Effective Lagrangians with higher derivatives and equations of motion

The problems that are connected with Lagrangians which depend on higher derivatives (namely, additional degrees of freedom, unbound energy from below, etc.) are absent if effective Lagrangians are considered because the equations of motion may be used to eliminate all higher time derivatives from the effective interaction term. The application of the equations of motion can be realized by making field transformations that involve derivatives of the fields. Using the Hamiltonian formalism for higher-order Lagrangians (Ostrogradsky formalism), Lagrangians that are related by such transformations are shown to be physically equivalent (at the classical and at the quantum level). The equivalence of Hamiltonian and Lagrangian path integral quantization (Matthews's theorem) is proven for effective higher-order Lagrangians. Effective interactions of massive vector fields involving higher derivatives are examined within gauge noninvariant models as well as within (linearly or non-linearly realized) spontaneously broken gauge theories. The Stueckelberg formalism, which relates gauge noninvariant to gauge invariant Lagrangians, becomes reformulated within the Ostrogradsky formalism.