  The translation invariant model in quantum field theory is considered by
functional integrations. Ultraviolet renormalization of the translation
invariant Nelson model with a fixed total momentum is proven by functional
integrations. As a corollary it can be shown that the Nelson Hamiltonian with
zero total momentum has a ground state for arbitrary values of coupling
constants in two dimension. Furthermore the ultraviolet renormalization of the
polaron model is also studied.
