  It has been established that Matrix Product States can be used to compute the
ground state and single-particle excitations and their properties of lattice
gauge theories at the continuum limit. However, by construction, in this
formalism the Hilbert space of the gauge fields is truncated to a finite number
of irreducible representations of the gauge group. We investigate
quantitatively the influence of the truncation of the infinite number of
representations in the Schwinger model, one-flavour QED$_2$, with a uniform
electric background field. We compute the two-site reduced density matrix of
the ground state and the weight of each of the representations. We find that
this weight decays exponentially with the quadratic Casimir invariant of the
representation which justifies the approach of truncating the Hilbert space of
the gauge fields. Finally, we compute the single-particle spectrum of the model
as a function of the electric background field.
