  This paper establishes a precise mapping between lattice and continuum
operators in theories of (1 + 1)D fermions. To reach the continuum regime of a
lattice theory, renormalization group techniques are here supplemented by a new
kind of decimation called smoothing out. Smoothing out amounts to imposing
constraints that make fermion fields vary slowly in position space; in momentum
space, this corresponds to introducing boundary conditions below a certain
depth of the Dirac sea. This procedure necessitates the introduction of a
second small parameter to describe the continuum limit. This length scale, much
larger than the lattice spacing but much smaller than the macroscopic system
size, controls the derivative expansions of fields, and hence plays the role of
the "string scale" in quantum field theory. Smoothing out a theory of Dirac
fermions on the lattice provides a transparent, fully lattice-based derivation
of the operator product expansion, current algebra, and Abelian bosonization
rules in the continuum. Nontrivial high-momentum effects are demonstrated in
all these derivations. In particular, the radius of the compact scalar dual to
the Dirac fermion is found to vary with momentum, smoothly shrinking to zero at
a cutoff set by the "string scale."
