# Singular Hamiltonians in models with spontaneous Lorentz symmetry breaking

## TL;DR

This paper investigates models with spontaneous Lorentz symmetry breaking, revealing that some seemingly well-posed models have singular Hamiltonians on the vacuum manifold, leading to potential ill-posedness.

## Contribution

It demonstrates that certain models with vector or tensor fields breaking Lorentz symmetry have singular Hamiltonians, identifying conditions for this pathology.

## Key findings

- Some models have singular Hamiltonians at the vacuum
- Ill-posedness arises despite well-posed Lagrangian formulations
- Conditions for Hamiltonian singularities are identified

## Abstract

Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. One common way to construct such a model is to posit a smooth potential for this field; the natural low-energy solution of such a model would then be excepted to have the tensor field near the minimum of its potential. It is shown in this work that some such models, while appearing well-posed at the level of the Lagrangian, have a Hamiltonian which is singular on the vacuum manifold and are therefore ill-posed. I illustrate this pathology for an antisymmetric rank-2 tensor field, and find sufficient conditions under which this pathology occurs for more general field theories.