# This theory, created by female mathematicians, reshaped the face of physics today In 1882, Amy nott was born into a wealthy Jewish family in Bavaria. Her father, Max Noether, was a mathematics professor famous for his research on algebraic geometry, but Amy was initially interested in linguistics. Only after she became a teacher of English and French did she begin to study mathematics at the university level, although women were not allowed to study in German universities at that time. The rule was amended in 1904, and nort received her doctorate in 1907. Her thesis deals with abstract algebra, especially the theory of invariants, the property that functions or groups of functions remain unchanged when they are transformed. It is in this field that she quickly established her reputation. In 1909, she was invited to join the German Mathematical Society. In 1915, mathematicians David Hilbert and Felix Klein found nott with a question. Einstein published the field equation of general relativity earlier that year, but there seems to be a worrying flaw in its theory. In some cases, the most fundamental principle in physics, the conservation of energy, is broken. In 1915, Hilbert and Klein found nott as an expert on invariants. If anyone can find a way to fill the gap in theory, it’s her. It turns out that their choice is right. Nott’s scheme became one of the most elegant and powerful results in theoretical physics. < / P > < p > if the Lagrangian of a system has some continuous symmetry, then there must be a conserved quantity related to it, and vice versa. Firstly, the conserved quantity is a property of the system which does not change with time. For example, if I tap a golf ball, the quality of the ball does not change between the time I hit the ball and it (hopes) goes into the hole. Therefore, the mass of the ball is the conserved quantity of the system. < / P > < p > on the contrary, the speed of the ball changes with time, whether through friction with grass or collision with the ground. In this case, the velocity of the ball is not a conserved quantity. < / P > < p > next comes the concept of continuous symmetry. Recall that in elementary school, we could say that a square has rotational symmetry when it rotates 90 degrees. This means that when we rotate a square 90 degrees, the final state looks like we’ve done nothing. However, this conclusion only holds for certain angles. If we change to 45 ° rotation, the final state will be significantly different from the original direction. So, we say that a square has discrete rotational symmetry. < / P > < p > now imagine doing the same thing on a circle. This time, however, no matter what angle we rotate the circle, the circle always looks exactly the same, even if the angle is very, very small. This means that the circle has continuous rotational symmetry. < / P > < p > finally, what is the Lagrangian of the system? To explain Lagrangian, we must first understand another basic concept in Physics: the principle of minimum action. < / P > < p > in fact, this shows that the universe is “lazy.”. Physical systems operate in such a way that the effort required to evolve from one state to another is minimized. We call this “effort” the action of the system. < / P > < p > for example, after I hit the ball, in general, the physical system evolved from “the ball under my feet” to “the ball in the hole.”. Nothing can stop the ball from going a long way under my feet to 10 feet from the hole, unless the system’s “action on this path” is much greater than “the amount of action the ball will play in our desired trajectory.”. The latter path, i.e. the trajectory of the ball, is the one corresponding to the minimization of action. This is exactly what the minimum action principle does. < / P > < p > What does this have to do with Lagrangian quantities? Lagrangian describes how the energy of the system should change in a process by minimizing the action of the whole process. By examining its behavior in space and time in a group of differential equations called equations of motion, we can determine how the system evolves from one state to another according to the principle of minimum action. < / P > < p > back to nott’s theorem. What does the continuous symmetry of Lagrangian mean? If the Lagrangian of a system is invariant when it is continuously transformed along a coordinate, then the system is considered to be continuous symmetric about that coordinate. Therefore, consider a classic physics examination question: the collision of two identical balls on the x-axis. Assuming that there is no friction or air resistance, it is easy to show that the dynamics of the system depends only on the relative values between the position and velocity of the ball, not on their absolute values. < / P > < p > if we translate two balls at the same time in the X direction at any same distance, then the difference in position and velocity between the two balls is constant. Therefore, the system must move in the same way as before, as if it had not been moved at all. Since this behavior is encoded by Lagrangian, it means that it cannot be changed by translation in the X direction. So the Lagrangian of the system must have continuous symmetry in the X direction! In the case of translational symmetry, the conserved quantity is momentum. This is the origin of the law of conservation of momentum! The tool we use to deal with the problem related to this assumption is the result of the symmetry of the Lagrangian of the system. Similarly, if the Lagrangian of the system is rotationally symmetric, the angular momentum is conserved in the system if the Lagrangian is rotated at any angle. The Lagrange equation describing the gravity of a planet is rotationally symmetric, so the angular momentum is conserved in the orbit of the planet. Due to some symmetry, the charge will also be conserved. This time, it involves the more profound concept of gauge symmetry of wave function, and its details need to refer to the corresponding professional articles. < / P > < p > recall that Hilbert and Klein realized that in some cases, energy is not conserved in general relativity. We know that energy is usually conserved, so according to nott’s theorem, the conservation of energy is related to some kind of symmetry – indeed, time translation symmetry. If a system is symmetric under time translation – if the Lagrangian of the system does not explicitly depend on time – then the system must be energy conserved. Time in general relativity is not an absolute quantity like that in Newtonian mechanics. It flows and distorts with the bending of space-time. Time translation symmetry only applies to general relativity in some special cases, that is, when space-time is flat or almost flat. Therefore, for curved space-time, energy does not need to be conserved! Thus, by combining her expertise in abstract algebra with her incredible analytical thinking, nott not only fills in one of the most important theories of the 20th century, but also reveals a real basic concept in theoretical physics. Her theorem lays the foundation for a large number of physical phenomena that we encounter every day in the classroom and in the world. Even in particle physics, nott’s theorem is still applicable. “We have to rely on theoretical insight and concepts of beauty, aesthetics and symmetry to guess how things work,” Wilczek said Nott’s theorem is very helpful in this respect. < / P > < p > in particle physics, the symmetry concerned is a kind of relatively secret symmetry called gauge symmetry. There is such symmetry in electromagnetism, which leads to the conservation of charge. Gauge symmetry appears in the definition of voltage. For example, the voltage between the two ends of a battery is the result of a potential difference. The actual value of the potential itself is not important, but the difference. < / P > < p > this results in the symmetry of the potential: its overall value can be changed without affecting the voltage. This feature explains why a bird can sit on a wire without getting electrocuted, but if it touches two wires at different potentials at the same time, rest, bird. < / P > < p > in the 1960s and 1970s, physicists expanded this idea and discovered other hidden symmetries associated with conservation laws, thus developing the standard model of particle physics. “It’s a conceptual connection – once you realize it – it’s like you have a hammer, and then you can find a nail to use it,” Wilczek said Physicists look for symmetry everywhere they discover conservation laws, and vice versa. The standard model based on symmetry explains the behavior of elementary particles and their interaction. Wilczek won the Nobel Prize in 2004 for his contribution to the development of the standard model. In terms of its ability to accurately predict experimental results, many physicists now consider it one of the most successful scientific theories ever. However, to a large extent, no one knows about nott. This is what Einstein described as “the most important and creative mathematical genius that has emerged since women’s higher education.” However, she has never been a permanent teacher. Hilbert was forced to promote her lecture course in Gottingen under her own name, which was the exclusion of female scholars in the University hierarchy at that time. With the rise of Nazi forces in 1933, she moved to the United States. Before her sudden death from cancer in 1935, she should have started working with Einstein at Princeton University. The report shows that the number of app store purchases soared in the first half of this year due to the impact of covid-19