# 'Variational' optimization in quantum field theory

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 Title: 'Variational' optimization in quantum field theory Mattingly, Alan Charles Stevenson, Paul M. Doctor of Philosophy thesis We examine two different techniques for studying quantum field theories in which a 'variational' optimization of parameters plays a crucial role. In the context of the O(N)-symmetric $\lambda\phi\sp4$ theory we discuss variational calculations of the effective potential that go beyond the Gaussian approximation. Trial wavefunctionals are constructed by applying a unitary operator $U = e\sp{-is\pi\sb{R}\phi\sbsp{T}{2}}$ to a Gaussian state. We calculate the expectation value of the Hamiltonian using the non-Gaussian trial states generated, and thus obtain optimization equations for the variational-parameter functions of the ansatz. At the origin, $\varphi\sb{c} = 0,$ these equations can be solved explicitly and lead to a nontrivial correction to the mass renormalization, with respect to the Gaussian case. Numerical results are obtained for the (0 + 1)-dimensional case and show a worthwhile quantitative improvement over the Gaussian approximation. We also discuss the use of optimized perturbation theory (OPT) as applied to the third-order quantum chromodynamics (QCD) corrections to $R\sb{e\sp+e\sp-}.$ The OPT method, based on the principle of minimal sensitivity, finds an effective coupling constant that remains finite down to zero energy. This allows us to apply the Poggio-Quinn-Weinberg smearing method down to energies below 1 GeV, where we find good agreement between theory and experiment. The couplant freezes to a zero-energy value of $\alpha\sb{s}/\pi = 0.26,$ which is in remarkable concordance with values obtained phenomenologically. Mattingly, Alan Charles. (1993) "'Variational' optimization in quantum field theory." Doctoral Thesis, Rice University. http://hdl.handle.net/1911/16649. http://hdl.handle.net/1911/16649 1993