  A free boson on a lattice is the simplest field theory one can think of. Its
partition function can be easily computed in momentum space. However, this
straightforward solution hides its integrability properties. Here, we use the
methods of exactly solvable models, that are currently applied to spin systems,
to a massless and massive free boson on a 2D lattice. The Boltzmann weights of
the model are shown to satisfy the Yang-Baxter equation with a uniformization
given by trigonometric functions in the massless case, and Jacobi elliptic
functions in the massive case. We diagonalize the row-to-row transfer matrix,
derive the conserved quantities, and implement the quantum inverse scattering
method. Finally, we construct two factorized scattering $S$ matrix models for
continuous degrees of freedom using trigonometric and elliptic functions. These
results place the free boson model in 2D in the same position as the rest of
the models that are exactly solvable \`a la Yang-Baxter, offering possible
applications in quantum computation.
