Non-Hermitian Interactions Between Harmonic Oscillators, with Applications to Stable, Lorentz-Violating QED

We introduce a new description of the simple harmonic oscillator Hamiltonian, using infinite-dimensional representations of the Lie algebra $su(2)$. This model, while physically equivalent to the ordinary harmonic oscillator, possesses a fundamentally different operator structure, with new raising and lowering operators, which are nonlinearly related to the standard $a^{\dag}$ and $a$. The new operators give rise to a natural family of two-oscillator couplings. These nonlinear couplings are not generally hermitian, but their low-energy limits are hermitian, exactly solvable, and stable. We discuss the structure of a theory involving these couplings. Such a theory might have as its ultra-low-energy limit a Lorentz-violating Abelian gauge theory, and we discuss the extremely strong astrophysical constraints on such a model