Penrose’s Space of Quantized Directions

Abstract

In the sixties, Roger Penrose came up with a radical new idea for a quantum geometry which would be entirely background independent, combinatorial, discrete (countable number of degrees of freedom), and involve only integers and fractions, not complex or real numbers. The basic structures are spin-networks. One reason we might believe that space or space-time might be discrete is that current physique tells us that matter is discrete and that matter and geometry are related through gravity. Once a discrete theory is decided on, it seems awkward that the dynamics would retain "continuous elements" in the form of real numbers (used for the probabilities for example). The great achievement of Penrose's theory is that there is a well defined procedure which gives the semi-classical limit geometry (always of the same dimension) without any input on topology (the fundamental theory does not contain a manifold).