  The variational method is a powerful approach to solve many-body quantum
problems non perturbatively. However, in the context of relativistic quantum
field theory (QFT), it needs to meet 3 seemingly incompatible requirements
outlined by Feynman: extensivity, computability, and lack of UV sensitivity. In
practice, variational methods break one of the 3, which translates into the
need to have an IR or UV cutoff. In this letter, I introduce a relativistic
modification of continuous matrix product states that satisfies the 3
requirements jointly in 1+1 dimensions. I apply it to the self-interacting
scalar field, without UV cutoff and directly in the thermodynamic limit.
Numerical evidence suggests the error decreases faster than any power law in
the number of parameters, while the cost remains only polynomial.
