abstract

It seems highly plausible that in changing from classical general relativity to the quantum gravity regime there is a stage at which in some regions classical regime still remains in power (and the size of these regions gradually shrinks to zero), whereas at some other places topological anomalies, changes of the metric signature, metric degeneracies, and so on, take over. The Lorentz metric behaviour in such situations is investigated in terms of the theory of differential spaces. It turns out that a Lorentz metric can locally exist on space-time (modeled by a differential space) only if the differential space in question can locally be immersed in a Minkowski space of a suitably large dimension. It is shown that in such a differential space, provided it is Hausdorff, the regions at which the manifold structure breaks down can have only the form of edges and vertices of a “lattice of crystals” on the faces of which the dimension is constant. The question of how locally defined Lorentz metrics could be “glued together” to form a “global metric field” is also considered. Some suggestions are discussed concerning the transition to the radically non-classical regime.

abstract

Equations of gravitation in the Lobachevsky space are formulated. The problem of the gravitational field of point mass in the Lobachevsky space is solved. In the Newtonian (nonrelativistic) case, this problem was posed and solved by Lobachevsky himself. In the relativistic case, one should first find adequate equations for the metric describing the gravitational field and then find their solutions. These equations are found by the author on the basis of the theory, developed by him, with two affine connections; one called Christoffel and the other, background. The latter is given by the equations of motion of free material particle in the Lobachevsky space. It is independent of the light velocity \(c\). The static spherically symmetric metric found here depends on the ratio of the gravitational radius \(\gamma Mc^{-2}\) of mass \(M\) to the Lobachevsky constant \(k\) for the visible world. In the limit \(k \to - \infty \) it turns into the well known Schwarzschild metric. The world line of a planet is geodesic with respect to this metric. The relativistic Kepler problem in the Lobachevsky space is reduced to a nonlinear differential equation.

abstract

Conventional approach to constructing a quantum model, consisting in canonical quantization of a related classical model, can be applied only if we know the classical model and its Hamiltonian formulation in advance. Approach based on a set of postulates reflecting properties of physical configuration space is more general. As an example, quantum mechanics of a particle on a space consisting of just two points is constructed.

abstract

We analyze ensembles of random matrices capable of describing the transitions between orthogonal, unitary and Poisson ensembles. Scaling laws found in complex Hermitian band random matrices and in additive random matrices allow us to apply them to represent the changes of the statistical properties of quantum systems under a variation of external parameters. The properties of spectrum and eigenvectors of an illustrative dynamical system are compared with the properties of ensembles of random matrices. To describe the motion of the eigenvectors of the matrix representing a dynamical system under a change of external parameters we define the relative localization length of the eigenvectors and analyze its properties. We propose a criterion for selection of generic basis, in which statistics of eigenvector components might be described by random matrices. The properties of products of unitary matrices, representing composed quantum systems, are investigated.

abstract

Multiparton correlations in a QCD jet are calculated analytically in the double logarithmic approximation for constant \(\alpha _s\). For a well developed cascade, i.e. far from the low energy cut-off, correlations show characteristic scale invariant power behaviour. Cumulant moments in restricted angular cells also follow the power dependence on the volume of a cell with known exponents.

abstract

We study the spin alignment of D\(^{*+}\)(2010) in its helicity frame for a very clean sample of 127 D\(^{*+}\)(2010) mesons produced in 230 GeV/\(c\) \(\pi ^-\)–Cu interactions. We measure the spin alignment parameter to be equal to \(\eta = 0.10^{+0.12}_{-0.11} \pm 0.01\). This parameter, within our statistics, does not depend on \(x_F\) or \(p_T\). We compare our results with statistical approach.