A $\lambda {\varphi }^4$ Quantum Field Theory without Cutoffs. I

We construct the dynamics for a local quantum field theory in two-dimensional space-time. Our model describes a scalar field $\varphi$ with a $\lambda {\varphi }^4$ self-interaction. We rely upon the fact that, in the Heisenberg picture, influence propagates at the speed of light. The resulting dynamics is independent of any cutoff, and hence the theory is formally Lorentz-covariant.