Group Theoretical Discussion of Relativistic Wave Equations

  • V. Bargmann
  • E. P. Wigner
Part of the The Scientific Papers book series (WIGNER, volume A / 3)


1 — The wave functions, ψ, describing the possible states of a quantum mechanical system form a linear vector space V which, in general,. is infinite dimensional and on which a positive definite inner product (φ, ψ) is defined for any two wave functions φ and ψ (i.e., they form a Hilbert space). The inner product usually involves an integration over the whole configuration or momentum space and, for particles of higher spin, a summation over the spin indices.


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  1. All the essential results of the present paper were obtained by the two authors independently, but they decided to publish them jointly.Google Scholar
  2. Wigner, E. P., Ann. Math., 40, 149–204 (1939).CrossRefMathSciNetGoogle Scholar
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  5. Gelfand, L., and Neumark, M., J. Phys. (USSR),X, 93–94 (1946); Harish-Chandra, Proc. Roy. Soc. (London), A, 189, 372–401 (1947); and Bargmann, V., Ann. Math.,48,568–640 (1947), have determined the representations of the homogeneous Lorentz group. These are representations also of the inhomogeneous Lorentz group. In the quantum mechanical interpretation, however, all the states of the corresponding particles are invariant under translations and, in particular, independent of time. It in very unlikely that these representations have immediate physical significance. Is addition, the third paper contains a determination of those representations for which the momentum vectors are space like. These are not considered in the present article as they also are unlikely to have a simple physical interpretation.Google Scholar
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  8. The literature on relativistic wave equations is very extensive. Besides the papers quoted in reference 11, we only mention the book by de Broglie, L., Théorie générale des particules à spin (Paris, 1943), and the following articles which give a systematic account of the subject: Pauli, WV.. Rev. Mod. Phys.,13, 203–232 (1941); Bhabha, H. J., Rev. Mod. Phys.,17, 203–209 (1945); Kramers, H. A., Belinfante, F. J., and Lubânski, J. K., Physica,VIII, 597–627 (1941). In this paper, the sum of (14) over all v was postulated; (14a) then has to be added as an independent equation (except for N = 1). Reference 11 uses these equations in the form given by Kramers, Belinfante and Lubânski.Google Scholar
  9. One may derive this result in a more elegant way, without specializing the coördinate system. For the sake of brevity, we omit this derivation.Google Scholar
  10. de Wet, J. S., Phys. Rev.,58, 236–242 (1940), in particular, p. 242.ADSCrossRefMathSciNetGoogle Scholar
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© Springer-Verlag Berlin Heidelberg 1997

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  • V. Bargmann
  • E. P. Wigner

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