Explicit θ-deformed interaction Hamiltonian with quon deformations

Determine the explicit expression of the θ-deformed interaction Hamiltonian density \(\mathcal{H}_I^\theta\) for quantum electrodynamics on Moyal spacetime when quon deformations are present (i.e., for \(|\eta|<1\) in the \(\mathcal{Q}_\theta\) oscillator algebra), in order to enable verification of whether the Bethe–Salpeter construction yields a closed equation for twisted-symmetry amplitudes in helium-like atomic systems.

Background

The paper develops a relativistic quantum field theory on Moyal spacetime based on a general $\mathcal{Q}_\theta$ oscillator algebra that combines the twist factor $f_\theta$ with a quon-like deformation $\eta(p,q)$. In the interacting theory (QED), quon deformations ($|\eta|<1$) imply that the interaction Hamiltonian in the interaction picture becomes an infinite-degree operator. This structure is required to preserve Lorentz invariance and conservation of statistics but complicates explicit constructions.

To analyze atomic bound states, the authors generalize the Bethe–Salpeter (BS) framework. For purely twisted statistics ($|\eta|=\pm1$), they can show how the BS equation generalizes. However, when quon deformations are included, establishing a closed BS equation for amplitudes with definite twisted permutation symmetry requires explicit knowledge of the θ-deformed interaction Hamiltonian density $\mathcal{H}_I^\theta$. The authors state that this explicit form is not known in the presence of quon deformations, leaving the construction of $\mathcal{H}_I^\theta$ as a concrete open task.