Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-22T13:28:46.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Irreducibility of some unitary representations of the Poincaré group with respect to the Poincaré subsemigroup, I

Published online by Cambridge University Press:  22 January 2016

Hitoshi Kaneta
Hitoshi Kaneta*
Affiliation:
Department of MathematicsNagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Since E. Wigner set up a framework of the relativistically covariant quantum mechanics, several aspects of unitary representations of the Poincaré group have been investigated (see [8], [16]). In this paper it will be shown that some unitary representations of the Poincaré group are irreducible, even if they are restricted to the Poincaré semigroup (Theorem 1, 2 and 3).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 78 , May 1980 , pp. 113 - 136
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[1] Agmon, S., Spectral properties of Schrödinger operators and scatttering theory, Annali dela Scuola Normal Superiore di Pisa, series 3, 2 (1975), 149–218.Google Scholar
[2] Angelopoulos, E., Decomposition sur le sou-groupe de Poincaré de la represéntation de mass positive et de spin nul du groupe de Poincaré. Ann. Inst. Henri Poincaré XV, no. 4 (1971), 303–320.Google Scholar
[3] Angelopoulos, E., Reduction on the Lorentz subgroup of UIR’s of the Poincaré group induced by a semisimple little group, Math. Phys. 15 (1974), 155–165.Google Scholar
[4] Dixmier, J., Les C*-algèbres et leurs represéntations. Gauthier-Villars Paris, 1969.Google Scholar
[5] Erdélyi, A., Higher transcendental functions 1, McGraw-Hill, 1955.Google Scholar
[6] Hormander, L., Linear partial differential operators, Springer, 1963.Google Scholar
[7] Itatsu, S. and Kaneta, H., Spectral properties of first order ordinary differential operators, to appear.Google Scholar
[8] Joos, H., Zur Darstellungstheorie per inhomogeneous Lorentzgruppe als Grundlage quantenmechanischer Kinematik, Fortschr. Phts. 10 (1962), 65–146.Google Scholar
[9] Kato, T., Growth properties of solutions of reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425.Google Scholar
[10] Mackey, G. W., Induced representations of locally compact group I, Ann. of Math. 55, no. 1 (1951), 101–139.Google Scholar
[11] Lax, P. D. and Phillips, R. S., Scattering theory, Academic Press, 1967.Google Scholar
[12] Naimark, M. A., Linear representation of Lorentz group, VEB Deucher Verlag der Wissenschaften Berlin, 1963.Google Scholar
[13] Radhakrishnan, B. and Mukunda, N., Spacelike representations of the inhomogeneous Lorentz group in a Lorentz basis, J. Math. Phys. 15 (1974), 477–490.Google Scholar
[14] Segal, I. E., A class of operator algebras which are determined by groups, Duke Math. J. 18 (1951), 221–265.Google Scholar
[15] Tatsuuma, N., Decomposition of representations of three-dimensional Lorentz group, Proc. Japan Acad. 38 (1962), 12–14.Google Scholar
[16] Tatsuuma, N., Decomposition of Kronecker products of representations of inhomogeneous Lorentz group, Proc. Japan Acad. 38 (1962), 156–160.Google Scholar
[17] Vilenkin, N., Special functions and the theory of group representations, AMS translation of monographs 22, 1968.CrossRefGoogle Scholar