A Gel'fand-Yaglom Spin-3/2 Equation with Three Invariant Electromagnetic Couplings

Relativistic equations for massive spin-3/2 particles face well-known subsidiary-constraint problems in external fields. Within the Gel’fand–Yaglom formalism, we construct a parity-invariant first-order theory based on an extended 28-component Lorentz representation with four repeated bispinor sectors. Nilpotency of the repeated spin-1/2 block removes nonzero additional spin-1/2 mass eigenvalues in the Gel’fand–Yaglom spectral sense, and differential reduction of the free parent equations yields a 16-component vector-bispinor equation reproducing the Pauli–Fierz–Rarita–Schwinger constraints. In an external electromagnetic field a corresponding differential consequence is derived consistently to linear order in the field strength, relevant to the one-photon response. Its parameter dependence collapses to three elementary symmetric invariants of the repeated-sector matrix. The σ- and η-sectors act through first and second ordered derivatives of a field-strength-dependent combination, so they remain present at the reduced-operator level even for a homogeneous field. Their simultaneous realization requires retaining the inequivalent signature assignments of the invariant bilinear form: the all-positive repeated-sector class maps to a lowerdimensional boundary in invariant space, whereas a mixed-signature class of the invariant bilinear form contains an open three-dimensional admissible domain. These are algebraically independent invariant coefficients; their identification with independent observable multipoles requires a current calculation from the parent Lagrangian. Higher orders in the external field, causal propagation, the nilpotent Jordan classification, and positivity of the quantized interacting theory are not established here. We also state the minimal curved-space covariantization of the parent first-order system while leaving the reduced curved-space constraint analysis for separate work.