Proceedings of the National Academy of Sciences · 1953 · 1.1K citations · 2 references
Classical mechanics and its Galilei symmetry arise as limiting cases of relativistic mechanics and the Poincaré group, and analogous limits exist between the inhomogeneous Lorentz and de Sitter groups. The note investigates the conditions under which one group can be regarded as a limiting case of another and how the representations of the limit group can be derived from those of the parent group. Section III outlines the transition from the inhomogeneous Lorentz group to the Galilei group. The analysis demonstrates that the Schrödinger‑equation representation of the Galilei group emerges as a limit of an inhomogeneous Lorentz representation, and explains why the real representations of the latter lack a physical interpretation.
Classical mechanics is a limiting case of relativistic mechanics. Hence the group of the former, the Galilei group, must be in some sense a limiting case of the relativistic mechanics’ group, the representations of the former must be limiting cases of the latter’s representations. There are other examples for similar relations between groups. Thus, the inhomogeneous Lorentz group must be, in the same sense, a limiting case of the de Sitter groups. The purpose of the present note is to investigate, in some generality, in which sense groups can be limiting cases of other groups (Section I), and how their representations can be obtained from the representations of the groups of which they appear as limits (Section II). Section III deals briefly with the transition from inhomogeneous Lorentz group to Galilei group. It shows in which way the representation up to a factor of the Galilei group, embodied in the Schrodinger equation, appears as a limit of a representation of the inhomogeneous Lorentz group and also gives the reason why no physical interpretation is possible for the real representations of that group.
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