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# Understanding the concept of symmetry using Neuter's theorem and the bird that didn't get electrocuted!

If you go back to the 1920s in Gttingen, Germany, on a hot summer evening you might hear the buzz of conversation coming from an apartment in Friedlander. With a glance through the room's window, you can see a gathering of students talking about the day's math problems. With more eavesdropping, you might finally hear the laughter of the woman who is hosting the party, and she is none other than Emmy Noether, the genius mathematician who gave Noether's theorem to the world of science!

that changed the way physicists study the world.
• Amy Nutter, the woman who changed the face of physics!li>
• Noether's theorem, laws of constancy and symmetries
• gender view in science; From Marie Curie to Amy Noether
• Symmetries are more important than you think!
• General relativity and Noether's theorem
• An overview of Amy Noether's life
• li>
• Scalar symmetries and Noether's theorem
• Summary

## Amy Noether, the woman who changed the face of physics!

About a century has passed since the day on July 23, 1918, when the famous Noether theorem was introduced to the world of science, with This has not diminished its importance one bit. In fact, this case was and is like a guiding star for the physicists of the 20th and 21st centuries. Noether was a leading mathematician of his time, and in addition to his famous theorem, which is now called "Noether's theorem", he also launched a branch of mathematics called abstract algebra.

Noether's theorem at its peak shows that any symmetry can represent a certain quantity keep More precisely, for each symmetry in the system (the existence of a symmetric and differentiable function), there is a physical quantity that is intractable. The above equation expresses the concept that the quantity in parentheses does not change over time.

But Noether's professional life did not go so well despite such genius. After getting his doctorate, he had to work for years without pay. Although she began working at the University of Gttingen in 1915, she was initially only allowed to teach as an assistant to her male colleague and received no salary until 1923. Ten years later, when the Nazis came to power, Notter was forced to leave his job by the government because he was not only Jewish but also suspected of having left-wing political views. "Bryn Mawr" in Pennsylvania left for the United States of America; But less than two years later, he died at the age of 52 due to complications from surgery! While the importance of his case was not fully known! Although most people have never heard of Amy Notter, physicists always praise her. In this regard, Ruth Gregory, a theoretical physicist at Durham University in England, admits that Noether's theorem is current in everything that physicists do.

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## Noether's Theorem, conservation laws and symmetries

Noether specified the connection between two important concepts in physics, i.e. conservation laws and symmetries. In general, conservation laws, such as conservation of energy, state that a certain quantity such as energy is neither created nor destroyed, but is always constant. The same certainty in the conservation of energy helps physicists to solve many problems, from calculating the speed of a ball rolling down a hill to Understanding the processes of nuclear fusion. , indicates the stability of a physical quantity related to a specific system during the evolution of that system. symmetry in physics includes physical or mathematical properties that remain unchanged under certain transformations (such as rotation).

On the other hand, symmetries describe changes that can be made without changing the appearance or function of an object. For example, a sphere is perfectly symmetrical, because if you rotate it in any direction, it still looks like the original shape. The same description includes the symmetry of the laws of physics. This means that the equations do not change at different times or places.

A picture of Amy Noether, whose Noether's theorem became one of the most important theories of modern physics in 1918!

Now, according to the explanation of these two important principles in physics, you should know that Noether's theorem admits that every A symmetry has an associated conservation law and vice versa (for every conservation law, there is an associated symmetry!).

Noether's theorem states that every symmetry has an associated conservation law and every The law of conservation is associated with a related symmetry!

Simply put, the law of conservation of energy states that the physical laws that govern today are the same as they were yesterday. Likewise, conservation of momentum is related to the fact that physics in this part of the world is similar to anywhere else in the world. This connection between symmetry and survival reveals the existence of a reason behind the characteristics of the universe, which seemed to be random before that relationship was known! And it became the basis of the standard model of particle physics. The Standard Model of particle physics examines the universe from a subatomic perspective and is the same model that predicted the existence of the Higgs boson, a particle that was discovered with much fanfare in 2012.

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## gender view in science; From Marie Curie to Amy Noether

When Amy Noether died, Albert Einstein wrote in the New York Times Amy Nutter was the most creative genius mathematician there has been since the beginning of higher education among women. Although this is an intimate compliment on Einstein's part, Einstein's praise referred to Amy's gender rather than her scientific prominence among her male colleagues!

Similarly, several other mathematicians who praised Amy They had brought, along with those compliments, they commented on her appearance, and even in one of these cases, comments were made about Amy's sex life! So as you can see, even those who admired Noether judged him by different standards than men.

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## Symmetries from They are more important than you think!

In general, the discussion of symmetry is one of the fascinating topics from the past. so far For example, some studies report that people with more symmetrical faces are more beautiful than asymmetrical faces. This means that the two halves of these people's faces are almost mirror images of each other (mirror symmetry). On the other hand, in most cases, art displays symmetry, you can understand this well by looking at Iranian designs and tiles. In addition to these things, nature also likes symmetry; Because when you rotate the snowflake by 60 degrees, it looks the same. Besides that, petals, flowers, spider webs and many other things are formed with the principles of symmetry.

But at the same time, you should know that Noether's theorem does not directly apply to these familiar examples; This is because the symmetries we see around us are discrete, meaning they only apply to certain values. For example, a rotation of exactly 60 degrees for a snowflake is discrete. In general, the symmetries related to Noether's theorem are continuous, which means that no matter how much you move in space or time, continuous symmetries are always preserved and do not depend on specific values.

The symmetries related to Noether's theorem They are of a continuous type and are not considered discrete like the symmetry of a snowflake with a 60-degree rotation.

Among these, one of the types of continuous symmetry, which is known as translation symmetry, implies this principle. It says that the laws of physics do not change with movement in the universe and remain the same.

Consistency laws are among the basic tools in physics. In physics classes, students are taught the principle that energy is always constant. Because when the first billiard ball hits other balls, the energy of the first ball's motion is divided during the collision. Part of the energy makes other balls move, part produces sound or heat, and some energy remains in the first ball. In such a case, no matter what happens, the total amount of energy always remains the same. This is also true of momentum.

Generally these laws are taught to students as undisputed and unequivocal truths, but you need to know that there is a mathematical reason behind all of them. According to Noether, the conservation of energy is due to transitive symmetry in time. Similarly, the principle of conservation of momentum is also in space due to translational symmetry. Angular momentum conservation is a feature that allows skaters to increase their spin speed by hugging their arms. This fast rotation comes from rotational symmetry and it means that by rotating in space, everything will be in the same original shape.

In Einstein's theory of general relativity, there is no absolute sense of time or space. It doesn't exist, so it's harder to understand the consistency laws, so as we said at the beginning of the article, this is where Noder's theorem comes in and overcomes the leading challenges.

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## General relativity and Noether's theorem

Although general relativity was introduced to the world as a new and fascinating theory in 1915, the situation was such that David Hilbert and Felix Klein, the German mathematicians, were overwhelmed by the ambiguities of this new view. In this regard, Hilbert in order The mathematical development of the theory, which described gravity as the result of the curvature of matter in spacetime, was in competition with Einstein.

It was in this atmosphere that Hilbert and Klein stumbled upon a new challenge. The problem ahead was that trying to write an equation for the conservation of energy in the framework of general relativity would lead to tautology! In other words, like writing the expression "0 equals 0", the equation of conservation of energy in the framework of general relativity had no physical significance. This was surprising, because no accepted theory had such energy conservation laws. So these two mathematicians hired Amy Notter, who had special expertise in various fields of mathematics, to join them in Gttingen and help solve the puzzle ahead. It belongs to a special class of theories that are generally called covariant. In such theories, the equations related to the theory, whether you are moving continuously and without acceleration or with acceleration; Remains constant. Because during movement, both sides of the theoretical equation change together. As a result, covariant theories, including general relativity, are always associated with these unusual conservation laws; This discovery is known as Noether's second theorem. On the way to proving the second theorem, Notter proved his first theorem on the connection between symmetries and conservation laws, and published both results on July 23, 1918, in an article called "Gttinger Nachrichten".

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## A review of Amy Noether's life

Amy Noether was born in 1882 and was the daughter of two mathematicians Max Noether and Ida Amalia Noether was "Ida Amalia Noether". Growing up with her three brothers in Erlangen, Germany, young Amy's mathematical talent was not obvious as a child. However, he was known for solving puzzles that baffled other children. At the University of Erlangen, where her father taught, women were not officially allowed to teach, although they could attend classes on an exceptional basis with the professor's permission. When this rule changed in 1904, Amy Nutter quickly took advantage of it and earned her doctorate in 1907. An academic position with stipend was sought. Although the University of Gttingen finally began paying Notter a salary in 1923, it never recognized him as a full-fledged professor! However, Noter soon became everyone's favorite with his warm and kind personality. He used to take long walks in the countryside with his students and colleagues and hold long and stimulating debates on mathematical topics. Noter even sometimes took his students to his apartment and the conversation continued until the remains of the dessert dried on the dishes. According to these interpretations, his death, less than two years after his entry into the academic community in 1935, saddened everyone.

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In the world of particle physics, hidden symmetries called scale symmetries There are gauge symmetries that appear in the electromagnetic world that lead to the conservation of electric charge. In general, scale symmetry appears in the definition of electric voltage in such a way that the voltage (for example, between the two ends of the battery) is the result of the electric potential difference. In fact, the amount of electric potential is not important in itself, but it is the potential difference that is important.

This explanation shows the symmetry in electric potential, because the overall amount of electric potential without affecting the voltage difference (electrical potential difference) can be changed In fact, this is the same problem that explains why a bird can sit on an electric wire without getting electrocuted, but if it touches two wires with different electrical potentials at the same time, it must say goodbye to the world!

In the 1960s And in the 1970s, physicists expanded on this idea and found other hidden symmetries associated with the conservation laws to develop the Standard Model of particle physics. Therefore, it can be said that paying attention to the laws of stability and its connection with symmetries in physics, scientists are like those who have a hammer in their hands and are looking for a nail. Because wherever physicists find the conservation law, they are looking for symmetry and vice versa.

It is also necessary to know that the development of the standard model also owes to Noder and his important theorem. The standard model is a theory proposed by Wilczek to explain a large number of particles and the interactions between them, and he won the Nobel Prize in 2004 for his role in developing this model. The Standard Model is currently considered by many physicists as one of the most successful scientific theories due to its ability to accurately predict the results of experiments.

In the same vein, the Large Hadron Collider is looking for new particles predicted using Noether's insight and Noether's theorem, because a hypothesized hidden symmetry, Called supersymmetry, it suggests a heavier sister to every known particle in the universe. Of course, despite the great hopes to identify these particles, so far no such particles have been found with the descriptions proposed in the theory of supersymmetry!

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## Summary

The physics we deal with every day relies on Noether's theorem; Because the laws of stability that we mentioned many times in this article help to explain systems such as waves on the surface of the oceans and air flow on the wings of an airplane. In fact, simulating such systems helps physicists to make predictions about weather patterns, bridge vibrations, or the effects of nuclear explosions.

Besides the importance of Noether and her theorem in physics, this woman's ideas are also somewhat important in mathematics. It is notable that his name is placed as an adjective behind many elements of mathematics (such as Notherian rings, Notherian groups, and Notherian modules). Finally, it should be noted that although women in science today still face challenges, no one has to fight to get paid for their scientific work. This is Amy Nutter's social legacy that she gave us with her efforts.

• The story of Einstein's brain that was stolen and had nothing special!

Original sources:

R. Gregory and P. Olver. Emmy Noether: Her life, work, and influence. Convergence, Perimeter Institute for Theoretical Physics, Waterloo, Canada, June 22, 2015.

Y. Kosmann-Schwarzbach. The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Translated by Bertram E. Schwarzbach, Springer, 2011.

K. Brading. A Note on General Relativity, Energy Conservation, and Noether's Theorems. In: The Universe of General Relativity. Birkhuser, 2005.

K.A. Brading Which symmetry? Noether, Weyl, and conservation of electric charge. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. Vol. 33, March 1 2002, p. 3. doi: 10.1016/S1355-2198(01)00033-8.

N. Byers. The Life and Times of Emmy Noether: Contributions of E. Noether to particle physics. arXiv:hep-th/9411110. Posted November 15, 1994.

N. Byers. E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws. arXiv: physics/9807044 Posted July 23, 1998.

A. Dick. Emmy Noether 1882-1935. Translated by H.I. Blocher. Birkhuser, 1981.

K. Brading and H.R. Brown. Noether's Theorems and Gauge Symmetries. arXiv:hep-the/0009058. Posted September 8, 2000

Source: ScienceNews

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