  A general analysis of squeezing transformations for two mode systems is given
based on the four dimensional real symplectic group $Sp(4,\Re)\/$. Within the
framework of the unitary metaplectic representation of this group, a
distinction between compact photon number conserving and noncompact photon
number nonconserving squeezing transformations is made. We exploit the
$Sp(4,\Re)-SO(3,2)\/$ local isomorphism and the $U(2)\/$ invariant squeezing
criterion to divide the set of all squeezing transformations into a two
parameter family of distinct equivalence classes with representative elements
chosen for each class. Familiar two mode squeezing transformations in the
literature are recognized in our framework and seen to form a set of measure
zero. Examples of squeezed coherent and thermal states are worked out. The need
to extend the heterodyne detection scheme to encompass all of $U(2)\/$ is
emphasized, and known experimental situations where all $U(2)\/$ elements can
be reproduced are briefly described.
