A Feynman diagram is a pictorial representation of a mathematical expression that describes the behavior of subatomic particles. The idea was devised in the 1940’s by Richard Feynman and consists of lines that represent particles that converge at a vertex and diverge from there to demonstrate how pieces emerge following a crash. The lines either shoot off on their own or converge again. Numbers are then added into the mix and the final result is one number called the Feynman probability which tells us how likely it is that the particle collision will occur as predicted.
But, there are a few limitations to the Feynman diagram. The first problem is that high-energy particle collisions require a high precision of measurement and as the precision increases so does the intricacy of the diagram. Secondly, these diagrams are based on the assumption that the more potential collisions there are the more accurate the prediction will be, but the more elaborate the diagram, the more misleading it tends to be. Francis Brown is a mathematician at the University of Oxford and he says on the matter, “We know for a fact that at some point it begins to diverge. What’s not known is how to estimate at what point one should stop calculating diagrams.”
However, over the past decade mathematicians and physicists have been exploring Feynman diagrams and they have discovered a connection between the values calculated from the Feynman diagram and the most important numbers found in algebraic geometry. These numbers are called periods of motives. But, as of yet, they are still unsure as to why these numbers are same. Periods have been studied for centuries, but are still said to be one of the most abstract subjects of mathematics.
Motives on the other hand are said to be fundamental building blocks of polynomial equations and have their own lot of data that’s associated with them. They have essential measurements and the most important of which is the motive’s periods. You know the motives of two systems are the same if the period of a motive in one set of polynomial equations is the same as that in a different system.
Feynman diagrams and geometry are very closely linked as are formed from vertices, rays, and line segments. Each possible route a particle collision could follow can be represented using a Feynman diagram. Each of these has its own integral, called a Feynman path integral which is the path that’s taken over the set of all paths. If you want to calculate the probability of an outcome using a certain set of starting conditions, you need to take into consideration all the possible diagrams that could explain it, take each one’s integral, and add them all together. This number is the diagram’s amplitude. Square this number and you get the probability.
To make use of these diagrams even more, physicists also use loops. These loops represent times at which particles emit and then reabsorb extra particles. Using loops increases the precision of the results, but also increases the number of Feynman diagrams that will be needed. Mathematicians have found a connection between the number of loops in a Feynman diagram and weight (the dimension of the space being integrated over) of its amplitude.
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