Complex conductivity of monolayer graphene and Zitterbewegung

A recently derived formula for complex conductivity of the monolayer graphene is analyzed. We show that the real and imaginary parts in this formula obey the Kramers and Kronig dispersion relations which are a good test for validity of the formula for complex conductivity of monolayer graphene. We consider also an additional test for this formula, sensitive to the integral characteristic of the conductance such as the famous f sum rule. We write it in the two dimensional form and show that it fulfils identically if we admit the cyclotron mass as an effective one and take the principal value of the integral. We find a deep relation between the graphene complex optical conductivity singularities and electrons Zitterbewegung in graphene. Namely, the value of Zitterbewegung frequency is related with the recently found magnitudes of the inductance L and capacitance C by the Thomson formula.