A Complex-Quaternionic Riemann–Silberstein Formalism for Single-Photon Electromagnetic Wave Packets

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Abstract
Single photons are elementary excitations of the quantized electromagnetic field, yet their electromagnetic structure is often described through separate electric and magnetic field operators. Here we develop a theorem-based complex-quaternionic Riemann-Silberstein representation of single-photon electromagnetic wave packets. The formalism embeds the transverse electric and magnetic components into one helicity-resolved biquaternionic field, so that transversality, electric-magnetic orthogonality, helicity projection, and mode-wise nullity are encoded within a single algebraic structure. We show that the source-free Maxwell equations reduce to one first-order quaternionic Maxwell equation and that canonical quantization yields the one-photon electromagnetic amplitude as a vacuum-to-one-photon transition matrix element. The resulting expression is a spectral superposition of transverse modes weighted by the photon wave-packet envelope. Beyond notational compactness, the construction separates three operational ingredients of a one-photon state: spectral localization, helicity selection, and mode matching to a detector or target mode. The framework is fully compatible with standard QED and clarifies that broadband photon wave packets are generally not pointwise null, although each Fourier component is null.

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