  The compact groups such as $SU(n)$ and $SO(n)$ groups have been heavily
studied and applied in the study of quantum many body systems. However, the
non-compact groups such as the real symplectic groups are less touched. In this
paper, we will reveal that the quantum dynamics of two-component Bose-Einstein
condensate can be described by a \emph{non-compact} real symplectic group
$Sp(4,R)$. With this group, we can give a explicit form for the wavefunction in
any time of the evolution, meanwhile, map this whole time evolution to a
trajectory in a six-dimensional manifold. By introducing a polar coordinate, we
can visualize this six-dimensional manifold in 2d unit disk and reveal the
relation between the behavior of the trajectory in this manifold and the
eigen-energies of the Hamiltonian. Furthermore, the time evolution of
expectation value of a physical observable such as number operator is proven
closely related to the behavior of the trajectory in this manifold.
