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On momentum states in quantum mechanics

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On momentum states in quantum mechanics ANNALES DE L’I. H. P., SECTION A M. FANNES A. VERBEURE R. WEDER On momentum states in quantum mechanics Annales de l’I. H. P., section A, tome 20, no 3 (1974), p. 291-296. <http://www.numdam.org/item?id=AIHPA_1974__20_3_291_0> © Gauthier-Villars, 1974, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A », implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/legal.php). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 291 On momentum states in quantum mechanics M. FANNES (1), A. VERBEURE R. WEDER (2) Katholieke Universiteit Leuven, Belgium (3) Ann, Henri Poincaré, Vol. XX, nO 3, 1974, Section A : Physique ’ théorique, ’ ABSTRACT. - In the algebraic formulation of quantum mechanics a class of states, called the momentum states are defined. They correspond to plane wave and wave packet states of ordinary quantum mechanics. A uniqueness theorem is proved for this class of states. Furthermore a purely algebraic proof if given of the fact that in any irreducible represen- tation of the CCR-algebra the von Neumann algebra generated by all translations is maximal abelian. 0 . INTRODUCTION The algebraic formulation of quantum mechanics for a system with n degrees of freedom consists in defining a dynamical system, given by the following triplet : i) the abstract set of observables as the elements of the CCR- C*-algebra 0394 [1] build on the sympletic space H = R2n and sympletic form ~ : (1 ) Aspirant NFWO, Belgium. (2) On leave of absence from Universidad de Buenos Aires (Argentina). e) Postal address : Instituut voor Theoretische Natuurkunde, Celestijnenlaan 200 D, B-3030 Heverlee, Belgium. Annales

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