On the degrees of freedom of spatially covariant vector field theory

TL;DR

This work addresses whether a Lorentz- and U(1)-violating, spatially covariant vector field theory can propagate only the two transverse DOFs. Using a Hamiltonian constraint analysis for Lagrangians up to quadratic order in derivatives, the authors derive two degeneracy conditions that reduce the naive $3$ DOFs to $2$ and classify the resulting theories into three types (I, II, III) based on their constraint algebra. Maxwell theory arises as a special Lorentz-invariant limit within this framework, while Branch 1 yields type-I theories and Branch 2 yields type-II or type-III theories depending on coefficient choices. The results provide a structured foundation for constructing Lorentz-violating vector theories with only two propagating DOFs and point toward future work incorporating gravity and higher-derivative terms.

Abstract

We investigate a class of spatially covariant vector field theories on a flat background, where the Lagrangians are constructed as polynomials of first-order derivatives of the vector field. Because Lorentz and $\mathrm{U}(1)$ invariances are broken, such theories generally propagate three degrees of freedom (DOFs): two transverse modes and one longitudinal mode. We examine the conditions under which the additional longitudinal mode is eliminated so that only two DOFs remain. To this end, we perform a Hamiltonian constraint analysis and identify two necessary and sufficient degeneracy conditions that reduce the number of DOFs from three to two. We find three classes of solutions satisfying these degeneracy conditions, corresponding to distinct types of theories. Type-I theories possess one first-class and two second-class constraints, type-II theories have four second-class constraints, and type-III theories contain two first-class constraints. The Maxwell theory is recovered as a special case of the type-III theories, where Lorentz symmetry is restored.