Discussion

If the mentioned ideas are taken seriously then the nature of space-time would re-appear in a radically new light. There would be some profound consequences to be addressed and some bizarre features would emerge. First of all space and time would be comprised of fundamental `atoms', i.e. quanta of Planck length ( $l_P \approx 1.6 \times 10^{-35}$ [m]) and time ( $t_P \approx 5.4 \times 10^{-44}$ [s]). Space would be built up from `boxes' of the volume $l_P^3$ [m$^3$] and time would be manifested as a series of strung together `time-frozen' frames separated by $\Delta t =t_P$, exactly like a cinematic film consists of a series of pictures running at a speed of 48 images per second, creating the illusion of a continuous passage of time. Movement would be quite magical in this scenario because an idealized particle localized in such a fundamental space volume would vanish at a certain time $t_i$ and reappear in the neighboring volume at $t_f = t_i + \Delta t$. There exists no reality `during' $\Delta t$. The closest picture visualizing this strange concept would be a 2-dimensional lattice of a LCD computer panel where movement is simulated by the individual pixels comprising the image switching on and off. An interesting hypothesis resting on the premise of the finite nature of reality is concerned with the transformation or evolution of the finite set of information describing the state of such a finite system (i.e. the universe) from one time frame to the next. Fredkin analyzes these processes within the context of information processing, programming and computer science; [Fre-i], [Rh00] and [Fre-ii]. These ideas propose that information is in fact the deepest layer of reality. $^{\text{\scriptsize\cite{apdsp5}, \cite{apdsp6}, \cite{apdsp7}, \cite{apdsp8}}}$

Further applying the ideas of discrete space time to different branches of physics:

• If the big bang is the origin of time then this instant would be labeled $t_0=0$. The next existing time would be at $t_1$, after the Planck interval $\Delta t$. At that time frame the universe would be one single `box' (or cell) of space containing the whole energy that is still seen to be conserved today. It is interesting to note that all cosmological calculations only start at $t_1$ (but then go on to describe the evolution for the next 15 billion years). So the quantum gravitational problem of times smaller than $t_P$ wouldn't actually exist in this finite reality.
• If the creation of the first space cell is accompanied by the creation of an equivalent space cell with negative energy, then the universe could be seen to emerge from nothing without violating any quantities. So the question where reality came from could simply be answered by the analogy of two waves with opposite amplitudes canceling each other. Hence anything which arises ex nihilio is necessarily dual. What causes the void to polarize is, of course, still unanswered. The possibility of creating the universe out of nothing was also noted on page of chapter 4.2.
• Is the expansion of space due to the fundamental space volumina undergoing a kind of division or multiplication, invoking the picture of (reproductive biological) cell division? What drives the expansion?
• Are the space-time cells in fact the compact dimensions? This would mean that a higher-dimensional space does not have a discrete structure. Only upon compactification do the higher-dimensions manifest themselves at every `point' in the non-compact dimensions as a small geometric space (e.g. Calabi-Yau manifold); recall the last paragraph of appendix A.4. These shapes would then not only influence the properties of the non-compact space (as seen in note []) they would also be the fundamental elements of space-times themselves. Is this idea compatible with the notion of expansion being generated by the increase of the number of space quanta?
• What other characteristics can these space-time cells display other than the ability to contain energy and mass?
• How exactly can a cell incorporate a string or generally a $p$-brane? How do D-branes fit into the picture? What about the loops in loop gravity?
• The notion of space-time cells (and their dynamics) calls for a thorough mathematical analysis. Is topology sophisticated (i.e. fundamental) enough to describe such phenomena? Does quantum geometry support these ideas?

The notion of an intrinsic energy density of space, as summarized in the first section, does in fact also induce the idea of a discrete nature of space-time. If space, i.e. the vacuum can incorporate energy then it is natural to ask why space is not a quantized entity. The idea of natural units, i.e. that energy is the only free parameter relating length and time (among other quantities) also hints at this question; see note []. Consider the two basic energy relations of special relativity and QM:

$$E=mc^2 \qquad E = \hbar \omega.$$ (A.3)

Energy is proportional to mass and energy is proportional to the frequency of massless particles. In natural units and observing that frequency is defined as an inverted time interval one finds

$$E=m = \omega = \frac{1}{\Delta t} = \frac{1}{L}.$$ (A.4)

Hence eq. (A.5) relates energy to time intervals and length scales. This also reflects the ideas of special relativity because zero mass implies an infinite time interval (i.e. no time), whereas a finite mass implies a finite time interval. Also, Lorentz-contraction seems to be incorporated in some sense. Recall that space contains energy by virtue of $v$ and $\Lambda$; eqs. (), (), () and eqs. () (or () in natural units), (). In conjunction with eq. (A.5):

$$E=m = \omega = \frac{1}{\Delta t} = \frac{1}{L} = v = \Lambda^{1/2}.$$ (A.5)

This is a very basic unifying scheme relating different fundamental phenomenas to energy. Hence the effect of linking all physical manifestations to one source which is quantized, i.e. energy, leads to the requirement that space-time should be discrete as well. Eq. (A.6) allows the introduction of at least four fundamental energy/mass scales: $E_{space-time}$ defined by $\Delta t$ and $L$, $E_{electro-weak}$ defined by $v$, $E_{inflation}$ defined by $\Lambda_{inf}$ (appendix A.2) and $E_{today}$ defined by $\Lambda_0$ (section 3.3). A loose description of physics is also present in eq. (A.6)

$$E= \overbrace{m = \omega}^{\text{fermions and bosons}} = \underbr... ...pace-time}} = \overbrace{v = \Lambda^{1/2}.}^{\text{Higgs mech. and inflation}}$$ (A.6)

The ideas presented in this section are, of course, extremely speculative and on a hypothetical scientific level. However, the continual progression of science is slowly reaching boarder-line subjects. Especially string/M-theory, if it is to be taken seriously, will require a lot of new physical interpretations. The above ideas are tentatively proposed as a small step in such a direction, and are concerned with the most basic constituent of reality any physical theory hopes to model: space-time.