  We extend the recently introduced continuous matrix product state (cMPS)
variational class to the setting of (1+1)-dimensional relativistic quantum
field theories. This allows one to overcome the difficulties highlighted by
Feynman concerning the variational procedure applied to relativistic theories,
and provides a new way to regularize quantum field theories. A fermionic
version of the continuous matrix product state is introduced which is
manifestly free from fermion doubling and sign problems. We illustrate the
power of the formalism with the simulation of free massive Dirac fermions, the
Gross-Neveu model, and the Casimir effect. We find that cMPS can capture chiral
symmetry breaking with absolute scaling of the chiral parameter, and that
boundary effects can be accommodated with modest computational effort.
