A.7.3

Probably the first mentioning of the idea of a finite, i.e. discrete, nature of reality goes back to the Greek philosopher Leukipp in 500 B.C. (and his scholar Demokrit), coining the term `atom' as the fundamental indivisible building block of matter. This picture still holds today where fermions (i.e. leptons and quarks) -- not showing any signs of a substructure -- account for the (SM) matter content of the universe. In the year 1900 Planck quantized, i.e. discretized (black body) electromagnetic radiation, in effect founding QM. Hence another phenomenon which appeared to be continuous was made finite. Indeed the whole structure of space-time itself could turn out to be discrete. This idea is in fact not new. Within the context of 0-point energy seen in note [] and section 3.3 it has been observed that a space-time lattice would give a cut-off rendering eqs. () and () naturally finite. Unfortunately these discrete structures destroy Lorentz invariance. In 1947 [Sn47] and [Ya47] introduced the idea of using non-commuting coordinates (appendix A.6) to mimic a lattice structure of space-time in a covariant way. $^{\text{\scriptsize\cite{apdsp3}}}$ The notion of discrete space-time has found various applications since and still appears in many different modern-day theories, most prominently within the context of quantum gravity, mentioned in appendix A.5. Loop gravity and spin networks were seen to yield a discrete structure of space-time. The whole issue of quantum geometry builds on this finite notion of space: the area/volume of a given physically defined surface/spatial region become operators which in turn are quantized and can be computed to yield a discrete spectrum of eigenvalues. In [Ro98] three traditional approaches to discrete quantum gravity are explained, namely Regge calculus, dynamical triangulations and Ponzano-Regge state sum models. The idea is that classical GR is regularized in terms of some lattice theory which is then quantized. If string/M-theory is correct, then the $p$-brane is the fundamental dynamical entity. In our universe strings (1-branes) play the most important role: depending on the excitational mode of the string it manifests itself as fermionic matter or bosonic force carriers. String/M-theory also rejects the notion of infinitely small sizes, thus bounding reality from below by defining a minimum size. $^{\text{\scriptsize\cite{apdsp0}}}$ This notion is however related to the subtle question of how we can measure (i.e. probe) small length scales. The issue is very close to the quantum mechanical measurement problem of the next section. Hence the reality or status of concepts we cannot perceive (but can intellectually deduce or comprehend) seems to be in limbo. The implication of string/M-theory is also that space is quantized or discretized. $^{\text{\scriptsize\cite{apdsp1}}}$ It seems as though the apparent continuous and infinite nature of reality is only an illusion based on the smallness of the quanta comprising it. $^{\text{\scriptsize\cite{apdsp2}}}$ Smolin also deduces a discrete nature of space-time from general quantum gravity research; [Sm00].

In [Lie00] a space-time lattice of harmonic oscillators is linked to Lorentz transformations and special relativity. Quantized space-time and classical gravity are brought into relation in reference [MM96]. The idea that the quanta of space-time posses physical reality is proposed in [Af00] with the intriguing result that the fundamental string is itself identified as such a space-time quantum. Space-time quantization and Matrix theory is discussed in [Tan00]. In an interesting series Sidharth discusses the idea that fermions are black holes $^{\text{\scriptsize\cite{apdsp4}}}$. He is then lead to a possible unification scheme and deduces that space-time must have discrete character; [Sid98i]. A somewhat exotic paper proposing a fractal theory of space and time with fractional dimensions is [Kob00]. An interesting result is the existence of an anti-gravitational force; recall section 3.1. In 1974 Wilson introduced the so-called lattice gauge theory which enables non-perturbative calculations of low-energy strong interactions in QCD; [Kak93]. In this approach space-time is replaced by a discrete lattice. Grady proposes that our universe is a giant crystal (lattice) growing in a 5-dimensional liquid; [NS99ii].