There is a branch of mathematics called Game theory that looks at how groups solve composite problems. As most of you know, the Schrödinger equation is the foundational equation of entire quantum mechanics realm. So apparently, there’s no reason to imagine any link among these two different subjects. But according to a group of French physicists, it’s conceivable to translate a vast number of problems in game theory into the vital language of quantum mechanics. In a recent research paper published in Physical Review Letters, physicists show that electrons and fish follow the precisly same mathematics.
Everyone knows about Schrödinger because of his popular weird cat experiment, but he’s famous among physicists for being the first person to write down an equation that completely defines the weird things that occur when you try to do experiments on the fundamental parts of matter. Schrödinger realized that you can’t label electrons or atoms or any of the other smallest bits of the Cosmos as billiard balls that will be precisely where you expect them to be accurately when you expect them to be there.
As an alternative, you have to assume that these tiny particles have locations that are spread out in the space, and that they only have some possibility of existing where you think they’re probably going to be at any point in given time. If you try to solve spread-out probabilities instead of with exact positions, you can precisely forecast the outcomes of a bunch of experiments that baffled scientists at the start of the 20th century.
On the other hand, the Game theory doesn’t seem to have completely anything to do with any of that. Overall, it looks at how a group of agents make choices to get closer to whatsoever goal they have in mind. That might mean people (optimistically) working together in traffic, or it can be people working against each other like they do in some sort of a board game.
In mean-field game theory, you’re analyzing what all of the different agents are doing on average - so it may freely apply to people in traffic, but it'd be a lot tougher to relate to a single game of Monopoly.
The example scientists directed by Igor Swiecicki from France's Laboratoire de Physique Théorique Orsay use is a group of fishes that want to stay near each other while also looking individually for food.
The fish normally travel as a lone group, with a team of individuals moving around quite arbitrarily within it. And every so often, a fish might see a bit of food away from everyone else, and move over on its own to seize it, before swimming back to its school for safety.
This means that the fish have some scattering; they’re focused in the group and rarer as you get farther away from it. Similarly, if you choose a specific spot in space, there’s some chance that you picked somewhere with a fish and some chance you picked somewhere without a fish. As the school sways past your selected spot, the possibility of discovering a fish there goes up.
The possibility of discovering a fish could have changed in any number of complex ways with equations that had never been written down before. But it doesn’t. The possibility of spotting a fish alters precisely like the possibility of finding an electron does. The fish moves according to Schrödinger’s equation, Swiecicki and his group report.
In the upcoming few years, we may see game theory progress in leaps and bounds as it takes benefit of this new linking. Scientists have been extending and twisting Schrödinger’s equation for nearly a century now, and they’ve acquired a really good approach at using it to crack even the most complicated problems. But mean-field game theory has only been known for nearly 10 years or so, signifying that there are plenty of wide-open questions showering the landscape.
Now, a vast series of those open problems might be adaptable into the context of quantum mechanics. Given how much work has gone into cracking every imaginable quantum mechanics problem, there’s a good chance those new problems will end up looking a lot like something physicists have seen before.