Rules of the Game

How to sort infinity and infinity, and what that means for our ability to understand the world we live in

Prof. Doron Gefner. The merger circle
Human curiosity sometimes leads us to areas that are hard to imagine. The attempts to understand the world in which we live do not only distinguish humans from animals. They also chart the evolution of human culture, sorting out the concepts and ideas that developed in different communities, at different times and in different places. As far as the objective function is concerned, there is no great difference between the philosophers, the theologians and the scientists. Everyone wants to know how the world will react to this or that fact. Or, what exactly will happen if we press a certain button. What, really, are the rules of the game according to which "the world" behaves. But what exactly do we mean when we say "the world": the universe in its entirety? To a simplified model that represents the universe? The answer to this question depends on how accurate you want to be. If you are willing to compromise on the level of accuracy, you can talk about the entire universe. If you insist on complete accuracy, it seems that there is no choice but to compromise on a certain reduction or simplification of the definition of "the world".
Physicists, it turns out, examine many different "worlds". Some of them are purely theoretical, but all of them can teach us something about the real world in which we live.
One type of such systems are field theories. For example, quantum field theory is a theory in which a field is divided into packets of photons (electromagnetic field). Field theory in four dimensions was tested and confirmed - in theory and experiments - with an accuracy of up to 13 digits after the point. In fact, this is the most precisely studied physical theory.
For comparison, two-dimensional field theories of a certain type ("rational conformations"), which do not include mass, can be solved accurately. This solution may be used as a model for phase transitions (like the transitions of matter from one state of aggregation to another state of aggregation), as well as a model for string theories in which the four forces acting in the universe can be calculated. We are not talking about one and only model, since there are an infinite number of such theories, and in fact, since any number chosen can have an infinite number of such theories, we can say that it is an infinite number of infinites. How is it possible to arrange such an infinite amount of turns? This is the question that occupied Prof. Doron Gefner from the Department of Particle Physics at the Weizmann Institute of Science. In an article he recently published in the International Journal of Modern Physics B, Prof. Gefner presents a way to sort all field theories (rational conformations) into two dimensions.

The merger circle
The method created by Prof. Gaffner is based on a concept called the "fusion circle", which is the merging of two fields to create a third field. It turns out that in every model (of the world) there exist several primary fields (for example, an electromagnetic field), from which all other types of fields are created. These initial fields maintain a "circle" of fields that are created by multiplying two such fields. A "circle" in this context is a group whose members have relations of addition and multiplication (when the order of multiplication does not change the result). For example, one member of a group is a product of two other members; Another member can be a product of three other members together. When it comes to numbers, this is always true, but when multiplying abstract components, such as fields, this property is neither automatic nor guaranteed.
In fact, different field theories can have different relationships between the fields. For example, the Ising model describes the relationships in a two-dimensional magnetic field. If we divide this field into lines of length and width into a sort of grid, because then, according to Ising's model, the energy of the model corresponds to the sum of the energy of all the nodes where small magnets pointing up or down are located. It is a statistical model suitable for a two-dimensional rational conformal theory of particles with spin (fermions), which spin in one direction (up or down) similar to the small magnets that are also directed up or down. All models of this type are suitable for certain field theories.

The theory of groups
To sort through these many teachings, Prof. Gefner used the tool called "Gloa Abelit Group". This tool is named after two mathematicians who died in the blood of their day, and in their lives there was no connection between them (although during a period of 10 months they were both in Paris): Brist Galois, and Nils Henri Abel. Galois was born in France in 1812. He was a passionate political activist, and was thrown into prison for his activities. His wild lifestyle caused a lack of appreciation for him, and articles he sent were not accepted by reputable journals. When he was only 20 years old, he got involved in a duel for the heart of a girl, and was killed (his opponent in the duel was the greatest sniper of the French army). The evening before the duel, he wrote the theory that later became known as the "Glua group", and sent it to a mathematician friend, who published it only 14 years after his death.
In this theory he proved that an equation of the fifth degree (which includes the fifth power) is not solvable using roots (equations of the second, third and fourth degrees can be solved using roots). To prove this assertion, he defined a group ("manifest group"). A bunch is a collection of elements that meet several mathematical conditions (for example, results of multiplication in a certain order). In this way Galois was able to show that every solvable group can be solved using a root, and that there is no solution using a root for a group that is not solvable.
Henry Abel was born in 1803 in Norway, as the son of a poor priest. He too, like Galois, did not receive the recognition he deserved in his lifetime, and only after his death, at the age of 26, from tuberculosis, did he become famous, among other things, for defining a type of bruise that bears his name ("mourning bruises"). A mourning group maintains a symmetric exchange between the multiplication arrays of its members. That is: the result you get from multiplying AZ by AZ is equal to the result you get when multiplying AZ by AZ.
In the first stage of the development of the method for sorting out conformal rational field theories, Prof. Gefner proved that a visible bundle of the merger circle in conformal theories is always abelian, and that all quantities that appear in conformal field theories can be calculated using roots of rational numbers.
The sorting method focuses on two quantities that appear in conformal field theories: the fusion circle, and a modular matrix that describes the "behavior" of the theory on a structure like a chach (this structure has two directions: the circumference of the chach, and a direction perpendicular to it, on the circumference of the cylinder that creates the chach), and the modular matrix switches between these directions. It turns out that between these two dimensions there is a mathematical connection discovered by the Dutch physicist Erich Verlinda. Because
Since the modular matrix is ​​symmetric, it is possible to use it to sort all field theorems. Since the relations between the "members" of the fusion circle can be described using polynomials, the roots of these polynomials hold an equation that, with the help of a visible group, creates a set of equations of the third degree that describe the relationships between the fields in the fusion circle. Solutions of these equations constitute a complete sorting of all conformal field theories according to their Galois bundles.
In this way Prof. Gaffner was able to sort out conformal field theories up to the six primary fields. This is, of course, a very small amount of field theories compared to the total amount (infinitely infinite), but basically, it is a method that can sort an unlimited amount of field theories, according to their properties.

A critical point
Here there will be those who ask: "What do we get out of this? Why should we go to such great lengths to sort out conformist rational field theories?" Well, it turns out that such a sorting ability has an important meaning in a unique material field: the study of the properties of all materials that exist in nature. The way materials go through phase transitions. Such transitions, first degree, are, for example, a transition of materials from one state of aggregation to another. Second order phase transitions
Take place, for example, when you heat a magnet. At a certain temperature it loses the property of magnetism, and passes from the aggregation state "magnet", to another aggregation state: "non-magnet" (this temperature is called the "Curie temperature", named after Marie Curie, who discovered it). A point where substances pass from one state of aggregation to another
"Critical point".
Transition between states of aggregation is a central property of materials. Therefore, understanding the way materials behave near "critical points" where phase transitions occur, may lead to the development of advanced and new uses for different materials. It turns out that as far as "behavior" near critical points is concerned, there is a great similarity between
different materials. Materials that "behave" similarly near critical points are considered members of the same "universality class". The behavior of each such "class" (near a critical point) is described by a unique conformal field theory. In other words, a sorting of field teachings is, in fact, a sorting of all materials
existing in nature, according to their properties and "behavior" near critical points, where phase transitions take place.
These models may also be used as a "playground" for more complex field theories,
which describe, for example, the "rules of operation" of the forces that trap the quarks and allow them to create other particles, such as protons, neutrons and more. Prof. Gefner says that solutions of these models may also help solve a physical theory (string theory) that will accurately describe the basic forces in nature.