On the degrees of freedom of spatially covariant vector field theory

TL;DR

This paper analyzes spatially covariant vector field theories, identifying conditions under which the theories propagate only two degrees of freedom by eliminating the longitudinal mode through Hamiltonian constraint analysis.

Contribution

It introduces degeneracy conditions that reduce the degrees of freedom in such theories and classifies three distinct classes of solutions with different constraint structures.

Findings

Identified two key degeneracy conditions for DOF reduction.

Classified three types of theories based on their constraint structures.

Recovered Maxwell theory as a special case within these classes.

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.