Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

TL;DR

This work resolves how Nambu–Goldstone bosons behave when Lorentz invariance is absent by proving the Brauner–Watanabe conjecture: $n_{BG}-n_{NGB}=\tfrac{1}{2}\mathrm{rank}\,\rho$, where $\rho_{ij}=-\frac{i}{\Omega}\langle0|[Q_i,Q_j]|0\rangle$. Using a general, non-Lorentz-invariant effective Lagrangian, the authors show that nonzero $\rho$ induces canonically conjugate pairs among NGB fields, yielding type B NGBs with quadratic dispersion and reducing the total NGB count relative to broken generators by half the rank of $\rho$. They provide a precise decomposition into type A (linear dispersion) and type B (quadratic dispersion) modes, discuss central extensions that contribute to $\rho$, and give a geometric interpretation in terms of a partially symplectic coset space $G/H$ with a fiber-bundle structure. The paper supports these results with explicit examples (ferromagnets, antiferromagnets, phonons, and spinor BECs) and highlights the underlying geometry: a base symplectic manifold $B$ and a fiber $F$ over which NGBs organize as conjugate pairs or independent modes. Overall, the work clarifies NGB counting and dispersion in non-Lorentz-invariant systems and connects physical spectra to coset geometry and potential central extensions.

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.