Ray representations of N(≤2)+1‐dimensional Galilean group

• Published in: Journal of mathematical physics 1995-07, Vol.36 (7), p.3210-3215
• Main Authors: Doebner, H.‐D., Mann, H.‐J.
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.531026

For a quantum mechanical system, modeled via unitary or antiunitary representations of a symmetry group G with Lie algebra g, the Hamiltonian H is of special interest. If H is an element of g or of the universal enveloping algebra U(g), a generic time dependence for any element in U(g) is given through the Heisenberg picture. As an example we consider a system with Gal(N) as symmetry group with H as one of the generators. For N≥3 one gets from ray representations the free Schrödinger equation. For N=1 a peculiarity occurs: the ‘‘free’’ equation has an interaction term, which results from the construction and parametrization of unitary ray representations. For N=2 there is another special feature: there exist ray representations of the universal covering group Gal(2)■, which induce no ray representations of Gal(2). Furthermore, for these representations it is not possible to construct a Schrödinger equation.