A supersymmetric Vista for quantum cosmology

A brief overview on the subject of Supersymmetric Quantum Cosmology (SQC) is presented here. Different approaches are described, all of them being inspired by the search of a square root of (quantum) General Relativity. Some new ideas in the form of a list of (still!) open problems and an extensive bibliography are included.     

A definition for time in quantum cosmology

A definition for the intrinsic time co-ordinate is proposed, using the phase of the wave function of the universe. This definition generalizes the notion of time co-ordinate which arises in the semiclassical cosmology. It also leads to acceptable results for the evaluation of expectation values of physical variables.     

Observational effects from quantum cosmology

The status of quantum cosmologies as testable models of the early universe is assessed in the context of inflation. While traditional Wheeler–DeWitt quantization is unable to produce sizable effects in the cosmic microwave background, the more recent loop quantum cosmology can generate potentially detectable departures from the standard cosmic spectrum. Thus, present observations constrain the parameter space of the model, which could be made falsifiable by near‐future experiments.     

Loop quantum cosmology: Recent progress

Aspects of the full theory of loop quantum gravity can be studied in a simpler context by reducing to symmetric models like cosmological ones. This leads to several applications where loop effects play a significant role when one is sensitive to the quantum regime. As a consequence, the structure of and the approach to classical singularities are very different from general relativity. The quantum theory is free of singularities, and there are new phenomenological scenarios for the evolution of the very early universe such as inflation. We give an overview of the main effects, focussing on recent results obtained by different groups.     

Noncommutative quantum cosmology

We consider noncommutative quantum cosmology in the case of the low-energy string effective theory. Exact solutions are found and compared with the commutative case.     

Non commuting quantum cosmology with scalar matter

We study noncommutative quantum and classical cosmology with a scalar field by means of deforming the minisuperspace. We analyze the homogeneous and isotropic model and compare our results with the usual commutative case. In particular the classical behaviour of the scalar field can dramatically change due to the presence of noncommutativity.We dedicate this work to Michael P. Ryan on the occasion of his sixtieth birthday. Around 30 years ago, Mike patiently introduced us to Quantum Cosmology. Since that time, we have had rich and friendly discussions and fruitful collaborations.     

Tomographic representation of minisuperspace quantum cosmology and noether symmetries

The probability representation, in which cosmological quantum states are described by a standard positive probability distribution, is constructed for minisuperspace models selected by Noether symmetries. In such a case, the tomographic probability distribution provides the classical evolution for the models and can be considered an approach to select “observable” universes. Some specific examples, derived from Extended Theories of Gravity, are worked out. We discuss also how to connect tomograms, symmetries and cosmological parameters.     

Loop quantum cosmology: an overview

A brief overview of loop quantum cosmology of homogeneous isotropic models is presented with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. These results bear out the remarkable intuition that John Wheeler had. Discussion is organized at two levels. The the main text provides a bird’s eye view of the subject that should be accessible to non-experts. Appendices address conceptual and technical issues that are often raised by experts in loop quantum gravity and string theory.     

Dust and radiation quantum perfect fluid cosmology: selection of time variable

We studied the expectation value of the scale factor in radiation and dust quantum perfect fluid cosmology. We used Schutzs variational formalism to describe the perfect fluid and selected the conjugate coordinate of the perfect fluid to be the dynamical variable. After quantization and solving the Wheeler-DeWitt equation we obtained an exact solution. By superposition of exact solutions, we obtained one wave packet and used it to compute the expectation value of the scale factor. We found that if one selects a different dynamical variable being the time variable in each of these two systems, the expectation value of the scale factor of these two systems can fit in with the prediction of General Relativity. Therefore we thought that the selection of a reference time can be different for different quantum perfect fluid systems.     

A quantum model of option pricing: When Black-Scholes meets Schrödinger and its semi-classical limit

The Black-Scholes equation can be interpreted from the point of view of quantum mechanics, as the imaginary time Schrödinger equation of a free particle. When deviations of this state of equilibrium are considered, as a product of some market imperfection, such as: Transaction cost, asymmetric information issues, short-term volatility, extreme discontinuities, or serial correlations; the classical non-arbitrage assumption of the Black-Scholes model is violated, implying a non-risk-free portfolio. From Haven (2002) [1] we know that an arbitrage environment is a necessary condition to embedding the Black-Scholes option pricing model in a more general quantum physics setting. The aim of this paper is to propose a new Black-Scholes-Schrödinger model based on the endogenous arbitrage option pricing formulation introduced by Contreras et al. (2010) [2]. Hence, we derive a more general quantum model of option pricing, that incorporates arbitrage as an external time dependent force, which has an associated potential related to the random dynamic of the underlying asset price. This new resultant model can be interpreted as a Schrödinger equation in imaginary time for a particle of mass 1/σ² with a wave function in an external field force generated by the arbitrage potential. As pointed out above, this new model can be seen as a more general formulation, where the perfect market equilibrium state postulated by the Black-Scholes model represent a particular case. Finally, since the Schrödinger equation is in place, we can apply semiclassical methods, of common use in theoretical physics, to find an approximate analytical solution of the Black-Scholes equation in the presence of market imperfections, as it is the case of an arbitrage bubble. Here, as a numerical illustration of the potential of this Schrödinger equation analogy, the semiclassical approximation is performed for different arbitrage bubble forms (step, linear and parabolic) and compare with the exact solution of our general quantum model of option pricing.     

A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics

In this paper we are concerned with the modeling of quantum dissipation and diffusion effects at the level of the multidimensional Schrödinger equation. Our starting point is the quantum Fokker-Planck master equation describing dissipative interactions (of mass and energy) of the particle ensemble with a thermal bath in thermodynamic equilibrium. When considering its associated hydrodynamic system, which rules the temporal evolution of the local density and the mean fluid-flow velocity, and imposing physically admissible closure relations, these equations can be seen as describing the fluid-mechanical evolution of the macroscopic amplitude and phase of an envelope wavefunction, thus giving rise to a family of dissipative Schrödinger equations of logarithmic type whose steady state and radial dynamics are analyzed. Also, numerical comparison with the exactly solvable models for the free particle and the damped harmonic oscillator is performed.     

A parallel algorithm for solving the 3d Schrödinger equation

We describe a parallel algorithm for solving the time-independent 3d Schrödinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ∝ (N_nodes)^{−0.95 ± 0.04}. This makes it possible to solve the 3d Schrödinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.     

A phenomenology analysis of the tachyon warm inflation in loop quantum cosmology

We investigate the warm inflation condition in loop quantum cosmology. In our consideration, the system is described by a tachyon field interacted with radiation. The exponential potential function, V(?)=V₀e−α?$V(?)=V_0{e}^{-\alpha ?}$, with the same order parameters V₀$V_0$ and α, is taken as an example of this tachyon warm inflation model. We find that, for the strong dissipative regime, the total number of e-folds is less than the one in the classical scenario, and for the weak dissipative regime, the beginning time of the warm inflation will be later than the tachyon (cool) inflation.     

Phantom cosmology with non-minimally coupled real scalar field

We find that the expansion of the universe is accelerating by analyzing the recent observation data of type Ia supernova (SN-Ia). It indicates that the equation of state of the dark energy might be smaller than -1, which leads to the introduction of phantom models featured by its negative kinetic energy to account for the regime of equation of state parameter w < -1. In this paper the possibility of using a non-minimally coupled real scalar field as phantom to realize the equation of state parameter w < -1 is discussed. The main equations which govern the evolution of the universe are obtained. Then we rewrite them with the observable quantities.     

Quantum information entropies for position-dependent mass Schrödinger problem

The Shannon entropy for the position-dependent Schrödinger equation for a particle with a nonuniform solitonic mass density is evaluated in the case of a trivial null potential. The position S_x$S_x$ and momentum S_p$S_p$ information entropies for the three lowest-lying states are calculated. In particular, for these states, we are able to derive analytical solutions for the S_x$S_x$ entropy as well as for the Fourier transformed wave functions, while the S_p$S_p$ quantity is calculated numerically. We notice the behavior of the S_x$S_x$ entropy, namely, it decreases as the mass barrier width narrows and becomes negative beyond a particular width. The negative Shannon entropy exists for the probability densities that are highly localized. The mass barrier determines the stability of the system. The dependence of S_p$S_p$ on the width is contrary to the one for S_x$S_x$. Some interesting features of the information entropy densities ρ_s(x)${\rho }_s\left(x\right)$ and ρ_s(p)${\rho }_s\left(p\right)$ are demonstrated. In addition, the Bialynicki-Birula–Mycielski (BBM) inequality is tested for a number of states and found to hold for all the cases.     

A theoretical model for quantum nanostructures electronic wave functions, magnetic field effects

Analytical solutions of electronic wave functions in symmetric quantum ring (QR), quantum wire (QWR) and quantum dots (QD) structures are given using a parabolic coordinates system. The solutions for low-energy states are combinations of Bessel functions. The density of states of perfect 1D QR and QWR are shown to be equivalent. The continuous evolution from a 0D QD to a perfect 1D QR can be precisely described. The sharp variation of electronic properties, related to the build up of a potential energy barrier at the early stage of the QR formation, is studied analytically. Paramagnetic and diamagnetic couplings to a magnetic field are computed for QR and QD. It is shown theoretically that magnetic field induces an oscillation of the magnetization in QR.     

Quasiparticle excitations in relativistic quantum field theory

We analyze the particle-like excitations arising in relativistic field theories in states different than the vacuum. The basic properties characterizing the quasiparticle propagation are studied using two different complementary methods. First we introduce a frequency-based approach, wherein the quasiparticle properties are deduced from the spectral analysis of the two-point propagators. Second, we put forward a real-time approach, wherein the quantum state corresponding to the quasiparticle excitation is explicitly constructed, and the time-evolution is followed. Both methods lead to the same result: the energy and decay rate of the quasiparticles are determined by the real and imaginary parts of the retarded self-energy, respectively. Both approaches are compared, on the one hand, with the standard field-theoretic analysis of particles in the vacuum and, on the other hand, with the mean-field-based techniques in general backgrounds.     

Magnetic field effects on THz quantum cascade laser: A comparative analysis of three and four quantum well based active region design

We consider the influence of additional carrier confinement, achieved by application of strong perpendicular magnetic field, on inter Landau levels electron relaxation rates and the optical gain, of two different GaAs quantum cascade laser structures operating in the terahertz spectral range. Breaking of the in-plane energy dispersion and the formation of discrete energy levels is an efficient mechanism for eventual quenching of optical phonon emission and obtaining very long electronic lifetime in the relevant laser state. We employ our detailed model for calculating the electron relaxation rates (due to interface roughness and electron–longitudinal optical phonon scattering), and solve a full set of rate equations to evaluate the carrier distribution over Landau levels. The numerical simulations are performed for three- and four-well (per period) based structures that operate at 3.9 THz and 1.9 THz, respectively, both implemented in GaAs/Al_0.15Ga_0.85As. Numerical results are presented for magnetic field values from 1.5 T up to 20 T, while the band nonparabolicity is accounted for.     

Time dependent Schrödinger equation for black hole evaporation: No information loss

In 1976 S. Hawking claimed that “Because part of the information about the state of the system is lost down the hole, the final situation is represented by a density matrix rather than a pure quantum state”. ¹ This was the starting point of the popular “black hole (BH) information paradox”.