Guide:
A false theory from 1918 to 1919 began a great journey, leading to a theoretical framework describing the three fundamental forces of nature, as well as many important physical and mathematical achievements. Many of the physicists who contributed to this journey later won the Nobel Prize in Physics, and others won the Fields Medal and the Abel Prize.
Written by | Shi Yu (Professor, Department of Physics, Fudan University)
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The full text is divided into eleven chapters:
Normative theory from 1918 to 1919
Quantum theory of salvation norms
The rebirth of gauge theory
Klein and Pauli
The Young-Mills theory
The Standard Model of Particle Physics and the Revival of the Young-Mills Theory
Experiments give life to the Young-Mills theory
The Young-Mills theory in nobel prize-winning speeches
Symmetry governs interactions
10. Normative Theory and Mathematics
11. The Beauty and Truth of Normative Theory
1. Normative theory from 1918 to 1919
Between 1918 and 1919, Hermann Weyl of the Swiss Federal Institute of Technology in Zurich published three articles in an attempt to incorporate electromagnetic forces into the framework of the theory of gravitational geometry [1-4]. This was the beginning of gauge theory. Weil was one of the most influential mathematicians of the 20th century. Sir Michael Atiyah, winner of the Fields Medal and the Abel Prize, once told me that Weyl was one of his idols.
Hermann S. Weil. Image source: wikipedia
Weil's three articles, dominated by Gravitation and electricity, were published in the Proceedings of the Prussian Academy of Sciences. It was in this journal that Einstein published the geometric theory of gravity in 1915, known as general relativity, revealing that gravity is essentially the curvature of space-time caused by matter [5].
In 1905, Einstein proposed the special theory of relativity. Minkowski noted that special relativity unifies time and space into four dimensions of space-time. Ten years later, Einstein's general theory of relativity stated that matter bends space-time, and bending space-time determines the motion of matter.
To grasp the meaning of the curvature of space-time, let's imagine a sphere. Place an arrow at a certain point on the sphere, and the arrow points in a certain direction from this point. Then we move the bottom end of the arrow on the sphere, and every moment of the movement process, the direction of the arrow is kept unchanged, which is called parallel movement. However, for movements of finite length (not infinitesimals), the arrow direction changes. For ease of comparison, we can move the bottom end of the arrow in a parallel circle along a closed curve, returning to its original position, and the direction of the arrow is no longer the original direction. This is what causes the curvature of the spherical surface. In curved space-time, the logic is similar, the parallel movement of any vector (both size and direction, like an arrow) may change its direction, but the size does not change.
At that time, it was understood that there are two basic forces in nature, in addition to gravity, that is, electromagnetic force, which is determined by the electromagnetic potential. So Weyl naturally wanted to incorporate electromagnetic forces into the framework of general relativity as well. He tried to popularize the concept of parallel movement, imagining that when vectors move in parallel, not only the direction changes, but also the size. For parallel movements of infinitesimals, the vector size changes infinitesimals, and Weil assumes that this infinitesimal multiple is proportional to the infinitesimal change of the electromagnetic potential. It is not difficult to deduce that for a parallel movement of a finite length, the vector size becomes the original size multiplied by an exponential factor, which is proportional to the integration of the electromagnetic potential along the path (that is, the electromagnetic potential of each point is accumulated). This exponential factor that changes the size of the vector depends on the path traveled by parallel movement, so it is called a non-product scale factor.
This conclusion is strange. In relativity , both the size of the ruler and the speed of the clock can be seen as the size of the vector. Then according to Weil's theory, the change in the size of the ruler and the speed of the clock between the two points actually depends on the path the ruler and the clock move!
This is what Einstein commented after Weier's "Gravity and Electricity." The comments were followed by Weil's long but unconvincing reply, but he admitted that he was "chasing the concept of madness like a wild goose."
In Weyl's article, there is a sentence that reflects Weyl's way of thinking (the direct quotations in this article are all translated from English by the author):
"The law of conservation of electromagnetism is linked to the new normative invariance, expressed through a fifth arbitrary function. In my opinion, this analogy with the principle of energy momentum is the strongest argument for the current theory – as long as the argument is allowed to be discussed in the case of pure conjecture. ”
It can be seen that Weier's confidence stems from the invariance, or symmetry, behind his proposal of the law of conservation of electromagnetism. The conservation quantity here is not the charge we know today (that would require a modified gauge theory). Weil here compares the conservation of symmetry to the conservation of energy momentum. These can all be used as examples of Nodorean's theory. Nodinal theory is that symmetry corresponds to a conservation quantity. It is not known whether Weyl was aware of Nordic Theory, which was also published in 1918 [6]. Nott was under Hilbert at the University of Göttingen, but Weil had left five years earlier.
2. Quantum theory of salvation norms
After the advent of quantum theory, in 1922, Weyl's friend Schrödinger speculated that it was possible to add the imaginary unit i [7] to the index of Weier's scale factor. Thus, the scale factor becomes a complex number of magnitude 1, which is the phase factor of the quantum mechanical wave function. But Schrödinger did not have the concept of a wave function at that time, but only said a certain "length" in general. It was later discovered that this work inspired Schrödinger's creation of wave dynamics in 1926 [8-10].
Also in 1922, in classical theory, Theodor Kaluza generalized 4-dimensional space-time to 5-dimensional space-time, where some of the degree gauge components represent the electromagnetic potential (the degree gauge is the relationship between the coordinates of two points and the distance between them). When the theory does not depend on 5-dimensional coordinates, 5-dimensional coordinate transformations degenerate into 4-dimensional coordinate transformations and 1-dimensional canonical transformations.
In 1926, Schrödinger created a wavedynamic formulation of quantum mechanics through four articles (hence the sharing of the 1933 Nobel Prize in Physics). In his fourth article, he pointed out that the momentum and energy operators of charged particles in an electromagnetic field must contain the electromagnetic potential [11]. In quantum mechanics, physical quantities are represented by operators, meaning to perform operations on wave functions.
Also in 1926, Oskar Klein and Vladimir Fock discussed the wave dynamics of The Caruza's theory [12,13]. In particular, Fokker pointed out that the wave function equation of motion has a gauge invariance, that is, the wave function is multiplied by the phase factor, and the electromagnetic potential makes a corresponding transformation, and the equation of motion remains true [13].
At the end of 1926, Fritz London wrote a letter to Schrödinger asking him about the connection between his 1922 revision of the canonical factor and his 1926 wave dynamics [8-10]. London then wrote two articles of its own, linking the usual wave dynamics to Weyer's gauge theory, changing Weier's non-productable scale factor to the non-productable phase factor of the wave function.[14-15] The london work did not have the additional burden of Fokker's 5-dimensional theory.
3. The rebirth of normative theory
Building on the work of London and Fokker, Weyl, in his 1928 book Group Theory in Quantum Mechanics [16] and two articles in 1929 [17-18], finally revised his theory from 1918 to 1919, formally changing the scale factor to a phase factor and scale invariance to phase invariance, but following the original term "eich". We deliberately call his non-productable factors between 1918 and 1919 scale factors, because both scale factors and phase factors are called gauge factors. 2019 also coincides with the ninetieth anniversary of the rebirth of normative theory.
The two articles in 1929 were written by Weil as a visiting professor at Princeton University in the United States. The first was the English summary "Gravitation and the electron," published in the Proceedings of the National Academy of Sciences (PNAS), in which the "principle of gauge invariance" appeared. The second is an exhaustive article in German titled Elektronund Gravitation. The two titles echo his 1918 book Gravitation and Electricity.
The principle of norm invariance is, of course, the most important content of Weyl's 1929 article. Within the framework of quantum mechanics, electromagnetism is deduced as a consequence of norm invariance, and norm invariance also leads to conservation of charges. The canonical transformation here is no longer a scale factor of a real number, but a phase factor (there is an imaginary unit on the exponent), and the canonical invariance is in fact phase invariance. This is the normative theory that we are familiar with today.
The norm that causes charge conservation to remain unchanged is under the overall gauge transformation (also known as the first type of canonical transformation), that is, the phase transformation is independent of the space-time coordinates, and the same phase transformation is made for the wave function of each time and place. This can be seen as an example of Noctic's theory, although Weyl does not directly mention Noctic's theorem. By 1929, Nordisk was widely known. The canonical invariance of derived electromagnetism is under the local canonical transformation (also known as the second type of canonical transformation), that is, the phase factor depends on the space-time coordinates.
Weyl's 1929 article still considers gravity and electromagnetic forces as a whole, but now it is necessary to consider quantum mechanics, starting from the spin of electrons in the gravitational field. Thus Weyl discussed the theory of two-component rotations ( describing electrons in terms of two wave functions ) , including the chiral weyl spins ( later known as weyl fermions ) with the signs of weal.
What I found particularly interesting is that in Waier's theory, local flat space-time depends on space-time coordinates, resulting in a kind of connection, which adds up to the curved Christopher contact of space-time, similar to the non-Abel gauge potential known later, and the corresponding Rieman tensor is similar to the non-Abel canonical field strength!
Wolfgang Pauli, one of the founders of quantum mechanics known as the "conscience of physics" and the "whip of God", who won the 1945 Nobel Prize in Physics for the Pauli principle of incompatibility, wrote a letter to Weyl [19-20]:
"I have in front of me the Proceedings of the National Academy of Sciences in April. Under the 'Physics' section, not only do you have an article, but it also indicates that you are now working in a 'physics lab': I heard that you were given a position as a professor of physics in the United States. I admire your courage; for the inevitable conclusion is that you wish to be evaluated not by pure mathematical success, but by your true but unpleasant love of physics. ”
Pauli. Image source: wikipedia
Weyl's article is understandably objectionable to Pauly: Weyl's 1918 article was published with Einstein's doubts, Pauli was also opposed to [4] at the time, and now it is making a comeback, plus the two-component theory in the article violates the conservation of cosmology (synonym conservation is equivalent to mirror symmetry). Later, in 1933, Pauli continued to criticize the two-component theory in his famous Handbook of Physics article. In January 1957, after Pauli heard about the experiment of Wu Jianxiong and others on the non-conservation of cosmology in weak interactions (weak interactions are the basic interactions that dominate particle decay), he first expressed disbelief that cosmology is not conserved, and after reading the paper ten days later, he changed his opinion.
But after Pauli read Weyl's second article on normative theory in 1929, his attitude changed completely, and he wrote another letter [19-20]:
"Contrary to my last ugly remark, the main part of my last letter has been negated, especially by your article in the Journal of Physics. So I later regretted writing you a letter. After studying your article, I would say that I really got the idea of what you wanted to do (which is not the case, according to your short article in the Proceedings of the National Academy of Sciences). Let me first emphasize what I fully agree with: you incorporate the rotation theory into the framework of gravitational theory. "Here I must admit your physical ability. Your early theory of using kik′=λgik was purely mathematical and had no physical significance. Einstein's criticism and blame for you is justified. Now your time for revenge has arrived. ”
In 1930, Weil went to teach at the University of Göttingen and became the successor of his mentor Hilbert.
Pauli became a proponent of gauge theory. In his 1933 Handbook of Physics article mentioned above, Weyl's gauge theory was also introduced [21].
Also in 1933, Weil received a letter of appointment from the Institute for Advanced Study in Princeton, which he did not accept. Later, the political situation in Germany deteriorated, and he accepted a second offer from the Princeton Institute for Advanced Study. He worked there until his retirement in 1951, where he later lived in Zurich and Princeton, mainly Zurich.
Although norms and canonical transformations are also discussed in classical electromagnetism, the term was used after Weyl modified the canonical theory with quantum mechanics in 1929 [22].
4. Klein and Pauli
Protons and neutrons are collectively referred to as nucleons and are combined into nuclei by strong nuclear forces. In 1922, considering that the strong nuclear force had nothing to do with whether each nucleon was a proton or a neutron, Heisenberg proposed the concept of an isospin, similar to spin, representing protons and neutrons as the two basic states of an isospine, while under a strong nuclear force, the isotopic cyclone is conserved. In 1936, Hideki Yukawa proposed the meson theory of strong nuclear forces, that is, the strong nuclear force is generated by the exchange of mesons between nucleons, just as the electromagnetic force between charged particles is achieved by exchanging photons. Photons have no mass, so the electromagnetic force is long-range, but the meson has mass, so the strong nuclear force can only be limited to a short distance.
Mathematically , weyl 's gauge transformation belongs to the U(1) transformation , with only one wave function as a phase transformation. The isospinic spin transformation of the nucleon belongs to the SU(2) transformation, which corresponds to the two wave functions of the two isospin states for transformations that meet certain conditions. For a certain transformation, if the result of two consecutive transformations is independent of the order, it is called Abel's, otherwise it is called non-Abel's. The U(1) transform is Abel's, and the SU(2) transform is non-Abel's.
In 1938, Klein proposed a unified theory of gravity, electromagnetism, and strong nuclear forces at a conference in Poland,[20,23] changing the aforemention of the 5th dimensional coordinates in the Kaluza-Klein theory to relying on a phase factor, in which the 5th dimensional coordinate appears on the index. In this new theory , certain degree gauge coefficients become matrices with SU(2) non-Abelian canonical structures. Klein applied this theory to nucleons. Klein does not require the theory to have SU(2) normative invariance. However, he commented that the mass term might not be needed, and that the mass might come from some kind of self-power, a conjecture that was a bit of the spirit of the later Higgs mechanism. Klein's report did not arouse much interest from the participants, and was later published only in the proceedings of the conference, and coupled with the outbreak of World War II, this work did not attract attention.
At the Lorentz-Onnes Conference in Leiden in 1953, Abraham Pais of the Institute for Advanced Study in Princeton, in order to study particle classification, proposed a field theory based on isospines, generalizing each space-time point into a 2-dimensional sphere, in an attempt to generalize the charge conservation of the weyer to the conservation of isospines [20,24]. Pais only considered the overall canonical transformation, that is, the SU(2) transformation had nothing to do with space-time coordinates. Pauli was present, commenting [20,24]:
"I have a specific question about the interaction of mesons and nucleons... I'm in favor of linking conservation laws and invariant properties to nature's mathematical transformation groups. If, in addition to conservation of energy and conservation of charge, the conservation of the number of nucleons and the charge of nuclear forces are unmistakable, then they do have to be linked to the group-theoretic properties of the laws of nature, as Pais is now trying to express in mathematical terms... In connection with this, I would like to ask whether this constant transformation group (isotopic rotation group) can be enlarged, similar to the gauge group of the electromagnetic potential, so that the meson-nucleon action is associated with this enlarged group..."
Pauli is asking if the isotopic spin SU(2) integral transformation, which is independent of space-time coordinates, can be changed to a SU(2) local canonical transformation that relies on space-time coordinates. He immediately studied the problem himself. Pauli was familiar with gauge theory, had previously studied the unified theory of gravity and electromagnetic forces [24], and favored Weil's combination of electron rotation theory and gravity theory.
In two letters to Pais in July and December 1953, Pauli described his theory, entitled "Meson-nucleon interaction and differential geometry" [20,24,25]. He also used the Karuza-Klein theory, but with two additional dimensions. In his theory, the electromagnetic potential does not come from the degree gauge, but from the Christopher contact. Pauli did not write down the Lagrange quantities of the gauge field and the field equations here, nor did he formally publish the theory. But he gave a lecture in the fall of 1953, and his students later compiled lecture notes . In a long letter to Yang Zhenning in 1954, Pauli stated that his students had discussed the Lagrange volume of the normative field in this lecture note . In another letter to Pais in December 1953, Pauli said [24]:
"If you try to give the exit equation,...... Then always get a vector meson with a rest mass of 0. ”
The emphasis mark was added by Pauli himself. Here and later Pauli's conversations with Yang Zhenning and his long letters to Yang Zhenning reflect Pauli's realization that regulating the mass of particles is a nuisance. This is because if the gauge particle has mass, the canonical invariance of the theory is lost; and if the gauge particle has no mass, it means that the gauge field can transmit the force infinitely farther, but this is not the case with the strong nuclear force.
5. The Young-Mills theory
Yang was a young member of the Institute for Advanced Study in Princeton at the time and a colleague of Pais, but from the summer of 1953 to the summer of 1954, he visited the Brookhaven laboratory. Yang Zhenning once recalled [27]:
"When I was a graduate student in Kunming and Chicago, I carefully studied Pauli's review articles on field theory. I was very impressed with the relationship between charge conservation and the theory that does not change in phase. Later I discovered that these ideas came from Weil. Even more impressive is that normative invariance determines the entire electromagnetic interaction. ”
Yang Zhenning went on to recall that when he was a graduate student in Chicago, he began to try to generalize the gauge theory to isotopes. The 2005 edition of 50 Years of Yang-Mills Theory includes Yang's three-page notes from 1947,[28] and the editor, Gerardus't Hooft, said, "This is the note of a graduate student who was working to standardize the concept of invariance at the time, and it is still a long way from the masterpiece of 1954." After 1947, Yang Zhenning also made many unsuccessful calculations.
Later, as more and more mesons were discovered in the experiment, Yang Zhenning believed that a principle of writing down interactions was needed. So in the summer of 1953 at Brookhaven Lab, Yang Returned to this question. At the time, he shared an office with Robert Mills, who was about to graduate with a Ph.D. at Columbia University. This time, the two finally completed the non-Abel promotion of gauge theory [27].
Yang Zhenning and Mills. Image source: Selected Works of Yang Zhenning
In February 1954, Yang Zhenning returned to the Institute for Advanced Study in Princeton to give an academic report introducing this theory [27]. Pauli was in the audience, and during that period he was "oscillating" between Zurich and the Institute for Advanced Study in Princeton, like his old friend Weyl. The sharp Pauli kept asking Yang Zhenning to regulate the mass of the particles. Yang Zhenning replied that he did not know, he had studied it, but there was no clear conclusion. Pauli said: "This is not a good reason. The next day, Yang Zhenning received a text message from Pauli and went to find Pauli. Pauli suggested that Yang Zhenning read Schrödinger's article on Dirac's equations in the gravitational field. Yang Found that the equations inside are related to Riemannian geometry on the one hand, and similar to those of him and Mills on the other hand [27]. This month, Pauli also wrote a long letter to Yang Zhenning, simplifying his previous results in flat space-time and other conditions, in line with Yang Mills' results, and saying that his students discussed the Lagrange of the normative field. Pauli concluded [26]:
"But I have been and still am disgusted by vector fields with zero rest masses of particles (I don't take your 'complex' seriously), and there are difficulties in groups caused by the properties of electromagnetic fields."
Yang Zhenning and Mills' work is completely unrelated to general relativity and has no burden of additional dimensions. Before their article, in addition to the London article and the Pauli review, papers on non-accumulative gauge factors and gauge theory all entangled the gauge field with gravity, which is somewhat misleading, but has certain technical advantages. Yang Zhenning and Mills proposed a clear non-Abel gauge theory with clear physical motivations. But because they did not understand the geometric significance of the canonical field at that time, they did not know that the field strength could be obtained directly from the pair of transposons of the covariate derivative. Klein and Pauli's work is within the framework of curved space-time theory, so it is natural to obtain field strength through the pair of transponds of covariate derivatives. However, the author notes that in Pauli's original review article on the Weyl gauge field, for the usual straight space, the field strength is also expressed by the pair of covariate derivatives [21]. Yang Zhenning did not pay attention to this, otherwise the promotion process would have been much smoother.
Despite pauli's criticism, Yang Zhenning still believes that the idea is beautiful and should be published. This shows great courage, because Pauli's criticism is powerful and lethal. In 1925, for example, George Uhlenbeck and Samuel Gouldsmit proposed that electrons have spins. The contribution did not receive the Nobel Prize because Ralph Kronig came up with the same idea more than half a year ago, but it was not published against the objections of Pauli, Heisenberg and Hendrik Kramers.
In 2012, Yang Zhenning said [29]:
"This article is the most important work of my life. Although it was not fully successful, the decision to publish at that time was extremely correct. ”
In 1954, Yang Zhenning and Mills actually published two articles proposing the Yang-Mills theory. The first is just an abstract of Yang Zhenning's report "Isotopic spin conservation and a generalized gauge invariance" delivered by Yang Zhenning at the April meeting of the American Physical Society that year, published in the Physical Review, which summarizes the physical ideas of the Young-Mills theory.[30]
"Conservation of isospines is similar to conservation of charge, showing the existence of a fundamental law of invariance. In the latter case, charge is the source of an electromagnetic field; an important concept here is gauge invariance, which is closely related to (1) the equation of motion of the electromagnetic field, (2) the presence of a flow density, and (3) the possible interaction of a charged field with an electromagnetic field. We try to generalize this concept of normative invariance to apply to isotropic cyclone conservation. The results show that this promotion is natural. A field similar to an electromagnetic field is a vector field that satisfies nonlinear equations even if other fields do not exist. (Unlike the electromagnetic field, this field has isotropic spins and is its own source.) The flow density exists automatically, and the interaction of this field with other fields of any isoisocy has a definite form (except for possible terms similar to the abnormal magnetic moment action in electrodynamics). ”
Another of their articles, "Conservation of isotopic spin and isotopic gauge invariance," was accepted by the Physical Review on June 28. The summary of this article emphasizes local transformation [31]:
"This paper points out that the principle of invariance, which is usually under isotopic spin rotation, does not harmonize with the concept of a local field. This article explores invariance under local isospin rotation. This leads to the establishment of a isocycle specification invariance principle and the presence of a b-field particle whose relationship with the isospin is similar to the relationship of an electromagnetic field to an electric charge. The b-field satisfies nonlinear differential equations. The quanta of the b field are particles with a spin of 1, an isocentric spin of 1, and a charge of ±e or 0. ”
The article concludes by stating that there is no satisfactory answer to the question of the mass of canonical particles, and points out that the option with zero mass faces divergence difficulties.
Incidentally, the integral symmetry is also legitimate, does not contradict the normative symmetry, and both symmetries exist, the former can also be seen as a special case of the latter, and in fact the conservation of charge is the consequence of the invariance of the overall phase transformation.
Although it is a beautiful theory, the Young-Mills theory does not immediately apply to physics. At that time, in addition to Princeton, Yang Zhenning only gave a speech at Harvard University to introduce the work.
In 2012, Yang Zhenning commented [32]:
"In recent years, I've often been asked why in 1954 Pauli didn't publish his calculations about gauge fields, while Mills and I did. I think the answer lies in our different value judgments about (A) the beauty and power of normative invariance, and (B) the quality of normative bosons. For Mills and me, the central motivation came from (A), as our short summary suggests. As for (B), Mills and I explored the possibilities, concluding in our 1954 article: 'Therefore we have not come to any conclusions about the mass of b-quanta.' That is, we think of (B) as a matter of the future. For Pauli, the beauty of normative immutability is clearly not fully understood. He has a persistent negative attitude towards the whole idea. See footnote 34 to article [85j]. Thus (B), the quality issue, for Pauli, becomes central and decisive. ”
The [85j] mentioned here is Yang Zhenning's Weyl's Contributions to Physics,[4] which notes 34 that Pauli had a negative attitude toward the idea of the gauge field in the last years of his life, and mentions that in 1956 Pauli wrote a series of afterwords to his 1921 essay "The Theory of Relativity", in which the afterword to "Weier's Theory" is no longer as positive as his original German text in 1921.
In short, Yang Zhenning and Mills became the founders of non-Abell gauge theory, which is also known as Yang-Mills theory. In 1999, on the occasion of the 100th anniversary of the Founding of the American Physical Society, Yang Zhenning's original colleague and witness Pais wrote a historical review of theoretical particle physics, in which he wrote [33]:
In 1954, two wonderful essays marked the beginning of non-Abelle gauge theory. They deal with a completely new strong interaction, passed by a zero-mass vector meson. This work was of great interest, but how to apply these esoteric ideas was another matter, when there were no vector bosons, let alone zero-mass vector bosons. The issue was shelved until the 1970s. ”
The publication of the Young-Mills theory led many physicists to pay attention to this problem as a candidate theory, although it did not immediately succeed in solving specific physical problems. Yang Recalls: "In the late 1950s, gauge theory was used for both strong and weak interactions. In 1960, Jun John Sakurai published a very ardent article proposing a non-Abel gauge theory of strong interactions. [27] However, Yang zhenning believed that Sakurai's approach undermined the most wonderful concept of normative invariance, the normative theory, and that normative theory should not be sloppily transformed into something that was symbolic. In 1961, S. Glashow, like Sakurai, proposed a model with U(1)× SU(2) symmetry to unify electromagnetic and weak interactions.
On the other hand, the Young-Mills theory has also aroused some pure theoretical interest. In the 1960s, R.P. Feynman, B. D. Vincent, and 1960s DeWitt), V. Popov N. Popov) and L. Fadeev D. Faddeev) studied the quantization of the Young-Mills theory.
6. The Standard Model of Particle Physics and the Revival of the Young-Mills Theory
Later, after years of efforts by many other physicists, after adding other elements of thought, the Young-Mills theory became the theoretical framework of the Standard Model of particle physics. The object of the description is different from what was envisaged in 1954. Protons and neutrons are no longer elementary particles, and isotropic rotation conservation is only approximate. Leptons and quarks are elementary particles. Nucleons are made up of quarks, and the strong nuclear force between nucleons comes from the strong interaction between quarks and gluons. Fundamental interactions include strong interactions and "electroweak interactions" that unify electromagnetic and weak interactions, each described by a Young-Mills theory.
Standard Model of Particle Physics. Source: wikipedia
In addition to gauge symmetry, another major idea of the Standard Model of particle physics is spontaneous symmetry breaking. This idea first appeared in condensed matter physics, especially in superconductivity theory, and particle physics was first introduced in 1960 by Yoichiro Nambu. He noted that vacuum does not necessarily have the symmetry of the energy function.
In 1963, P. Anderson W. Anderson) draws on superconductivity theory to argue that spontaneous breakouts of gauge symmetry lead to mass acquisition of gauge particles. In 1964, R. Braute Brout) and F. Inglade Englert), P. Higgs W. Higgs), and later G. Guralnik, C. R. Hagen, and T. Kibble) three groups of researchers point out that in relativistic field theory, the spontaneous breaking of normative symmetry allows normative particles to acquire mass. This mechanism also leads to the production of massive Higgs particles. As Yang Zhenning once commented[34]:
"The idea of spontaneous defection solves the mass problem of normative particles without breaking the spirit of symmetry."
In 1967, based on the spontaneous breaking of gauge symmetry, as a model of leptons, Weinberg proposed an electroweak theory with U(1) ×SU(2) gauge symmetry. Salam also independently proposed this theory at a conference, and for the first time adopted the name "electroweak theory". Spontaneous symmetry breaking allows the electrodependent canonical particles (W± with 1 positive or negative unit charge and uncharged Z0 particles) to gain mass, and the subsequent parts of the theory are consistent with Glashow's 1961 theory based on anthropogenic assumptions. In 1971, Weinberg applied this theory to quarks. From 1971 to 1972, Teffert and Viltermann (M. Berger) Veltman) demonstrated the reformibility of the Young-Mills theory, a major breakthrough that shows that electroweak theory is a self-consistent quantum field theory.
Another major idea of the Standard Model of particle physics is the so-called progressive freedom. In 1973, David Gross and Frank Wilczek, as well as David Politzer, discovered that the Young-Mills theory had the nature of progressive freedom, that is, the shorter the distance, the weaker the interaction. Terhofert had obtained this result the year before, but had not published it. The discovery of progressive freedom determined the physical significance of quantum chromodynamics. This is H. Fritzsch Frizsch) and M. Gell-Mann Gell-Mann's 1972 Yang-Mills theory, which has SU(3) gauge symmetry, describes strong interactions through what is called the degrees of freedom of color, and its canonical particles are gluons. In the SU(3) transformation, three wave functions are transformed to meet certain conditions. Later, Gross and Verczek and Weinberg proposed that the gauge symmetry of quantum chromodynamics was not broken, so the gluon mass was zero. Leptons and Higgs particles are colorless and therefore do not participate in strong interactions.
In 1972, Makoto Kobayashi and Toshihide Maskawa proposed a model of a weak mixture of quarks of 3 generations and 6 species (this species is called taste) in the framework of electro-weakness theory, explaining the CP symmetry failure of the weak interaction observed experimentally (the symmetry failure here is not the spontaneous breaking of the previously discussed). C refers to charge conjugate, that is, mathematically transforming a particle into its antiparticle, P refers to the cosmic transformation, and CP refers to two transformations at the same time. In 1956, the theoretical analysis of Li Zhengdao and Yang Zhenning showed that whether the universe in the weak interaction was conserved by experiments to be tested experimentally, and then Wu Jianxiong found that P and C in the weak interaction were asymmetrical through the β decay experiment of cobalt 60, and Li and Yang won the 1957 Nobel Prize in Physics. In 1964, James Cronin and Val Fitch discovered in the decay of neutral K mesons that CP was also asymmetrical, for which they won the 1980 Nobel Prize in Physics.
In 1974, Kenneth Geddes Wilson invented lattice point gauge field theory, which performs non-perturbative computations by separating space-time. He won the Nobel Prize in Physics in 1982 for his study of phase transitions with reintegration groups.
Around 1970, Yang Zhenning realized the geometric significance of normative fields and the importance of non-accumulative phase factors. In 1974 he published the integral form of gauge theory and proposed that gravity is a kind of gauge field [36]. In 1975, Yang Zhenning and Wu Dajun gave a holistic description of the gauge field using non-accumulatable phase factors, and corresponded the gauge theory with the basic concepts of fiber cluster theory [37], which promoted the successful cooperation between mathematics and physics in this regard. In this context, yang zhenning discussed the relevant history at the centenary commemoration of the birth of Waier and Schrödinger in 1985 and 1987 [4,9].
7. Experiments give life to the Jan-Mills theory
Weyl derived the electromagnetic theory from gauge invariance and peeked into a corner of the theoretical structure of nature, but electromagnetic theory had long been summarized by physicists from a large number of experiments and was summed up by Maxwell. Whether the Young-Mills theory can become a theoretical framework for describing fundamental interactions, whether the Standard Model is correct or not, needs to be tested by experiments, so experiments give life to the Young-Mills theory. Experimental validation has also given many physicists who have made important contributions to the establishment of the Standard Model the Nobel Prize in Physics.
The work of Yang Zhenning and Mills was driven by the discovery of a large number of mesons in experiments. Mesons were first found in cosmic rays, and many were found after the Proton Synchrotron COSMOTRON in Brookhaven Laboratory was put into operation in 1953. The Young-Mills theory was questioned after it was proposed because the gauge particles in the theory did not exist experimentally at the time.
In the 1970s, the neutral flow predicted by electroweak theory was confirmed in many experiments by the European Organization for Nuclear Research (CERN), FermiLab and stanford Linear Accelerator Laboratory (SLAC), so Glashow, Salam and Weinberg shared the 1979 Nobel Prize in Physics.
Glashow, Salam, Weinberg.
Image source: http://www.boiledbeans.net
In 1983, the normative W± and Z0 particles in electroweak theory were discovered at CERN, so Carlo Rubbia and Simon Van der Meer shared the following year's Nobel Prize in Physics. Calculations of the specific properties of W±, Z0, and other particles (including the masses of the top quarks discovered only in Fermilab in 1995) rely on Tefofft and Wiltmann's theory of gauge field reformulation, so they shared the 1999 Nobel Prize in Physics "for clarifying the quantum results of the theory of electroweaks" [38].
Experiments have shown that quarks cannot exist in isolation and can only be observed at high energies, so a strong interaction theory with progressive free properties is needed. Gross and Verczek, along with Pulitzer' calculations, showed that the Young-Mills theory had such properties, and they shared the 2004 Nobel Prize in Physics. Quantum chromodynamics has been tested by many experiments, such as explaining the behavior of J/ψ particles in the bound state of cannabis and their antiparticles, predicting electron-proton deep inelastic scattering deviations from the James Bjorken scale and experimentally confirming them, and explaining the phenomenon of quark jetting. In 1968, Feynman explained the Björkan scale with a partial submodel, which was quickly considered to be a quark [39].
Yoichiro Minami's work with Kobayashi Makoto and Toshihide Maskawa all enabled the Standard Model under the framework of gauge symmetry to explain the experimental facts of asymmetry, so they shared the 2008 Nobel Prize in Physics. In 1972, when the model of Makoto Kobayashi and Toshiei Maekawa was proposed, there were only 3 known quarks. In 1964 Bjorkon and Glashow proposed a fourth type of quark, called a cantonal quark. In 1970, Glashow, Jean Iliopoulos, and Luciano Maiani showed that the existence of a cannibal can be explained by the change of the singular number by no more than 1 under the weak interaction. In 1974, Ding Zhaozhong and B. Richter discovered J/ψ particles, confirming the existence of cantonal quarks and even the authenticity of quark models, known as the "November Revolution". Ding Zhaozhong and Richter shared the 1976 Nobel Prize in Physics. In 1977, the Martin Perl group discovered the τ lepton (Böhl shared the 1995 Nobel Prize in Physics), indicating that there are 3 generations of leptons, and leon Lederman's group discovered the fifth quark. In this way, the models of Makoto Kobayashi and Toshiei Mashikawa began to receive attention. The sixth type of quark is the top quark that was discovered in 1995.
In 2012, CERN discovered the Higgs particle, and the following year Inglet and Higgs shared the Nobel Prize in Physics (Braut died in 2011).
Experimental discoveries of the Higgs particle. Source: wikipedia
8. The Young-Mills Theory in nobel prize-winning speeches
Many of the relevant Nobel Prize-winning speeches in physics mention the Young-Mills theory or non-Abel gauge theory. At the International Conference on the 60th Anniversary of the Jan-Mills Theory, I presented these fragments [40] in a concentrated way, and they are enough to paint a fascinating history.
Glashow (1979 award):
"Today we have a 'standard theory' called elementary particle physics, in which the interaction of strength, weakness, and electromagnetics is given from the principle of local symmetry... Electromagnetic forces are not only transmitted by photons, but also derived from the requirements of local normative invariance. This concept was used in 1954 for non-Abelian local symmetry groups. ”
Weinberg (1979 award):
"The generalization to more complex groups was made in 1954 in an important article by Young and Mills, who showed how to construct a strongly interacting SU(2) gauge theory... To a large extent, our current theory of the details of elementary particle interactions can be understood through deduction, as a consequence of the symmetry principle and dealing with the infinitely large reformatable principle. ”
Wiltman (1999 award):
"So I conclude that the Jan-Mills theory is probably the best theory for reorganization... Then starting with exploring the Feynman diagrams in the Jan-Mills theory, I determined that many divergent figures disappeared as long as the outer legs of these diagrams were on the mass shell. ”
Terhofert (1999 Award):
"In 1971, I calculated the scale properties of field theory, and the first theory I tried was the Young-Mills theory... Quantum chromodynamics is a Yang-Mills theory of the canonical group SU(3), which can be used as a theory of strong interactions. ”
Gross (2004 award):
"Terhofert's outstanding work on the reformibility of the Young-Mills theory reintroduces non-Abel gauge theory to the industry... We judge that the beta function of the Young-Mills theory can be computed... Pulitzer made his calculations about the beta function of the Jan-Mills theory... Our abstract is as follows: We have demonstrated a large class of non-Abel gauge theories up to computable logarithmic corrections, asymptotics with free fields... His summary is as follows: The calculations show that perturbation theory is good for any Yang-Mills theory and many Yang-Mills theories with fermions..."
Yoichiro Minami (2008 Award):
"Thus the beautiful properties of electromagnetism are extended to SU(2) non-Abelian canonical fields."
Makoto Kobayashi (2008 Award):
"In the framework of gauge theory, taste mixing stems from the inconsistency between gauge symmetry and particle states."
Toshiei Maskawa (2008 Award):
"We began to study CP destruction based on the quadruple quark model in the unified gauge theory of electroweaks."
Inglater (2013 Award):
"Since the correct theory of short-range interactions clearly requires quantum self-consistency, we are naturally drawn to the generalization of quantum electrodynamics, the Young-Mills theory, as a model for the corresponding long-range action... In order to transform the long-range action into a short-range action within the Framework of the Young-Mills theory, it is only necessary to assign this generalized photon mass, which, as we have just mentioned, is ostensibly forbidden by local symmetry... Thus the disappearance of the southern-Goldstone boson is a consequence of local symmetry ... Thus the coupling of the original Southern-Goldstone boson to the canon law field must have allowed the latter to gain mass. This is the gist of the Braut-Inglet-Higgs mechanism. ”
Higgs (2013 award):
"Anderson says 'Goldstone's zero-mass difficulties are not serious because we might be able to offset it with the Jan-Mills zero-mass problem'... Schwenger wrote an article in 1962 to overturn a legend that normative immutability alone requires zero photon mass... He offers the nature of a gauge theory containing mass 'photons'... On July 18-19, it occurred to me that Schwenger's formulation of gauge theory had dug into the axioms used to prove Goldstone's theorem. Thus normative theory might be able to salvage the Southern plan. ”
9. Symmetry governs interactions
Weil died in Zurich in December 1955. Shortly before that, Weyl included gravity and electricity from 1918 in his anthology and wrote a review [20]:
"The strongest argument for my theory is that gauge invariance corresponds to charge conservation, just as coordinate invariance corresponds to conservation of energy and momentum. Later quantum theory introduced the Schrödinger-Dirac potential ψ of electron-positron fields; it has the principle of normative invariance with experimental basis, guarantees charge conservation, and associates ψ with the electromagnetic potential φi, just as I suspect that theory associates gravitational potential gik with φi, and uses known atomic units rather than cosmological units as measures of φi. I think there is no doubt that the right area for the principle of normative invariance is here, not in the fusion of electromagnetic forces and gravity, as I believed in 1918. ”
Here Weyl mentions the correspondence between normative invariance and charge conservation. This was not mentioned in his 1918 article, but was mentioned in his 1929 article and in Pauli's introduction to normative invariance, which was well known at the time.
It is even more interesting to contrast this summary of Weil with the summary of Yang Zhenning and Mills' articles. Deriving electromagnetism with normative principles is to clarify the theoretical structure after already understanding the laws of electromagnetism, but just as the invariance of coordinate transformations determines the unknown gravitational force, the non-Abel norm invariance determines the unknown normative interaction.
In 1979, at a conference to commemorate the 100th anniversary of Einstein's birth, Yang Zhenning summarized the principle of "symmetry dominating interactions", pointing out that the first example was Einstein's general relativity from coordinate transformation invariance, the second example was Weyl's electromagnetics from the invariance of Abel's canonical transformation, and the third example was obtaining non-Abelian canonical fields from non-Abelian canonical transformation invariance [41]. This year also marks the 40th anniversary of the generalization of "symmetry dominating interactions".
In a speech commemorating the centenary of Weil's birth, Yang Zhenning said [4]:
"After theoretical and experimental developments, symmetry, Lie groups, and gauge invariance are now confirmed to play an indispensable role in determining the fundamental forces of the physical universe. I call this principle symmetry dominating interactions. ”
Freenman J. Dyson, who spent a long career at the Institute for Advanced Study in Princeton, was familiar with both Weil and Yang Zhenning, and was another witness to the Young-Mills theory. Dyson once said [42]:
Shortly after Weyl left Princeton, Yang Zhenning came from Chicago and moved into Weil's old house. Young replaced Weyl as the leading bird of my generation of physicists. ”
Dyson went on to comment:
The idea that symmetry governs interactions is a generalization of Yang Zhenning's comment on Wai'er's statement that "normative invariance corresponds to conservation of charge, just as coordinate invariance corresponds to conservation of energy and momentum"). Weyl observed that gauge invariance is closely related to the law of conservation of physics. Weyl could not have gone further, for he knew only the normative invariance of the Abel field of Yi. Young made much stronger connections by introducing non-Abelian norm fields. Non-mediocre Lie algebras are produced by non-Abelian gauge fields, and the forms of interaction between the fields are uniquely determined, so symmetry governs the interactions. This idea is Yang Zhenning's greatest contribution to physics. ”
In 2002, Yang Zhenning summarized the three main themes of theoretical physics in the 20th century: quantization, symmetry, and phase factors.[43,44]
Mr. Yang Zhenning once told this writer:
In the 1950s, Weil came to the Institute for Advanced Study in Princeton for only a few weeks a year. My contact with him was limited to cocktail parties. I'm sure he didn't know I published an article on gauge theory. On my side, I didn't know at the time that he was still interested in gauge theory. Apparently neither Oppenheimer nor Pauli told Weil what I and Mills had written. ”
Three examples of "symmetry dominating interactions" embody the beauty of the theory of physics. On the other hand, all three examples reflect the close relationship between theoretical physics and mathematics. The mathematical Riemannian geometry (geometry of curved spaces) corresponding to general relativity was founded in the first half of the nineteenth century by Karl Friedrich Gauss and Bernhard Riemann, who then apprenticed Gregorio Ricci and Tullio Levi-Civita to invent tensor calculus, a tool for analyzing Riemannian geometry. When Einstein began to develop general relativity, his classmate Marcel Grossmann told him that the proper mathematics was Riemann geometry.
Regarding the mathematics of normative theory, Yang Zhenning once wrote [41]:
"In 1975, impressed by the fact that the gauge field was a link on the fiber bush, I drove to the Chan province of Berkeley in El Cerrito... As our conversation shifted to the normative field, I told him I had finally learned the beauty of fiber bundle theory and Weil's theorem from Jim Simons. I said I was amazed that the gauge field is precisely the connection between the fiber bundles that mathematicians have studied without considering the physical world. I went on to say, 'It's exciting and confusing because you mathematicians came up with these concepts out of thin air.' He immediately protested, 'No, no. These concepts were not conceived out of thin air. They are natural and real. ’ ”
Atiyah once wrote [45]:
"After 1977 my interests turned to gauge theory and the interaction of geometry and physics... The inspiration of 1977 came from two sources. On the one hand, Isadore Singer told me that the Jan-Mills equation, through Young's influence, is infiltrating into the mathematical circle. ”
In Yang Zhenning and Wu Dajun's 1975 "The Concept of Non-Accumulatable Phase Factors and the Integral Form of Gauge Fields" [37], there is a "dictionary" that "translates" the basic concepts of normative theory into the basic concepts of fiber cluster theory. Among them, the source in gauge theory corresponds to a question mark, because mathematicians have not yet studied the concept corresponding to the source. One solution to the passive case is the self-dual solution, also known as the instantaneous solution, which is a minimal solution of action with only a single singularity, and is defined by Belafan (A. Bellavan). Belavin), A. Pulyakov Polyakov), A. Schwartz Schwartz) and Tupkin (Y. Tyupkin) got it first.
Atiyah and his collaborators studied the classification of transient solutions. He worked with Nigel Hitchin and Singer to calculate the dimensionality of the instantaneous module space using the Atiyah-Singh indicator theorem (for which the two won the 2004 Abel Prize).
Atiyah's series of academic reports on Jan-Mills and gauge theory sparked karen Keskulla Uhlenbeck (Keskulla' maiden name, whose father was one of the spin-spin proposers, Ulenbeck), whose first husband was one of the proponents of spin. She represents the Jan-Mills equation as an elliptic system, leading to her theory of compactness that "curvature is limited by the contact of Lp" and the theory that "the removable singularity of the Jan-Mills equation on a punctured 4-dimensional sphere". She also showed that any solution with a finite amount of action and isolated singularities is on a SU(2) fiber bundle. Clifford Taubes studied the boundaries of the transient module space and the bonding of self-dual 4-dimensional manifolds.
Based on the work of Artia, Karen Ullenbeck and Taubs, Simon Donaldson studied the topology of 4-dimensional differential manifolds using transient modular space, obtained Donaldson's theorem, and combined with Freedman's theorem to discover the existence of singular differential structures on 4-dimensional Euclidean space, thus winning the 1986 Fields Medal. Karen Ullenbeck and Chengtong Yau discovered the presence of Analyn-Mills contacts on stable, holomorphic vector bundles on complex n-dimensional manifolds.
Edward Witten, who used gauge field theory (especially the supersymmetric Yang-Mills theory) to study mathematical problems such as low-dimensional topologies, was the only physicist to have won the Fields Medal (for proving the positive energy theorem in general relativity). This shows that both general relativity and gauge theory can feed back into mathematics.
Karen Ulenbeck has an interesting comment about the mathematics of the Young-Mills theory [46]:
"How did gauge theory emerge and succeed in mathematics within a few years?" The basic mathematical elements are present (fibers and vector bundles, conjunctions, Chan-Wey theory, Dram upper cohomology, Hodge theory). In hindsight, the Jan-Mills equation was waiting to be discovered. But mathematicians can't create them themselves. Normative field theory is an adopted child. ”
During the typesetting of this article (March 19, 2019), Karen Ulenbeck received the 2019 Abel Prize for his pioneering contributions to geometric partial differential equations, gauge theory, and integrable systems, as well as his fundamental impact on analysis, geometry, and mathematical physics.
Weyl once half-jokingly said to Dyson [42]:
"I always try to unify truth with beauty, but if I have to choose one or the other, I usually choose beauty."
Weyl's dedication to normative symmetry profoundly reflects this. Fortunately, normative symmetry is beautiful and true. Anderson once said [48]:
"A truly beautiful scientific group is the normative principle of particle physics: all known interactions are normative, and the forces that couple particles together come from symmetry, not the other way around."
Einstein's successful creation of general relativity under the guidance of the generalized covariance principle, and even his later dedication to the unified field theory, fully reflected his confidence in the beauty of the laws of nature. In his paper on the creation of general relativity, he wrote [49]:
"No one who masters it can escape its charm."
The pursuit of the beauty of physics runs through Yang Zhenning's research work [50] and reflects his dedication to non-Abel gauge theory. Yang Zhenning once lamented [34]:
"When a scientist does research, when he finds that there are some very wonderful phenomena in nature, when he finds that there are many natural structures that can be said to be incredible charms, I think the way to describe it is that he has a vibration that touches the soul."
The evolution of Pauli's attitude toward gauge theory reflects his meticulousness in his truth-seeking. On the occasion of the 60th anniversary of Pauli's death, we pay a deep tribute to this genius master.
The beauty of non-Abel gauge theory led Yang Zhenning in 1954 to believe that it contained a certain degree of truth, although it could not solve the problem of canonical particle mass for the time being. Yang Zhenning and Mills's paper concludes with a discussion and conclusion on this issue that "we cannot draw any conclusions about the mass of the b-quanta" [31] because it was later discovered that canonical particles can obtain mass by spontaneously breaking through gauge symmetry (W± and Z0) or massless (gluon). In the future, after the Standard Model is surpassed, it is likely that gauge theory will remain important.
Normative theories were immediately rejected and later reborn after they were first proposed; non-Abell gauge theories were immediately questioned and later revived after adding other elements of thought to become the framework of the Standard Model of particle physics. History shows that whether beauty and truth can be unified or not is determined by experimentation. Similarly, further theoretical developments, such as clarifying the mystery of the spontaneous breaking of normative symmetry, exploring the origin of normative symmetry, understanding why neutrino masses are not zero, and so on, also require experimental progress.
This article is reproduced in chapters of Physical Culture and the Physicalculture.