  Using path integral formalism, we show that the Abrikosov-Tkachenko vortex
lattice may equivalently be understood as an array of Josephson junctions. The
Tkachenko modes are found to be basically equivalent to the low energy
excitations (Goldstone modes) of an ordered state. The calculated frequencies
are in very good agreement with recent experimental data. Calculations of the
fluctuations of the relative displacements of the vortices show that vortex
melting is a result of quantum fluctuations around the ordered state due to the
low energy excitations (Tkachenko modes)and occurs when the ratio of the
kinectic energy to the potential energy of the vortex lattice is 0.001.
