Classical and quantum dynamics for an extended free rigid body

In this paper, a free rigid body of dimension three is extended and analysed both in classical and quantum mechanics. The extension is performed by bringing the inverse inertia tensor, which is a positive-definite symmetric matrix for the ordinary rigid body, into an arbitrary real symmetric one. With an arbitrary real symmetric matrix chosen, associated is a Lie–Poisson structure on the Euclidean space of dimension three, through which the classical dynamics for an extended free rigid body is defined, and characterized by two first integrals. In parallel to this, the quantum dynamics is formulated as the problem of simultaneous spectral resolution of the two operators which are viewed as the quantization of the two classical first integrals. Intensive use is made of the unitary representation theory for Lie groups concerned. The explicit spectral resolution is obtained, in particular, when the extended free rigid body is an extended free symmetric top.