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The question: whether quantum coherent states can sustain decoherence, heating and dissipation over time scales comparable to the dynamical timescales of brain neurons, has been actively discussed in the last years. A positive answer on this question is crucial, in particular, for consideration of brain neurons as quantum computers. This discussion was mainly based on theoretical arguments. In the present paper nonlinear statistical properties of the Ventral Tegmental Area (VTA) of genetically depressive limbic brain are studied in vivo on the Flinders Sensitive Line of rats (FSL). VTA plays a key role in the generation of pleasure and in the development of psychological drug addiction. We found that the FSL VTA (dopaminergic) neuron signals exhibit multifractal properties for interspike frequencies on the scales where healthy VTA dopaminergic neurons exhibit bursting activity. For high moments the observed multifractal (generalized dimensions) spectrum coincides with the generalized dimensions spectrum calculated for a spectral measure of a quantum system (so-called kicked Harper model, actively used as a model of quantum chaos). This observation can be considered as a first experimental (in vivo) indication in the favor of the quantum (at least partially) nature of brain neurons activity.

Quantum mechanical states are normally described by the Schrödinger equation, which generates real eigenvalues and quantizable solutions which form a basis for the estimation of quantum mechanical observables, such as momentum and kinetic energy. Studying transition in the realm of quantum physics and continuum physics is however more difficult and requires different models. We present here a new equation which bears similarities to the Korteweg–DeVries (KdV) equation and we generate a description of transitions in physics. We describe here the two- and three-dimensional form of the KdV like model dependent on the Plank constant $\hslash$ and generate soliton solutions. The results suggest that transitions are represented by soliton solutions which arrange in a spiral-fashion. By helicity, we propose a conserved pattern of transition at all levels of physics, from quantum physics to macroscopic continuum physics.

This article studies quantum games with imprecise payoffs simulated by means of fuzzy numbers. Three two-person game-types are scrutinized via the iterated confronting of a large number of players laying in a two-dimensional lattice. In every iteration, every player interacts with his nearest neighbours and adopts the strategy of his best paid mate. Variable degree of quantum entanglement and of optimism in the fuzzy payoffs are taken into consideration in the study.

We study the asymptotic behavior of a singular potential that arises under several frequently occurring analytic behaviors of the eigenfunctions (of the Schrödinger eigenvalue problem) within the C²-class of functions. We find that the asymptotic behavior of the singular potential crucially depends on the analytic property of the eigenfunction near the singular point.

We consider an anti de Sitter universe filled by quantum CFT with classical phantom matter and perfect fluid. The model represents the combination of a trace-anomaly annihilated and a phantom driven anti de Sitter universes. The influence exerted by the quantum effects and phantom matter on the AdS space is discussed. Different energy conditions in this type of universe are investigated and compared with those for the corresponding model in a de Sitter universe.

This paper explores the idea that within the framework of three-dimensional quantum gravity one can extend the notion of Feynman diagram to include the coupling of the particles in the diagram with quantum gravity. The paper concentrates on the non-trivial part of the gravitational response, which is to the large momenta propagating around a closed loop. By taking a limiting case one can give a simple geometric description of this gravitational response. This is calculated in detail for the example of a closed Feynman loop in the form of a trefoil knot. The results show that when the magnitude of the momentum passes a certain threshold value, non-trivial gravitational configurations of the knot play an important role.

It is a commonplace to note that in a world governed by special or general relativity, an observer has access only to data within her past lightcone (if that). The significance of this for prediction, and thus for confirmation, does not however seem to have been appreciated. In this paper we show that what we regard as our most well-confirmed relativistic theory, Maxwell's theory of electromagnetism, is not at all well-confirmed in the absence of an additional assumption, the assumption that all fields have sources in their past. We conclude that we have reason to believe that there is a lawlike time-asymmetry in the world.

An application of the quantum N-portrait to the Universe is discussed, wherein the spacetime geometry is understood as a Bose–Einstein condensate of N soft gravitons. If near or at the critical point of a quantum phase transition, indications are found that the vacuum energy is partly suppressed by 1/N, as being due to quanta not in the condensate state. Time evolution decreases this suppression, which might have implications for cosmic expansion.

Recently, some authors showed that a classical collapse scenario ignores this richness of information in the resulting spectrum and a consistent quantum treatment of the entire collapse process might allow us to retrieve much more information from the spectrum of the final radiation. We confirm these results and show that by considering the quantum entanglement between metrics, we can uncover information of black holes. In our model, a density matrix is defined for the spaces, both inside and outside of the event horizon. These inside and outside spaces of black holes are obtained by tracing from a bigger space. An observer that lives in this big space can recover total information regarding the inside and outside of black hole.

The supremacy of quantum approach is able to solve the problems which are not practically feasible on classical machines. It suggests a significant speed up of the simulations and decreases the chance of error rates. This paper introduces a new quantum model for time series data which depends on the appropriate length of intervals. To provide effective solution of this problem, this study suggests a new graph-based quantum approach. This technique is useful in discretization and representation of logical relationships. Then, we divide these logical relations into various groups to obtain efficient results. The proposed model is verified and validated with various approaches. Experimental results signify that the proposed model is more precise than existing competing models.

This paper presents a new model using optimization approach for efficient prediction of load in real-life environment. Monte Carlo simulation and Schrödinger equations provide the effective number of solutions. This technique is useful in representation of relationships between different models. The proposed algorithm is verified and validated with various state-of-the-art approaches for solving economic load power dispatch problem to demonstrate its efficiency. Experimental results signify that the proposed algorithm is more precise than existing competing models.

The supremacy of quantum approach is able to provide the solutions which are not practically feasible on classical machines. This paper introduces a novel quantum model for time series data which depends on the appropriate length of intervals. In this study, the effects of these drawbacks are elaborately illustrated, and some significant measures to remove them are suggested, such as use of degree of membership along with mid-value of the interval. All these improvements signify the effective results in case of quantum time series, which are verified and validated with real-time datasets.

Quantum holonomies of closed paths on the torus ${𝕋}^2$ are interpreted as elements of the Heisenberg group $H_1$. Group composition in $H_1$ corresponds to path concatenation and the group commutator is a deformation of the relator of the fundamental group ${\pi }_1$ of ${𝕋}^2$, making explicit the signed area phases between quantum holonomies of homotopic paths. Inner automorphisms of $H_1$ adjust these signed areas, and the discrete symplectic transformations of $H_1$ generate the modular group of ${𝕋}^2$.

In the context of (2+1)–dimensional gravity, we use holonomies of constant connections which generate a q–deformed representation of the fundamental group to derive signed area phases which relate the quantum matrices assigned to homotopic loops. We use these features to determine a quantum Goldman bracket (commutator) for intersecting loops on surfaces, and discuss the resulting quantum geometry.

Motivated by the generalized uncertainty principle, we derive a discrete picture of the space that respects Lorentz symmetry as well as gauge symmetry by setting an equivalency between the linear Generalized Uncertainty Principle (GUP) correction term and electromagnetic interaction term in the Dirac equation. We derived a wave function solution that satisfies this equivalency. This discreteness may explain the crystal and quasicrystal structures observed in nature at different energy scales.

Since the Wigner function (WF) is related to a Lindard-constant type linear dielectric function derived in the symmetric gauge,⁷ it is expected to show de Haas-van Alphen (dHvA) oscillations. Starting with the symmetric eigenfunctions, we derived the pure-state WF in a magnetic field, whose plots in phase space and in term of B^-1 for increasing n are consistent with the dHvA effect. Furthermore the asymptotic expansion of WF at large n show periodic oscillations with a period related to the Fermi energy. The phase space plots of WF also show that dHvA and similar oscillations could be a consequence of Nature's strategy for increasing the effective spatial range without violating the uncertainty principle. Properties of the symmetric eigenfunctions were derived. The dynamics of WF can be obtained from the solution of the time-dependent Schrödinger equation (SE). A new method to solve the SE in a magnetic field in the interaction picture based on expansion in term of symmetric eigenfunctions has been developed. The matrix element for a Gaussian potential were derived explicitly, plotted against B^-1, and showed oscillations. The total WF was shown to be a linear combination of the diagonal pure-state WF's by using the orthogonality for symmetric eigenfunctions. The no-special-point property for WF was confirmed, which is important for the construction of a numerical algorithm to solve the SE in a magnetic field.

The Wigner function is shown related to the quantum dielectric function derived from the quantum Vlasov equation (QVE), with and without a magnetic field, using a standard method in plasma physics with linear perturbations and a self-consistent mean field interaction via Poisson's equation. A finite-limit-of-integration Wigner function, with oscillatory behavior and negative values for free particles, is proposed. In the classical regimes, where the problem size is huge compared to the particle wavelength, these limits go to infinity, and for free particles, the Wigner function becomes a positive delta function as expected. For the harmonic oscillator potential, there is no distinction between finite and infinite limits of integration when these are larger than the eigenfunction localization length.

We present an approach that allows quantifying decoherence processes in an open quantum system subject to external time-dependent control. Interactions with the environment are modeled by a standard bosonic heat bath. We develop two unitarity-preserving approximation schemes to calculate the reduced density matrix. One of the approximations relies on a short-time factorization of the evolution operator, while the other utilizes expansion in terms of the system-bath coupling strength. Applications are reported for two illustrative systems: an exactly solvable adiabatic model, and a model of a rotating-wave quantum-computing gate function. The approximations are found to produce consistent results at short and intermediate times.

Kohn proved in 1961 that interactions between electrons did not change the de Haas–van Alphen (dHvA) oscillation frequency for single electrons in the nondegenerate ground-state [Phys. Rev.123(4), 1242 (1961)]. It was proved recently that the pure-state Wigner function for an electron in a magnetic field carries this quantum and physical oscillation, and a quantum dielectric function, so the conductance can be calculated from the Wigner function [Int. J. Mod. Phys. B17(25), 4555 (2003)], [Int. j. Mod. Phys. B17(26), 4683 (2003)]. We present the first complete proof that at a finite temperature, the mixed-state Wigner function also shows dHvA oscillations with the same frequency. The Wigner function is a fundamental quantity, the fact that it carries observable physical information shows a great potential in the design of new quantum materials at the nanoscale.

The definition of the mixed-state Wigner function involves a grand canonical partition function (GCPF). Although dHvA is a well-known phenomenon, we present the first complete proof of it happening in degenerate mixed-states, based on a GCPF, which requires reconciliation between the dHvA experimental condition of a fixed number of particles and the GCPF's sum over number of particles. The GCPF is applied to one of the two spin species, while both the spin and spin-magnetic moment interaction are considered. We show that the contour integration in ω(ε) leads to a non-oscillatory term that is much larger than an oscillatory term, in the dHvA experimental conditions of high fields and low temperatures. This dominance of the non-oscillatory term explains the constancy of the chemical potential, allowing it to reduce to the Fermi energy in the limit of zero temperature. The obtained mixed-state Wigner function shows a fundamental period of oscillation with respect to B^-1 that reduces to the Onsager's period for dHvA oscillations. This indicates that in mixed-states, dHvA oscillations depend on electrons of one spin species, this means the population of electrons of each spin species oscillates with the magnetic field.

The temperature dependence in the Wigner function will allow a combination of phase-space and thermodynamics information for mesoscopic structures, and the study of phase-space density holes such as BGK modes in the quantum domain.

The predictions of the Quantum Theory have been verified so far with astonishingly high accuracy. Despite of its impressive successes, the theory still presents mysterious features such as the border line between the classical and quantum world, or the deep nature of quantum nonlocality. These open questions motivated in the past several proposals of alternative and/or generalized approaches. We shall discuss in the present paper alternative theories that can be infered from a reconsideration of the status of time in quantum mechanics. Roughly speaking, quantum mechanics is usually formulated as a memory free (Markovian) theory at a fundamental level, but alternative, nonMarkovian, formulations are possible, and some of them can be tested in the laboratory. In our paper we shall give a survey of these alternative proposals, describe related experiments that were realized in the past and also formulate new experimental proposals.