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Limitable Dynamical Groups in Quantum Mechanics. I. General Theory and a Spinless Model

### Abstract

A pure group‐theoretical description of nonrelativistic interacting systems in terms of irreducible representations *U*(*D*) of a so‐called dynamical group *D* is investigated. The description is assumed to be complete in the sense that all observable quantities of the system can be calculated from *U*(*D*) in the same way as the nonrelativistic free particle can be identified with an irreducible representation *U*(*G*_{E} ) of the central extension of the inhomogeneous Galilei group *G*_{E}. D depends on the interaction. It is a noninvariance group and it contains a spectrum‐generating algebra. Our problem is to connect a representation of an arbitrary abstract group with a complete description of an interacting system. This needs some physically motivated principles. Some such principles are proposed. We assume that the interaction can be turned off, which implies that *U*(*D*) and the physical representation *U*(*G*_{E} ) of the free‐particle group *G*_{E} can be limited into each other. If this limitation can be formulated without violating super‐selection rules, i.e., mass and spin conservation in nonrelativistic systems, the group *D*^{t} is called a limitable group. Properties of these groups are derived. An explicit construction of a limitable *D*^{t} is given by embedding the free‐particle group *G*_{E} into a larger group. A discussion of all embeddings leads to the special choice. is the central extension of the pure inhomogeneous Galilei group in *N* dimensions and *Sp*(2*N, R*) the noncompact real form of the symplectic group. A representation theory for *D*^{t} is established using the technique of Nelson extensions, together with some properties of the universal enveloping algebra of the Lie algebra of . Our main success is that *D*^{t} is a limitable dynamical group and that the physical system described by *D*^{t} and the physical representation can be calculated uniquely from the proposed principles. The group‐theoretical description is equivalent to nonrelativistic quantum mechanics for a spinless particle in *N* dimensions with an arbitrary second‐order polynomial in *P*_{i}, Q_{i}, i = 1, ⋯, *N* as Hamiltonian. The possibility of further models is discussed.

© 1968 The American Institute of Physics

Received 25 July 1967
Published online 28 October 2003

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/content/aip/journal/jmp/9/10/10.1063/1.1664494

2003-10-28

2016-09-27

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