Abstract
text of abstract (We present a utilitarian review of the family of matrix
groups $Sp(2n,\Re)$, in a form suited to various applications both in optics
and quantum mechanics. We contrast these groups and their geometry with the
much more familiar Euclidean and unitary geometries. Both the properties of
finite group elements and of the Lie algebra are studied, and special attention
is paid to the so-called unitary metaplectic representation of $Sp(2n,\Re)$.
Global decomposition theorems, interesting subgroups and their generators are
described. Turning to $n$-mode quantum systems, we define and study their
variance matrices in general states, the implications of the Heisenberg
uncertainty principles, and develop a U(n)-invariant squeezing criterion. The
particular properties of Wigner distributions and Gaussian pure state
wavefunctions under $Sp(2n,\Re)$ action are delineated.)
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