Diagonalizing the Hamiltonian of $\lambda \phi^4$ Theory in 2 Space-Time Dimensions

We propose a new non-perturbative technique for calculating the field-theory S-matrix directly from the eigenstates of the Hamiltonian. Our method involves a discretized momentum space and a momentum cutoff, thereby truncating the Hilbert space and making numerical diagonalization of the Hamiltonian achievable. We show how to do this in the context of a simplified $\lambda \phi^4$ theory in two space-time dimensions. We present the results of our diagonalization, its dependence on time, its dependence on the parameters of the theory and its renormalization.