We present a complete, step‑by‑step algorithm to determine whether a given linear differential operator D is physically admissible as a free kinetic operator in a local, relativistic, unitary field theory. The algorithm formalizes the notion of the D‑operator space D₇ₘₒ and provides explicit computational criteria for locality, hyperbolicity, gauge invariance, degree of freedom counting, and positivity of energy. Each step is illustrated with fully worked examples (Klein–Gordon, Dirac, Maxwell, Proca, linearized Einstein, and counterexamples such as tachyons and higher‑derivative operators). The presentation is self‑contained and mathematically rigorous, while honestly acknowledging the limitations and the precise scope of the admissibility conditions.
Timmermans et al. (Tue,) studied this question.