Non-Trivial Vacuum Solutions of Low Dimensional Scalar Field Theories in the Oscillator Representation Method.

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Title: Non-Trivial Vacuum Solutions of Low Dimensional Scalar Field Theories in the Oscillator Representation Method.
Author: Shalaby, Abouzeid Mohammed
Advisors: Prof. Chueng-Ryong Ji, Committee Chair
Prof. Dean Lee, Committee Member
Prof. Lung O. Chung, Committee Member
Prof. Thomas Schaefer, Committee Member
Abstract: This dissertation presents a study of self-organizing nature of quantum field theories. In particular, we study the low-dimensional scalar field theories using a nonpperturbative approach known as the Oscillator Representation (OR) method which appears to be simpler than other well-known nonp-perturbative approaches such as the Hartree Approximation (HA) and the Gaussian Effective Potential (GEP) method. The key idea of the OR method is to make canonical transformations between the original particle theory and the quasi-particles theory and find non-trivial vacuum solutions which satisfy the self-consistency conditions required by a desired form of the quasi-particles effective Hamiltonian. In the low-dimensional scalar field theories, we find the duality property between the original particle theory and the quasi-particles theory, i.e. the original nonp-perturbative strong interaction theory is equivalent to the weakly interacting quasi-particles theory. We present explicit examples of the duality which allows the conversion of the original nonp-perturbative strong interaction problem into a weekly interacting quasi-particles problem that can be solved by the usual perturbative analysis. We also make a link between the OR method and the Effective Action approach with a specific canonical transformation of field shifts. However, we point out a difficulty of these nonp-perturbative methods (OR, HA, GEP) in describing the order of the phase transition near the critical coupling region where the nontrivial vacuum condensations occur. For the case of OR method, we present a detailed prescription how one can overcome this difficulty using the Borel summation technique and the Kleinert algorithm. In the example of (4)1+1 theory, we show the recovery of correct second order phase transition with this improvement. The OR applications to the 2+1 dimensional scalar field theories of $phiˆ4$ and $phiˆ6$ interactions are also detailed with the description of the renormalization procedure for the different levels of loop calculations such as involving one-loop, normal-ordering and two-loop regularizations. Various numerical results of nontrivial vacuum solutions, including their energy densities, quasi-particles mass spectra and classical effective potentials, are discussed along with their symmetry properties.
Date: 2005-08-13
Degree: PhD
Discipline: Physics
URI: http://www.lib.ncsu.edu/resolver/1840.16/4664


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