Relativistic continuous matrix product states for quantum fields without cutoff

I introduce a modification of continuous matrix product states (CMPS) that makes them adapted to relativistic quantum field theories (QFT). These relativistic CMPS can be used to solve genuine 1+1 dimensional QFT without UV cutoff and directly in the thermodynamic limit. I explain the intuition for the modification, and show all the computations needed for the optimization of the ansatz. I illustrate the method on the self-interacting scalar field, a.k.a. the $\phi^4_2$ model. The numerical results obtained are truly variational, and thus provide rigorous energy upper bounds.