Elements of the Continuous Renormalization Group

Abstract

These two lectures cover some of the advances that underpin recent progress in deriving continuum solutions from the exact renormalization group. We concentrate on concepts and on exact non-perturbative statements, but in the process will describe how real non-perturbative calculations can be done, particularly within derivative expansion approximations. An effort has been made to keep the lectures pedagogical and self-contained. Topics covered are the derivation of the flow equations, their equivalence, continuum limits, perturbation theory, truncations, derivative expansions, identification of fixed points and eigenoperators, and the role of reparametrization invariance. Some new material is included, in particular a demonstration of non-perturbative renormalizability, and a discussion of ultraviolet renormalons.

Invited lectures at the Yukawa International Seminar '97. 20 pages including 6 eps figs. LaTeX. PTPTeX style files included

Cited by in corpus (158)

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• Renormalization group methods and the 2PI effective action
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• Scaling and superscaling solutions from the functional renormalization group
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• Asymptotic freedom and safety in quantum gravity
• BRST in the Exact RG
• Triviality from the Exact Renormalization Group
• Exploring new corners of asymptotically safe unimodular quantum gravity
• Brans-Dicke theory in the local potential approximation
• Remarks on Exact RG Equations
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• Asymptotically free scaling solutions in nonabelian Higgs models
• Nonabelian Higgs models: paving the way for asymptotic freedom
• A Manifestly Gauge Invariant and Universal Calculus for SU(N) Yang-Mills
• Holographic renormalisation group flows and renormalisation from a Wilsonian perspective
• Renormalization group flow equations connected to the $n$PI effective action
• On Exact Proper Time Wilsonian RG Flows
• The fate of non-polynomial interactions in scalar field theory
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• Momentum dependence of quantum critical Dirac systems
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• An exact RG formulation of quantum gauge theory
• Quantum gravity, renormalizability and diffeomorphism invariance
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• General Computations Without Fixing the Gauge
• Provable properties of asymptotic safety in $f(R)$ approximation
• A Primer for Manifestly Gauge Invariant Computations in SU(N) Yang-Mills
• Universality From Very General Nonperturbative Flow Equations in QCD
• Higgs scalar potential coupled to gravity in the exponential parametrization in arbitrary gauge
• Towards Functional Flows for Hierarchical Models
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• The zero-dimensional O(N) vector model as a benchmark for perturbation theory, the large-N expansion and the functional renormalization group
• Asymptotic freedom in $Z_2$-Yukawa-QCD models
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• The local potential approximation in the background field formalism
• New fixed points of the renormalisation group for two-body scattering
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• UV completion of extradimensional Yang-Mills theory for Gauge-Higgs unification
• Non-renormalization for the Liouville wave function
• Scaling solutions for asymptotically free quantum gravity
• Exact renormalization group for wave functionals
• An Extension of Pohlmeyer's Theorem
• Functional renormalization group study of a four-fermion model with $CP$ violation
• Preferred foliation effects in Quantum General Relativity
• Renormalization Flow of Nonlinear Electrodynamics
• Higher derivative extension of the functional renormalization group
• The Conformal Anomaly and a new Exact RG
• A Lindbladian for exact renormalization of density operators in QFT
• Sensitivity of Nonrenormalizable Trajectories to the Bare Scale
• Universal scaling dimensions for highly irrelevant operators in the Local Potential Approximation
• Functional Renormalization Group Analysis of $O(3)$ Nonlinear Sigma Model and Non-Abelian Bosonization Duality
• On the wave function renormalization for Wilson actions and their 1PI actions
• Smoothing of Singular Legendre Transforms in Renormalization Group Flows
• Naturalness, Wilsonian Renormalization, and "Fundamental Parameters" in Quantum Field Theory