The formalism behind quantum field theory requires one to construct physical models satisfying a number of technical precepts known as Wightman axioms. In the 1970's it was discovered that the Wightman axioms can be recast into a more malleable form known as the Osterwalder-Schrader axioms. The purpose of the thesis is to demonstrate that scattering processes can be explicitly realized in this new formalism by utilizing classical tools from mathematical analysis.