Interplay Between Approximation Theory and Renormalization Group

Abstract

The review presents general methods for treating complicated problems that cannot be solved exactly and whose solution encounters two major difficulties. First, there are no small parameters allowing for the safe use of perturbation theory in powers of these parameters, and even when small parameters exist, the related perturbative series are strongly divergent. Second, such perturbative series in powers of these parameters are rather short, so that the standard resummation techniques either yield bad approximations or are not applicable at all. The emphasis in the review is on the methods advanced and developed by the author. One of the general methods is {\it Optimized Perturbation Theory} now widely employed in various branches of physics, chemistry, and applied mathematics. The other powerful method is {\it Self-Similar Approximation Theory} allowing for quite simple and accurate summation of divergent series. These theories share many common features with the method of renormalization group, which is briefly sketched in order to stress the similarities in their ideas and their mutual interconnection.

review, 138 pages

References in corpus (131)

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• Theory of the Weakly Interacting Bose Gas
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• Holographic and Wilsonian Renormalization Groups
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• Specific Heat of Liquid Helium in Zero Gravity very near the Lambda Point
• The Holographic Renormalization Group
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• Critical Ruptures
• Principal problems in Bose-Einstein condensation of dilute gases
• Series Expansions for the Massive Schwinger Model in Hamiltonian lattice theory
• Financial ``Anti-Bubbles'': Log-Periodicity in Gold and Nikkei collapses
• Numbers and Functions in Quantum Field Theory
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• Non-Ground-State Bose-Einstein Condensates of Trapped Atoms
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• Convergence of the Linear Delta Expansion in the Critical O(N) Field Theory
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• Non-universal Critical Quantities from Variational Perturbation Theory and Their Application to the BEC Temperature Shift
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• Extrapolation of power series by self-similar factor and root approximants
• Bose-Einstein condensation temperature of weakly interacting atoms
• Optimal trap shape for a Bose gas with attractive interactions
• On the Convergence of the Linear Delta Expansion for the Shift in T_c for Bose-Einstein Condensation
• Chiral Symmetry Breaking in QCD: A Variational Approach
• High-density correlation energy expansion of the one-dimensional uniform electron gas
• Renormalization Group Optimized Perturbation Theory at Finite Temperatures
• Estimate of the Critical Exponents from the Field-Theoretical Renormalization Group: Mathematical Sense of the "Standard Values"
• Scale Invariant Resummed Perturbation at Finite Temperatures
• Variational Quark Mass Expansion and the Order Parameters of Chiral Symmetry Breaking
• Ground state of a homogeneous Bose gas of hard spheres
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• Three-loop critical exponents, amplitude functions, and amplitude ratios from variational perturbation theory
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• Resummation Methods for Analyzing Time Series
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• Critical Temperature in Weakly Interacting Multicomponent Field Theory
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• A recursion technique for deriving renormalized perturbation expansions for one-dimensional anharmonic oscillator
• Non-perturbative calculations of a global U(1) theory at finite density and temperature
• Equation of State in Quantum Chromodynamics
• Evaporation and Condensation of Clusters
• Cooperative Electromagnetic Effects
• Gell-Mann - Low Function in the Phi^4 Theory
• Spectrum of Coherent Modes for Trapped Bose Gas
• Conditions for Nuclear-Matter Lasers

Cited by in corpus (6)

• Evolutionary Processes in Quantum Decision Theory
• Continued functions and perturbation series: Simple tools for convergence of diverging series in $O(n)$-symmetric ${\varphi }^4$ field theory at weak coupling limit
• Constrained extrapolation problem and order-dependent mappings
• Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit
• From Asymptotic Series to Self-Similar Approximants
• Symmetry breaking patterns for two coupled complex scalar fields at finite temperature and in an external magnetic field