  We consider critical dense polymers ${\cal L}_{1,2}$, corresponding to a
logarithmic conformal field theory with central charge $c=-2$. An elegant
decomposition of the Baxter $Q$ operator is obtained in terms of a finite
number of lattice integrals of motion. All local, non local and dual non local
involutive charges are introduced directly on the lattice and their continuum
limit is found to agree with the expressions predicted by conformal field
theory. A highly non trivial operator $\Psi(\nu)$ is introduced on the lattice
taking values in the Temperley Lieb Algebra. This $\Psi$ function provides a
lattice discretization of the analogous function introduced by Bazhanov,
Lukyanov and Zamolodchikov. It is also observed how the eigenvalues of the $Q$
operator reproduce the well known spectral determinant for the harmonic
oscillator in the continuum scaling limit.
