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Table of Contents
Preface; Acknowledgements; Contents; 1 Quantum Field Theory (QFT) on the Lattice; 1.1 A Brief History of Quarks and Gluons ; 1.2 Classical Fields and Gauge Invariance ; 1.3 Hamiltonian and Path Integral Formulations of Quantum Mechanics ; 1.3.1 One Bosonic Oscillator ; 1.3.2 One Fermionic Oscillator ; 1.4 Quantum Fields on a Lattice ; 1.4.1 From One Oscillator to a Field ; 1.4.2 Correlation Functions ; 1.4.3 Analytic Continuation to Minkowski Space; 1.4.4 Gauge Invariance on the Lattice ; 1.4.5 The Wilson Gauge Action; 1.4.6 Strong Coupling Expansions; 1.4.7 The Wilson Fermion Action
1.4.8 Chiral Symmetry and the Nielsen
Ninomiya No-Go Theorem1.4.9 Other Fermion Formulations; 1.5 Recovering Continuum QCD ; 1.5.1 Classical Continuum Limit; 1.5.2 Renormalisation Group Equation; 1.5.3 Isospin Symmetry and Ward Identities; 1.5.4 Improvement of the Continuum Limit; 1.5.5 Improvement of Wilson Fermions ; 1.5.6 Twisted Mass Fermions; 1.6 Further Reading; References; 2 Monte Carlo Methods; 2.1 Markov Chain Monte Carlo; 2.2 Sampling Yang
Mills Gauge Fields; 2.2.1 Random Numbers and the Rejection Method ; 2.2.2 Heatbath Algorithms; 2.2.3 Overrelaxation Algorithms
2.3 Hybrid Monte Carlo2.3.1 Detailed Balance Condition; 2.3.2 Hamilton's Equation of Motion; 2.4 Symplectic Integration Schemes; 2.5 Summary and Further Reading; References; 3 Handling Fermions on the Lattice; 3.1 Wick Contractions; 3.2 Sparse Linear Algebra; 3.2.1 Krylov Subspace Techniques; 3.2.2 Iterative Solvers for Linear Systems; 3.2.3 Algebraic Multigrid Methods; 3.2.4 Other Uses of Krylov Subspaces; 3.3 Fermion Determinant; 3.3.1 Pseudofermions; 3.3.2 Factorisations; 3.4 HMC with Fermions Revisited ; 3.4.1 Equations of Motion; 3.4.2 Multi-rate Integration Schemes
3.4.3 A Single Dynamical Quark Flavour: The RHMC Algorithm3.5 The Quark Propagator from a Point Source ; 3.5.1 Quark Observables from a Single Point Source; 3.5.2 Reducing Variance in Point Propagator Calculations; 3.6 All-to-All Quark Propagators; 3.6.1 Stochastic Estimators; 3.6.2 Exploiting Low Eigenmodes of the Dirac Operator; 3.7 Summary and Further Reading; References; 4 Calculating Observables of Quantum Fields; 4.1 Symmetry Properties of Creation and Annihilation Operators; 4.1.1 Gauge-Invariant Observables; 4.1.2 Charge Conjugation, Isopin and Flavour Symmetry; 4.1.3 Spin and Parity
4.1.4 Translation Invariance and Momentum4.1.5 Reducing Representations of Symmetries; 4.2 Techniques for Hadron Spectroscopy; 4.2.1 Variational Methods; 4.2.2 Scale Setting; 4.2.3 The Wilson Flow; 4.2.4 Scattering and the Lüscher Method; 4.3 Gluons, Wilson Loops and Glueballs; 4.3.1 Gauge Smearing; 4.3.2 Glueballs; 4.3.3 The Static Potential and Strong Coupling Constant; 4.3.4 Topological Charge; 4.4 Quarks and Hadron Physics; 4.4.1 Quark Smearing; 4.4.2 Hadron Physics with Quarks; 4.4.3 Hadron Scattering in Lattice Monte Carlo Calculations
1.4.8 Chiral Symmetry and the Nielsen
Ninomiya No-Go Theorem1.4.9 Other Fermion Formulations; 1.5 Recovering Continuum QCD ; 1.5.1 Classical Continuum Limit; 1.5.2 Renormalisation Group Equation; 1.5.3 Isospin Symmetry and Ward Identities; 1.5.4 Improvement of the Continuum Limit; 1.5.5 Improvement of Wilson Fermions ; 1.5.6 Twisted Mass Fermions; 1.6 Further Reading; References; 2 Monte Carlo Methods; 2.1 Markov Chain Monte Carlo; 2.2 Sampling Yang
Mills Gauge Fields; 2.2.1 Random Numbers and the Rejection Method ; 2.2.2 Heatbath Algorithms; 2.2.3 Overrelaxation Algorithms
2.3 Hybrid Monte Carlo2.3.1 Detailed Balance Condition; 2.3.2 Hamilton's Equation of Motion; 2.4 Symplectic Integration Schemes; 2.5 Summary and Further Reading; References; 3 Handling Fermions on the Lattice; 3.1 Wick Contractions; 3.2 Sparse Linear Algebra; 3.2.1 Krylov Subspace Techniques; 3.2.2 Iterative Solvers for Linear Systems; 3.2.3 Algebraic Multigrid Methods; 3.2.4 Other Uses of Krylov Subspaces; 3.3 Fermion Determinant; 3.3.1 Pseudofermions; 3.3.2 Factorisations; 3.4 HMC with Fermions Revisited ; 3.4.1 Equations of Motion; 3.4.2 Multi-rate Integration Schemes
3.4.3 A Single Dynamical Quark Flavour: The RHMC Algorithm3.5 The Quark Propagator from a Point Source ; 3.5.1 Quark Observables from a Single Point Source; 3.5.2 Reducing Variance in Point Propagator Calculations; 3.6 All-to-All Quark Propagators; 3.6.1 Stochastic Estimators; 3.6.2 Exploiting Low Eigenmodes of the Dirac Operator; 3.7 Summary and Further Reading; References; 4 Calculating Observables of Quantum Fields; 4.1 Symmetry Properties of Creation and Annihilation Operators; 4.1.1 Gauge-Invariant Observables; 4.1.2 Charge Conjugation, Isopin and Flavour Symmetry; 4.1.3 Spin and Parity
4.1.4 Translation Invariance and Momentum4.1.5 Reducing Representations of Symmetries; 4.2 Techniques for Hadron Spectroscopy; 4.2.1 Variational Methods; 4.2.2 Scale Setting; 4.2.3 The Wilson Flow; 4.2.4 Scattering and the Lüscher Method; 4.3 Gluons, Wilson Loops and Glueballs; 4.3.1 Gauge Smearing; 4.3.2 Glueballs; 4.3.3 The Static Potential and Strong Coupling Constant; 4.3.4 Topological Charge; 4.4 Quarks and Hadron Physics; 4.4.1 Quark Smearing; 4.4.2 Hadron Physics with Quarks; 4.4.3 Hadron Scattering in Lattice Monte Carlo Calculations