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Renormalization methods a guide for beginners by McComb in 2004

books

Three sections

I. What is renormalization

II. Renormalized perturbation theories

III. Renormalization group

I: What is renormalization

1. The bedrock problem: why we need renormalization methods

2. Easy applications of renormalization group to simple models

3. Mean-field theories for simple models

II: Renormalized perturbation theories

4. Perturbation theory using a control parameter

5. Classical nonlinear systems driven by random noise

  • 5.1 The generic equation of motion

    • Consider the Navier-Stokes equations, Burgers equation, and Kardar-Parisi-Zhang equation

      • All these equations can be represented as special cases of a generic nonlinear Langevin equation given in Eq. 5.2
    • 5.1.1 The Navier-Stokes equation : NSE
    • 5.1.2 The burgers equation
    • 5.1.3 The KPZ equation
  • 5.2 The moment closure problem
  • 5.3 The pair-correlation tensor
  • 5.4 The zero-order “model” system
  • 5.5 A toy version of the equation of motion
  • 5.6 Perturbation expansion of the toy equation of motion
  • 5.7 Renormalized transport equations for the correlation function
  • 5.8 Reversion of power series
  • 5.9 Formulation in Wyld diagrams
  • 6.1 the real and idealized versions of the turbulence problem
  • 6.2 Two turbulence theories: the DIA and LET equations
  • 6.3 Theoretical results: free decay of turbulence
  • 6.4 Theoretical results: stationary turbulence
  • 6.5 Detailed energy balance in wave number
  • 6.6 Application to other systems

III: Renormalization group

7. Setting the scene: critical phenomena

  • 7.1 Some background material on critical phenomena
  • 7.2 Theoretical models
  • 7.3 Scaling behavior
  • 7.4 Linear response theory
  • 7.5 Serious mean-field theory
  • 7.6 Mean-field critical exponents \[\alpha, \beta, \gamma\], and \[\delta\] for the Ising model
  • 7.7 The remaining mean-field critical exponents for the Ising model
  • 7.8 Validity of mean-field theory
  • 7.9 Upper critical dimension

8. Real-space renormalization group

  • 8.1 A general statement of the RG transformation
  • 8.2 RG transformation of the Hamiltonian and its fixed points
  • 8.3 Relations between critical exponents from RG
  • 8.4 Applications of the linearized RGT

9. Momentum-space renormalization group

  • 9.1 Overview of chapter
  • 9.2 Statistical field theory
  • 9.3 Renormalization group transformation in wave number space
  • 9.4 Scaling dimension: anomalous and normal
  • 9.5 Restatement of our objectives: numerical calculation of the critical exponents
  • 9.6 The Gaussian zero-order model
  • 9.7 Partition function for the Gaussian model
  • 9.8 Correlation functions
  • 9.9 Fixed points for the Gaussian model
  • 9.10 Ginsberg-Landau (GL) theory

10. Field-theoretic renormalization group

  • 10.1 Preliminary remarks
  • 10.2 The Ginsburg-Landau model as a quantum field theory
  • 10.3 Infrared and ultraviolet divergences
  • 10.4 Renormalization invariance
  • 10.5 Perturbation theory in x-space
  • 10.6 Perturbation expansion in x-space
  • 10.7 Perturbation expansion in k-space
  • 10.8 The UV divergence and renormalization
  • 10.9 The IR divergence and the epsilon-expansion
  • 10.10 The pictorial significance of Feynman diagrams

11. Dynamical renormalization group applied to classical nonlinear systems

  • 11.1 The dynamical RG algorithm

  • 11.2 Application to the Navier-Stokes equation

    • 11.2.1 The RG transformation: the technical problems
    • 11.2.2 Overview of perturbation theory
    • 11.2.3 The application of RG at small wave numbers
    • 11.2.4 The application of RG at large wave numbers

  • 11.3 Application of RG to stirred fluid motion with asymptotic freedom as \[k \rightarrow 0\]

    • 11.3.1 Differential RG equations
    • 11.3.2 Application to other systems

  • 11.4 Relevance of RG to the large-eddy simulation of turbulence

    • 11.4.1 Statement of the problem
    • 11.4.2 Conservation equations for the explicit scales \[k \leq k_c\]

  • 11.5 The conditional average at large wave numbers

    • 11.5.1 The asymptotic conditional average

  • 11.6 Application of RG to turbulence at large wave numbers

    • 11.6.1 perturbative calculation of the conditional average
    • 11.6.2 Truncation of the moment expansion
    • 11.6.3 The RG calculation of the effective viscosity
    • 11.6.4 Recursion relations for the effective viscosity

IV: Appendices

A. Statistical ensembles

  • A.1 statistical specification of the N-body assembly
  • A.2 The basic postulates of equilibrium statistical mechanics
  • A.3 Ensemble of assemblies in energy contact
  • A.4 Entropy of an assembly in an ensemble
  • A.5 Principle of maximum entropy
  • A.6 Variational method for the most probably distribution

B. From statistical mechanics to thermodynamics

  • B.1 The canonical ensmble

    • B.1.1 Identification of the Lagrange multiplier
    • B.1.2 General thermodynamic processes
    • B.1.3 Equilibrium distribution and the bridge equation

  • B.2 Overview and summary

    • B.2.1 The canonical ensemble

C. Exact solutions in one and two dimensions

  • C.1 The one-dimensional Ising model
  • C.2 Bond percolation in d=2

D. Quantum treatment of the Hamiltonian N-body assembly

  • D.1 The density matrix \[\rho_{mn}\]
  • D.2 Properties of the density matrix
  • D.3 Density operator for the canonical ensemble

E. Generalization of the Bogoliubov variational method to a spatially varying magnetic field

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