Lee Model and Quantisation of Non Linear Field Equations

  • W. Heisenberg
Part of the Gesammelte Werke / Collected Works book series (HEISENBERG, volume A / 3)


In a model for the theory of elementary particles an unconventional method of quantisation had been applied which makes use of an indefinite metric in Hilbert space. This method is analysed with the help of the Lee model. The constants of the Lee model can be adjusted in such a manner that, after the renormalisation has been carried out, the mathematical structure is in many ways analogous to that of the method of quantisation mentioned above (§ 2). The total Hilbert space can be divided into two parts, the one of which has a positive metric and comprises the physical states of the system, while the other one contains ‘dipole-ghosts’ and is needed for the commutation relations and the convergence of the theory. The S-matrix is unitary for the states of Hilbert space I, since the states of Hilbert space II are separated from those of Hilbert space I in a manner similar to the case of “non-combining term systems” (§§3 and 5). Some general results concerning the Hilbert space of indefinite metric are discussed in § 4. The questions of causality are briefly raised in § 6. The results seem to show that a Hilbert space with indefinite metric can be used for the description of real phenomena.


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  1. 1).
    G. Kállén and W. Pauli, Mat. Fys. Medd. Dan. Vid. Selsk. 30 (1955) no. 7Google Scholar
  2. 2).
    T. D. Lee, Phys. Rev. 95 (1954) 1329CrossRefzbMATHGoogle Scholar
  3. 3).
    P. A. M. Dirac, Proc. Roy. Soc. A 180 (1942) 1.CrossRefMathSciNetGoogle Scholar
  4. 3a).
    Compare also W. Pauli, Rev. of Modern Physics 15 (1943) 175CrossRefzbMATHMathSciNetGoogle Scholar
  5. 4).
    W. Heisenberg, Nachr. Gött. Akad. Wiss. (1953) 111;Google Scholar
  6. 4a).
    W. Heisenberg, Zs. f. Naturf. 9a (1954) 292;zbMATHMathSciNetGoogle Scholar
  7. 4b).
    W. Heisenberg, F. Kortel u. H. Mitter, Zs. f. Naturf. 10a (1955) 425;zbMATHMathSciNetGoogle Scholar
  8. 4c).
    W. Heisenberg, Zs. f. Physik 144 (1956) 1; Nachr. Gött. Akad. Wiss (1956) 27;CrossRefzbMATHMathSciNetGoogle Scholar
  9. 4d).
    R. Ascoli, W. Heisenberg, Zs. f. Naturf. 12a (1957) 177zbMATHMathSciNetGoogle Scholar
  10. 5).
    V. Glaser u. G. Källén, Nuclear Physics 2 (1956) 706CrossRefGoogle Scholar
  11. 6).
    Chr. Møller, Mat. Fys. Medd. Dan. Vid. Selsk. 23 (1945) no. 1; 24 (1946) no. 19Google Scholar
  12. 7).
    H. Lehmann, K. Symanzik u. W. Zimmermann, Nuovo Cimento (to be published)Google Scholar
  13. 8).
    W. Heisenberg, Zs. f. Naturf. 6a (1951) 281zbMATHMathSciNetGoogle Scholar
  14. 9).
    R. Haag, lecture at the international conference in Lille (June 1957)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • W. Heisenberg
    • 1
  1. 1.Max-Planck-Institut für PhysikGöttingenDeutschland

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