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Proof of the most general multiple-scalar field theory in Minkowski space-time free of Ostrogradski Ghost

We give a proof of the most general multiple scalar field theory in flat space-time free of Ostrogradski ghosts. We start from the assumption that the action is a functional of the scalar fields and their derivatives of order upto two (which in general can have field equations of derivative order up to four), and arrive at the most-general field theory that can be constructed such that the field equations are of derivative order up to two. Naturally, this theory includes all physically viable multi-scalar field theories in flat space-time, in the sense that they don't suffer from Ostrogradski instability \cite{ostro}.