Finite-dimensional approximation of time evolution operators: A nonperturbative approach to quantum field theories

We suggest a time-dependent, finite-dimensional approximation scheme for time evolution operators and the S matrix in relativistic quantum field theories. It can be solved by diagonalization of finite-dimensional Hermitian matrices. The method is presented for a ${\phi }^4$-field theory. In the weak-coupling regime, where perturbation theory is applicable, our method tends to the results of perturbation theory in all orders, when the approximation parameters tend to infinity. Renormalization is achieved by introducing counterterms in the Lagrangian in the usual way. This method does not involve vertices and propagators and is a nonperturbative scheme. It is applicable to arbitrary coupling constants.