Calculation path integrals statistical mechanics field theory variational techniques

Author(s)

Cristiane Moura Lima de Aragão

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2002-06-06

Examining board members

Carlos Eugenio Imbassahy Carneiro; Antônio José Accioly; Bruto Max Pimentel Escobar; Josif Frenkel; Maria Teresa Climaco dos Santos Thomaz

• Advisor: Carlos Eugenio Imbassahy Carneiro

Abstract

We have extended the Kleinert variational technique to field theory. This method was first used in quantum mechanics and provides a convergent cumulate expansion that is extremely accurate. Its extension to field theory is non-trivial because of the ultraviolet divergences that appear when the space dimension is greater than 2. Due to these divergences the theory has to be regularized and renormalized. In addition to the usual difficulties associated with renormalization, one has to decide whether one calculates the optimum value of the variational parameter before or after renormalization. In this thesis we deal with the renormalization of the variational effective potential. Firstly, we show that the zero temperature regularized variational potential coincides with the post-Gaussian effective potential introduced by Stancu and Stenvenson. Secondly, we present a renormalization scheme that enables one to renormalize the theory before calculating the optimum variational parameter. Using this scheme we show that the usual 1-loop effective potential can be obtained from the Kleinert variational scheme by interacting only once the equation that determines the variational parameter. In this sense, the 1-loop expansion is contained within the variational scheme. For the 2-loop effective potential the same approximation is not so good. The renormalization of the theory before the calculation of the variational parameter allows one to study the variational effective potential numerically and in a non-pertubative way, as it was done in quantum mechanics by Kleinert. (AU)