Institute of Fundamental Technological Research

There are various models of gravitation: the metrical Hilbert‐Einstein theory, a wide class of intrinsically Lorentz‐invariant tetrad theories (generally covariant in the space‐time sense), and many gauge models based on various internal symmetry groups (Lorentz, Poincare, GL(n,R), SU(2,2), GL(4,C), etc). The gauge models are usually preferred but nevertheless it is an interesting idea to develop the class of GL(4,R)‐invariant (or rather GL(n,R)‐invariant) tetrad (n‐leg) generally covariant models. This is done below and motivated by our idea of bringing back to life the Thales of Miletus concept of affine symmetry. Formally, the obtained scheme is a generally covariant tetrad (n‐leg) model, but it turns out that generally covariant and intrinsically affinely invariant models must have a kind of nonaccidental Born‐Infeld‐like structure. Let us also mention that they, being based on tetrads (n‐legs), have many features common with continuous defect theories. It is interesting that they possess some group‐theoretical solutions and more general spherically symmetric solutions, discussion of which is the main new result presented in this paper, including the applications of the 't Hooft‐Polyakov monopoles in the generally covariant theories, which enables us to find some rigorous solutions of our strongly nonlinear equations. It is also interesting that within such a framework, the normal‐hyperbolic signature of the space‐time metric is not introduced by hand but appears as a kind of solution, rather integration constants, of differential equations. Let us mention that our Born‐Infeld scheme is more general than alternative tetrad models. It may be also used within more general schemes, including also the gauge ones.

micromorphic medium, modified gravity, relativistic continuum, spherically symmetric solutions, theory of fundamental interactions.

In this paper we discuss certain dynamical models of affine bodies, including problems of partial separability and integrability. There are some reasons to expect that the suggested models are dynamically viable and that on the fundamental level of physical phenomena the "large" affine symmetry of dynamical laws is more justified and desirable than the restricted invariance under isometries.

homogeneous deformation, structured media, affinely-invariant dynamics, elastic vibrations encoded in kinetic energy, Calogero-Moser and Sutherland integrable lattices

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configurationmanifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group GL(n,R) and the space of translations 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 2-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 elastic media and microstructured continua.

Homogeneously deformable body, Peter-Weyl analysis, Schrödinger quantization.

We propose, for a finite crystal, that the entire system be expressed in terms of the phase space, including momentum, configuration position and spin. This is to be done both classically and quantum mechanically. We illustrate this with a silicon crystal. Then, quantum mechanically measuring with a wire containing electrons, we obtain a theoretically good approximation for an electron in one semiconductor. As well, we outline what has to be done for any crystal.

Quantum mechanics in Phase space, Symmetry groups for a crystal, Groups representation, Quantization

The aim of this paper is to perform a deeper geometric analysis of problems appearing in dynamics of affinely rigid bodies. First of all we present a geometric interpretation of the polar and the two-polar decomposition of affine motion. Later on some additional constraints imposed on the affine motion are reviewed, both holonomic and non-holonomic. In particular, we concentrate on certain natural non-holonomic models of the rotation-less motion. We discuss both the usual d’Alembert model and the vakonomic dynamics. The resulting equations are quite different. It is not yet clear which model is practically better. In any case they both are different from the holonomic constraints defining the rotation-less motion as a time-dependent family of symmetric matrices of placements. The latter model seems to be non-geometric and non-physical. Nevertheless, there are certain relationships between our non-holonomic models and the polar decomposition.

Affine motion, polar and two-polar decompositions, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

Discussed is the problem of the mutual relationship of differentially first-ordr and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order Lagrangians are primary, and in case more adequate. It is shown that in principle, the Schodinger idea about Lagrangians which are quadratic in derivatives, and leading to second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal or rather its universal covering SU(2,3) as a gauge group. This has also some influence on the theory of defect in continua.

field theory, gauge group, Lagrangians

We present new mathematical foundations of classical Maxwell–Lorentz electrodynamic models and related charged particles interaction-radiation problems, and analyze the fundamental least action principles via canonical Lagrangian and Hamiltonian formalisms. The corresponding electrodynamic vacuum field theory aspects of the classical Maxwell–Lorentz theory are analyzed in detail. Electrodynamic models of charged point particle dynamics based on a Maxwell type vacuum field medium description are described, and new field theory concepts related to the mass particle paradigms are discussed. We also revisit and reanalyze the mathematical structure of the classical Lorentz force expression with respect to arbitrary inertial reference frames and present new interpretations of some classical special relativity theory relationships.

Lagrangian and Hamiltonian Formalism, Least Action Principle, Vacuum Field Theory, Lorentz Force Problem, Feynman Approach, Maxwell Equations, Lorentz Constraint, Electron Radiation Theory

The main leitmotivs of this paper are the essential nonlinearities, symmetries and the mutual relationships between them. By essential nonlinearities we mean ones which are not interpretable as some extra perturbations imposed on a linear background deciding about the most important qualitative features of discussed phenomena. We also investigate some discrete and continuous systems, roughly speaking with large symmetry groups. And some remarks about the link between two concepts are reviewed. Namely, we advocate the thesis that the most important non-perturbative nonlinearities are those implied by the assumed “large” symmetry groups. It is clear that such a relationship does exist, although there is no complete theory. We compare the mechanism of inducing nonlinearity by symmetry groups of discrete and continuous systems. Many striking and instructive analogies are found, e.g., analogy between analytical mechanics of systems of affine bodies and general relativity, tetrad models of gravitation, and Born-Infeld nonlinearity. Some interesting, a bit surprising problems concerning Noether theorem are discussed, in particular in the context of large symmetry groups.

Essential nonlinearity, dynamical symmetries, affine bodies, general relativity, tetrad models and micromorphic continua

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’.

Weyl–Wigner–Moyal–Ville formalism, topological groups, classical momentum mapping, unitary projective representations, Pontryagin duality, ‘Umklapp–Prozessen’

Discussed are some problems of two (or more) mutually coupled systems with gyroscopic degrees of freedom. First of all, we mean the motion of a small gyroscope in the non-relativistic Einstein Universe RXS3(0;R); the second factor denoting the Euclidean 3-sphere of radius R in R4. But certain problems concerning two-gyroscopic systems in Euclidean space R3 are also mentioned. The special stress is laid on the relationship between various models of the configuration space like, e.g., SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R) etc. They are locally diffeomorphic, but globally different. We concentrate on classical problems, nevertheless, some quantum aspects are also mentioned.

Rigid body, gyroscopic degrees of freedom, Einstein Universe, Euclidean space, SU(2)xSU(2), SO(4;R), SO(3;R)xSO(3;R)

Discussed are certain strange properties of the Galilei group, connected first of all with the property of mechanical energy-momentum covector to be an affine object, rather than the linear one. Its affine transformation rule is interesting in itself and dependent on the particle mass. On the quantum level this means obviously that we deal with the projective unitary representation of the group rather than with the usual representation. The status of mass is completely different than in relativistic theory, where it is a continuous eigenvalue of the Casimir invariant. In Galilei framework it is a parameter characterizing the factor of the projective representation, in the sense of V. Bargmann. This “pathology” from the relativistic point of view is nevertheless very interesting and it underlies the Weyl-Wigner-Moyal-Ville approach to quantum mechanics.

Galilei group, affine transformation, particle mass, projective unitary representation, Weyl-Wigner-Moyal-Ville formalism

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.

Systems of affine bodies, mutual deformation tensors, affine and Euclidean invariance, structured media, elastic vibrations, geometric structures

We use the mathematical structure of group algebras and $H^+$-algebras for describing certain problems concerning the quantum dynamics of systems of angular momenta, including also the spin systems. The underlying groups are SU(2) and its quotient SO(3, R). The proposed scheme is applied in two different contexts. Firstly, the purely group-algebraic framework is applied to the system of angular momenta of arbitrary origin, e.g., orbital and spin angular momenta of electrons and nucleons, systems of quantized angular momenta of rotating extended objects like molecules. Secondly, the other promising area of applications is Schroedinger quantum mechanics of rigid body with its often rather unexpected and very interesting features. Even within this Schroedinger framework the algebras of operators related to group algebras are a very useful tool. We investigate some problems of composed systems and the quasiclassical limit obtained as the asymptotics of “large” quantum numbers, i.e., “quickly oscillating” wave functions on groups. They are related in an interesting way to geometry of the coadjoint orbits of SU(2).

Systems of angular momenta, models with symmetries, quantum dynamics, spin systems, Schroedinger quantum mechanics, quasiclassical limit

In Part I of this series we have presented the general ideas of applying group-algebraic methods for describing quantum systems. The treatment there was very “ascetic” in that only the structure of a locally compact topological group was used. Below we explicitly make use of the Lie group structure. Relying on differential geometry one is able to introduce explicitly representation of important physical quantities and to formulate the general ideas of quasiclassical representation and classical analogy.

Systems of angular momenta, models with symmetries, quantum dynamics, Lie group structures, quasiclassical representation, classical analogy

This is the third part of our series “Quasiclassical and Quantum Systems of Angular Momentum”. In two previous parts we have discussed the methods of group algebras in formulation of quantum mechanics and certain quasiclassical problems. Below we specify to the special case of the group SU(2) and its quotient SO(3,R), and discuss just our main subject in this series, i.e., angular momentum problems. To be more precise, this is the purely SU(2)-treatment, so formally this might also apply to isospin. However. it is rather hard to imagine realistic quasiclassical isospin problems.

Systems of angular momenta, models with symmetries, quantum dynamics, quasiclassical isospin problems

Discussed is the structure of classical and quantum excitations of internal degrees of freedom of multiparticle objects like molecules, fullerens, atomic nuclei, etc. Basing on some invariance properties under the action of isometric and affine transformations we reviewed some new models of the mutual interaction between rotational and deformative degrees of freedom. Our methodology and some results may be useful in the theory of Raman scattering and nuclear radiation.

Internal degrees of freedom, isometric and affine invariance, classical and quantum excitations, multiparticle objects, Raman scattering, nuclear radiation

Discussed is a model of the two-dimensional affinely-rigid body with the double dynamical isotropy. We investigate the systems with potential energies for which the variables can be separated. The special stress is laid on the model of the harmonic oscillator potential and certain anharmonic alternatives. Some explicit solutions are found on the classical, quasiclassical (Bohr–Sommerfeld) and quantum levels.

affinely-rigid body, harmonic oscillator potential, Sommerfeld polynomial method

Considered is the Schrödinger 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 “Schrödinger” 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”.

Hamiltonian systems on manifolds of scalar products, finite-level quantum systems, finite-dimensional Hilbert space, Hermitian forms, scalar product as a dynamical variable, Schroedinger equation, Dirac formalism, essential nonperturbative nonlinearity, quantum paradoxes, conservation laws, GL_n(C)-invariance

Discussed is mechanics of objects with internal degrees of freedom in generally non- Euclidean spaces. Geometric peculiarities of the model are investigated in detail. Discussed are also possible mechanical applications, e.g. in dynamics of structured continua, defect theory and in other fields of mechanics of deformable bodies. Elaborated is a new method of analysis based on nonholonomic frames. We compare our results and methods with those of other authors working in nonlinear dynamics. Simple examples are presented.

affine invariance, affinely-rigid bodies, collective modes, internal degrees of freedom, nonlinear elasticity, Riemannian manifolds

Described is n-level quantum system realized in the n-dimensional “Hilbert” space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The usual, first- and second-order modified Schroedinger equations are obtained.

Schroedinger equation, Hamiltonian systems on manifolds of scalar products, n-level quantum systems, scalar product as a dynamical variable, essential non-perturbative nonlinearity, conservation laws, GL_n(C)-invariance

Discussed is the quantized version of the classical description of collective and internal affine modes as developed in Part I. We perform the Schroedinger 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.

Collective modes, affine invariance, Schroedinger quantization, quantum many-body problem

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.

Collective modes, affine invariance, integrable lattices, nonlinear elasticity

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.

Lie groups, homogeneous spaces, defomable bodies, left and right affine invariance, geodetic models, classical and quantum mechanics

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.

Affinely-rigid body, Riemannian metrics, geodetic problems, two-polar decomposition, Hamiltonian systems

Discussed is the Klein-Gordon-Dirac equation, i.e. a linear differential equation with constant coefficients, obtained by superposing Dirac and d'Alembert operators. A general solution of KGD equation as a superposition of two Dirac plane harmonic waves with different masses has been obtained. The multiplication rules for Dirac bispinors with different masses have been found. Lagrange formalism has been applied to receive the energy-momentum tensor and 4-current. It appears, in particular, that the scalar product is a superposition of Klein-Gordon and Dirac scalar products. The primary approach to canonical formalism is suggested. The limit cases of equal masses and one zero mass have been calculated.

Klein-Gordon-Dirac equation, plane harmonic waves with different masses, Dirac bispinors, Lagrange formalism, canonical formalism

The classical mechanics of structured test particles in a manifold with affine connection is studied. Gyroscopic rotations and homogeneous deformations are taken into account as internal degrees of freedom. Hence, in addition to the orbital motion of the centre of mass, the body undergoes affine rotations about the centre (“affinely-rigid body”). Configurations of particles are described mathematically by linear frames in an underlying manifold (physical space).

Symmetries of the theory are discussed and in some special cases the equations of motion are derived. The orbital motion is found to be influenced by internal degrees of freedom which are dynamically coupled to the geometry of the manifold. For example, in a Riemann-Cartan space ([4], [8], [9], [25]), internal degrees of freedom interact with curvature and torsion tensors. Imposing then some holonomic constraints (orthonormal frames only), one gets the theory of test rigid bodies in a curved space with torsion. In a Riemannian case (no torsion) such a theory seems to coincide with non-relativistic (although spatially non-Euclidean) limit of theories studied by Dixon, Künzle, Tulczyjew and Papapetrou [5], [13], [26], [15].

As for all standard techniques and notations of differential geometry, we follow mainly Kobayashi-Nomizu [12] and Sternberg [23].

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.

This paper is motivated by the Weyl-Wigner-Moyal approach to quantum mechanics [8], [12]. The nonlocal multiplication of phase space functions is generalized to functions over locally compact Abelian groups or over homogeneous spaces of such groups. It is shown that such nonlocal algebras of functions are closely related to the ray-representations of the corresponding dual groups. An arbitrary continuous solution of the factor- equation on Ĝ gives rise to some translationally invariant nonlocal algebra over G or over its homogeneous space. This enables us to find the group-theoretical interpretation of the well-known symmetrization rules (e.g. the standard, antistandard and Weyl rules [2]). Some properties of the nonlocal products are studied (e.g. the existence of the identity). It should be noted that some of our results (e.g. the theory of weighted convolutions of measures) are valid in the non-Abelian cases, as well, and they generalize some ideas of Edwards and Lewis [14]. When working with the Fourier analysis on the locally compact Abelian groups, we mostly use the language of W. Rudin's book [10].

The aim of this paper is to find classical counterparts of pure quantum states. It is shown that these are singular probability distributions concentrated on the so-called maximal null manifolds in a phase space. They are equivalent to densities studied by Van Vleck and Schiller and to WKB solutions (cf. Van Vleck, 1928; Schiller, 1962). Properties of such distributions and their relativistic generalisations have been studied in previous papers (Sławianowski, 1971; Slawianowski, 1972). However, it has not been shown there that such distributions arise actually in the limith→0. When working with the standard apparatus of differential geometry we mostly use the language of Kobayashi & Nomizu (1963).

The paper is a second part of [9], where the basic concepts and ideas used here, were introduced. We present an approach to the quasiclassical problems, based on the formalism given by Van Vleck [15] and Schiller [5], [6]. We show that the phase-space geometry contains a priori some information about the probability distribution for quasi-classical ensembles. That approach is generalized to homogeneous mechanics and, in particular, to relativistic mechanics. Van Vleck relativistic ensembles are constructed.

Reviewed are ideas underlying our concept of affinely-rigid body. We do this from the perspective of the Leonhard Euler two main achieve- ments, known under his name: the mechanics of rigid body and the dynamics of incompressible ideal fluid. But we formulate the theory which is some-how placed between those two models. Our scheme is a finite-dimensional dynamical system, but admitting deformative degrees of freedom. We also stress the connection with the general idea of affine invariance in physics.

igid body, incompressible ideal fluid, affine invariance, finite-dimensional dynamical systems, deformative degrees of freedom

In the previous lecture we have introduce and discussed the concept of affinely-rigid, i.e., homogeneously deformable body. Some symmetry problems and possible applications were discussed. We referred also to our motivation by Euler ideas. Below we describe the general principles of the quantization of this theory in the Schroedinger language. The special stress is laid on highly-symmetric, in particular affinely-invariant, models and the Peter-Weyl analysis of wave functions.

Homogeneously deformable body, Schroedinger quantization, affine invariance, highly symmetric models, Peter-Weyl analysis

Discussed are some classical and quantization problems of the affinely-rigid body in two dimensions. Strictly speaking, we consider the model of the harmonic oscillator potential and then discuss some natural anharmonic modifications. It is interesting that the considered doubly-isotropic models admit coordinate systems in which the classical and Schrödinger equations are separable and in principle solvable in terms of special functions on groups.

affinely-rigid body, quantization problems, two-dimensional models, harmonic oscillator potential, anharmonic modifications, doubly-isotropic models, Schrödinger equation, special functions, separability problem

Studied is the problem of degeneracy of mechanical systems the configuration space of which is the three-dimensional sphere, the elliptic space, i.e., the quotient of that sphere modulo the antipodal identification, and finally, the three-dimensional pseudo-sphere, namely, the Lobatchevski space. In other words, discussed are systems on groups SU(2), SO(3,R), and SL(2,R) or its quotient SO(1,2). The main subject are completely degenerate Bertrand-like systems. We present the action-angle classical description, the corresponding quasi-classical analysis and the rigorous quantum formulas. It is interesting that both the classical action-angle formulas and the rigorous quantum mechanical energy levels are superpositions of the flat-space expression, with those describing free geodetic motion on groups.

action-angle description, Bertrand systems, completely degenerate problems, elliptic space, Lobatchevski space, quasi-classical analysis, sphere

Discussed are classical and quantized models of affinely rigid motion with degenerate dimension, i.e., such ones that the geometric dimensions of the material and physical spaces need not be equal to each other. More precisely, the material space may have dimension lower than the physical space. Physically interesting are special cases m = 2 or m = 1 and n = 3, first of all m = 2, n = 3, i.e., roughly speaking, the affinely deformable coin in three–dimensional Euclidean space. We introduce some special coordinate systems generalizing the polar and two–polar decompositions in the regular case. This enables us to reduce the dynamics to two degrees of freedom. In quantum case this is the reduction of the Schrödinger equation to multicom ponent wave functions of two deformation invariants.

affinely rigid motion, degenerate dimension, polar and two-polar decompositions, multicomponent wave function, Schrödinger equation, quantized models, deformation invariants

Discussed is the problem of the mutual relationship of differentially first-ordr and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order Lagrangians are primary, and in case more adequate. It is shown that in principle, the Schodinger idea about Lagrangians which are quadratic in derivatives, and leading to second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal or rather its universal covering SU(2,3) as a gauge group. This has also some influence on the theory of defect in continua.

field theory, gauge group, Lagrangians

Our work has been inspired among others by the work of Arnold, Kozlov and Neihstadt. Our goal is to carry out a thorough analysis of the geometric problems we are faced with in the dynamics of affinely rigid bodies. We examine two models: classical dynamics description by d'Alembert and vakonomic one. We conclude that their results are quite different. It is not yet clear which model i practically better.

Affine motion, non-holonomic constraints, d’Alembert and Lusternik variational principles, vakonomic constraints

The paper is motivated by gauge theories of gravitation and condensed matter, tetrad models of gravitation and generalized Born-Infeld type nonlinearity. The main idea is that any generally-covariant and GL(n,R)-invariant theory of the n-leg field (tetrad field when n=4) must have the Born-Infeld structure. This means that Lagrangian is given by the square root of the determinant of some second-order twice covariant tensor built in a quadratic way of the field derivatives. It is shown taht there exist interesting solutions of the group-theoretical structure. Some models of the interaction between gravitation and matter are suggested. It turns out that in a sense the space-time dimension n=4, the normal-hyperbolic signature and velocity of light are integration constants of our differential equations.

gauge gravitation, linear frames, condensed matter

Schrödinger equation as a self-adjoint differential equation of mathematical physics is discussed. For simplicity, a finite-level system is considered. A modified Schrödinger equation with the second time derivatives is described and some direct nonlinearity is admitted. The key of our idea is the assumption that the scalar product is not fixed once for all, but is a dynamical quantity mutually interacting with the state vector. We assume that the Lagrangian term describing its dynamics has the large symmetry group, the total complex linear group. This implies the strong essential nonlinearity. There is a hope that this geometrically implied nonlinearity may explain the decoherence and measurement paradoxes in quantum mechanics.

Schrödinger equation, Krawietz-type matrices, affine rigid body

The concept of n-dimensional projectively-rigid body is introduced and its connection to the concept of (n+ 1)-dimensional incompressible affinely-rigid body is analysed. The equations of geodetic motion for such a projectively-rigid body are obtained. As an instructive example, the special case of n = 1 is investigated.

Projectively-rigid bodies, incompressible affinely-rigid bodies, left and right invariant problems, geodetic motion