Hamiltonian Constraints on Spontaneous Lorentz Symmetry Breaking in the Bumblebee Model

TL;DR

This paper clarifies the correct Hamiltonian approach to spontaneous Lorentz symmetry breaking in the Bumblebee model, challenging common potential minimization methods and identifying viable potentials for stable VEVs.

Contribution

It establishes the Hamiltonian framework for analyzing Lorentz violation, showing standard quadratic potentials are insufficient and proposing cubic potentials as viable alternatives.

Findings

Quadratic potentials cannot generate consistent VEVs.

Cubic potentials are the simplest viable alternative.

Stable timelike or lightlike VEVs only supported by smooth potentials.

Abstract

This study demonstrates that the common practice of determining spontaneous Lorentz violation via the minimum of a Lagrangian potential is generally incorrect. By analyzing the Hamiltonian structure and constraints of vector fields, we show that the true vacuum must be derived from the Hamiltonian density. We prove that the standard quadratic potential cannot consistently generate a vacuum expectation value (VEV), identifying a cubic potential as the simplest viable alternative. Furthermore, we prove that smooth potentials only support stable timelike or lightlike VEVs. These conclusions extend to higher-rank tensor fields and impose rigorous consistency constraints on higher-rank tensor fields and Lorentz-violating effective field theories.