Quantum Origin of a Rotating Universe of the Bianchi IX Type

A closed cosmological model with rotation of the Bianchi IX type is constructed. Λ-term and anisotropic liquid are the sources of gravitational field for the model. A quantum origin of the Universe is examined. The Wheeler — De Witt equation is derived. A tunneling coefficient for the Universe is calculated. It is found for a particular case that the probability of quantum birth of a rotating Universe is higher than for the model without rotation. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 71–75, June, 2005.     

Operator ordering, ellipticity and spurious solutions in k · p calculations of III-nitride nanostructures

We analyze the ellipticity of the standard k · p wurtzite model for the symmetrized and the Burt–Foreman operator ordering. We find that for certain situations the symmetrized Hamiltonian is unstable, leads to unplausible results and can cause spurious solutions. We show that the operator ordering in wurtzite must be completely asymmetric to be stable. The asymmetric operator ordering is elliptic and consequently no spurious solutions are obtained. Therefore we recommend the use of a complete asymmetric operator ordering for nitride device simulation.     

Behavior of a charged particle in a schwarzschild field

The influence of the De Witt self-action force on the motion of and electromagnetic emission from a charged particle in a Schwarzschild field is considered. It is shown that a charged particle in a Schwarzschild field is equivalent to a neutral particle of the same mass in a certain Reissner-Nordstrom field. A relationship is found between the power of the electromagnetic emission from an accelerated charge and the power of the thermal emission generated in a reference frame with the same acceleration at the event horizon. The quantum-mechanical problem of the motion of and emission from a charge in the field of a minihole is considered. Wave functions, the energy spectrum, and the widths of quasi-stationary levels are found with allowance for the De Witt self-action force. It is shown that the latter is important for large charges, when the solution becomes oscillatory. "Brainstorm" Little Science and Technology Enterprise. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 75–82, July, 1998.     

Quantum Birth of the Rotating Universe

Quantum birth of the Universe of the Bianchi type IX filled by a rotating anisotropic liquid is studied. The Wheeler–De Witt equation is derived for the model under consideration. The tunneling wave function as a solution to the equation is found using the WKB method. The tunneling coefficient of the Universe is calculated. The probabilities of quantum birth of the Universe with and without rotation are compared for different formulations of the problem.     

Static solutions of the Brans-Dicke equations

It is shown that, given any static solution of the Einstein vacuum equations, a corresponding family of static vacuum solutions of the Brans-Dicke equations can be written down by inspection. Spherically and axially symmetric fields are considered explicitly. It is demonstrated how some solutions of the Brans-Dicke equations may be obtained without having to solve any field equations explicitly at all.     

Mathieu and Heun Solutions to the Wheeler–De Witt Equation for Hyperbolic Universes

For the spatially open Friedmann–Robertson–Walker (FRW) Universe with stiff matter and radiation as non-interacting matter sources, the scale function coming from the integration of the Friedmann equation is expressed in terms of elliptic integrals. For a negative cosmological constant, we identify the allowed ranges for the model’s parameters. Within the quantum analysis, the Wheeler–De Witt (WDW) equation turns into a modified Morse equation whose solutions are Mathieu and Heun functions.     

Perfect fluid metrics conformal to the Schwarzschild exterior spacetime

We construct perfect fluid spacetimes by performing a conformal transformation on a non-conformally flat vacuum solution, namely the Schwarzschild exterior metric. It should be noted that conformally Ricci flat perfect fluid solutions, except those that are conformally flat, are rarely reported explicitly. In this article it is demonstrated that perfect fluid metrics conformal to the Schwarzschild exterior line element are necessarily static. The Einstein field equations for the static case reduce to a fully determined system of three differential equations in three unknowns and the conformal factor is uniquely determined in closed form. The solution is analysed for physical plausibility by establishing the positivity of the energy density and pressure profiles graphically. Additionally, the solution is observed to be causal in an appropriate limit and both the energy density and pressure is shown to be decreasing outwards towards the boundary. Finally, the weak, strong and dominant energy conditions are found to be satisfied in the region under investigation. Accordingly, the most common elementary physical conditions are met and the model is seen to be suitable for a core-envelope stellar model.     

Wave Functions for Arbitrary Operator Ordering in the de Sitter Minisuperspace Approximation

We derive exact series solutions for the Wheeler–DeWitt equation corresponding to a spatially closed Friedmann–Robertson–Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The resulting wave functions are those relevant to the approximation which has been widely used in two-dimensional minisuperspace models with an inflationary scalar field for the purpose of predicting the period of inflation which results from competing boundary condition proposals for the wave function of the universe. The problem that Vilenkin's tunneling wave function is not normalizable for general operator orderings, is shown to persist for other values of the spatial curvature, and when additional matter degrees of freedom such as radiation are included.     

Quantization of the space－time with topological defect

We present the classical solution of Lagrange equations for the black hole with a global monopole or with a cosmic string. Then we obtain the wavefunction of the space－time by solving the Wheeler－De Witt equation. De Broglie－Bohm interpretation applied to the wavefunction gives the quantum solution of the space－time. In the end, the quantum effect on Hawking radiation is studied.     

Evolution of quantum states via Weyl expansion in dissipative channel

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.     

On integrating Einstein's equations using the Bondi metric

It is demonstrated that a closed form, time-dependent, expression for the axisymmetric Bondi metric may be obtained through a straightforward integration of Einstein's equations. The solution is shown to be a special case of Petrov Type I solutions having shearing geodesic rays (Newman-Tamburino). The evolution equation is expanded in powers of 1/r and it is found that the expansions can be useful in identifying other potential solutions.     

Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

We propose a model of a spinless electrical charge as a self-consistent field configuration of the electromagnetic (EM) field interacting with a physical vacuum effectively described by the logarithmic quantum Bose liquid. We show that, in contrast to the EM field propagating in a trivial vacuum, a regular solution does exist, and both its mass and spatial extent emerge naturally from dynamics. It is demonstrated that the charge and energy density distribution acquire Gaussian-like form. The solution in the logarithmic model is stable and energetically favourable, unlike that obtained in a model with a quartic (Higgs-like) potential.     

Self-consistent random phase approximation within the O(5) model and Fermi transitions

Self-consistent quasiparticle random phase approximation (SCQRPA) is considered in application to the Fermi transitions within the O(5) model. It is demonstrated that SCQRPA improves on renormalized QRPA (RQRPA), a method that has recently become rather popular in this context. The analytical form of the SCQRPA vacuum is used to evaluate all the matrix elements. The SCQRPA results show a general trend similar to the exact solutions. The necessity to change the single particle basis beyond the transition point, and to include the proton-proton and neutron-neutron channels in the QRPA operator, in addition to the proton-neutron one, is pointed out.     

Instantons and vacuum expectation values in the Schwinger model

Vacuum expectation values for the Schwinger model are computed within the path integral formalism for an instanton vacuum. The calculations are done in flat Euclidean space and use the wellknown anomaly result. It can be demonstrated that the instanton vacuum is absolutely necessary to obtain a consistent quantization. At last the results are compared to a similar treatment on the sphere and to the operator solution of the Schwinger model.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Projekt Nr. P8444-TEC     

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.     

Operator product expansions in the massless Schwinger model

The known operator solution of the massless Schwinger model is used to calculate exactly, in three operator product expansions, the coefficient functions of the first few operators of low dimension which contribute when vacuum matrix elements are to be taken. A comparison of the results provides a test of the procedure used by M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov [Nucl. Phys. B147 (1979), 385–447] in their study of QCD. It is found that the shift in vacua does not affect the calculation of coefficient functions. The vacuum insertion approximation yields somewhat misleading estimates of vacuum expectation values of some composite operators; however, the standard method used to estimate the errors of vacuum insertion indicates that the approximation is unreliable in this model.     

Photon-Number Distribution and Wigner Function of Generalized Squeezed Thermal State

In this paper, we present the density operator of the generalized squeezed thermal state (GSTS) and obtain its normal ordering form by virtue of the technique of integration within an ordered product of operators and the Weyl ordering invariance under similarity transformations. Some significant quantum statistical properties of the GSTS are investigated, such as the photon-number distribution (PND) and the Wigner function (WF). It is found that the PND of the GSTS is a Legendre polynomial, and its squeezed vacuum oscillations imply the nonclassicality, as well as the GSTS whose WF has no negative region is indeed a special type of nonclassical state.     

Cosmological Friedman models with the charged vacuum

Exact solutions corresponding to the stable charged vacuum are obtained for cosmological Friedman models with a charged scalar field. It is demonstrated that the presence of a massive vector field for the vacuum changes the law of expansion at early stages of evolution of the universe. A method for construction of solutions with the charged vacuum for an O(N) invariant scalar multiplet is given.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 19–24, December, 1985.     

The Fourth-Order Non-Linear Sigma Models and Asymptotic Freedom in Four Dimensions

Starting with the most general Lagrangian of the fourth-order non-linear sigma model in four space-time dimensions, we calculate the one-loop, on shell ultra-violet-divergent part of the effective action. The formalism is based on the background field method and the generalised Schwinger De Witt technique. The multiplicatively renormalisable case is investigated in some detail. The renormalisation group equations are obtained, and the conditions for a realisation of asymptotic freedom are considered.