In quantum optics, the Hilbert space of a mode of light corresponds to functions on a plane called the phase space (so called because it reminded Boltzmann of oscillators in 2-d real space.) This correspondence offers three important features: it can autonomously handle quantum theoretical calculations, it allows for the infinite-dimensional Hilbert space to be easily visualized, and it is intimately related to a basic experimental measurement (the so-called heterodyne detection). Continuous phase space correspondences exist naturally for many types of Hilbert space besides this particular infinite-dimensional one. Specifically, the two-sphere is a natural phase space for quantum spin systems. Although well studied on the theoretical and visualization fronts, the corresponding measurement (theoretically referred to as the spin-coherent-state positive-operator-valued measure or SCS POVM) has yet to find a natural way to be experimentally performed. In this talk, I will review the history of phase space, it’s connection to representation theory, quantization, coherent states, and continuous measurement. Finally, I will explain how the SCS POVM can be simply performed, independent of the quantization. Such a demonstration is a fundamental contribution to the theory of continuous quantum measurement which revives several differential-geometric ideas from the classical and modern theory of complex semisimple Lie groups.
