Summary
While the transition to a moving coordinate system (x→ x + vt, t → t) commutes in classical mechanics with displacements (x → x + a, t → t), the corresponding operations in Schrödinger’s non-relativistic wave mechanics do not commute. In fact, the two operations, taken in different orders, differ by a factor exp [imv·a/h]. The present article considers the possibility of a nonrelativistic wave mechanics in which the transformations of the wave functions obey the same commutation relations as the transformations themselves. It shows that in such a mechanics position and momentum operators can exist only if it is reducible, i.e. only if the set of all states can be decomposed into subsets which are themselves invariant with respect to all permissible transformations of classical mechanics (rotations, displacements, proper Galilei transformations).
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© 1993 Springer-Verlag Berlin Heidelberg
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Inönü, E., Wigner, E.P. (1993). Representations of the Galilei Group. In: Wightman, A.S. (eds) The Collected Works of Eugene Paul Wigner. The Collected Works of Eugene Paul Wigner, vol A / 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02781-3_29
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DOI: https://doi.org/10.1007/978-3-662-02781-3_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08154-5
Online ISBN: 978-3-662-02781-3
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