A Special Guest Post by Richard ("Dick") Flatman
"I call our world Flatland,not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space."
This is how my great-grandfather started his memoirs. Memoirs he wrote in solitary confinement. Years later he died, imprisoned and alone, and not aware that his ideas were already starting to change not only the views and imagination of us Flatlanders, but also the perspectives of you Spacelanders.
How time has progressed! My great-grandfather is now considered a hero of science. A visionary thinker who has shown us Flatlanders the route towards scientific enlightenment. An unstoppable growth of science got built upon his legacy.
It is the latest of this progress that I would like to share with you: the remarkable new insights gained into the inner workings of our microscopic world. I have played no minor part in this development. How proud my great-grandfather would have been!
===Cells and Hexons===
Our universe is two dimensional. A planar sheet that stretches endlessly. However, when thinking about Flatland, you should not imagine an infinite structureless sheet, but rather an infinite grid of microscopically small hexagonal cells. Elementary particles, called hexons, populate this grid. All objects, all matter, our own bodies, everything we Flatlanders see, is build from hexons. So if you understand the interactions and behaviors of hexons, you understand Flatland.
When hexons were discovered to be the elementary building block of matter, it was understood that clumps of matter move due to the hexons that constitute the clump jumping from cell to cell on the hexagonal grid. But how does this hopping dynamics work? How can we predict the movements of hexons? Years ago my investigations started with these questions. Little did I know what surprises I would encounter.
// visualization of a small (eight hexon) clump of matter moving along the Flatland grid.//
===The strange behavior of caged Hexons===
I spent years fruitlessly investigating the behavior of hexons. Real progress came when I managed to control planar space at the level of individual cells. This allowed me to isolate and confine single hexons. My first successful experiment is depicted below. I call this set-up a hexon cage, but you Spacelanders might prefer the term hexon billiard. By blocking a few individual cells from the planar grid (the grey cells referred to as scatter centers) I created an area in which a single hexon can rattle around. The figures below depicts what I expected to see: a hexon trapped forever in a cage formed by the scatter centers. As the dynamics would be too fast to follow, I expected not to see the time-resolved dynamics, but rather the various hexon trails as depicted.
//A hexon rattling in a trap (grey cells depict scatter cells that can not accommodate any hexons)//
//time average of the rattling dynamics//
//Alternative hexon dynamics//
//And its time averaged trace//
To my surprise I observed only one of the various hexon traces predicted. No matter how hard I tried, I could not get any trace different from the hexagon loop. This was puzzling. Things became even more puzzling when I realized I had forgotten about other hexon traces. I had overlooked the simple possibility of a hexon at rest anywhere in its cage. I never observe any of these.
Was something wrong with the experiment? To find out, I started building alternative hexon cages. I immediately hit upon further surprises. Against expectations, some of the cages build did not result in any trapped particles. Other experiments were even more puzzling. An example is shown in the picture below. The traces shown in the top give a few examples of what one would expect to see. What I observed (lower part of the figure) was markedly different. One of the expected traces, a triangular loop hitting the centers of the cage walls, did indeed reveal itself. I also encounter other traces, but none of them I could identify as one of the traces expected. One such observed trace was very enigmatic. It vaguely resembled one of the expected traces consisting of a triple loop. However, while I was sure at any time there was no more than one single hexon in the trap, the pattern consisted of three separate traces. Did I witness a hexon splitting in three?*
One other pattern was also unexpected. It consisted of one giant loop along the edges of the trap. It was as if the hexon skated along the outer cells without feeling the obstacles along the walls.
//Triangular hexon trap. Some expected paths depicted in the top part of the figure, observed paths in the lower part.//
As I started contemplating these results, I remembered a recent theory from a fellow flatlander. The theory was supposed to be an alternative description of the known laws of mechanics and states that particles going from A to B always select the shortest path possible. It is a remarkable statement, and it seems to work for large scale objects. When applied to my microscopic experiments, the principle seemed capable not only of describing the traces I expected to see, but also some of the surprising observed traces. The strange trace along the edges of the triangular trap is a case in point. That path could be described as the shortest loop that hits all corner scatterers of the cage. It was not clear to me how a shortest path rule could be morphed to describe all observed traces including the one consisting of three loops, but maybe it all depended on looking at these matters from the right perspective.
Yet I didn't really like the minimum path theory. How would a particle know what lays ahead? And if it doesn't, how can it be able to decide at each step which one would lead to a shorter route? Shortest path selection can only happen if the particle would 'sniff out' the various possible paths. Somehow this idea struck an inner chord, and I tried to work out scenarios for hexons sniffing out their environment.
===Quantum Hexodynamics===
What if a hexon tries all possibilities and 'sniffs out' all neighboring cells, and in the next step all neighbors of these neighboring cells, etc? An avalanche of paths would result. A remarkable thought occurred to me. What if these path to some degree cancel each other, and thereby largely wipe out the avalanche resulting from the sniffing process?
For cancellation to occur, hexons must have associated with them a number that can be positive as well as negative. Let's assume this number, which I will refer to as the amplitude of the hexon, can take on only two values, +1 and -1.
So we have a grid of hexagonal cells, and each cell carries an amplitude. This amplitude can be +1 (a cell occupied by an hexon with positive amplitude), it can be -1 (a cell occupied by an hexon with negative amplitude), or it can be 0 (an empty cell). In discrete time steps the hexons jump from cell to cell and thereby carry amplitude across cells. At each time step each hexon sniffs out all seven possibilities by carrying it's amplitude to each of the six neighboring cells as well as to its own cell. When all hexons have carried their amplitudes, for each cell the resulting amplitude is determined simply by adding up all the amplitudes that have arrived at that cell.
I started drawing grids and experimenting with different configuration. A cell with a positive amplitude hexon I marked red, a negative amplitude hexon cell blue, and an empty (zero ampIitude) cell faint yellow. As before, the scatter cells used to trap the hexons I colored dark grey. When I started experimenting with different patterns, the configurations invariably resulted in a runaway process. In all the cases that I tried, the sniffing out led to avalanches that filled the grid with ever larger positive and negative amplitudes. Just as I was about to abandon this pointless 'sniffing idea', I realized that for stable patterns to result, empty (yellow) cells need to be surrounded by an equal number of red and blue cells. This simplified the trial and error process, and almost immediately I stumbled upon a stable pattern. The pattern is depicted below. Recognize this? Yes, it is the observed circular trace observed in my first experiment.
//Eureka! A stable path that results when allowing positive and negative amplitude hexons to follow all possible paths. The alternation between red and blue colors represent amplitude sign flipping.//
I immediately turned to the larger triangular cage geometry. I again had to experiment a bit with different hexon configurations. I soon found a stable configuration: the puzzling three loop configuration. And then I found another one, and another one (see picture). I knew I had struck gold. The enigmatic hexon dynamics had succumbed. Quantum Hexodynamics (QHD) was born.
//Stable paths for the triangular cage.//
The simple 'sum over all possibilities algorithm' not only delivers the right hexon paths, it also excludes those paths I initially expected but never observed. A case in point is provided by any of the 'hexon-at-rest' configuration. Each of these lead to a runaway process and don't form a stable configuration.
===Quantum Pandora's box===
It is now several years after the discovery of QHD. It seems the whole Flatland physics community has started working on QHD. Many are working on practical problems and are involved in the computation of stable hexon patterns for geometries of interest. Others are investigating the implications of QHD at a more fundamental level. My friend Werner is mostly intrigued by the sum over all possibilities approach preventing hexons to be localized in one cell. Key issue is the fact that while the hexon traces make sense, the occurrence of a hexon at a particular cell is much more enigmatic. QHD predicts that if one attempts to measure the specific location of a hexon, randomness enters the picture. In fact, any cell with +1 or -1 amplitude has equal probability of the hexon showing up. I refer to this measurement principle as the 'collapse of the hexon trace'. Albert, probably the most famous Flatland physicist of all times, abhors this aspect of QHD. He insists that "God doesn't play dice".
Yet, wether Albert's God likes it or nor, fact of the matter is: QHD works. Each and every of its predictions that has been put to a test has been confirmed. Maybe QHD describes a world for which we can make successful predictions, yet we can not really understand. What is 'understanding' anyway? So let's be pragmatic on this issue. The advise I give to my fellow flatlanders is: do not keep saying to yourself, if you can possibly avoid it, "But how can it be like that?" because you will get down the drain, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
I just wonder: have you, Spaceland physicists, also landed on a QHD-like theory to describe your microscopic world? And if so, how do you handle its philosophical implications?
-Richard P. Flatman-
===Notes===
* As far as we know, Hexon splitting doesn't occur, and it is widely believed the hexon is our most fundamental building block. Despite huge efforts that culminated in the building of the Large Hexon Collider (LHC), no one has ever witnessed a hexon splitting up in more fundamental elementary particles.
