  In the Reststrahlen region, between the transverse and longitudinal phonon
frequencies, polar dielectric materials respond metallically to light and the
resulting strong light-matter interactions can lead to the formation of hybrid
quasiparticles termed surface phonon polaritons. Recent works have demonstrated
that when an optical system contains nanoscale polar elements these excitations
can acquire a longitudinal field component as a result of the material
dispersion of the lattice, leading to the formation of secondary quasiparticles
termed longitudinal-transverse polaritons. In this work we build on previous
macroscopic electromagnetic theories developing a full second-quantised theory
of longitudinal-transverse polaritons. Beginning from the Hamiltonian of the
light-matter system we treat distortion to the lattice introducing an elastic
free energy. We then diagonalise the Hamiltonian, demonstrating the equations
of motion for the polariton are equivalent to the those of macroscopic
electromagnetism and quantise the nonlocal operators. Finally we demonstrate
how to reconstruct the electromagnetic fields in terms of the polariton states
and explore polariton induced enhancements of the Purcell factor. These results
demonstrate how nonlocality can narrow, enhance and spectrally tune near field
emission with applications in mid-infrared sensing.
