Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

168 some considerations on infinity in physicsQuantum gravity is more the name of a problem than the name of a theory. Theproblem is viewed by many as the major open problem in fundamental physics. Itoriginates from two momentous advances in the physics of the last century: quantumtheory and general relativity. These two theories have proved extraordinarily effectiveand ground most of what we know today about the physical world. However, they arecurrently formulated on the basis of assumptions that largely contradict one another.The effort to find a coherent scheme in which these two theories can make sensetogether, and therefore the effort to get back to a coherent picture of the physical world,is the task of the research in quantum gravity.The problem is not yet satisfactorily resolved, but there exist today well-developedtheories that are possible tentative solutions. The two best developed of these theoriesare string theory and loop quantum gravity (see, for instance, Rovelli 2004a). Otherdirections of research include, among others, noncommutative geometry and causal settheory. All these theories have much to say about the infinite divisibility of space, andquite remarkably, they are consistent with one another in this regard. They open a novelperspective on the possibility of thinking about the problem of the infinite divisibilityof space.In the following section, I illustrate this new perspective, with particular emphasis onthe loop quantum gravity (or loop gravity) approach, given that this is the developmentin which I have participated. In a later section, I discuss the relevance of a theory thatis still tentative, in the context of the discussion of a general problem such as the oneposed by the notion of infinity.General relativity is Einstein’s final theory of spacetime. It has successfully replacedthe Newtonian conceptualization of space and time. The empirical success of generalrelativity has been triumphal, and today the theory is one of the best-confirmed scientifictheories ever. In this theory, Newtonian space, which for Newton was a “continuoussubstratum” without dynamical properties, is reinterpreted as a continuous “substance”with dynamical properties, that is, a continuous substance that can be stretched and bentlike a rubber sheet. Space stretches and bends following some equations that Einsteinwrote, which are today called Einstein’s equations.In other words, general relativity is formulated in a background-independent way:it does not presuppose from the very beginning an arena for physical processes todevelop, but rather it shows that such an arena is born out of a dynamical entity.On the other hand, according to quantum theory, any entity with dynamical propertiesof this kind is always “quantized.” This means, in particular, that, when observedat sufficiently small scale, this entity manifests itself in the form of “packets” or small“chunks,” or “quanta.” A well-known example of this “quantization” phenomenon isthe fact that electromagnetic waves, when observed on a sufficiently small scale, turnout to be formed by a cloud of tiny particles: the photons. Quantum theory also has hadspectacular confirmations. It is at the basis of much of the current technology, such asall the computer technology. It is one of the best theories of the world we have ever had,and it therefore encodes the basis of our present knowledge about the physical world.Now, if we rely on quantum theory and general relativity, we are necessarily led toconclude that space, being an entity with dynamical properties like the electromagneticfield, is itself made by small “quanta” as well. In other words, to our best knowledgeabout the natural world, it is quite likely that space is not infinitely divisible.