Quantized Mechanics of Affinely-Rigid Bodies

TL;DR

This paper develops a quantum mechanical framework for affinely-rigid bodies using the Schrödinger equation on the affine group, with potential applications in nuclear physics, astrophysics, and materials science.

Contribution

It introduces a quantized model of affinely-rigid motion based on the affine group, extending previous classical theories with a new quantum approach.

Findings

Formulation of quantum mechanics on the affine group.

Analysis of kinetic energy invariance under affine transformations.

Potential applications in nuclear dynamics and astrophysics.

Abstract

In this paper we develope the main ideas of the quantized version of affinely-rigid (homogeneously deformable) motion. We base our consideration on the usual Schr\"odinger formulation of quantum mechanics in the configuration manifold which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group ${\rm GL}(n,\mathbb{R})$ and the space of translations $\mathbb{R}^{n}$, where $n$ equals the dimension of the "physical space". In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the two-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of nuclear dynamics or even in apparently exotic phenomena in vibrating neutron stars. And, of course, some more prosaic applications in macroscopic elasticity, structured continua, molecular dynamics, dynamics of inclusions, suspensions, and bubbles are also possible.