Quasi-Set-Theoretical Foundations of Statistical Mechanics: A Research Program

Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell–Boltzmann statistics, it is not necessary to assume that the particles are distinguishable or individuals. In other words, Maxwell–Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of view. The main goal of this paper is to provide the mathematical grounds of a quasi-set theoretical framework for statistical mechanics.     

Solution of the Dirac Equation on the Bertotti–Robinson Metric

It is shown that each component of the Dirac field satisfies a decoupled equation, which admits separable solutions, when the background spacetime is the Bertotti–Robinson metric, which is a solution of the Einstein vacuum field equations with a cosmological constant. Furthermore, the seperated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigenfunctions of a certain differential operator.     

A Dicke Type Model for Equilibrium BEC Superradiance

We study the effect of electromagnetic radiation on the condensate of a Bose gas. In an earlier paper we considered the problem for two simple models showing the cooperative effect between Bose–Einstein condensation and superradiance. In this paper we formalize the model suggested by Ketterle et al. in which the Bose condensate particles have a two level structure. We present a soluble microscopic Dicke type model describing a thermodynamically stable system. We find the equilibrium states of the system and compute the thermodynamic functions giving explicit formulæ expressing the cooperative effect between Bose–Einstein condensation and superradiance.     

Bose–Einstein Condensation for Homogeneous Interacting Systems with a One-Particle Spectral Gap

We prove rigorously the occurrence of zero-mode Bose–Einstein condensation for a class of continuous homogeneous systems of boson particles with superstable interactions. This is the first example of a translation invariant continuous Bose-system, where the existence of the Bose–Einstein condensation is proved rigorously for the case of non-trivial two-body particle interactions, provided there is a large enough one-particle excitations spectral gap. The idea of proof consists of comparing the system with specially tuned soluble models.     

Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

We apply the method of moving anholonomic frames with associated nonlinear connections to the (pseudo) Riemannian space geometry and examine the conditions when locally anisotropic structures (Finsler like and more general ones) could be modeled in the general relativity theory and/or Einstein–Cartan–Weyl extensions [1]. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einstein gravity theory on na–backgrounds provided with a set of na–map invariant conditions and local conservation laws. There are illustrated some examples when vacuum Einstein fields are generated by Finsler like metrics and chains of na–maps.     

Particle Creation in the Oscillatory Phase of Inflaton

A thermal squeezed state representation of inflaton is constructed for a flat Friedmann–Robertson–Walker (FRW) background metric and the phenomenon of particle creation is examined during the oscillatory phase of inflaton, in the semiclassical theory of gravity. An approximate solution to the semiclassical Einstein equation is obtained in thermal squeezed state formalism perturbatively and is found obey the same power-law expansion as that of classical Einstein equation. In addition to that the solution shows oscillatory in nature except on a particular condition. It is also noted that, the coherently oscillating nonclassical inflaton, in thermal squeezed vacuum state, thermal squeezed state, and thermal coherent state, suffers particle production and the created particles exhibit oscillatory behavior. The present study can account for the postinflation particle creation due to thermal and quantum effects of inflation in a flat FRW universe.     

Curvature Vacuum Correlations in N-Dimensional Einstein Gravity

Under the flat Euclidean space–time background, the expressions for the leading terms of several two-point curvature vacuum correlation functions in N-dimensional Einstein gravity are calculated by using the perturbative expansion of the metric. It is shown that the contributions of the leading terms of such two-point curvature vacuum correlation functions are all vanishing.     

The Conformal Metric Associated with the U(1) Gauge of the Stueckelberg–Schrödinger Equation

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates.     

Locally Anisotropic Structures and Nonlinear Connections in Einstein and Gauge Gravity

We analyze locally anisotropic configurations modeled by anholonomic frames with associated nonlinear connections in general relativity, affine–Poincarè and/or de Sitter gauge gravity and Kaluza–Klein theories. A suitable geometrical formalism for theories with higher order anisotropies and non compactified extra dimensions is introduced. We give a mostly self–containing review of some aspects of gauge models of gravity and discuss their anholonomic generalizations and the conditions of equivalence with the Einstein gravity in arbitrary dimensions. New classes of cosmological solutions describing Friedmann–Robertson–Walker like universes with resolution ellipsoid or torus symmetry.     

Measurement of genuine three-particle Bose–Einstein correlations in hadronic Z decay

We measure three-particle Bose–Einstein correlations in hadronic Z decay with the L3 detector at LEP. Genuine three-particle Bose–Einstein correlations are observed. By comparing two- and three-particle correlations we find that the data are consistent with fully incoherent pion production.     

A New Way of Solving the Wheeler–DeWitt Equation as Applied to Point Sources

A new way of solving the Wheeler–DeWitt equation is proposed which is based on quantization over free parameters of metrics satisfying the Einstein equations. This technique is applied to two point sources described in the classical case by the Tangherlini metric (in an n-dimensional space) and the Reissner–Nordström metric (in the case of the presence of a charge). The results obtained clarify the sense of the Wheeler hypothesis about statistical weights of small dimensionalities and make possible a new approach to the problem of variation of fundamental constants.     

A continuation BSOR-Lanczos–Galerkin method for positive bound states of a multi-component Bose–Einstein condensate

We develop a continuation block successive over-relaxation (BSOR)-Lanczos–Galerkin method for the computation of positive bound states of time-independent, coupled Gross–Pitaevskii equations (CGPEs) which describe a multi-component Bose–Einstein condensate (BEC). A discretization of the CGPEs leads to a nonlinear algebraic eigenvalue problem (NAEP). The solution curve with respect to some parameter of the NAEP is then followed by the proposed method. For a single-component BEC, we prove that there exists a unique global minimizer (the ground state) which is represented by an ordinary differential equation with the initial value. For a multi-component BEC, we prove that m identical ground/bound states will bifurcate into m different ground/bound states at a finite repulsive inter-component scattering length. Numerical results show that various positive bound states of a two/three-component BEC are solved efficiently and reliably by the continuation BSOR-Lanczos–Galerkin method.     

An investigation of the dynamical evolution of a class of Bianchi VI−1/9 cosmological models

We consider the evolution of a subclass of the orthogonal spatially homogeneous cosmologies of Bianchi type VI_–1/9. Expansion normalized variables are introduced to write the Einstein field equations for these models as a three-dimensional autonomous system of ordinary differential equations. This system is analyzed qualitatively using the techniques of dynamical systems, and a cosmological interpretation of the phase portraits is given.     

Two-spinor Formulation of First-Order Gravity Coupled to Dirac Fields

A two-spinor formalism for the Einstein Lagrangian is developed. The gravitational field is regarded as a composite object derived from soldering forms. Our formalism is geometrically and globally well-defined and may be used in virtually any 4m-dimensional manifold with arbitrary signature as well as without any stringent topological requirement on space-time, such as parallelizability. Interactions and feedbacks between gravity and spinor fields are considered. As is well known, the Hilbert–Einstein Lagrangian is second order also when expressed in terms of soldering forms. A covariant splitting is then analysed leading to a first-order Lagrangian which is recognized to play a fundamental role in the theory of conserved quantities. The splitting and thence the first-order Lagrangian depend on a reference spin connection which is physically interpreted as setting the zero level for conserved quantities. A complete and detailed treatment of conserved quantities is then presented.     

Closed time-like curves in flat Lorentz space–times

We consider the region of closed time-like curves (CTCs) in three-dimensional flat Lorentz space–times. The interest in this global geometrical feature goes beyond the purely mathematical one. Such space–times are lower-dimensional toy models of sourceless Einstein gravity or cosmology. In three dimensions all such space–times are known: they are quotients of Minkowski space by a suitable group of Poincaré isometries. The presence of CTCs would indicate the possibility of “time machines”, a region of space–time where an object can travel along in time and revisit the same event. Such space–times also provide a testbed for the chronology protection conjecture, which suggests that quantum back reaction would eliminate CTCs. In particular, our interest in this note will be to find the set free of CTCs for , where is modeled on Minkowski space and γ is a Poincaré transformation. We describe the set free of CTCs where γ is hyperbolic, parabolic, and elliptic.     

Nonsingular Collapse of a Perfect Fluid Sphere Within a Dilaton-Gravity Reformulation of the Oppenheimer Model

A generic four-dimensional dilaton gravity is considered as a basis for reformulating the paradigmatic Oppenheimer–Synder model of a gravitationally collapsing star modelled as a perfect fluid or dust sphere. Initially, the vacuum Einstein scalar-tensor equations are modified to Einstein–Langevin equations which incorporate a noise or micro-turbulence source term arising from Planck scale conformal, dilaton fluctuations which induce metric fluctuations. Coupling the energy-momentum tensor for pressureless dust or fluid to the Einstein–Langevin equations, a modification of the Oppenheimer–Snyder dust collapse model is derived. The Einstein–Langevin field equations for the collapse are of the form of a Langevin equation for a non-linear Brownian motion of a particle in a homogeneous noise bath. The smooth worldlines of collapsing matter become increasingly randomised Brownian motions as the star collapses, since the backreaction coupling to the fluctuations is non-linear; the input assumptions of the Hawking–Penrose singularity theorems are then violated. The solution of the Einstein–Langevin collapse equation can be found and is non-singular with the singularity being smeared out on the correlation length scale of the fluctuations, which is of the order of the Planck length. The standard singular Oppenheimer–Synder model is recovered in the limit of zero dilaton fluctuations.     

Black Holes in Bose–Einstein Condensates

It is shown that there exist both dynamically stable and unstable dilute-gas Bose–Einstein condensates that, in the hydrodynamic limit, exhibit a behavior completely analogous to that of gravitational black holes. The dynamical instabilities involve creation of quasiparticle pairs in positive and negative energy states. We illustrate these features in two qualitatively different one-dimensional models. We have also simulated the creation of a stable sonic black hole by solving the Gross–Pitaevskii equation numerically for a condensate subject to a trapping potential that is adiabatically deformed. A sonic black hole could in this way be created experimentally with state-of-the-art or planned technology.     

Charge in cosmological models

The balanced field equations due to Penney are used to find solutions for cosmological models in the presence of charge. Herein, it is found that the introduction of charge adds additional terms to the Einstein conservation equation and distribution expressions. The curvature parameter is affected and it is concluded that whereas matter affects it positively, the charge does so negatively. There then arises also the possibility of an evolution of local systems against the background of a global expansion.     

Quasi-gravity in Branes

In contrast with pseudo-gravitational effects that are mathematically analogous but physically quite distinct from gravity, this presentation deals with a kind of quasi-gravitational effect that can act in an asymmetrically moving brane worldsheet in a manner that approximates (and in a crude analysis might be physically indistinguishable from) the effect that would arise from genuine gravitation, of ordinary Newtonian type in nonrelativistic applications and of scalar–tensor (Jordan–Brans–Dicke rather than pure Einstein) type in relativistic applications.     

Effective Vilcovisky-De witt action in mutidimensional R2 gravitation

The gauge- and parametrization-invariant Vilcovisky-De Witt effective action (EA) in multidimensional R² gravitation against R₄×T_d–4 background, where R₄ is a four-dimensional flat space and T_d is a d-dimensional torus, is calculated in the one-loop approximation. It is shown that with the help of a limiting procedure the Vilcovisky-De Witt EA in multidimensional Einstein gravitation against the background R₄×T_d–4 can be constructed from the EA found. The question of the stability of spontaneous compactification is analyzed taking into account quantum effects in d=5 R² gravitation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 106–111, October, 1989.I thank I. L. Bukhbinder and I. V. Tyutin for useful discussions.