Geometry underlying no-hidden-variable theorems

The set of orientations of a measuring device (e.g., a Stern-Gerlach magnet) produced by the action of a Lie group constitutes a honmogeneous space S (e.g., a sphere). A hidden-variable measure determines a metric D on S, the triangle inequality being Bell’s inequality. But identification of S with Hilbert-space projectors induces a locally convex metric d on S. The Einstein-Podolsky-Rosen (EPR) hypotheses imply that D=d², which is impossible because the square of a locally convex metric cannot be a metric. This proves the Bell-EPR theorem. Classical systems avoid the contradiction by allowing only values d=0,1. The ‘‘watchdog’’ effect is shown to result from the form of the quantum-mechanical metric.