  A theory of nonunitary-invertible as well as unitary canonical
transformations is formulated in the context of Weyl's phase space
representations. Exact solutions of the transformation kernels and the phase
space propagators are given for the three fundamental canonical maps as
fractional-linear, gauge and contact (point) transformations. Under the
nonlinear maps a phase space representation is mapped to another phase space
representation thereby extending the standard concept of covariance. This
extended covariance allows Dirac-Jordan transformation theory to naturally
emerge from the Hilbert space representations in the Weyl quantization.
