- Title Pages
- Dedication
- Preface
- 1 The random walk: Universality and continuum limit
- 2 Functional integration: From path to field integrals
- 3 The essential role of functional integrals in modern physics
- 4 From infinities in quantum electrodynamics to the general renormalization group
- 5 Renormalization Group: From a general concept to numbers
- 6 Critical phenomena: The field theory approach
- 7 Stability of renormalization group fixed points and decay of correlations
- 8 Quantum field theory: An effective theory
- 9 The non-perturbative renormalization group
- 10 O(N) vector model in the ordered phase: Goldstone modes
- 11 Gauge invariance and gauge fixing
- 12 The Higgs boson: A major discovery and a problem
- 13 Quantum chromodynamics: A non-Abelian gauge theory
- 14 From BRST symmetry to the Zinn-Justin equation
- 15 Quantum field theory: Asymptotic safety
- 16 Symmetries: From classical to quantum field theories
- 17 Quantum anomalies: A few physics applications
- 18 Periodic semi-classical vacuum, instantons and anomalies
- 19 Field theory in a finite geometry: Finite size scaling
- 20 The weakly interacting Bose gas at the critical temperature
- 21 Quantum field theory at finite temperature
- 22 From random walk to critical dynamics
- 23 Field theory: Perturbative expansion and summation methods
- 24 Hyper-asymptotic expansions and instantons
- 25 Renormalization group approach to matrix models
- Bibliography
- Index

# Periodic semi-classical vacuum, instantons and anomalies

# Periodic semi-classical vacuum, instantons and anomalies

- Chapter:
- (p.319) 18 Periodic semi-classical vacuum, instantons and anomalies
- Source:
- From Random Walks to Random Matrices
- Author(s):
### Jean Zinn-Justin

- Publisher:
- Oxford University Press

Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.

*Keywords:*
band structure, barrier penetration, cosine potential, Bogomolnyi’s inequality, instanton, homotopy class, topological charge, winding number, strong CP problem

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- Title Pages
- Dedication
- Preface
- 1 The random walk: Universality and continuum limit
- 2 Functional integration: From path to field integrals
- 3 The essential role of functional integrals in modern physics
- 4 From infinities in quantum electrodynamics to the general renormalization group
- 5 Renormalization Group: From a general concept to numbers
- 6 Critical phenomena: The field theory approach
- 7 Stability of renormalization group fixed points and decay of correlations
- 8 Quantum field theory: An effective theory
- 9 The non-perturbative renormalization group
- 10 O(N) vector model in the ordered phase: Goldstone modes
- 11 Gauge invariance and gauge fixing
- 12 The Higgs boson: A major discovery and a problem
- 13 Quantum chromodynamics: A non-Abelian gauge theory
- 14 From BRST symmetry to the Zinn-Justin equation
- 15 Quantum field theory: Asymptotic safety
- 16 Symmetries: From classical to quantum field theories
- 17 Quantum anomalies: A few physics applications
- 18 Periodic semi-classical vacuum, instantons and anomalies
- 19 Field theory in a finite geometry: Finite size scaling
- 20 The weakly interacting Bose gas at the critical temperature
- 21 Quantum field theory at finite temperature
- 22 From random walk to critical dynamics
- 23 Field theory: Perturbative expansion and summation methods
- 24 Hyper-asymptotic expansions and instantons
- 25 Renormalization group approach to matrix models
- Bibliography
- Index