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Sistemas gaussianos não-autônomos: aplicação à cosmologia

Fernando Nogueira

Published Aug 14, 2009• 0 citations
OpenAlex

Abstract

In this work a connection between dynamical quantum mechanical systems and scalar quantum field theory is built. Is verified that the Covariance Matrixs determinant is a time conserved quantity for Gaussian Systems under non autonomous Hamiltonians, and the generalizations for N and infinity dimensions is presented. Two cosmological models for the pre-heating period are treated, and a close connection between the Wigner function, squeezing and particle creation is found. A geometrical interpretation for the dynamics of the pre-heating is made possible, just as a experimental set up for testing cosmological hypothesis.

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In this work a connection between dynamical quantum mechanical systems and scalar quantum field theory is built. Is verified that the Covariance Matrixs determinant is a time conserved quantity for Gaussian Systems under non autonomous Hamiltonians, and the generalizations for N and infinity dimensions is presented. Two cosmological models for the pre-heating period are treated, and a close connection between the Wigner function, squeezing and particle creation is found. A geometrical interpretation for the dynamics of the pre-heating is made possible, just as a experimental set up for testing cosmological hypothesis.
During the last decades, there have been many theoretical and experimental advances both in the extension of thermodynamics to comprise microscopic systems out-of-equilibrium and in the understanding of quantum mechanics. Along with the state-of-the-art capability of controlling fragile quantum systems in a wide variety of physical platforms, this context has paved the way for the current strategic efforts to develop a thermodynamic theory of quantum systems. In this sense, the research field coined as quantum thermodynamics (QT) already plays a key role in the design and development of future quantum-based technologies. More specifically, QT aims both to apply the usual thermodynamic concepts and notions to describe arbitrary non-equilibrium quantum systems and to understand the emergence of classical thermodynamic behaviour from the underlying fundamentally quantum dynamics. However, despite all current progress, there is still no consolidated formalism for a general thermodynamic description of fully autonomous quantum objects. Besides, the lack of consensus on some central aspects, such as the definitions of quantum counterparts of thermodynamic quantities, is particularly notorious. In this thesis, we focus on the energetic analysis within autonomous quantum systems. To this aim, we propose a novel and general formalism for a dynamic description of the energy exchanges between interacting subsystems. From the Schmidt decomposition approach, we identify effective Hamiltonians as the representative operators for characterizing the local internal energies, whose expectation values satisfy the usual thermodynamic notion of energy additivity. In contrast to the currently used methodologies, such procedure treats the subsystems with equal footing and do not rely on any sort of approximations and additional hypotheses, e.g., semi-classical description, weak-coupling regime, strict energy conservation and Markovian dynamics. In short, our proposal contributes to the development of QT by providing a new formalism that does not suffer from the usual restrictive shortcomings and establishes a new and exact route for defining other general thermodynamic quantities to the quantum regime.
In this paper we study quantum dynamics of the bouncing cosmological model. We focus on the model of the flat Friedmann-Robertson-Walker universe with a free scalar field. The bouncing behavior, which replaces the classical singularity, appears due to the modification of general relativity using the methods of loop quantum cosmology. We show that there exists a unitary transformation that enables one to describe the system as a free particle with a Hamiltonian equal to the canonical momentum. We examine properties of the various quantum states of the universe: the boxcar state, the standard coherent state, and the soliton-like state, as well as Schr\"odinger's cat states constructed from these states. Characteristics of the states---such as quantum moments and Wigner functions---are investigated. We show that each of these states have, for some range of parameters, a proper semiclassical limit fulfilling the correspondence principle. The decoherence of the superposition of two universes is described and possible interpretations in terms of triad orientation and the Belinsky-Khalatnikov-Lifshitz conjecture are given. We also examine negative-part sectors of the Wigner functions for the considered states. Their respective areas and localizations in the phase space feature an adequate estimate of nonclassicality. In particular, we show that regions where the Wigner function for the Glauber coherent cat state assumes its negative values have area equal to one fourth of the Planck constant. Based on the examined examples, we also conjecture that regions with negative values for the Wigner function are never smaller than one fourth of the Planck constant for states satisfying the uncertainty principle.
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
The quantized Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) model minimally coupled to a free massless scalar field is studied and interpreted in the Bohm-de Broglie framework. We analyze the quantum bohmian trajectories corresponding to a certain class of gaussian packets, solutions of the Wheeler-DeWitt equation. We show that these bohmian trajectories undergo an accelerated expansion in the middle of its evolution due to the presence of quantum cosmological effects in this period. It is shown that the luminosity-redshift relation in the quantum cosmological model can be made close to the corresponding relation coming from the classical model suplemented by a cosmological constant, for z<1z<1z<1. In this way we have the posibility of interpreting the present observations of high redshift supernovae as the consequence of a quantum cosmological effect.
This paper develops a simple dynamical framework in which rest mass arises from the collapse of an initially coherent “point” in an abstract selector space. The central idea is that a localized selector state expands isotropically into many degrees of freedom, with symmetry-enforced equal-and-opposite reactions that maintain vanishing net vector quantities while a positive scalar functional (the selector load) at the origin increases. When this functional reaches a critical threshold, the extended selector configuration is postulated to collapse into a localized state in spacetime, carrying an effective rest mass determined by the accumulated selector load. Mathematically, the model is formulated as a classical field theory on a compact selector manifold with a symmetry-generated conservation law, a positive scalar functional encoding the selector load, and a deterministic collapse rule that maps the pre-collapse configuration to a pointlike degree of freedom with a well-defined invariant mass. The construction is contrasted with standard invariant-mass definitions in special relativity and with objective collapse models, and it is shown how, in principle, the selector-load functional could lead to experimentally testable deviations in effective inertial mass or collapse thresholds for highly coherent mesoscopic systems. This revised version integrates the selector-space collapse mechanism with the Trinity Geometric Model, interpreting rest mass as the outcome of record-driven collapse in an auxiliary selector manifold coupled to a scalar-clock, expansion–locking geometry. It introduces a record-load functional and deterministic collapse rule, connects mass assignment to stabilized inner expansion and record formation, and refines the phenomenology for mesoscopic and cosmological tests of this geometric, record-sustaining mass-creation framework