  We develop a field theory with dissipation based on a finite range of wave
propagation and associated gapped momentum states in the wave spectrum. We
analyze the properties of the Lagrangian and the Hamiltonian with two scalar
fields in different representations and show how the new properties of the
two-field Lagrangian are related to Keldysh-Schwinger formalism. The proposed
theory is non-Hermitian, and we discuss its properties related to
$\mathcal{PT}$ symmetry. The calculated correlation functions show a decaying
oscillatory behavior related to gapped momentum states. We corroborate this
result using path integration. The interaction potential becomes short-ranged
due to dissipation. Finally, we observe that the proposed field theory
represents a departure from the harmonic paradigm and discuss the implications
of our theory for the Lagrangian formulation of hydrodynamics.
