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Fermions and associated bosons of one-dimensional model

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Abstract

The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fock-representation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly selfadjoint operators. It provides the key to the solubility ofLuttinger's model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.

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References

  1. Pryce, M. H. L.: Proc. Roy. Soc. (London), Ser. A165, 247 (1938). This article also reviews the earlier work and the relevant references are to be found there. Among these the following are particularly relevant to our discussion:Born, M., andN. Nagendra-Nath: Proc. Ind. Acad. Sci.3, 318 (1936),N. Nagendra-Nath, ibid. p. 448.

    Google Scholar 

  2. Perkins, W. A.: Phys. Rev.137, B1291 (1965).

    Google Scholar 

  3. Wightman, A. S.: Introduction to some aspects of the relativistic dynamics of quantical fields, I.H.E.S. Bures-sur-Yvette, S.-O., France, May 1964.

  4. Luttinger, J. M.: J. Math. Phys.4, 1154 (1963).

    Google Scholar 

  5. Mattis, D. C., andE. H. Lieb: J. Math. Phys.6, 304 (1965).

    Google Scholar 

  6. Tomonaga, S.: Progr. Theoret. Phys. (Kyoto)5, 544 (1950).

    Google Scholar 

  7. Engelsberg, S., andB. B. Varga: Phys. Rev.136, A 1582 (1964).

  8. Schroer, B.: Fortschr. Physik11, 1 (1963).

    Google Scholar 

  9. For example:Araki, H.: J. Math. Phys.1, 492 (1960).

    Google Scholar 

  10. J. R. Klauder, andJ. McKenna: J. Math. Phys.6, 68 (1965). AlsoKlauder, J. R., J. McKenna, andE. J. Woods: J. Math. Phys.7, 822 (1966).

    Google Scholar 

  11. Cook, J. M.: Trans. Am. Math. Soc.74, 222 (1953).

    Google Scholar 

  12. Araki, H., andE. J. Woods: J. Math. Phys.4, 637 (1963).

    Google Scholar 

  13. Araki, H., andW. Wyss: Helv. Phys. Acta37, 136 (1964).

    Google Scholar 

  14. Bourbaki, N.: Eléments de mathématique. Algèbre, Ch. 2, p. 18; Paris: Hermann 1964.

    Google Scholar 

  15. Bogoliubov, N. N.: J. Phys. (U.S.S.R.)11, 23 (1947).

    Google Scholar 

  16. Ezawa, H.: J. Math. Phys.6, 380 (1965).

    Google Scholar 

  17. Segal, I. E.: Math. problems of Rel. Physics. Am. Math. Soc. 1963.

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Authors and Affiliations

  1. The Institute for Advanced Study, Princeton, New Jersey

    D. A. Uhlenbrock

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  1. D. A. Uhlenbrock
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Work supported by the National Science Foundation.

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Uhlenbrock, D.A. Fermions and associated bosons of one-dimensional model. Commun.Math. Phys. 4, 64–76 (1967). https://doi.org/10.1007/BF01645177

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