Goldstone and Pseudo-Goldstone Bosons in Nuclear, Particle and Condensed-Matter Physics

TL;DR

This work presents a comprehensive, field-theoretic treatment of Goldstone and pseudo-Goldstone bosons via effective Lagrangians, emphasizing nonlinear realizations of continuous symmetries, and the systematic power counting that makes EFTs predictive at low energies. It starts with general theorems (Noether and Goldstone), develops abelian and nonabelian constructions in both relativistic and nonrelativistic settings, and then applies the framework to pions in QCD, magnons in magnets, and an ambitious SO(5) scheme for high-T_c superconductors. The pionic application demonstrates how chiral symmetry and its explicit breaking yield precise low-energy predictions (e.g., pion-nucleon couplings, Goldberger-Treiman relation, and soft-pion theorems) without requiring a full QCD solution. The magnets and SO(5) sections illustrate the geometric and phenomenological power of EFTs in diverse domains, highlighting how symmetry-breaking patterns dictate the spectrum and dispersion relations of Goldstone and pseudo-Goldstone modes and how experimental probes (neutron scattering, beta decays) can determine fundamental EFT parameters. Overall, the notes argue that symmetry, geometry, and power counting together furnish a highly predictive, model-independent toolkit for understanding low-energy dynamics across nuclear, particle, and condensed-matter physics.

Abstract

These notes review the effective lagrangian treatment of Goldstone and pseudo-Goldstone bosons, taking examples from high-energy/nuclear and condensed-matter physics. The contents are: 1. Goldstone Bosons 2. Pions: A Relativistic Application 3. Magnons: Nonrelativistic Applications 4. SO(5) Invariance and Superconductors 5. Bibliography

Figures (5)

• Figure 1: The Feynman graphs which describe ${\cal R}-{\cal I}$ scattering at tree level. Solid lines denote ${\cal R}$ and dashed lines represent ${\cal I}$.
• Figure 2: The Feynman graphs which give the dominant nucleon matrix elements of the Noether currents in the low-energy effective theory. Solid lines represent nucleons, and dashed lines represent pions.
• Figure 3: The Feynman graphs which give the dominant contributions to pion-pion scattering in the low-energy pion-nucleon theory. The first graph uses a vertex involving two derivatives. The second involves the pion mass, but no derivatives.
• Figure 4: The Distinction Between Ferromagnets (F) and Antiferromagnets (AF).
• Figure 5: A typical Temperature-- vs--Doping phase diagram for a high-$T_c$ system.