Jan J. Sławianowski

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.

Lie-algebraic solutions of affinely-invariant equations for the field of linear frames

Abstract We present a certain class of natural solutions of affinely-invariant and generally-covariant variational field equations postulated in [11] for the field of linear frames ( n -leg field, tetrad field). These solutions are interesting from the geometric point of view; for example they induce in manifolds local structures of homogeneous spaces with trivial isotropy groups. The corresponding Lie groups are either semisimple or have the direct product forms R × G , U (1)× G , G being a semisimple Lie group.

The mechanics of an affinely-rigid body

The mechanics of an affinely-rigid body is investigated on both the classical and the quantum level. An affinely-rigid body is defined as a system of material points or a continuous medium in which all affine relations are frozen. Our treatment is based on the general theory of systems with closed teleparallelisms, presented in Section 2 of this paper.

Global invariance and Lie-algebraic description in the theory of dislocations

The relationships between the Lie-algebraic description of continuously distributed dislocations and the global affine invariance of their tensorial density are studied. The affinely-invariant Lagrange description of static self-equilibrium distributions of dislocations is proposed and field equations describing distributions of internal stresses and couple stresses are formulated. The analogy between the proposed theory of dislocations and “3-fold electrodynamics” is formulated.

Invariant geodetic problems on the affine group and related hamiltonian systems

Abstract Discussed are (pseudo-)Riemannian metrics on the affine group. A special stress is laid on metric structures invariant under left or right regular translations by elements of the total affine group or some of its geometrically distinguished subgroups. Also some non-geodetic problems in corresponding Riemannian spaces are discussed.

Classical and Quantized Affine Physics: A Step towards it

Abstract The classical and quantum mechanics of systems on Lie groups and their homogeneous spaces are described. The special stress is laid on the dynamics of deformable bodies and the mutual coupling between rotations and deformations. Deformative modes are discretized, i.e., it is assumed that the relevant degrees of freedom are controlled by a finite number of parameters. We concentrate on the situation when the effective configuration space is identical with affine group (affinely-rigid bodies). The special attention is paid to left- and right-invariant geodetic systems, when there is no potential term and the metric tensor underlying the kinetic energy form is invariant under left or/and right regular translations on the group. The dynamics of elastic vibrations may be encoded in this way in the very form of kinetic energy. Although special attention is paid to invariant geodetic systems, the potential case is also taken into account.

Linear frames in manifolds, riemannian structures and description of internal degrees of freedom

Abstract Discussed is affine model of internal degrees of freedom, i.e. infinitesimal Affnely rigid body in a manifold. The treatment is nonrelativistic and classical. The special stress is laid on geodetic models and invariance classification of a priori possible kinetic energies. Geometrically this is equivalent to deriving natural Riemannian structures in the bundle of linear frames from some geometry on the base manifold (affine connection, metric). It turns out that the general structure of dynamical balance laws resembles that derived in general relativity for the pole-dipole particle. Linear momentum and (affine) spin are coupled to the torsion and curvature tensors.

Schrödinger and related equations as hamiltonian systems, manifolds of second-order tensors and new ideas of nonlinearity in quantum mechanics

Considered is the Schrodinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting example of “mechanics” with singular Lagrangians, effectively treatable within the framework of Dirac formalism. We discuss also some modified “Schrodinger” equations involving second-order time derivatives and introduce a kind of nondirect, nonperturbative, geometrically-motivated nonlinearity based on making the scalar product a dynamical quantity. There are some reasons to expect that this might be a new way of describing open dynamical systems and explaining some quantum “paradoxes”.

Geodetic Systems on Linear and Affine Groups. Classics and Quantization.

Abstract Described are classical and quantized systems on linear and affine groups. Unlike the traditional models applied in astrophysics, nuclear physics, molecular vibrations and elasticity, our models are not only kinematically ruled by the affine group, but also their kinetic energies are affinely invariant. There are geodetic SL(n, R)-invariant models with an open family of bounded solutions and with discrete spectra on the quantized level. They seem to be applicable in nuclear physics, theory of defects in solids, astrophysics, dynamics of inclusions, small droplets of fluids and gas bubbles. Independently of these hypothetical applications, they are interesting in themselves.