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

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 $$D^t{\mathrm{\approx G}}_E^0\text{(N)[multiplication in left half circle]Sp(2N,\hspace{0.17em}R)}$$ . $G_E^0\mathrm{(N)}$ is the central extension of the pure inhomogeneous Galilei group in N dimensions and Sp(2N, 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 $G_E^0\mathrm{(N)}$ . 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.