  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. The main idea is to work
directly in the basis that diagonalizes the free part of the model considered,
which allows to fit its short distance behavior exactly. This makes
computations slightly less trivial than with standard CMPS. However, they
remain feasible and I present all the steps needed for the optimization. The
asymptotic cost as a function of the bond dimension remains the same as for
standard CMPS. I illustrate the method on the self-interacting scalar field,
a.k.a. the $\phi^4_2$ model. Aside from providing unequaled precision in the
continuum, the numerical results obtained are truly variational, and thus
provide rigorous energy upper bounds.
