Symplectic groups in quantum optics

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.