Conditions for the emergence of gauge bosons from spontaneous Lorentz symmetry breaking

TL;DR

This paper demonstrates that gauge bosons like photons and gravitons can emerge as Goldstone bosons from spontaneous Lorentz symmetry breaking within a non-abelian Nambu model, establishing their equivalence to Yang--Mills theory.

Contribution

It proves the equivalence between a non-abelian Nambu model and Yang--Mills theory, showing gauge bosons as Goldstone bosons from spontaneous Lorentz symmetry breaking.

Findings

Yang--Mills theory is equivalent to the NANM with initial current and Gauss law conditions.

A specific parametrization simplifies the NANM analysis and reveals a canonical transformation.

Imposing Gauss laws as initial conditions suffices to recover full gauge invariance.

Abstract

The emergence of gauge particles (e.g., photons and gravitons) as Goldstone bosons arising from spontaneous symmetry breaking is an interesting hypothesis which would provide a dynamical setting for the gauge principle. We investigate this proposal in the framework of a general $% SO(N)$ non-abelian Nambu model (NANM), effectively providing spontaneous Lorentz symmetry breaking in terms of the corresponding Goldstone bosons. Using a non-perturbative Hamiltonian analysis, we prove that the $SO(N)$ Yang--Mills theory is equivalent to the corresponding NANM, after current conservation together with the Gauss laws are imposed as initial conditions for the latter. This equivalence is independent of any gauge fixing in the YM theory. A substantial conceptual and practical improvement in the analysis arises by choosing a particular parametrization that solves the non-linear constraint defining the NANM. This choice allows us to show that the relation between the NANM canonical variables and the corresponding ones of the YM theory, $A_{i}^{a}$ and $E^{bj}$, is given by a canonical transformation. In terms of the latter variables, the NANM Hamiltonian has the same form as the YM Hamiltonian, except that the Gauss laws do not arise as first class constraints. The dynamics of the NANM further guarantees that it is sufficient to impose them only as initial conditions, in order to recover the full equivalence. It is interesting to observe that this particular parametrization exhibits the NANM as a regular theory, thus providing a substantial simplification in the calculations.