History of discrete [models of] space

The idea that matter might be made up of discrete particles existed in antiquity (see page 876), and occasionally the notion was discussed that space might also be discrete—and that this might for example be a way of avoiding issues like Zeno's paradox. In 1644 René Descartes proposed that space might initially consist of an array of identical tiny discrete spheres, with motion then occurring through chains of these spheres going around in vortices—albeit with pieces being abraded off. But with the rise of calculus in the 1700s all serious fundamental models in physics began to assume continuous space. In discussing the notion of curved space, Bernhard Riemann remarked in 1854 that it would be easier to give a general mathematical definition of distance if space were discrete. But since physical theories seemed to require continuous space, the necessary new mathematics was developed and almost universally used—though for example in 1887 William Thomson (Kelvin) did consider a discrete foam-like model for the ether (compare page 988). Starting in 1930, difficulties with infinities in quantum field theory again led to a series of proposals that spacetime might be discrete. And indeed by the late 1930s this notion was fairly widely discussed as a possible inevitable feature of quantum mechanics. But there were problems with relativistic invariance, and after ideas of renormalization developed in the 1940s, discrete space seemed unnecessary, and has been out of favor ever since. Some non-standard versions of quantum field theory involving discrete space did however continue to be investigated into the 1960s, and by then a few isolated other initiatives had arisen that involved discrete space. The idea that space might be defined by some sort of causal network of discrete elementary quantum events arose in various forms in work by Carl von Weizsäcker (ur-theory), John Wheeler (pregeometry), David Finkelstein (spacetime code), David Bohm (topochronology) and Roger Penrose (spin networks; see page 1055). General arguments for discrete space were also sometimes made—notably by Edward Fredkin, Marvin Minsky and to some extent Richard Feynman—on the basis of analogies to computers and in particular the idea that a given region of space should contain only a finite amount of information. In the 1980s approximation schemes such as lattice gauge theory and later Regge calculus (see page 1054) that take space to be discrete became popular, and it was occasionally suggested that versions of these could be exact models. There have been a variety of continuing initiatives that involve discrete space, with names like combinatorial physics—but most have used essentially mechanistic models (see page 1026), and none have achieved significant mainstream acceptance. Work on quantum gravity in the late 1980s and 1990s led to renewed interest in the microscopic features of spacetime (see page 1054). Models that involve discreteness have been proposed—most often based on spin networks—but there is usually still some form of continuous averaging present, leading for example to suggestions very different from mine that perhaps this could lead to the traditional continuum description through some analog of the wave-particle duality of elementary quantum mechanics. I myself became interested in the idea of completely discrete space in the mid-1970s, but I could not find a plausible framework for it until I started thinking about networks in the mid-1980s.