Functional Integral Approach to $C^*$-algebraic Quantum Mechanics II: Symplectic Quantum Mechanics

We propose $Sp\,(8,\R)$ and its Langlands dual $SO(9,\R)$ as dynamical groups for closed quantum systems. Restricting here to the non-compact group $Sp\,(8,\R)$, the quantum theory is constructed and investigated. The functional Mellin transform plays a prominent role in defining the quantum theory. It provides a bridge between the quantum algebra of observables and the algebra of operators on Hilbert spaces furnishing unitary representations that are induced from a distinguished parabolic subgroup of $Sp\,(8,\C)$. As well, the parabolic subgroup renders a fiber bundle construction that models what can be described as a matrix quantum gauge theory. The formulation is strictly quantum mechanics: no \emph{a priori} space-time is assumed and the only geometrical input comes indirectly from the group manifold. But what appears on the surface to be a fairly simple-minded model turns out to have a capacious structure suggesting some compelling physical interpretations regarding space-time and fundamental interactions.