Generalized Wavefunctions for Correlated Quantum Oscillators II: Geometry of the Space of States

In this second of a series of articles, a pair of quantized free oscillators is transformed into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, creating dynamical representations of numerous semi-groups from the Hamiltonian free system. The free oscillators are the quantum analogue of action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodate in a well defined formalism, which 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. 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. The space of states is explicitly constructed using a realization of the Lie algebra of the group of dynamical transformations as connections