Complex and spinorial techniques of general relativity are used to determine all the states of SU(2) invariant quantum mechanical systems in which the equality holds in uncertainty relations for the components of the angular momentum vector operator in two given directions. The expectation values depend on a discrete quantum number and two parameters, one of them is the angle between two angular momentum components and the other is the quotient of two standard deviations. Allowing the angle between two angular momentum components to be arbitrary, a new genuine quantum mechanical phenomenon emerges: it is shown that although standard deviations change continuously, one of the expectation values changes discontinuously on this parameter space. Since physically neither of the angular momentum components is distinguished over the other, this discontinuity suggests that the genuine parameter space must be a double cover of this classical one: it must be diffeomorphic to a Riemann surface known in connection with the complex function z. Moreover, the angle between angular momentum components plays the role of the parameter of an interpolation between the continuous range of expectation values in the special case of orthogonal angular momentum components and the discrete point spectrum of one angular momentum component. The consequences in the simultaneous measurements of these angular momentum components are also discussed briefly.
Elementary particles are found in two different situations: (i) bound to metastable states of matter, for which angular momentum is quantized, and (ii) free, for which, due to their high energy-momentum and leaving aside inner a.m. or spin, the
h-quantization step is completely harmless. Perhaps Quantum Mechanics can be seen just as the simplest mathematical formalism where angular momentum (the magnitude of each of its three orthogonal projections) is by construction quantized: all possible values are taken from a discrete set. Indeed: (i) This idea finds support in very reasonable, completely classical physical arguments, if we place ourselves in the framework of Stochastic Electrodynamics (SED): there, all sustained periodic movement of a charge must satisfy a power balance that restricts the value of the average angular momentum, on each of its projections. (ii) It gives a natural explanation of the concept of "photon", as a constraint on the observable spectrum of energy-momentum exchanges between metastable physical states, in particular also for its discreteness. QM would be, in this picture, a semi-static theory, transparent to all the (micro)-dynamics taking place between apparently "discrete" events (transitions in the state of the system). For instance, (the magnitude of the projections of) quantum angular momentum would only reflect average values over a (classical) cyclic trajectory, a fact that we regard as almost obvious given the particularity of the corresponding addition rules in QM.