Quantum Mechanics as a Theory of Incompatible Symmetries

It is increasingly becoming realized that incompatible variables, which play an essential role in quantum mechanics (QM), are not in fact unique to QM. Here we add a new example, the "Arrow" system, to the growing list of classical systems that possess incompatible variables. We show how classical probability theory can be extended to include any system with incompatible variables in a general incompatible variables (GIV) theory. We then show how the QM theory of elementary systems emerges naturally from the GIV framework when the fundamental variables are taken to be the symmetries of the states of the system. This result follows primarily because in QM the symmetries of the Poincare group play a double role, not only as the operators which transform the states under symmetry transformations but also as the fundamental variables of the system. The incompatibility of the QM variables is then seen to be just the incompatibility of the corresponding space-time symmetries. We also arrive at a clearer understanding of the Born Rule: although not primarily derived from symmetry - rather it is simply a free Pythagorean construction for accommodating basic features of classical probability theory in Hilbert spaces - it is Poincare symmetry that allows the Born Rule to take on its familiar form in QM, in agreement with Gleason's theorem. Finally, we show that any probabilistic system (classical or quantal) that possesses incompatible variables will show not only uncertainty, but also interference in its probability patterns. Thus the GIV framework provides the basis for a broader perspective from which to view QM: quantal systems are a subset of the set of all systems possessing incompatible variables (and hence showing uncertainty and interference), namely the subset in which the incompatible variables are incompatible symmetries.