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

TL;DR

This paper explores non-Hermitian interactions between harmonic oscillators using a novel Lie algebra representation, leading to stable, exactly solvable models with potential applications to Lorentz-violating quantum electrodynamics and astrophysical constraints.

Contribution

It introduces a new family of nonlinear, non-Hermitian oscillator couplings derived from $su(2)$ representations, with implications for Lorentz-violating gauge theories.

Findings

Low-energy limits are self-adjoint and exactly solvable.

The models are stable under certain conditions.

Astrophysical constraints strongly limit the parameter space.

Abstract

We examine a new application of the Holstein-Primakoff realization of the simple harmonic oscillator Hamiltonian. This involves the use of infinite-dimensional representations of the Lie algebra $su(2)$. The representations contain nonstandard raising and lowering operators, which are nonlinearly related to the standard $a^{\dag}$ and $a$. The new operators also give rise to a natural family of two-oscillator couplings. These nonlinear couplings are not generally self-adjoint, but their low-energy limits are self-adjoint, 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.