We present a modification of the Berkovits superparticle. This is firstly in order to covariantly quantize the pure spinor ghosts, and secondly to covariantly calculate matrix elements of a generic operator between two states. We proceed by lifting the pure spinor ghost constraints and regaining them through a BRST cohomology. We are then able to perform a BRST quantization of the system in the usual way, except for some interesting subtleties. Since the pure spinor constraints are reducible, ghosts for ghosts terms are needed, which have so far been calculated up to level 4. Even without a completion of these terms, we are still able to calculate arbitrary matrix elements of a physical operator between two physical states. * M.J.Chesterman@qmul.ac.uk 1

We describe the complete cohomology of the Berkovits BRST operator for the superparticle. It is non-zero at eight ghost numbers, splitting into two quartets, the members of each quartet being completely isomorphic. Based only on considerations of the isomorphisms of the cohomology, and using only the standard inner product, we derive the inner product appropriate for string amplitudes. It is in agreement with Berkovits' conjectured prescription, which is one element of an equivalence class. We discuss the Chern-Simons style action for D=10 super Yang-Mills, which is now manifestly superspace covariant.

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We reformulate the conditions of Liouville integrability in the language of Gozzi et al.'s quantum BRST anti-BRST description of classical mechanics. The Das-Okubo geometrical Lax equation is particularly suited to this approach. We find that the Lax pair and inverse scattering wavefunction appear naturally in certain sectors of the quantum theory.