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The dressed oscillator approach and particle creation in two simple models of a Friedmann-Robertson-Walker Universe Bruskiewich, Patrick

Abstract

In the First part of this thesis I look at the Algebraic Method which is a very straightforward technique. The idea behind the Algebraic Method is to generate all the states of a quantum system beginning with a well defined base state, generally the lowest energy state, through successive application of a creation operator (also known as a raising operator) which modifies the lowest energy state in such a fashion as to then characterize the rest of the spectrum of the system. The lowest energy state is defined as the state that is annihilated by the annihilation operator (also known as the lowering operator). Several examples of the Algebraic technique are presented including Landau Levels. In the Second part of this thesis I look at several examples of Unitary Similarity Transformations and how they can be used to simplify Hamiltonians describing quantum systems. Examples of the Similarity Transformation Method discussed in this thesis include a method to determine the ground state eigen-function using a generating function, Electron-Spin Resonance, the Foldy and Wouthuysen Transformation and an approach first proposed by Wentzel and applied by Schwinger to describe the non-relativistic interaction of an electron with a field. Schwinger used this approach to solve for the Lamb shift of the electron in a central coulombic potential. In the Third Part of this thesis I look at the Bogoliubov Transformation which can be used for Diagonalizing a Quadratic Bosonic Hamiltonian. In the Fourth Part I describe the coupling between a non-relativistic system of oscillators coupled linearly to a scalar field in ordinary Euclidean 3-space. From a physical point of view we give a nonperturbative treatment to the oscillator radiation introducing some coordinates that permit us to divide the coupled system into two parts, the "dressed" oscillator and the field. I also look at how one can describe transitions due to a forcing function. The first four sections of this thesis build up the mathematical tools, namely the Algebraic Method, the Bogoliubov transformation and the "dressed" oscillator approach, for Part Five in which I look at uniform acceleration n Rindler space, particle creation in two simple models of a Friedmann-Robertson-Walker Universe, as well as a hypothesis that Gravity is an Induced Quantum Effect.

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