Young-Mills theory is the basis of modern gauge field theory and an important physical breakthrough in the second half of the 20th century, aiming at describing the behavior of elementary particles with non-Abelian Lie groups. It was first proposed by physicists Yang Zhenning and Mills in 1954. This theory has played an important role in the formation of today's standard model. The Young-Mills equation in this theory is a set of nonlinear partial differential equations that have never been considered in mathematics.
First, let's briefly introduce these two scientists. Yang Zhenning, 1922 10/October 1 day was born in Hefei, Anhui Province, and is a world-famous physicist. He is currently a professor at the Chinese University of Hong Kong, a professor at Tsinghua University and an emeritus professor at the State University of New York at Stony Brook. He is also a member of China Academy of Sciences, National Academy of Sciences, Academia Sinica of Taiwan Province Province, Russian Academy of Sciences and Royal Society. He is the first China scientist to visit China after the loosening of Sino-US relations, and actively promotes Sino-US cultural exchanges and mutual understanding between Chinese and American people. He has made great contributions to the establishment of diplomatic relations, talent exchange and scientific and technological cooperation between China and the United States.
Another scientist's full name is Robert Laurence Mills (English: Robert Laurence Mills,1927 April15-1999 June 5438+001October 27th), an American physicist, who was born in Englewood, New Jersey. 1956 became a professor of physics at Ohio state university. The main contribution is the Yang-Mills theory of cooperation.
1954 Yang-Mills gauge field theory (that is, non-Abelian gauge field theory) was published. This theory, which was not valued by physics at that time, developed into a standard model through the concept of spontaneous symmetry breaking introduced by many scholars in the 1960s and 1970s. This is generally regarded as all the achievements of basic physics in the second half of the 20th century.
Yang Zhenning and Mills' paper, from a mathematical point of view, is a generalization from Abelian gauge field theory to non-Abelian gauge field theory. From the point of view of physics, it is to use this generalization to develop new basic rules of interaction.
Among the four basic interactions that dominate the world, the weak current interaction and the strong interaction are described by Young-Mills theory, and Einstein's general theory of relativity describing gravity is also similar to Young-Mills theory. Yang Zhenning called it "symmetrical dominant force". Young-Mills theory is a great physical achievement in the second half of the 20th century. Young-Mills equation, Maxwell equation and Einstein equation * * * have extremely important historical position.
Sorry, I can't find a picture of Robert Lawrence Mills. He is not as famous as Yang Zhenning because he didn't discover any other theories later. But we still should not forget him.
From the above introduction, we all know that Yang-Mills theory is also a nonlinear wave equation. This is very important. So this theory was originally designed to solve the above problems? It originated from the analysis of electromagnetic interaction, and the unified theory of weak interaction and electromagnetic interaction established by it was confirmed by experiments, especially the intermediate boson predicted by this theory was found in the experiments.
Young-Mills theory provides a powerful tool for studying the structure of hadrons (elementary particles involving strong interaction).
Yang Zhenning and Mills found that quantum physics revealed the amazing relationship between elementary particle physics and geometric object mathematics. The prediction based on Young-Mills equation has been confirmed in the following high-energy experiments in laboratories all over the world: Brockhaven, Stanford, CERN and Tsukuba.
However, they describe heavy particles and mathematically strict equations have no known solutions. In particular, the "mass gap" hypothesis has not been confirmed by most physicists, nor has it been applied to explain the invisibility of quarks, and it has never been satisfactorily proved mathematically. The progress on this issue needs to introduce basic new concepts into physics and mathematics.
I don't know if you have noticed that friends who have read my physics popular science book "Change" will definitely think of Einstein's field equation. Eikhfeld equation is also a nonlinear wave equation, so it is difficult to have an exact solution. This is very similar to Yang-Mills theory.
Because of the consistent concept, I don't think this is accidental. All the most basic equations in the universe show their profound volatility and openness. This has a profound impact on many laws mentioned above, such as wave-particle duality and uncertainty principle.
Let's look at this equation, although we can't understand it. But watching and not watching are two concepts. Never look, never understand.
Send another Einstein's field equation. Let's have a look. For the basic understanding of Eikhfeld equation, you can see my changes, which will be enlightening.
It is worth mentioning that Yang Zhenning's Yang-Mills theory and Einstein's theory of relativity have great influence, but neither of them won the Nobel Prize. Both of them won the Nobel Prize for other theories. Yang won the prize for putting forward the theory of parity non-conservation of weak interaction. Ai Shi won the prize because of the photoelectric effect theory.
As you can see, the equation of this theory is a nonlinear wave equation, so the equation has no ability to tell you, which is also a worldwide problem. I expect students from China to study more.
But in order to make everyone understand this theory, it needs to be supplemented. As mentioned above, Yang Zhenning and Mills' paper, from a mathematical point of view, is a generalization from Abelian gauge field theory to non-Abelian gauge field theory.
Non-Abelian gauge field is a kind of force field, which is supposed to describe why the nucleons (protons and neutrons at that time) in the nucleus are tightly pulled together without being blasted by the strong repulsive force between positive charges (protons are positively charged and repel each other).
Electromagnetic force propagates through electromagnetic field. The electric charge and the current formed by its movement produce electromagnetic fields, which can act on distant charges and currents after transmission. Therefore, Yang Zhenning and Mills also envisaged another kind of field similar to electromagnetic field to transmit nuclear force, that is, non-Abelian gauge field.
The idea seems easy, but it's actually very difficult. Most importantly, Yang Zhenning and Mills greatly expanded the meaning of field and charge at this time. They imagined a more complex charge (which can't be called charge, of course) and the field they generated: these charges and fields are not represented by ordinary real numbers, they are matrices. Matrix multiplication is noncommutative, and the noncommutativity of this multiplication is called "non-Abelian", so it is also called non-Abelian gauge field. It should be explained to those who have studied physics. Anyone who has studied quantum mechanics knows that mechanical variables can be represented by matrices in quantum theory. But the representation of field and charge as a matrix here is not the result of quantization, but in the sense of classical physics, they are matrices.
Yang Mills' article was published in 1954. At that time, there were still several key problems in this theory that could not be solved (such as quality, quantization and renormalization, which are relatively professional terms that only a doctor of theoretical physics who studies this direction can learn). In addition, in the 1960s, physicists were pessimistic about describing nuclear forces from the perspective of utility, when other viewpoints were dominant among physicists.
But after nearly 20 years of continuous exploration by several physicists, all the problems that could not be solved were solved. Moreover, it was later found that other different loads (unknown at that time) could produce strong and weak forces respectively (nuclear forces can be divided into these two forces). But the mathematical forms they use are similar, all of which are Yang-Mills theory.
The difference is that the size of the matrix is different: there are 2 times 2 and 3 times 3; If the matrix used is 1 multiplied by 1, it is Faraday and Maxwell's electromagnetic theory. By the end of 1970s, people knew that all the basic forces in nature-gravity, electromagnetic force, weak force and strong force-could be described by Yang-Mills field except gravity. This is how much influence this theory has on physics.
This strange charge conceived by Yang Zhenning and Mills is called isospin. They made a beautiful and profound extension of Maxwell's theory of describing electromagnetic field and got a new equation, which was later called Yang-Mills equation. Of course, it is much more common and complicated than Maxwell's equation. As for the popularization method, it is based on a principle from electromagnetic theory, which is called "localized gauge invariance principle". I will introduce this to you in the next chapter.
In fact, the process is like this. Yang Zhenning was inspired by Wyle's work, that is, Wyle established the U( 1) group through gauge invariance, and easily got Maxwell's equation. Yang Zhenning thought, can I explain strength, weakness and even gravity through a gauge group? Then we must first find gauge symmetry. Yang Zhenning found that strong interaction has rotation invariance in isospin space. Isn't this gauge symmetry? The concept of isospin is abstract. It is a viewpoint put forward by Heisenberg, which probably means that neutrons and protons are the same thing, that is, protons and neutrons are basically the same particle-two different states of nucleon, and they * * * together form an isospin dipole. In the abstract isospin space, protons can be "rotated" into neutrons, and neutrons can also be "rotated" into protons, because protons and neutrons are the same under strong interaction. In order to explain the strong interaction with this isospin conservation, SU(2) group needs to have a local gauge invariance.
To extend SU( 1) to SU(2), we must first solve the non-Abelian norm problem. Abel refers to Abel Group (named after Abel, a talented mathematician in Norway). One of his most famous theories is the first complete proof that there is no general algebraic solution for a general algebraic equation higher than quartic. This problem was one of his most famous unsolved problems at that time, and it was a suspense for more than 250 years. ), also called exchange group. Generally speaking, the operations in this group satisfy the commutative law. For example, 1+5 equals 5+ 1. Non-Abelian groups naturally refer to groups whose operations do not satisfy the commutative law, and the most common is the multiplication of matrices.
It's worth talking about Abel alone here. Although Abel achieved great success, he was not recognized before his death. He lived in poverty and died at the age of 27. It's a pity to be in the prime of life!
Anyone who reads my book, you must remember him, his story, because I am teaching you a truth-you must remember yourself! Don't think about living long, think about living wonderfully.
Niels henrik abel (1August 5th, 802-1April 6th, 829) was the greatest Norwegian mathematician in the 9th century. His father is a priest in the small village of Findu in kristiansand parish, and his family lives in poverty. 18 15 years, when he entered a Catholic school in Oslo, his talent in mathematics was revealed. Under the guidance of his teacher Holmbo, he studied the works of many famous mathematicians at that time, including Newton, Euler, Lagrange and Gauss. He not only understood their theory, but also found some tiny loopholes.
1820, Abel's father died, and the burden of taking care of a family of seven suddenly fell on his shoulders. Nevertheless, Abel was able to study at Christinania University in Oslo at 182 1 with the help of Hombiao, and obtained a pre-awarded degree at 1822, and continued his studies with the help of Hombiao. At school, he almost taught himself and spent a lot of time doing research.
After Abel's first paper was published in 1823, his friends urged the Norwegian government to fund him to study in Germany and France. While waiting for the government's reply, he published his paper "One-dimensional quintic equation without algebraic general solution" in 1824, hoping to bring him a positive position. The paper not only used tens of thousands of words instead of more than 500 pages to prove that Ruffini was also a mathematician, but also made many supplements.
But he really couldn't afford to print, so the manuscript of tens of thousands of words was compressed into six pages. Abel sent these six pages to famous mathematicians all over the country, hoping to get affirmation. Among them, he sent the paper to Gauss, a famous mathematician at that time. Unfortunately, Gauss missed the paper and didn't know that this famous algebraic problem had been solved.
Abel waited hard, but got no answer. Fortunately, the first paper published a year ago won him government funding for further research. Abel took the money and began to visit famous artists from all over the world. He met his lifelong friend Creaer when he visited Gauss, the prince of mathematics, in Berlin.
Gauss doesn't believe Abel can solve this super math problem in six pages. Only Creaer was very surprised. Creaer is a civil engineer, and he is very enthusiastic about mathematics. He and Abel became very good friends. 1826, with Abel's encouragement, Creaer founded the journal of pure mathematics and applied mathematics. The first issue of the magazine published Abel's work on quintic equations, as well as papers on equation theory, functional equations and theoretical mechanics. In Berlin, the new math guide enabled him to continue his research independently, and Abel later went to different parts of Europe.
Gauss
1in the summer of 826, he visited the top mathematicians at that time in Paris and finished a research report on transcendental functions. These works show an algebraic function theory, now called Abel theorem, which is also the theoretical basis of Abel integral and Abel function in the later period. He was snubbed in Paris. He sent his research report to the Academy of Sciences, hoping to get favorable comments, but his efforts were also in vain. Before leaving Paris, he suffered from a chronic disease. At first he thought it was just a cold, but later he knew it was tuberculosis.
Helpless, he returned to Norway, but he owed a lot of money and debts. He had to make a living by teaching and receiving a meager allowance from the university. However, his poverty and illness did not reduce his enthusiasm for mathematics. During this period, he wrote many papers, mainly about the theory of equations and elliptic functions, that is, the theory of Abel equation and Abel group. He finished the theory of elliptic function faster than Jacoby. At this point, Abel's reputation has spread all over the Mathematics Center, and people from all walks of life also hope to find a suitable professorship for him, among which clear hopes to find a professorship for him in Berlin.
1in the winter of 828, Abel's condition became more and more serious. During Christmas, he went to Finland to visit his fiancee, Clarie Kyoto Electronics Industry Co., Ltd., and his condition deteriorated. By 1829, 1 month, he knew that his days were numbered and the symptoms of bleeding could not be denied. Until1April 6, 829, Abel died, and his fiancee insisted on taking care of Abel without other people's help, occupying this last moment alone.
The photo on the left is Abel's fiancee, Clarie Kyoto Electronics Industry Co., Ltd..
On the third day after the funeral, Abel's family received a letter from his former friend Creaer and an offer letter from Berlin University, which said, "Dear Mr Abel, please don't refuse to hire you as a professor of mathematics! University of Berlin. "
After Abel died, his teacher Holmboe published a collection of works for him in 1839.
Back to the title of the article, Yang Zhenning's promotion work was not smooth sailing. Until 1954, he and Mills (who were in the same office with Mr. Yang Zhenning at that time and were doctoral students of Professor kroll) wrote epoch-making papers "Isospin Conservation and Isospin gauge invariance" and "Isospin Conservation and an Extended gauge invariance", which established his status as a master of physics.
Some physicists have pointed out that the basic laws of physics can be written on a cultural shirt (meaning very concise). Since Newton's time, the main goal of physics has been to describe the structure, interaction and motion of matter. With the development of physics, this simple goal has been further simplified. Because I later learned that matter is actually a field, but it is a so-called "quantum field". Therefore, the most basic law of physics is to describe the movements of various fields and their interactions. Now, if we want to make a physical T-shirt, we can definitely say that the Young-Mills equation should be written like this.
Some people ask this question: Some people say that many of the advances behind Young-Mills theory were made by other physicists instead of them. How can they say that their contribution is so great? Others say that the initial starting point of Yang-Mills equation is "wrong", so why is it so important?
The former problem is easy to explain. It is true that Yang-Mills theory has achieved such a great success in physics, which includes the contributions of many physicists before and after. Such as Higgs, Weinberg and Ichiro gherman. Several of them won the Nobel Prize.
Some mathematicians, such as Russian mathematicians fadeev and popov, quantized the Young-Mills field. They didn't win the Nobel Prize either, because he proposed the Young-Mills field.
The contribution of so many people, if the whole is made by one person, then his achievements are as great as Einstein, even more than Einstein! Einstein once established the theory of relativity by himself. Other physical theories, such as quantum mechanics, especially the later quantum field theory, were established by many people, sometimes after years of continuous efforts by generations.
Young-Mills theory provides us with a mathematical framework and train of thought. In this framework, as long as a certain symmetry is selected, the following interaction is almost completely determined, and the number of gauge bosons is also completely determined. This is why we can directly predict so many undiscovered particles from the theory of strong and weak electricity.
That is to say, in Young-Mills theory, those particles that transfer interactions are called gauge bosons, and each group has its corresponding gauge boson. For example, in the U( 1) group, there is only one gauge boson, and that is photon; In SU(3) group, no less than eight gauge bosons are theoretically calculated, and then experimental physicists search for them, and then they really find eight gluons.
However, it is not isospin but color charge that finally explains the strong interaction. Because the concept of quark was put forward by gherman and Zweig in 1964, Yang Zhenning could not have thought that the intensity was determined by the color symmetry of quarks even if he wanted to break his head in 1954.
Later, scientists also unified the weak force and electromagnetic force through this idea, that is, the aforementioned unified theory of weak electricity.
In addition, the study of the mathematical properties of Young-Mills theoretical equations is still one of the most significant mathematical problems in the world. For example, you can search "Millennium Mathematical Problems", which are seven mathematical problems published by Clay Institute of Mathematics in 2000. According to the rules formulated by Clay Institute of Mathematics, the prize for each question is $6,543,800+0,000. These questions are:
1, P/NP problem. 2. Hodge conjecture
3 Poincare conjecture. 4. Riemann hypothesis
5. There is a quality gap between Yang and Mills.
6. Existence and smoothness of Naville-Stokes equation.
7. Bell and Swenetton-Dale conjecture
The fifth is the quality problem of Young Mills problem, which is probably that bosons that transmit strong and weak forces should have mass, and only mass can correspond to short-range forces. For example, photons without mass are long-range forces, and gravitons without mass are long-range forces. The contradiction comes out, Yang Zhenning's theory, in order to make the system have local gauge invariance, the mass of the gauge boson that transfers force must be zero. Haha, do you see the contradiction?
How can Mr. Yang Zhenning not know this contradiction? But he thought it was a technical problem that could be solved, so he decisively published his own theory. Later, everyone knew that the Higgs mechanism was the solution to this problem. Young-Mills theory requires the gauge boson to have zero mass, but in the end we measured that both the W boson and the Z boson have mass, so why don't we try to give them mass? That is to say, their quality is not innate but acquired, so it does not conflict with Yang-Mills theory or actual measurement.
If you solve any of the above seven problems, you are a person with a super brain. It is never a lie that knowledge changes fate. Only if you believe in it can it change you.
Excerpted from Dunzhe Ling's popular science book "See the Micro-knowledge Book", an independent scholar and popular science writer.