Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

TL;DR

This paper clarifies the properties, counting, and dispersion relations of Nambu-Goldstone bosons in systems lacking Lorentz invariance, highlighting the role of algebraic structures and symplectic geometry.

Contribution

It provides a unified framework for understanding Nambu-Goldstone bosons without Lorentz invariance, including counting rules and geometric insights.

Findings

Number of NG bosons can be less than broken generators due to conjugate pairing.

Nonzero commutator expectation values lead to pairing of generators.

The coset space geometry is generally partially symplectic.

Abstract

Using the effective Lagrangian approach, we clarify general issues about Nambu-Goldstone bosons without Lorentz invariance. We show how to count their number and study their dispersion relations. Their number is less than the number of broken generators when some of them form canonically conjugate pairs. The pairing occurs when the generators have a nonzero expectation value of their commutator. For non-semi-simple algebras, central extensions are possible. The underlying geometry of the coset space in general is partially symplectic.