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arXiv · 2510.26639

On the degrees of freedom of spatially covariant vector field theory

Shu-Yu Li·Xian Gao
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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.

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Shu-Yu Li, Xian Gao. 2025-10-30. On the degrees of freedom of spatially covariant vector field theory. https://doi.org/10.1088/1572-9494%2Fae3bdf

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