Starting from the symplectic Lie groups Sp(2n, R) of an n-mode system in cl
assical and quantum mechanics we discuss, in particular, Sp(2, R) similar t
o SO(2, 1) for a single mode and Sp(4, R) similar to SO(3, 2) with some of
its important subgroups such as SU(I, 1) similar to Sp(2, R) and SU(2). We
then look at their applications in quantum optics (squeezing, phase states,
beamsplitter and polarization). We explicitly construct the root scheme of
the Lie algebra sp(4, R) in two different bases as it is an important mean
s for the investigation of the group properties and the construction of the
irreducible representations. Apart from the well known realizations of Sp(
2, R) by single-mode and two-mode systems, we discuss some other realizatio
ns of this group by fractal combinations of boson operators.