  The dimensionless universal coefficient $\xi$ defines the ratio of the
unitary fermions energy density to that for the ideal non-interacting ones in
the non-relativistic limit with T=0. The classical Thomson Problem is taken as
a nonperturbative quantum many-body arm to address the ground state energy
including the low energy nonlinear quantum fluctuation/correlation effects.
With the relativistic Dirac continuum field theory formalism, the concise
expression for the energy density functional of the strongly interacting limit
fermions at both finite temperature and density is obtained. Analytically, the
universal factor is calculated to be $\xi={4/9}$. The energy gap is
$\Delta=\frac{{5}{18}{k_f^2}/(2m)$.
