927).zbMATHADSCrossRefGoogle Scholar
  31. 31.
    Dirac, P.A.M.: The quantum theory of the electron, Proc. Roy. Soc. London A 117 (1928) 610–624.zbMATHADSCrossRefGoogle Scholar
  32. 32.
    Eckmann, J. P., Osterwalder, K.: An application of Tomita's theory of modular algebras to duality for free Bose algebras, Journ. Func. Anal. 13 (1973) 1–12.zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Emch, G.: Algebraic methods instatistical mechanics and quantum field theory, Wiley-Interscience 1972.Google Scholar
  34. 34.
    Fetter, A. L., Walecka, J. D.: Quantum theory of many-particle systems, McGraw-Hill Book Company 1971.Google Scholar
  35. 35.
    Fock, V.: Konfigurationsraum und zweite Quantelung, Z. Phys.: 75 (1932) 622–647.zbMATHCrossRefADSGoogle Scholar
  36. 36.
    Folland, G.: Harmonic analysis in phase space, Princeton University Press, Princeton, 1989.zbMATHGoogle Scholar
  37. 37.
    Friedrichs, K. O. Mathematical aspects of quantum theory of fields, New York 1953.Google Scholar
  38. 38.
    Gaarding, L. and Wightman, A. S.: Representations of the commutations and anticommutation relations, Proc. Nat. Acad. Sci. USA 40 (1954) 617–626.CrossRefADSGoogle Scholar
  39. 39.
    Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, second edition, Springer-Verlag, New-York, 1987.zbMATHGoogle Scholar
  40. 40.
    Haag, R.: Local quantum physics, Springer 1992.Google Scholar
  41. 41.
    Haag, R., Hugenholtz, N. M., Winnink, M.: On the equilibrium states in quantum mechanics, Comm. Math. Phys.5, 215–236.Google Scholar
  42. 42.
    Haagerup, U.: The standard form of a von Nemann algebra, Math. Scand. 37, 271 (1975).MathSciNetGoogle Scholar
  43. 43.
    Halmos, P. R.: Two subspaces, Trans. Amer. Math. Soc. 144 (1969) 381.zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Jakšić, V., Pillet, C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. 108, 787 (2002).CrossRefzbMATHGoogle Scholar
  45. 45.
    Jordan, P. and Wigner, E.: Pauli's equivalence prohibition, Zetschr. Phys. 47 (1928) 631.zbMATHCrossRefADSGoogle Scholar
  46. 46.
    Kato, T.: Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin 1976.zbMATHGoogle Scholar
  47. 47.
    Lounesto, P.: Clifford algebras and spinors, second edition, Cambridge University Press 2001.Google Scholar
  48. 48.
    Lawson, H. B., Michelson, M.-L.: Spin geometry, Princeton University Press 1989.Google Scholar
  49. 49.
    Lundberg, L.E. Quasi-free “second-quantization”, Comm. Math. Phys. 50 (1976) 103–112.zbMATHMathSciNetCrossRefADSGoogle Scholar
  50. 50.
    Neretin, Y. A.: Category of Symmetries and infinite-Dimensional Groups, Clarendon Press, Oxford 1996. Introduction to Representations of the CCR and CAR 143Google Scholar
  51. 51.
    von Neumann, J.: Die Eindeutigkeit der Schrödingerschen operatoren, Math. Ann. 104 (1931) 570–578.zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Pauli, W.: Zur Quantenmechanik des magnetischen Elektrons, Z. Physik 43 (1927) 601–623.CrossRefADSGoogle Scholar
  53. 53.
    Pedersen, G. K.: Analysis Now, revised printing, Springer 1995.Google Scholar
  54. 54.
    Powers, R.: Representations of uniformly hyperfine algebras and their associated von Neumann rings, Ann. Math. 86 (1967) 138–171.zbMATHMathSciNetCrossRefGoogle Scholar
  55. 55.
    Powers, R. and Stoermer, E.: Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970) 1–33.zbMATHMathSciNetCrossRefADSGoogle Scholar
  56. 56.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I. Functional Analysis, London, Academic Press 1980.zbMATHGoogle Scholar
  57. 57.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV. Analysis of Operators, London, Academic Press 1978.zbMATHGoogle Scholar
  58. 58.
    Robinson, D.: The ground state of the Bose gas, Comm. Math. Phys. 1 (1965) 159–174.MathSciNetCrossRefADSGoogle Scholar
  59. 59.
    Ruijsenaars, S. N. M.: On Bogoliubov transformations for systems of relativistic charged particles, J. Math. Phys. 18 (1977) 517–526.MathSciNetCrossRefADSGoogle Scholar
  60. 60.
    Ruijsenaars, S. N. M.: On Bogoliubov transformations II. The general case. Ann. Phys. 116 (1978) 105–132.MathSciNetCrossRefADSGoogle Scholar
  61. 61.
    Shale, D.: Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962) 149–167.zbMATHMathSciNetCrossRefGoogle Scholar
  62. 62.
    Shale, D. and Stinespring, W.F.: States on the Clifford algebra, Ann. Math. 80 (1964) 365–381.zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Segal, I. E.: Foundations of the theory of dynamical systems of infinitely many degrees of freedom (I), Mat. Fys. Medd. Danske Vid. Soc. 31, No 12 (1959) 1–39.Google Scholar
  64. 64.
    Segal, I. E.: Mathematical Problems of Relativistic Physics, Amer. Math. Soc. Providence RI 1963.Google Scholar
  65. 65.
    Simon, B.: The P(φ)2 Euclidean (quantum) field theory, Princeton Univ. Press 1974.Google Scholar
  66. 66.
    Slawny, J.: On factor representations of the C*-algebra of canonical commutation relations, Comm. Math. Phys. 24 (1971) 151–170.MathSciNetCrossRefADSGoogle Scholar
  67. 67.
    Stratila, S.: Modular Theory in Operator Algebras, Abacus Press, Turnbridge Wells 1981.zbMATHGoogle Scholar
  68. 68.
    Summers, S. J.: On the Stone - von Neumann Uniqueness Theorem and its ramifications, to appear in “John von Neumann and the Foundations of Quantum Mechanics, eds M. Redei and M. Stoelzner.Google Scholar
  69. 69.
    Takesaki, M.: Theory of Operator Algerbras I, Springer 1979.Google Scholar
  70. 70.
    Takesaki, M.: Theory of Operator Algerbras II, Springer 2003.Google Scholar
  71. 71.
    Trautman, A.: Clifford algebras and their representations, to appear in Encyclopedia of Mathematical Physics, Elsevier.Google Scholar
  72. 72.
    Varilly, J. C. and Gracia-Bondia, The metaplectic representation and boson fields Mod. Phys. Lett. A7 (1992) 659.MathSciNetADSGoogle Scholar
  73. 73.
    Varilly, J. C. and Gracia-Bondia, QED in external fields from the spin representation, arXiv:hep-th/9402098 v1.Google Scholar
  74. 74.
    Weil A. Sur certains groupes d'operateurs unitaires, Acta Math. 111 (1964) 143–211.zbMATHMathSciNetCrossRefGoogle Scholar
  75. 75.
    Weyl H. The Theory of Groups and Quantum Mechanics, Meuthen, London 1931.zbMATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • J. Dereziński
    • 1
  1. 1.Department of Mathematical Methods in PhysicsWarsaw UniversityWarszawaPoland

Personalised recommendations