α∗α∗-cohomology, and classification of translation-invariant non-commutative quantum field theories

Translation-invariant ⋆⋆ products are studied in the setting of α∗α∗-cohomology. It is explicitly shown that all quantum behaviors including Green’s functions and the scattering matrix of translation-invariant non-commutative quantum field theories are thoroughly characterized by α∗α∗-cohomology classes of the star products.