Agnieszka Martens

Affine symmetry in mechanics of collective and internal modes. Part II. Quantum models

Abstract Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schrodinger quantization and reduce effectively the quantized problem from n 2 to n degrees of freedom. Some possible applications in nuclear physics and other quantum many-body problems are suggested. Discussed is also the possibility of half-integer angular momentum in composed systems of spinless particles.

Affine symmetry in mechanics of collective and internal modes. Part I. Classical models

Discussed is a model of collective and internal degrees of freedom with kinematics based on affine group and its subgroups. The main novelty in comparison with the previous attempts of this kind is that it is not only kinematics but also dynamics that is affinely-invariant. The relationship with the dynamics of integrable one-dimensional lattices is discussed. It is shown that affinely-invariant geodetic models may encode the dynamics of something like elastic vibrations.

Quantization of the planar affinely-rigid body

Abstract This paper is a continuation of [1] where the classical model was analyzed. Discussed are some quantization problems of two-dimensional affinely rigid body with the double dynamical isotropy. Considered are highly symmetric models for which the variables can be separated. Some explicit solutions are found using the Sommerfeld polynomial method.

Quantization of an affinely rigid body with constraints

Abstract Discussed are some quantization problems of an affinely-rigid body with additional Weyl constraints. We investigate the systems with potential energies for which the variables can be separated. The Sommerfeld polynomial method is used to perform the quantization of such problems.

Dynamics of holonomically constrained affinely-rigid body

Abstract We discuss the dynamics of an affinely-rigid body with additional constraints described by Weyl group. Admitted is also the spinorial extension given by the group R + SU (2).

Mechanics of systems of affine bodies. Geometric foundations and applications in dynamics of structured media

Discussed are geometric structures underlying analytical mechanics of systems of affine bodies. Presented is detailed algebraic and geometric analysis of concepts like mutual deformation tensors and their invariants. Problems of affine invariance and of its interplay with the usual Euclidean invariance are reviewed. This analysis was motivated by mechanics of affine (homogeneously deformable) bodies, nevertheless, it is also relevant for the theory of unconstrained continua and discrete media. Postulated are some models where the dynamics of elastic vibrations is encoded not only in potential energy (sometimes even not at all) but also (sometimes first of all) in appropriately chosen models of kinetic energy (metric tensor on the configuration space), like in Maupertuis principle. Physically, the models may be applied in structured discrete media, molecular crystals, fullerens, and even in description of astrophysical objects. Continuous limit of our affine-multibody theory is expected to provide a new class of micromorphic media. Copyright

Hamiltonian dynamics of planar affinely-rigid body

Abstract We discuss the dynamics of an affinely-rigid body in two dimensions. Translational degrees of freedom are neglected. The special stress is laid on completely integrable models solvable in terms of the separation of variables method.

TEST RIGID BODIES IN RIEMANNIAN SPACES AND THEIR QUANTIZATION

Discussed are some classical and quantization problems of rigid bodies of infinitesimal size moving in Riemannian spaces. The rigorous meaning of “infinitesimal size” consists in replacing an extended body by the structured material point with internal degrees of freedom (co-moving orthonormal frame). The special case of constant curvature two-dimensional spaces is discussed. The Sommerfeld polynomial method is used to perform the quantization of such problems.

Quantization of two-dimensional affine bodies with stabilized dilatations

Discussed are some quantization problems of two-dimensional affine bodies. Quantum dilatational motion is stabilized by some appropriately chosen model potentials. Isochoric part of the dynamics is geodetic, i.e. potential-free. Surprisingly enough, this is compatible with the existence of discrete spectrum (bounded quantum motion). The Sommerfeld polynomial method is used to perform the quantization of such problems.

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright