Affine models of collective and internal degrees of freedom. Problems of dynamical affine invariance
The linear group GL(3, R) and its unimodular subgroup SL(3,R) have been used for a long time in various physical problems, e.g. in nuclear physics, theory of molecular vibrations, macroscopic elasticity and even in astrophysics. In all these problems geometry of degrees of freedom was ruled by the linear or affine groups, but the dynamics itself was invariant under smaller groups, usually SO(3,R). In nuclear physics SL(3,R) was used as a dynamical non-invariance group.
We present some models where the kinetic energy itself is also invariant under SL(3,R). So, without potential, we use geodetic models on SL(3,R) invariant under this group or even under the direct product SL(3,R) x SL(3,R). For mathematical generality we consider the n-dimensional case SL(n,R). It turns out that in spite of the non-compact character of this group, there exists an open family of bounded classical motions. On the quantum level this is expressed by the existence of discrete energy spectra. One can also consider more general case, when some potential term is also admitted (at least the one stabilizing the volume object). On the classical level the problem is analyzed with the use of appropriate Poisson structures.
There are some reasons to expect that such models may be physically applicable in nuclear physics, astrophysics and even in certain non-standard elastic problems. Besides, they present some interest from the purely geometrical point of view.