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Translation-invariant products are studied in the setting ofalpha^star-cohomology. It is explicitly shown that all quantum behaviorsincluding the Green's functions and the scattering matrix oftranslation-invariant non-commutative quantum field theories are thoroughlycharacterized by alpha^star-cohomology classes of the star products.

alpha^star-Cohomology and Classification of Translation-Invariant Non-Commutative Quantum Field Theories