Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.     

Horizon thermodynamics and gravitational field equations in quasi-topological gravity

In this paper we show that the gravitational field equations of $(n+1)$ -dimensional topological black holes with constant horizon curvature, in cubic and quartic quasi-topological gravity, can be recast in the form of the first law of thermodynamics, $dE=TdS-PdV$ , at the black hole horizon. This procedure leads to extract an expression for the horizon entropy as well as the energy (mass) in terms of the horizon radius, which coincide exactly with those obtained in quasi-topological gravity by solving the field equations and using the Wald’s method. We also argue that this approach is powerful enough to be extended to all higher order quasi-topological gravity for extracting the corresponding entropy and energy in terms of horizon radius.     

More qualitative cosmology

Standard geometric techniques of differential equation theory are employed to determine the qualitative behaviour of a set of non-rotating perfect-fluid cosmologies, whose spatially homogeneous hypersurfaces admit a 3-parameter group of isometries of Bianchi types I, II, III, V, or VI. In this way we are led to some new exact solutions of the field equations.The field equations for a broad class of cosmological models are presented in a regularised form, limitations on the use of this procedure are examined, and some suggestions are made of ways of avoiding the difficulties that arise.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

A class of new LRS Bianchi type-I perfect fluid universes with decaying vacuum energy density Λ

A class of new LRS Bianchi type-I cosmological models with a variable cosmological term is investigated in presence of perfect fluid. A procedure to generate new exact solutions to Einstein’s field equations is applied to LRS Bianchi type-I space-time. Starting from some known solutions a class of new perfect fluid solutions of LRS Bianchi type-I are obtained. The cosmological constant Λ is found to be positive and a decreasing function of time which is supported by results from recent supernovae Ia observations. The physical and geometric properties of spatially homogeneous and anisotropic cosmological models are discussed.     

Why does shear banding behave like first-order phase transitions? Derivation of a potential from a mechanical constitutive model

Numerous numerical and experimental evidence suggest that shear banding behavior looks like first-order phase transitions. In this paper, we demonstrate that this correspondence is actually established in the so-called non-local diffusive Johnson-Segalman model (the DJS model), a typical mechanical constitutive model that has been widely used for describing shear banding phenomena. In the neighborhood of the critical point, we apply the reduction procedure based on the center manifold theory to the governing equations of the DJS model. As a result, we obtain a time evolution equation of the flow field that is equivalent to the time-dependent Ginzburg-Landau (TDGL) equations for modeling thermodynamic first-order phase transitions. This result, for the first time, provides a mathematical proof that there is an analogy between the mechanical instability and thermodynamic phase transition at least in the vicinity of the critical point of the shear banding of DJS model. Within this framework, we can clearly distinguish the metastable branch in the stress-strain rate curve around the shear banding region from the globally stable branch. A simple extension of this analysis to a class of more general constitutive models is also discussed. Numerical simulations for the original DJS model and the reduced TDGL equation is performed to confirm the range of validity of our reduction theory.     

Palatini formulation of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity theory,and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type \(f=R-\alpha ^2/R+g(T)\) and \(f=R+\alpha ^2R^2+g(T)\) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.     

Lovelock gravity from entropic force

In this paper, we first generalize the formulation of entropic gravity to ( $n+1$ )-dimensional spacetime and derive Newton’s law of gravity and Friedmann equation in arbitrary dimensions. Then, we extend the discussion to higher order gravity theories and propose an entropic origin for Gauss–Bonnet gravity and more general Lovelock gravity in arbitrary dimensions. As a result, we are able to derive Newton’s law of gravitation as well as the corresponding Friedmann equations in these gravity theories. This procedure naturally leads to a derivation of the higher dimensional gravitational coupling constant of Friedmann/Einstein equation which is in complete agreement with the results obtained by comparing the weak field limit of Einstein equation with Poisson equation in higher dimensions. Our strategy is to start from first principles and assuming the entropy associated with the apparent horizon given by the expression previously known via black hole thermodynamics, but replacing the horizon radius $r_+$ with the apparent horizon radius $R$ . Our study shows that the approach presented here is powerful enough to derive the gravitational field equations in any gravity theory and further supports the viability of Verlinde’s proposal.     

The class of exact cosmological models with a scalar-field nonlinearity induced by the Yang-Mills field

A new class of exact solutions of the system of Einstein-Yang-Mills equations and the equations of a nonlinear scalar field interacting with the Yang-Mills SO(3) field is obtained on the basis of the Friedman spatially homogeneous cosmological models.     

Some cosmological models in the bimetric theory of gravitation

It is shown that the field equations of the bimetric theory of gravitation have solutions corresponding to a class of homogeneous isotropic cosmological models with negative spatial curvature (k=–1). Some examples are given.     

On Quantization in Light-cone Variables Compatible with Wavelet Transform

Canonical quantization of quantum field theory models is inherently related to the Lorentz invariant partition of classical fields into the positive and the negative frequency parts u(x) = u⁺(x) + u^?(x), performed with the help of Fourier transform in Minkowski space. That is the commutation relations are being established between nonlocalized solutions of field equations. At the same time the construction of divergence free physical theory requires the separation of the contributions of different space-time scales. In present paper, using the light-cone variables, we propose a quantization procedure which is compatible with separation of scales using continuous wavelet transform, as described in our previous paper (Altaisky, M.V., Kaputkina, N.E.: Phys. Rev. D 88, 025015 2013).     

A Valid Finite Bounded Expanding Carmelian Universe Without Dark Matter

The solution of Einstein’s field equations in Cosmological General Relativity (CGR), where the Galaxy is at or cosmologically near the center of a finite yet bounded spherically symmetrical isotropic gravitational field, is identical with the unbounded solution. I show that this leads to the conclusion that the Universe may be viewed as a finite expanding “white hole.” This is not the white hole solution of Einstein’s spacetime but has similarities to it. The fact that CGR has been successful in describing the distance modulus verses redshift data of the high-redshift type Ia supernovae means that the data cannot be used to distinguish between unbounded models and those with finite bounded radii of at least cτ ( ${\approx}c H_{0}^{-1}$ ). According to Carmelian theory, whether or not the Universe is finite bounded or unbounded it is spatially flat all epochs where it was matter dominated. As a consequence in a finite bounded universe no gravitational redshift in the light from distant source galaxies would be observed.     

A criticism of the standard treatment of phonon drag

By the study of a simple example, namely the evolution in timet of an electron-phonon system with fixed, total momentum, it is shown that the “standard” treatment of “phonon drag”, which involves solving the (linearized and spatially homogeneous) coupled electron and phonon Boltzmann equations by an iteration procedure, is not always correct. In the asymptotic limit (t→∞), the iteration or “standard” procedure does not give the “correct” (i.e. the equilibrium statistical mechanical) result for the distribution of momentum between electrons and phonons. However, a proper treatment of the Boltzmann equations does lead to the “correct” sharing of momentum between electrons and phonons fort→∞. All the calculations in this paper are performed for metals at high temperatures (i.e.,T>Θ_D, the Debye temperature).     

A fragment-based approach towards <Emphasis Type="Italic">ab-initio</Emphasis> treatment of polymeric materials

The present study deals with hypersurface-homogeneous cosmological models with anisotropic dark energy in Saez–Ballester theory of gravitation. Exact solutions of field equations are obtained by applying a special law of variation of Hubble’s parameter that yields a constant negative value of the deceleration parameter. Three physically viable cosmological models of the Universe are presented for the values of parameter K occurring in the metric of the space–time. The model for K = 0 corresponds to an accelerating Universe with isotropic dark energy. The other two models for K = 1 and ?1 represent accelerating Universe with anisotropic dark energy, which isotropize for large time. The physical and geometric behaviours of the models are also discussed.     

Discrete isotropies in a class of cosmological models

It is shown that a certain class of cosmological models admits discrete isotropies. These models are solutions of Einsteins field equations, characterised by: (1) the matter is described as a perfect fluid, and (2) there exists a group of motions simply transitive on three-surfaces orthogonal to the fluid flow vector.     

Binary Mixture of Perfect Fluid and Dark Energy in Modified Theory of Gravity

A self consistent system of Plane Symmetric gravitational field and a binary mixture of perfect fluid and dark energy in a modified theory of gravity are considered. The gravitational field plays crucial role in the formation of soliton-like solutions, i.e., solutions with limited total energy, spin, and charge. The perfect fluid is taken to be the one obeying the usual equation of state, i.e., p = γρ with γ∈ [0, 1] whereas, the dark energy is considered to be either the quintessence like equation of state or Chaplygin gas. The exact solutions to the corresponding field equations are obtained for power-law and exponential volumetric expansion. The geometrical and physical parameters for both the models are studied.     

Invariant Solutions of Inhomogeneous Universe with Electromagnetic Field in Lyra Geometry

The present study deals with the cylindrically symmetric inhomogeneous cosmological models for perfect fluid distribution with electro-magnetic field in Lyra geometry. Lie group analysis has been used to identify the generators (symmetries) that leave the given system of partial differential equations (field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of partial differential equations is reduced to an ordinary differential equations whose simple solutions provide nontrivial solutions of the original system. They obtained a new class of invariant (similarity) solutions by considering the potentials of metric and displacement field are functions of coordinates t and x. The physical behavior of the derived models are also discussed.     

Exactly solvable models of conformal-invariant quantum field theory in D-dimensional space

The method for exact solution of a certain class of models of conformal quantum field theory in D-dimensional Euclidean space is proposed. The method allows one to derive closed differential equations for all the Green functions and also algebraic equations to scale dimensions of all field. A scalar field P of a scale dimension d_p = D − 2 is needed for nontrivial solutions to exist. At D ≠ 2 this field is converted to a constant that coincides with the central charge of two-dimensional theories. A new class of D = 2 models has been obtained, where the infinite-parametric symmetry is not manifest. The two-dimensional Wess-Zumino model is used to illustrate the method of solution.