  We find the full interacting Lagrangian and Hamiltonian for quantum strings
in a near plane wave limit of AdS_4 x CP^3. The leading curvature corrections
give rise to cubic and quartic terms in the Lagrangian and Hamiltonian that we
compute in full. The Lagrangian is found as the type IIA Green-Schwarz
superstring in the light-cone gauge employing a superspace construction with 32
grassmann-odd coordinates. The light-cone gauge for the fermions is non-trivial
since it should commute with the supersymmetry condition. We provide a
prescription to properly fix the kappa-symmetry gauge condition to make it
consistent with light-cone gauge. We use fermionic field redefinitions to find
a simpler Lagrangian. To construct the Hamiltonian a Dirac procedure is needed
in order to properly keep into account the fermionic second class constraints.
We combine the field redefinition with a shift of the fermionic phase space
variables that reduces Dirac brackets to Poisson brackets. This results in a
completely well-defined and explicit expression for the full interacting
Hamiltonian up to and including terms quartic in the number of fields.
