  We adopt the Dirac model for graphene and calculate the Casimir interaction
energy between a plane suspended graphene sample and a parallel plane perfect
conductor. This is done in two ways. First, we use the Quantum Field Theory
(QFT) approach and evaluate the leading order diagram in a theory with 2+1
dimensional fermions interacting with 3+1 dimensional photons. Next, we
consider an effective theory for the electromagnetic field with matching
conditions induced by quantum quasi-particles in graphene. The first approach
turns out to be the leading order in the coupling constant of the second one.
The Casimir interaction for this system appears to be rather weak. It exhibits
a strong dependence on the mass of the quasi-particles in graphene.
