  In this second in a series of four articles we create a mathematical
formalism sufficient to represent nontrivial hamiltonian quantum dynamics,
including resonances. Some parts of this construction are also mathematically
necessary. The specific construction is the transforming of a pair of quantized
free oscillators into a resonant system of coupled oscillators by analytic
continuation which is performed algebraically by the group of complex
symplectic transformations, thereby creating dynamical representations of
numerous semi-groups. The quantum free oscillators are the quantum analogue of
classical action angle variable solutions for the coupled oscillators and
quantum resonances, including Breit-Wigner resonances. Among the exponentially
decaying Breit-Wigner resonances represented by Gamow vectors are hamiltonian
systems in which energy transfers from one oscillator to the other. There are
significant mathematical constraints in order that complex spectra be
accommodated in a well defined formalism which represents dynamics, and these
may be met by using the commutative real algebra $\C (1,i)$ as the ring of
scalars in place of the field of complex numbers. These mathematical
constraints compel use of fundamental (spinor) representations rather than
UIR's. By including distributional solutions to the Schr{\"o}dinger equation,
placing us in a rigged Hilbert space, and by using the Hamiltonian as a
generator of canonical transformation of the space of states, the
Schr{\"o}dinger equation is the equation for parallel transport of generalized
energy eigenvectors, explicitly establishing the Hamiltonian as the generator
of dynamical time translations in this formalism.
