  We present the details of the novel framework for Lagrangian field theories
that are Lorentz-invariant and lead to at most second order equations of
motion. The use of antisymmetric structure is of crucial importance. The
general ghost-free Lagrangians are constructed and then translated into the
language of differential forms. The ghost-freeness has a geometric nature. A
novel duality is proposed which generalizes the Hodge duality in Maxwell's
theory. We discuss how the well-established theories are reformulated and
propose many new theories.
