General probabilistic theories in phase space, the harmonic oscillator and the hydrogen atom

We show that the framework of general probabilistic theories can be extended to describe continuous-variable theories on phase space and that the phase-space formulation of general probabilistic theories can include a generalized time-evolution.

This extended framework is then used to describe toy models of a harmonic oscillator and a nonquantum hydrogen-like system. The nonquantum harmonic oscillator has discrete spectrum and exhibits tunneling behavior.  The nonquantum hydrogen-like system is stable, has discrete energy levels, and includes the Zeeman effect. This allows us to also study dynamical effects such as excitations of the hydrogen-like system by a resonant laser and Rutherford scattering. Our construction demonstrates that classical theory and quantum theory can be seen as specific choices of general probabilistic theory in phase space and that other general probabilistic theories also lead to measurable predictions.