A formalism is developed whereby dispersion relations may be obtained for all meson-nucleon scattering processes, including those in which mesons are created or destroyed. A suitable kinematical description of the processes is first given and a new amplitude is defined in terms of it. This amplitude has two important properties. Firstly it coincides with the Feynman amplitude for positive energies of the incoming and outgoing particles. Secondly it satisfies a certain causality condition. These enable one to obtain dispersion relations. The significance of these results in affording a new way of formulating quantum field theory is discussed.