Louis Crane has contributed a great paper to Dan Oriti's "QG Approaches" book. I started a thread on Crane's paper when it made arxiv in February, but to get a better focus on it I want to start a fresh thread. Around 1 March, Crane corrected an error in the abstract and reposted on arxiv. Earlier he said the book's publisher was Oxford, instead of Cambridge U. P. This was confusing because Oriti's book is the obvious place for the essay. I recommend this paper to anyone interested in Quantum Gravity. Why? Because of something Carlo Rovelli said in his book. He said that physics can get along fine for extended periods of time----sometimes for many decades-----ignoring philosophical issues (like "what is time?" what is "space?") and working physicists will often act like they think considering basic questions are a waste of effort-----but sometimes you reach a point where you dont make progress unless you ask deep philosophical questions and check to see if your math foundations are actually right (or are they just what you inherited from the previous occupant of your office or picked up in grad school?) And Rovelli said that both Newton and Einstein were people who asked philosophical questions and changed the framework, or altered the foundations. They came at a time when there was a philosophical river to cross----so to make progress they had to be MORE than just routine physicists. Rovelli suggests that if people are having an extraordinary protracted struggle to make a GENERAL RELATIVISTIC QUANTUM FIELD THEORY that is may be because they are philosophically lax. They neglect to tackle philosophical questions and get an up-to-date mathematical model of spacetime to work with. So here is Crane ( of the Barrett-Crane spinfoam model) and HE MAY BE WRONG or he may be right but at least he grapples the foundation issues. http://arxiv.org/abs/gr-qc/0602120 Categorical Geometry and the Mathematical Foundations of Quantum General Relativity Louis Crane "We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites." In this paper he cites a paper with Dan Christensen that we had a thread about over a year ago IIRC http://arxiv.org/abs/gr-qc/0410104 Causal sites as quantum geometry J. Daniel Christensen, Louis Crane 21 pages, 3 figures; v2: added references; to appear in JMP J.Math.Phys. 46 (2005) 122502 "We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural 'tangent 2-bundle', analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well." Even if Crane is wrong this 2006 paper is a good place to start thinking about QG and I'm inviting people to discuss it. My thought just at the moment is that Georg Riemann invented the continuum (around 1850) that Einstein used (around 1915) to build Gen Rel. The continuum (a diff manif) was good because you could have one WITHOUT GEOMETRY. it didnt have a rigid metric structure, Riemann showed how to put any number of metrics on it. So Einstein could make the geometry itself, the metric, be a dynamical variable thing----the X that was not given to start with but that you solved his equation to get, and interacted with matter. So that is already very good, and using Riemann's continuum allowed Einstein to make a BACKGROUND INDEPENDENT theory of spacetime that didnt have a prior-chosen background metric. But Crane says THIS IS NOT GOOD ENOUGH and he says that working relativists do not believe that the Riemann continuum (the diff manif) is realistic. It is not a realistic model: it has arbitrarily small distances and uncountably many points arbitrarily close together and the same integer dimension all the way down-----a lot of stuff that has no operational meaning and doesnt correspond to realworld measurements. Crane suggests that we will never succeed in getting a good QG----a good quantum physics of spacetime and matter-----until we THROW OUT RIEMANN!!! And he suggests to replace Riemann with GROTHENDIECK!!! Alexander Grothendiek was the greated mathematician circa 1950 in the same way that Georg Riemann was the greatest circa 1850. Grothendiek is the immortal and awesome eminence of mid-20th century math. To me this sounds both crazy and remarkably reasonable. If, at this point, an idea of how to do QG has to be crazy enough to be right then this has a chance. Also this Crane paper is very well written and gives lots of intuitive/conceptual verbal description. It is not all formulas and technical stuff. Crane has thought about what the difficulties are and come to some general conclusions. So I hope a bunch of us read it and can discuss it a little.