  We present an interacting theory of a $U(1)$ gauge boson with a quadratic
dispersion relation, which we call the "nonlinear Lifshitz photon theory.'' The
Lifshitz photon is a three-dimensional generalization of the Tkachenko mode in
rotating superfluids. Starting from the Wigner crystal of charged particles
coupled to a dynamical $U(1)$ gauge field, after integrating out gapped degrees
of freedom, we arrive at the Lagrangian for the nonlinear Lifshitz photon. The
symmetries of the theory include a global $U(1)$ 1-form symmetry and
nonlinearly realized "magnetic" translation and rotation symmetries. The
interaction terms in the theory lead to the decay of the Lifshitz photon, the
rate of which we estimate. We show that the Wilson loop, which plays the role
of the order parameter of the spontaneous breaking of the 1-form global
symmetry, deviates from the perimeter law by an additional logarithmic factor.
We explore potential connections to other condensed matter systems, with a
particular focus on quantum spin ice and ferromagnets. Finally, we generalize
our theory to higher dimensions.
