Shulou(Shulou.com)11/24 Report--

In the 20th century, physicists made great progress on one of the most basic problems. The question is, what are the basic elements of ∶ that make up everything in the universe? In this process, they have to make many assumptions about the solutions of some mathematical problems. As these hypotheses and their inferences are supported by solid experimental evidence, scientists are very confident of the complete correctness of the theory. However, the rapid progress in physics faces a huge challenge ∶ solving the mathematics behind physics. The solution of Yang-Mills gauge field existence and mass interval hypothesis will be an important step to meet this challenge and will increase our understanding of the nature of matter. This makes the problem the latest step in a long-term human quest to understand the universe, which has relied heavily on mathematics for the past 2000 years.

God is a geometrist the ancients believed that the world was made up of four basic elements ∶ earth, water, air and fire. Around 350 BC, the ancient Greek philosopher Plato theoretically explained in his book Timaeus that these four elements were all formed by the aggregation of tiny solids. He argued that as the basic blocks of matter, these four elements must have perfect geometric shapes, that is, the five regular polyhedrons that fascinate Greek mathematicians-regular tetrahedron, cube, octahedron, dodecahedron and icosahedron with perfect symmetry.

On every polyhedron, every face is a regular polygon, in which all sides are equal and all angles are equal. Plato believes that fire is composed of regular tetrahedron, earth is made up of cube, water is regular icosahedron, and gas is regular octahedron. The last regular polyhedron represents the shape of the entire universe.

To Plato, God must be a geometrist. Or Galileo said ∶ in order to understand the universe, you must first know the language that describes it. The language is mathematics.

In Kepler's time, there were six known planets ∶ Mercury, Venus, Earth, Mars, Jupiter and Saturn. And Copernicus has proposed that all these planets move in a circular orbit centered around the sun. Kepler later proved that these orbits were not circles but ellipses. Kepler developed a theory to explain why there happen to be six planets and why they are in orbits at a specific distance from the sun, because every two adjacent orbits must be suitable for embedding a regular polyhedron. And each kind of regular polyhedron happens to be used once.

After some experiments, he finally found that the outermost sphere of the nested regular polyhedron and sphere ∶ (on which Saturn moves) contains an inscribed cube, in which is an inscribed sphere in which Jupiter orbits, followed by a regular tetrahedron on which Mars moves. The sphere in which Mars orbits is followed by a regular dodecahedron, and the sphere in which the Earth's orbit is located is tangent to it. The sphere in which the earth orbits is followed by a regular icosahedron, while the sphere in which Venus orbits is tangent to it. Finally, the sphere in which Venus orbits is connected with a regular octahedron, while the sphere in which Mercury orbits is tangent to it.

This is a model drawn by Kepler himself to describe his theory. Like Plato, Kepler used the same basic philosophy that ∶ the universe operates according to mathematical laws.

At the beginning of the 20th century, atomism became a widely accepted theory of matter. It believes that everything is made up of atoms, which are like miniature "solar systems" in which some electrons ("planets") move around the central core ("sun") along a certain orbit.

Later, various phenomena observed by scientists forced them to abandon the atomic model. It is replaced by a more complex mathematical theory-quantum theory. Quantum theory was born in the 1920s, it includes the idea that ∶ matter contains inherent uncertainty.

The common belief that connects Plato, Kepler, Einstein, quantum theory and today's string theorists is that the understanding of the basic matter of the universe will eventually come from mathematics. The research progress of mathematics sometimes surpasses physics.

Einstein discovered that Riemannian geometry could be used to explain gravity, so Riemannian geometry developed rapidly and quickly integrated into mainstream mathematics. Quantum theory itself also needs the development of some new branches of mathematics, such as functional analysis and group theory.

The existence and mass interval hypothesis of Yang-Mills gauge field is a specific problem that mathematicians must solve in order to respond to the challenge of physicists. This is a real mathematical problem. But in order to understand its origin and meaning, we have to start with physics.

The Holy Grail of modern physics for describing the common physical phenomena in life, Newton got quite good results as early as the 17th century. Newton's physics can calculate the time it takes for an object to fall from the table to the ground, as well as the round trip time of Apollo between the earth and the moon.

But we soon realized that Newtonian physics was becoming less and less accurate on an astronomical scale; similarly, it did not apply to the microscopic world within the atom.

In the early 20th century, physicists created new theories to explain these two "extreme" worlds. Einstein's theory of relativity describes the universe on an astronomical scale, while quantum theory describes the world on a subatomic scale.

Both theories have achieved great success. Which of the two is better? Relativity and quantum theory contradict each other. In a certain field, one is completely wrong, but the other is very accurate.

In most cases, the two theories perform their respective functions and do not interfere with each other. General relativity is responsible for the macrocosm and quantum field theory is responsible for the microcosm.

But there is one place where these two theories collide: black holes (massive stars collapse due to massive gravity). So far, there is no satisfactory theory to describe events in black holes. The solution of the equation obtained by combining the theory of relativity with quantum theory is infinite, so it is meaningless. Physicists are also unable to explain what happened at the moment of the Big Bang (all cosmic matter was huddled in a tiny area).

When physicists study the properties of matter at the smallest possible scale, there is also a contradiction between relativity and quantum theory. According to today's theory, the elementary particles (such as photons, quarks and bosons) studied by quantum theory are actually "quantum bubbles in space-time". To study them in this way, you need to use Einstein's mass-energy equation to establish the relationship between mass and energy.

It is obvious that there is a strong conflict between relativity and quantum theory. It strongly proves that neither of them is the ultimate theory of matter. So, if you want to really understand the universe, you must find a single, all-inclusive super theory, so that both relativity and quantum theory are approximations of this theory. But what is such a theory? Physicists have been looking for it since Einstein, but so far without success. The study of the Great Unification Theory of matter (GUT for short) is called the Holy Grail of modern physics.

In the final analysis, the grand unification theory is to find a framework to explain the four basic forces known at present: ∶ electromagnetic force, gravity, strong nuclear force and weak nuclear force. At present, electromagnetic forces are easily described by Newtonian physics; Einstein's general theory of relativity explains gravity; and quantum theory explains two kinds of nuclear forces. The purpose of the grand unification theory is to use a theory to accurately describe these four basic forces.

Since about 1925, most efforts to find a unified theory have been an extension of quantum theory, known as "quantum field theory" (QFT), which is the theoretical framework of the standard model of elementary particle physics.

The leader of quantum field theory is Witten, a physicist at the Princeton Institute of Advanced Studies. Witten describes quantum field theory as

The 20th century scientific theory of 21st century mathematics is used.

Witten was fed up with the slow progress of mathematics. Much of Newton's scientific achievements depended on the method of calculus, which he invented for this purpose, but it was not until 200 years later that calculus was fully established as a mathematical theory.

Yang-Mills theory, which was put forward as early as 1950s, is the first step towards such a great unified theory. The quality gap hypothesis is a special mathematical problem derived from the Yang-Mills framework.

To explain the Yang-Mills theory and the mass gap hypothesis, we will go back to the early 19th century, when scientists were trying to understand electrical phenomena.

An experiment changed the world.

One day in 1820, while working in the laboratory, the Danish physicist Oster discovered that the magnetic needle was deflected when an electric current passed through a wire near a magnetic needle. Oster reported his observations to the Royal Danish Academy of Sciences as a scientific discovery. He said there seemed to be a connection between electricity and magnetism.

A year later, the Frenchman Ampere made a similar discovery. He noticed that when currents pass through two parallel wires, they behave like magnets. When two wires flow in the same direction, they attract each other; when the current goes in the opposite direction, they repel each other. In 1831, Faraday of Britain and Henry of the United States independently discovered the corresponding effect ∶ put the coil into the alternating magnetic field, the coil will produce electric current.

Although the two cases seem to be different, electricity and magnetism seem to be closely related and can be converted to each other in some cases.

Hearing these news, the great British physicist Thomson began to wonder whether ∶ electricity and magnetism are different manifestations of the same potential phenomenon. As a mathematical theory of fluid motion was established not long ago, Thomson proposed that electricity and magnetism could be explained in a similar way. In a fluid, the force of the fluid is its own gravity. What is the force of electricity and magnetism?

A previous view was that space was filled with a continuous substance called "ether". The ether is assumed to be uniform, static, and pervasive throughout space. Stars and planets move in this constant ether, and light flows in it. Thomson proposed that electricity and magnetism are produced by some form of "force field" in the ether.

As long as there is a specific force at each point in an area, its magnitude and direction vary with the position. In many force fields, the force at each point also changes over time. Mathematicians study the force field by establishing an equation of the magnitude and direction of the force at each point (and possibly at each moment). The mathematical structure generated by this method is called a vector field.

Maxwell equation before 1850, the Scottish mathematician Maxwell decided to study electricity and magnetism. Fifteen years later, he had a result and called this unified force "electromagnetic force". In 1865 he published his research results in a paper called the dynamic Theory of electromagnetic Fields (A Dynamical Theory of the Electromagnetic Field).

The formula Maxwell uses to describe the electromagnetic field is now known as the Maxwell equations. Like the Navier-Stokes equation, they are all partial differential equations. The Maxwell equations describe the relationship between the electric field E and the magnetic field B, where E is a vector function, which gives the current (vector) at every point and moment in the field. B gives the magnetic force (also a vector) at every point and moment in the field.

The Maxwell equations unify two different forces (electrical and magnetic) in one frame. Then Maxwell began to seek to unify the four forces of nature.

The Maxwell equations mean that if the current fluctuates back and forth in the conductor, the electromagnetic field, which changes alternately with the current, will break away from the conductor and flow into space in the form of electromagnetic waves. The frequency of this wave is the same as the frequency of the current that produces it. Maxwell calls an electromagnetic field a wave only because the equations that describe the flow are very similar to the wave equations that describe the fluid. )

Wave or not, Maxwell can calculate that the speed of the electromagnetic current is about 300, 000 kilometers per second, which is very close to the speed of light. This led him to guess that light was electromagnetic radiation at a particular frequency. Today we know that he is right that light is indeed a form of electromagnetic radiation.

Frequency is an important feature of electromagnetic waves. The waves with the lowest frequency are radio waves, which are used to transmit radio and television signals. The higher frequencies are microwave and infrared, which can transmit heat. Then there is light, the visible part of the electromagnetic spectrum. The lowest frequency end of the spectrum is red and the highest frequency end is purple. The light frequencies of the spectrum are arranged in ∶ orange, yellow, green, blue and indigo in the following color order. The radiation with a higher frequency than purple is ultraviolet light, which is invisible to the human eye, but can make photographic negatives sensitive. Beyond the ultraviolet area is invisible X-rays, which not only make photographic negatives photosensitive, but also pass through the human body. The combination of these two characteristics makes it widely used in medical treatment. Finally, the uppermost end of the electromagnetic spectrum is gamma rays, which are emitted during the decay of radioactive material.

One of the questions that Maxwell's equations did not answer is, what is the medium through which electromagnetic waves travel? Many scientists are trying to find out this medium. Among them, the experiment conducted by Michelson and Morey in 1887 is the most famous. But all the efforts ended in failure. However, as a young man working in the Swiss patent office stepped onto the scientific stage, the direction of attention changed dramatically.

Special relativity in 1905, Einstein founded the famous special theory of relativity. In 1909, he quit his job in the patent office and became a professor of physics at the University of Zurich. Ten years later, his general theory of relativity was confirmed by experiments.

Special relativity says that the motion of an object is relative to another object. The absolute motion we feel is relative to the frame of reference in which we live. For Aristotle, the earth is stationary, so the motion relative to the earth is "absolute". Newton believed that there was a fixed "space" in which any object was either absolutely stationary or absolutely moving. Einstein took a step forward by denying Aristotle and Newton's view of motion, declaring that all movements were relative. This is Einstein's narrow principle of relativity.

This principle leads to a surprising conclusion that electromagnetic radiation has a very special property ∶, regardless of the reference frame, the speed of light is always the same. For Einstein, the only absolute thing is not the ether, but the speed of electromagnetic waves.

Because the speed of light is the same in all reference systems, Einstein solved another thorny problem, the problem of ∶ simultaneity. Einstein solved the problem of simultaneity by proclaiming that time is not absolute, but depends on the frame of reference for measuring time. It is clear that mathematical analysis of the universe is beginning to lead scientists into a counterintuitive field.

Gravitational ∶ General Relativity although Einstein's special theory of relativity is powerful, it only applies to cases where there are two or more reference frames moving at a uniform speed. And while this theory describes the properties of space and time, it makes no mention of ∶ mass and gravity, the other basic components of the universe. In 1915, Einstein found a way to extend his theory of relativity to take the latter two into account. His new theory is called general relativity.

The basis of this new theory is the generalized relativity principle ∶ in all reference frames, regardless of whether it is in accelerated motion or not, all phenomena occur in the same way. In general relativity, a natural process affected by gravity occurs when there is no gravity but the whole system is accelerating. That is, gravity and acceleration can be converted into each other.

One surprising conclusion of general relativity is that light seems to have mass. When light waves pass near a massive object such as a star, the gravitational field of the object will deflect the light. In 1919, the British astronomer Eddington confirmed Einstein's general theory of relativity.

General relativity not only accurately predicts the phenomenon of astronomical gravity, but also explains the nature of gravity. Surprisingly, this is a geometric explanation. According to Einstein's theory, any object will bend space-time. The degree of bending (curvature) is maximum near an object, and the farther away it is, the smaller it is. This curvature attracts two objects to each other. In other words, in Einstein's framework, what we call gravity is nothing more than the curvature of space-time caused by the existence of an object. The larger the volume and mass of an object, the greater the spatio-temporal deformation around it.

Unfortunately, Einstein's theory of relativity did not win him the Nobel Prize. But he acquired Norbert physicist for discovering the photoelectric effect, which promoted the development of quantum mechanics.

Quantum theory what is ∶ matter? Although general relativity describes the geometric structure of the universe and illustrates how matter interacts with this structure, it still does not answer the question: what on earth is ∶ matter? In order to solve this problem, physicists have to turn to another theory, ∶ quantum theory.

The accepted view in the early 20th century was that matter was made up of atoms. Each atom has a heavy nucleus around which one or more electrons move much lighter. The nucleus itself is thought to be made up of two types of elementary particles, ∶ protons and neutrons. Each proton carries one unit of positive charge and each electron carries one unit of negative charge. The electromagnetic attraction between these positive and negative charges binds electrons to nuclear orbits. (chemists still rely on this theory.)

But what exactly are the elementary particles (protons, neutrons, electrons) that make up the atom?

This is the problem studied by physicists Bohr, Heisenberg and Schrodinger before and after 1920. In their search for a solution, they have to consider some strange experimental results, one of which is that light behaves like a continuous wave in some cases, while in others it behaves as a stream of discrete particles. Their quantum theory solves all the problems, but also introduces the concept of "uncertainty" (which runs counter to the law of causality).

At the end of the 19th century, the first important step towards quantum theory was to try to solve a difficult problem caused by Maxwell's electromagnetic theory. According to this theory, the energy generated in a blackbody should be infinite.

In 1900, Planck, a German physicist, proposed a solution to this problem. He assumed that the energy radiated (or absorbed) by matter was not carried out continuously, but one by one, and could only be taken as an integer multiple of a minimum value. This minimum value is called energy quantum. The quantum number of energy carried by a given electromagnetic wave is proportional to the frequency of the wave. The higher the frequency, the more quantum energy. This immediately solves the problem of infinite energy. The ratio of wave energy to frequency is now called the Planck constant. In 1918, Planck won the Nobel Prize in Physics for this work. But Planck is still unable to answer the question: why does ∶ appear in the form of "energy quantum"?

In 1905, Einstein gave the answer, which was a result of his explanation of another strange physical phenomenon called the photoelectric effect. In 1887, the German physicist Hertz noticed that certain metals emit electrons when electromagnetic radiation hits them. Interestingly, increasing the intensity of light will increase the number of electrons emitted, but the energy of electrons will not increase, while increasing the frequency of light will make the electrons emitted have more energy.

To explain this phenomenon, Einstein assumed that light waves are made up of discrete energy packets (photons), and the energy of photons is proportional to the frequency of light waves (the ratio is Planck constant). When a photon with enough energy hits a metal, it can bump out an electron. The number of electrons knocked out depends on the number of energetic photons and thus on the intensity of light. The energy of each released electron depends on the energy of the photon that hits it (and thus on the frequency of light).

Einstein's new theory of light provides an explanation for the "quantum energy" in electromagnetic waves proposed by Planck. These "energy quanta" are photons, and according to Einstein, they make up waves.

What exactly is the meaning of waves made up of particles? Here we have encountered a thorny problem that no one has ever been able to deal with completely. Feynman once said

It was once said that only 12 people understood the theory of relativity. After people read Einstein's paper, many people understood the theory of relativity in one way or another, of course, far more than 12. But I can safely say that no one understands quantum mechanics.

When it comes to quantum mechanics, physicists have to give up their intuition and rely on mathematics to tell them what happened. With the advent of quantum theory, mathematics has become the only way for us to understand the world.

When light is studied in the laboratory, sometimes it behaves like waves, sometimes it behaves like particles. In the face of such behavior, the only way for scientists to make progress in the study of electromagnetic radiation falls on mathematics. So mathematics introduces scientists into a field completely different from the experience of daily life. According to mathematics, the inherent uncertainty of matter (including photons) causes light to behave like waves (a claim that the latest research seems to deny).

Quantum theory tells us that we must abandon the familiar image of particles (particles have a definite position and velocity). The most accurate information that can be obtained about the position and velocity of a particular particle is a probability distribution. Uncertainty is a basic aspect of physical reality.

The force of nature we know what keeps electrons in orbit around the nucleus ∶ is the electromagnetic force that causes particles with opposite charges to attract each other. So what binds the protons in the nucleus together? After all, the same charges repel each other. There must be some force that binds the nucleons together. Physicists call it a strong nuclear force, or strong interaction. The strong interaction must be large enough to bind protons in a nucleus together. On the other hand, a strong nuclear force can only act at a very short distance, an order of magnitude of the size of the nucleus itself, because it cannot attract protons in the nuclei of two different atoms.

People think that strong nuclear force, like gravity and electromagnetic force, is the basic force of nature. Just as photons are the carriers of electromagnetic forces, the carriers of strong nuclear forces are elementary particles called gluons.

Physicists believe that there is a fourth fundamental force. In order to explain the nuclear decay of radioactive elements such as uranium, there must be a second nuclear force, which can make protons fly away from ∶, which is weak nuclear force. The particles that carry this force are called bosons.

What is needed next is the quantum theoretical explanation of the field (quantum field theory, QFT), which includes the description of the four basic force fields of nature. The traditional description of quantum theory is usually limited to the theory originally conceived in the 1920s. In quantum mechanics, the behavior of a particle can be described by an equation that has been fully understood by the ∶ Schrodinger equation.

Where psi is a wave function, H is a Hamiltonian, and h is a Planck constant. The key fact to know is that there is little difficulty for mathematicians to grasp such equations. The early study of quantum mechanics produced results that the human brain could not fully understand, but on the whole, the relevant mathematics was quite simple.

By contrast, in quantum field theory, matter is regarded as some kind of field, a basic continuous medium that exists everywhere in space. The so-called "particles" in classical physics are merely the accumulation of energy in a quantum field. In quantum field theory, mathematics has also become difficult.

In 1973, the property of so-called "asymptotic freedom" (in quantum non-Abel gauge theory) was discovered, which has enabled physicists to understand what kind of results mathematics will produce (including the mass gap hypothesis).

Please note that Maxwell's electromagnetic theory is a representative of classical (non-quantum) field theory because it does not regard electromagnetism as a flow of particles, but as a field. Therefore, there is no substantial difficulty in using mathematics to describe the field. All the difficulties lie in trying to integrate quantum theory and field theory.

This view of matter as a certain property in the space-time field leads to some surprising conclusions. One is the prediction about antimatter ∶ that every particle has a corresponding antiparticle, that is, a particle with the same mass but opposite charge. Because matter and antimatter cannot coexist normally-if a particle encounters its antiparticle, the two particles immediately annihilate each other. In 1931, Dirac, the pioneer of quantum theory, predicted the existence of antimatter. Soon after, the antiparticles (positrons) of electrons were observed in cosmic rays, making Dirac's conclusion one of the first successful predictions of quantum field theory.

Interestingly, although quantum field theory seems to be counterintuitive, it puts physicists on the road to this theory, and they believe that the key mathematical ideas that eventually lead them to the desired grand unified theory, is the concept of symmetry at the center of our aesthetic sense.

The Symmetry of Nature in everyday language, if an object has some form of balance, then we say it is symmetrical. For example, a human face is symmetrical because the left side is very similar to the right side, and the snowflake is symmetrical because each part of it is very similar to the part directly opposite it.

Mathematically, if an object remains the same after a certain transformation, then it is said to be symmetrical with respect to the transformation. For example, the face is symmetrical about the left and right reversal, the snowflake is symmetrical about the center reflection, the square is symmetrical about the center 90 °rotation, and so on.

Mathematicians in the 19th century found that all symmetrical sets of objects (sets of all transformations that make the object look exactly the same as before) have some interesting structural properties independent of the object. In particular, it has a kind of "arithmetic"-you can "add" two symmetries of an object to get a third kind of symmetry, and this "addition" has the properties of some common addition of numbers that we are familiar with.

Mathematicians call these new types of arithmetic "groups". Over the years, they have created an important new branch of mathematics-group theory. Nowadays, group theory has become an important part of mathematics major. At the same time, we will see that it is also an important tool in physics.

The set of all symmetric transformations of an object, together with the arithmetic that combines them in this way, is the symmetry group. Knowing the arithmetic properties of the symmetry group, you will know a lot of information about the object, its shape and various other properties. Importantly, objects that can use group theory are not necessarily material entities such as faces and snowflakes, they can also be abstract mathematical objects, such as equations, or force fields.

In the early 20th century, physicists began to realize that many of their conservation laws came from symmetries in the structure of the universe. For example, many physical properties remain unchanged under translation or rotation. The results of the experiment do not depend on the location of the laboratory or the orientation of the equipment. These invariants mean the law of conservation of momentum and angular momentum in classical physics.

The German mathematician Eminot proved that this idea is correct. ∶ every conservation law can be regarded as the result of some kind of symmetry. In this way, each conservation law has a related group (corresponding symmetry group), which describes the related symmetry at every point in space-time. For example, the classical law of conservation of charge has a related symmetry group. Similarly, in quantum physics, the conservation laws of properties such as "singularity" and "spin" also have related symmetry groups.

Normative theory

In 1918, mathematician Hermann Weyl tried to unify special relativity with electromagnetism by the concept of symmetry. His idea is to take full advantage of the fact that the electromagnetic field has some kind of mathematical symmetry that keeps the equation constant at every point. For example, the Maxwell equations remain unchanged for the change of scale, which is a symmetrical form. In order to take advantage of this fact, Weir regards an electromagnetic field as a relativistic length distortion caused by moving along a closed curve. In order to do this mathematically, he must assign a symmetry group to every point in four-dimensional space-time.

Weir's basic idea is good, but his method doesn't work completely. The important thing in Maxwell's equations is not the scale but the phase. Weir adopted the wrong symmetry, thus the wrong symmetry group. The key symmetry of the electromagnetic field is now called "canonical symmetry". It means that even if the electromagnetic potential is multiplied by some quantum mechanical phase factor or gauge, the form of the field equation remains the same.

In daily life, gauge is a kind of measuring instrument. In a Jacuzzi, you can get a sense of the force field by moving your hand in the water. Similarly, in an electromagnetic field, you can also get a picture of the electromagnetic field by moving some kind of "measuring gauge".

In this way, a very important new discipline "gauge theory" is born, in which the symmetry group assigned to every point in space-time is called the gauge group.

When Weir focused on the scale, the symmetry group he studied was the multiplication of positive real numbers. When he shifts the attention from the scale to the phase, the group which is of great significance to the Maxwell equations becomes the "one-dimensional unitary group" U (1), which can be regarded as a set of rotational motion in the plane.

Following Weir's research, materialists modified Maxwell's theory into a normative theory. Their strategy of extending Maxwell's theory to a quantum field theory containing one or more nuclear forces (or even gravity) is to replace the gauge group U (1) with a more complex symmetry group. so that the resulting field theory may be a quantum field first, and then may include its local force field. They succeeded in this expansion (but not gravity) through several steps to win the Nobel Prize.

The first step appeared in the 1930s, when Dirac and others put forward an extremely accurate new theory, called quantum electrodynamics, or QED. QED provides a quantum description of electromagnetic phenomena, making it essentially an electronic version of Maxwell's theory.

In the 1940s, Feynman, Schwinger, Chaoyong Zhenichiro, Dyson and others created an extremely effective method of accurate calculation in this theory, which made it the most accurate scientific theory. The experimental results show that the theoretical calculation results are correct up to 11 decimal places. Feynman, Schwinger and Chaoyong Zhenichiro won the 1965 Nobel Prize for this work.

In 1954, physicists Yang Zhenning and Mills established a system of equations similar to the Maxwell equations in quantum theory. This is the crucial second step. Yang Zhenning and Mills adopted an excellent strategy to replace the group U (1) with a "compact lie group", a set of rigid body motion in a multi-dimensional complex space. The Maxwell equations are completely classical, that is, non-quantum theory, while the Yang-Mills equations have two characteristics: ∶ classics and quantum theory. In this way, Yang-Mills theory can establish a quantum field treatment method for matter, thus extending the QED.

Using the Yang-Mills equations requires more complex and sophisticated mathematics than Maxwell's theory. In particular, the group U (1) associated with Maxwell's equations is an "Abelian group" (that is, commutative). But this is not the case with the group used by Yang and Mills. Their theory is the "non-Abel" normative theory. The lack of exchangeability makes the corresponding mathematics more esoteric.

With the development of Yang-Mills theory, physicists began to try to use non-Abelian gauge theory to seek the great unified theory they have been looking forward to. The main idea is to find the correct gauge group so that they can capture both nuclear forces and gravity.

There is a problem when using the quantum version of Yang-Mills theory to unify electromagnetic force and weak force or strong force, that is, the classical (non-quantum) version of Yang-Mills equations describes zero-mass waves propagating at the speed of light. However, in quantum mechanics, each particle can be regarded as a special type of wave, so the characteristic of "massless" has become the main crux. Studies have shown that nuclear forces are carried by particles with non-zero mass.

For weak forces, this difficulty was solved by Glashow, Salam and Weinberg in 1967. They use a gauge theory in which the professional name of the symmetry group is SU (2) × U (1). The theory they founded is called the electroweak theory. By introducing an extra force, the Higgs field, they avoid masslessness. There is a large number of proofs to support the theory of electrical weakness. The goal of many experiments is to detect the Higgs boson, which carries the Higgs field, and the problem of finding it is completely solved.

In 2012, ATLAS and CMS experiments were carried out at CERN's large Hadron Collider (LHC) near Geneva, Switzerland, and subatomic particles with expected properties were found. It is later confirmed that the new particle matches the expected properties of the Higgs boson. Physicists Peter Higgs and Francois Englert won the 2013 Nobel Prize in physics for their theoretical predictions.

The electroweak theory not only covers the electromagnetic force and the weak nuclear force, but also shows that at a sufficiently high energy level, such as the first very short time after the Big Bang, these two forces combine into one, which is called the electroweak force. (the process by which electrically weak forces are divided into two seemingly different forces is called symmetry breaking.) for this achievement, Glashow, Salam and Weinberg won the 1979 Nobel Prize.

The next step is the discovery of the important property of the so-called asymptotic freedom of quantum Yang-Mills theory. Gross and Wilchek discovered this property in 1973, and Politzer discovered it independently. Generally speaking, the interaction between quark and gluon fails when the distance is very close, and the quantum effect will be shown only when the distance is large. Asymptotic freedom not only explains some mysterious experimental results, but also derives a unique quantum field theory including strength, which Gross and Wilchek call quantum chromodynamics, or QCD for short.

QCD is a gauge theory based on another symmetry group SU (3). This is the theory that gluons of eight colors correspond to and transform into three "color charges" after interacting with quarks. Quarks are elementary particles with a spin of 1 stroke 2, which combine to form protons, neutrons, and other previously known particles. The existence and properties of gluons predicted by QCD were soon discovered by experiments, which further confirmed the correctness of this new theory.

The mass gap hypothesis is produced in the derivation of quantum chromodynamics. Compared with quantum electrodynamics, many predictions of quantum chromodynamics have been experimentally confirmed with unprecedented accuracy in science. As a result, physicists are confident that they are on the right path. But our mathematical explanation for this theory is far from being formed. For example, no one can solve the Yang-Mills equations, let alone generalize them. Instead, physicists are using these equations to establish rules for calculating key values in an "approximate" way.

It is incredible that the most accurate scientific theory in the world today is based on a system of equations that no one can solve. This millennium problem about Yang Yi-Mills' theory poses a challenge to the mathematical circle to solve this problem. The first is to find a solution of Yang-Mills equations, and the second is to determine a special property of the solution, which is called the mass gap hypothesis.

Wilchek (one of the discoverers of asymptotic freedom and QCD), says ∶

We believe that QCD's equation fully describes the properties of protons and other strongly interacting particles, including their masses, and now we have to prove mathematically that this beautiful mathematical theory (QCD) does accomplish this task. In particular, this theory must cleverly use massless blocks to produce massless particles.

The basic mechanism of producing mass is Einstein's mass-energy equation. Experiments, computer simulations and some theoretical calculations make physicists believe that there must be a "mass gap" for vacuum excitation, that is, there is a non-zero minimum energy level (that is, there is no massless particle wave). The nature of the mass gap also explains why strength works only over such short distances.

So far, no one has been able to strictly prove this nature. The quality gap hypothesis requires an accurate mathematical version. In particular, proof of ∶ is required

For any compact, simple gauge group, the solution of the quantum Yang-Mills equations in four-dimensional Euclidean space has a mass gap.

The solution to this problem is not only a major breakthrough in theoretical physics, but also a major breakthrough in the larger pursuit of developing quantum field theory into a mathematical (not just physical) theory.

Although it originated from physics, this problem is essentially expounded as a mathematical problem. Indeed, many physicists think that most of the problem has been solved.

The solution of Yang-Mills theory and the mass gap hypothesis will mark the beginning of another important new field in mathematics, which will have a profound relationship with our current understanding of the universe.

Witten sees the millennium problem as a major challenge to mankind, he said

The understanding of natural science has been an important source of mathematical inspiration in history. Therefore, at the beginning of the new century, it is very frustrating that the main framework used by physicists to describe natural theorems cannot be dealt with mathematically.

Finding a general solution to the Yang-Mills equations essentially means that people understand the standard model of particle physics. If so, it will be an important achievement in mathematics.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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