11.3 Solitons and instantons in field theory

Solitons and instantons are fascinating solutions in field theory that help us understand complex phenomena. Solitons are stable, particle-like waves that keep their shape, while instantons describe tunneling between different states. They're key to grasping non-perturbative aspects of quantum field theory.

These solutions play crucial roles in various areas of physics. Solitons appear in things like superconductors and quantum Hall systems. Instantons help explain tricky problems in quantum chromodynamics and give insights into the structure of the vacuum in gauge theories.

Solitons in Field Theories

Concept and Role of Solitons

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• Solitons are stable, localized, non-dispersive solutions to nonlinear partial differential equations that maintain their shape and velocity upon interaction with other solitons
• In classical field theories, solitons represent particle-like excitations with finite energy and spatial extent, arising from the nonlinear nature of the field equations
  • Kinks in one-dimensional scalar field theories (φ^4 theory)
  • Vortices in two-dimensional scalar theories
  • Magnetic monopoles in three-dimensional gauge theories
• In quantum field theories, solitons can be treated as extended objects with quantum properties, such as quantized energy levels and scattering amplitudes
  • Quantum solitons play a crucial role in understanding non-perturbative aspects of field theories, such as confinement in quantum chromodynamics and the dynamics of supersymmetric gauge theories

Stability and Classification of Solitons

• The stability of solitons is related to the existence of conserved topological charges, which prevent the soliton from decaying into the vacuum state
• Solitons can be classified based on their topological properties
  • Winding number for kinks
  • Magnetic charge for monopoles
• The topological properties of solitons ensure their stability and distinguish them from other field configurations
• The conservation of topological charges is a fundamental aspect of soliton physics and underlies their particle-like behavior

Properties of Solitons

Dependence on Field Theory Models

• The properties of solitons depend on the specific field theory model and the dimensionality of the system
• In one-dimensional scalar field theories, such as the φ^4 theory, solitons appear as kink solutions that interpolate between two degenerate vacuum states
  • The stability of kinks is ensured by the topological winding number, which distinguishes between different vacuum configurations
• In two-dimensional scalar theories, such as the sine-Gordon model, solitons manifest as localized, non-dispersive excitations called breathers
  • The stability of breathers is related to the integrability of the sine-Gordon model, which allows for an infinite number of conserved quantities

Soliton Interactions and Moduli Space

• In three-dimensional gauge theories, such as the 't Hooft-Polyakov model, solitons appear as magnetic monopoles with quantized magnetic charge
  • The stability of magnetic monopoles is guaranteed by the topological properties of the gauge field configuration, characterized by the magnetic charge
• The interaction between solitons can be studied using various methods
  • Inverse scattering transform
  • Hirota bilinear method
• The moduli space of soliton solutions describes the space of all possible soliton configurations, parametrized by their collective coordinates, such as position, size, and orientation
  • The geometry of the moduli space encodes important information about the dynamics and interactions of solitons

Solitons vs Instantons

Relationship between Solitons and Instantons

• Instantons are classical solutions to the Euclidean field equations that describe tunneling processes between different vacuum states in quantum field theories
• In Euclidean spacetime, solitons and instantons are related by a dimensional reduction procedure, where the time dimension is treated as a spatial dimension
  • For example, a kink solution in a one-dimensional scalar theory can be interpreted as an instanton in a zero-dimensional quantum mechanical system
• The connection between solitons and instantons provides a unified framework for studying non-perturbative aspects of field theories, combining topological and dynamical properties of the system

Role of Instantons in Non-Perturbative Field Theory

• Instantons play a crucial role in understanding non-perturbative effects in gauge theories
  • U(1) problem in QCD
  • Calculation of the theta vacuum
• The instanton gas approximation treats the quantum field theory as a dilute gas of instantons, allowing for the calculation of non-perturbative contributions to correlation functions and partition functions
• Instanton techniques have been successfully applied to various field theory models, revealing important insights into the structure of the vacuum and the dynamics of strongly coupled systems

Instanton Techniques for Gauge Theories

Instantons in Quantum Chromodynamics (QCD)

• Instantons are essential for understanding the structure of the vacuum in non-Abelian gauge theories, such as QCD
• In QCD, instantons are responsible for the violation of the U(1) axial symmetry and the resolution of the U(1) problem, which explains the large mass of the η' meson
  • The instanton-induced effective interaction between quarks, known as the 't Hooft interaction, breaks the U(1) axial symmetry and gives rise to the η' mass
• Instanton contributions to the path integral can be calculated using semiclassical methods
  • Saddle-point approximation
  • Collective coordinate method
  • The collective coordinates of instantons, such as their position, size, and orientation, parametrize the moduli space of instanton solutions

Instantons in Supersymmetric Gauge Theories and Beyond

• Instanton effects can be studied in supersymmetric gauge theories, where the instanton calculus is greatly simplified due to the presence of fermion zero modes and the constraints imposed by supersymmetry
  • In N=2 supersymmetric Yang-Mills theory, the exact low-energy effective action can be determined by summing over instanton contributions, leading to the Seiberg-Witten solution
• The study of instanton effects in gauge theories has led to important developments in mathematics
  • Discovery of Donaldson invariants
  • Classification of four-manifolds
• Instanton techniques have also been applied to other areas of physics
  • Study of quantum chaos
  • AdS/CFT correspondence in string theory