There is reason to suppose that this, all this, is a simulation. Not the least enticing reason, but perhaps a misleading one, is that quantum mechanics tells us that at a small scale the world is described by probabilities and statistics. Not only is it described by a statistical distribution but unless we probe closely the statistical distribution is not rendered for us to notice. This is efficient and puts us in mind of a simulation or video-game where an algorithm decides what will be displayed on screen: if we zoom up close the texture of the simulation breaks down, but how it breaks down into pixels is determined by the rules of the program.

Heisenberg took seriously the notion that only what we can measure exists. Not so long ago I enjoyed Quantum: Einstein, Bohr and the Great Debate about Reality by Manjit Kumar and learnt that Heisenberg had been inspired, long before he knew much about matrices or anti-commuting variables, by the track of a charged particle through a cloud chamber. As the particle passes through the vapour of the cloud chamber it ionises molecules around which other molecule condense. Thus a track of condensed vapour is formed following the path of the charged particle/atom/molecule. But, reasoned Heisenberg, although the path appears to be continuous it is actually only a sequence of points which occur where each ion in the vapour is formed - unless it is measured the particle's postion is not known for certain. Of course faced with this thought Heisenberg opted for the wild solution that when the particle's position is not measured it does not exist. In the film version of Michael Frayn's play Copenhagen, Heisenberg is shown being inspired while walking beneath a series of pools of light and then vanishing in the shadows between. It all leads very temptingly to the idea that, just like in a computer game, wheresoever we do not look is not rendered and that the fine-grain detail could be displayed by according to an algorithm. So is this a simulation?

Let me take a different tack and wonder whether it might be possible one day (let t tend towards infinity...) to construct a simulator. We might dream of something into which we may project all the information in our most advanced brain-state and where all its functions can be replicated as if in the real world, or perhaps I should say the world we conceive as real presently. It will be informative to think about this notion of the real world. To do so permit me the assumption that there is some grand unified theory possessing a large symmetry and let the simulator we are considering be a perfect simulation. That is there are no flaws in the simulation that would allow the insider, the brain in the vat, to deduce that he/she/it exists inside the simulator (the cat from The Matrix is back in the bag, so to speak, and there are no rounding error problems inside the simulator for a messianic figure to take advantage of...). Note that I am taking this to mean this means that machine must encode all there is to know about M-theory in one form or another. Finally let us introduce a new constraint, let the machine be localised, i.e. it is not infinite in any direction - it can fit in a bounded volume, nutshell or even teapot, a big enough teapot. All the assumptions that have been made can be recast as saying we have a quantum gravity simulator in a box, let us call it M-box.

M-box buzzes, fizzes, gurgles and pops, and by assumption recreates all the information content of a universe. It must be one hell of a machine. If we suppose it could exist what are the corollaries for the physical description of the universe it exists within and simulates perfectly? Since we assumed it is finite in extent and yet contains all of M-theory must we throw out of M-theory any non-local effects that occur? After all the machine is localised and can recreate perfectly M-theory. By localised I had better emphasise that I mean that its internal workings do not rely on any non-local physical effects. This point is worth some more words. Our usual picture of a computer simulation involves information encoded in 1's or 0's in bits of information, more ambitious quantum computers would build information upon data units that can be in a superposition of two states - these are called qubits. Qubits could be built on electron spin for example. An electron is a relatively simple solution of quantum field theory. For the M-box we would like to build its circuitry on solutions of M-theory; we would like the M-box to be able to excite membranes extending into the compact higher-dimensional space. Membranes can be spun and vibrated, but membranes can also be U-dualised into fivebranes. To take this to its limit the clever M-box we imagine will encode data by U-dualising membranes. We try hard not to think about how it might do this. But we emphasise that the M-box is imagined to encode the full set of M-theory solutions by U-dual rotations of a single membrane. Mathematically the object which encodes all these solutions is the coset $\frac{E_{11}}{{\cal R}(E_{11})}$ where ${\cal R}(E_{11})$ is a real-form of $E_{11}$. Now we may ask the question is there a single solution to M-theory which itself possesses the full symmetries of M-theory. The closest we know of is the BKL cosmological singularity which is expected to carry the symmetries of $\frac{E_{10}}{{\cal R}(E_{10})}$. But awe-inspiring as a cosmological singularity may be it is still not enough to be part of the circuit board of our conjectured M-box. If we did construct such an $E_{10}$ M-box then once we embedded our brain-state inside the simulation we would be able to deduce that some parts of the full $E_{11}$ symmetry are missing (we would probably need some simulated psychiatric help after our simulated selves came to this conclusion). So if one could identify a solution to M-theory which itself possessed a full $E_{11}$ symmetry then we would be in business and M-boxes could start to be rolled off the manufacturing line. However as things stand if we deduce that there is an $E_{11}$ symmetry to M-theory and we wondered if we were simulated we might imagine that the simulator is not an M-box but rather an M'-box (surely this is a catchier name than M-box 360?) which is running in a universe which is governed by M'-theory - which has a larger symmetry than $E_{11}$. The argument is then repeated so that M-box $\subset$ M'-box $\subset$ M''-box $\subset \ldots $. One would agree that this is not an enticing picture for someone looking for a closed description of the universe and hoping that we do live in a simulation. One of the attractions of a Kac-Moody algebra is that it has infinite generators and one might hope yet that a clever way to embed $E_{11}$ inside itself could be found and associated with an M-theory (bound-state) solution. Alternatively keen simulationists might prefer to spend their time in Skyrim instead.

Michael Atiyah 1929-2019

21 hours ago