Athanase Papadopoulos

A BIOGRAPHY FOR HIS SELECTED WORKS EDITION

Herbert Busemann 1 was born in Berlin on May 12, 1905 and he died in Santa Ynez, County of Santa Barbara (California) on February 3, 1994, where he used to live. His first paper was published in 1930, and his last one in 1993. He wrote six books, two of which were translated into Russian in the 1960s.

Initially, Busemann was not destined for a mathematical career. His father was a very successful businessman who wanted his son to become, like him, a businessman. Thus, the young Herbert, after high school (in Frankfurt and Essen), spent two and a half years in business. Several years later, Busemann recalls that he always wanted to study mathematics and describes this period as "two and a half lost years of my life. " Busemann started university in 1925, at the age of 20. Between the years 1925 and 1930, he studied in Munich (one semester in the academic year 1925/26), Paris (the academic year 1927/28) and Göttingen (one semester in 1925/26, and the years 1928/1930). He also made two 1 Most of the information about Busemann is extracted from the following sources:

(1) An interview with Constance Reid, presumably made on April 22, 1973 and kept at the library of the Göttingen University. [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF] Other documents held at the Göttingen University Library, published in Volume II of the present edition of Busemann's Selected Works.

(3) Busemann's correspondence with Richard Courant which is kept at the Archives of New York University. A report on this correspondence, by Manfred Karbe, is published in Volume II of the present edition. (4) Anikó Szábo's Vertreibung, Rückkehr, Wiedergutmachung (Expulsion, return, reparation), Wallstein Verlag, Göttingen, 2000, pp. 482-484. The German original as well as an English translation by M. Karbe of the article on Busemann that appeared in this book is re-edited in Volume II of the present edition.

The author of the present biography is most grateful to the personnel of the Elmer Holmes Bobst Library of New York University for giving access to the correspondence between Busemann and Courant, to Manfred Karbe and Irene Zimmermann who kindly made available to him translations of this correspondence and of various other documents related to Busemann, to Hubert Goenner and the librarians of the University of Göttingen for providing a copy of the recording of the Reid interview and to Wallstein Verlag for the permission to re-publish the article on Busemann in the volume Vertreibung, Rückkehr, Wiedergutmachung, as well as an English translation of it.

stays of several months in Rome, in the summer of 1930 and the winter semester of the academic year 1931/32. When Busemann enrolled in Göttingen University in 1926, the mathematics and physics departments there were at their full glory. This was due to people like David Hilbert, Emmy Noether, Bartel Leendert van der Waerden, Edmund Landau, Hermann Weyl, Gustav Herglotz, Felix Bernstein, Carl Runge, James Franck, Max Born and others. There were regular distinguished visitors at the mathematics department, like Pavel Alexandroff and Andreï Kolmogorov. Busemann's meeting with Alexandroff was crucial to him, we shall see why below. Göttingen had also an excellent group of German and foreign PhD students and young researchers, and by the time of Busemann, these included William Feller, Hans Lewy, John von Neumann, Robert Oppenheimer, Werner Heisenberg, Paul Dirac, Franz Rellich, Werner Fenchel, Otto Neugebauer, Oswald Teichmüller, Harald Bohr, Norbert Wiener, Saunders Mac Lane, and there were others.

Several years after he left Göttingen, Busemann shared his recollections on the teaching he received there. According to his account, Courant had so many administrative duties that he was always unprepared for his lectures. He remembers him trying to reconstruct some proofs at the blackboard, and he found this very disturbing. For Busemann, "the aesthetic lecture was the main thing" and in this sense the ideal lecturer was represented by Herglotz. Busemann adds that Alexandroff was also a "first rate lecturer. [...] His German was amazing. You could hear that he had an accent, but he mastered the German language perfectly." Hilbert retired in 1930, but he continued giving a course entitled "Introduction to philosophy on the basis of modern science." Weyl was his successor, and he lectured on differential geometry, topology and the philosophy of mathematics.

Busemann obtained his doctoral degree on February 25, 1931. In a document he filled concerning his application for his PhD examination (see [START_REF] Goenner | Documents concerning Busemann from the library of the University of Göttingen[END_REF]), Busemann states that during his years of study in the three cities he visited, he attended courses by Pavel Alexandroff, Bessel-Hagen, Bohr, Borel, Born, Carathéodory, Cartan, Courant, Franck, Graetz, Grandjot, Herglotz, Hilbert, Julia, Landau, Montel, Noether, Nordheim, Ostrowski, Perron, Walther, van der Waerden, Wegner, Weyl and Wien. In the last paper that Busemann wrote, [START_REF] Busemann | Novel results in the geometry of geodesics[END_REF] (1993), in collaboration with Phadke, Busemann mentions the influence of Minkowski. We read the following (p. 181):

Busemann has read the beginning of Minkowski's Geometrie der Zahlen in 1926 which convinced him of the importance of non-Riemannian metrics. At the same time he heard a course on point set topology and learned Fréchet's concept of metric spaces. The older generation ridiculed the idea of using these spaces as a way to obtain results of higher differential geometry. But it turned out that a few simple axioms on distance suffice to obtain many non-trivial results of Riemannian geometry and, in addition, many which are quite inaccessible to the classical methods.

The title of Busemann's doctoral dissertation is On geometries in which the circles with infinite radius are the shortest lines, and this is also the subject of his paper [START_REF] Busemann | Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind[END_REF]. The subject of the dissertation is the axiomatic characterization of Minkowski geometries (metric spaces associated to norms on finite-dimensional vector spaces). The characterization of Minkowski geometries is a theme that accompanied Busemann during his whole life. In document [START_REF] Goenner | Documents concerning Busemann from the library of the University of Göttingen[END_REF] which we mentioned, Busemann declares that the choice of the topic of his dissertation was inspired by the courses he followed by Alexandroff, who was then a frequent visitor in Göttingen. Busemann recounts the following: 2

Officially, I took my degree with Courant. This was only officially, in the sense that I was really inspired by Paul Alexandroff, who visited Göttingen regularly. He gave me the idea of the subject of the thesis. I wrote it, but of course he could not be the official reviewer of my thesis, so he was my co-referee.

I must say that my thesis was partly in protest against Courant. I went to Courant originally, he gave me something and it turned out to be much easier than what he thought. I did it and it became a small paper (see [START_REF] Busemann | Die Vollständigkeit der Minimalfolgen von Eigenwertproblemen[END_REF]), but not enough for a thesis. Then he gave me something else, I did this too, but then Kolmogorov came to Göttingen, Courant showed it to him, and Kolmogorov said: Oh that's all very fine, but it's well known. So I became really mad, and I went away to Rome. I was angry with Courant, and wrote my thesis there, on my own.

Busemann says that he had very close contacts in Rome, and that furthermore, Courant had given him introduction letters to Levi-Civita and Enriques. About his stay in Rome, Busemann says: "This was the most useful thing in the sense that I decided I would never live in a fascist state. Mussolini was there since 1923, and we were in 1930."

In a letter dated August 8, 1930, Courant informs Busemann that his "geometric work" done in Rome is sufficient for a "reasonable" dissertation. 3 The final dissertation was submitted in December 1930. After obtaining his doctorate, Busemann returned to Rome. In a letter to 2 All of Busemann's quotations are from the recording of the interview with Reid. The recording is corrupted at some places and we tried to reconstruct some words. We have also eliminated the questions of the interviewer and skipped some hesitations, incomplete sentences and repetitions and we have transformed some oral expressions into written English. Lefschetz, dated November 15, 1935, he mentions a lecture by Enriques he was attending in the winter of the academic year 1931/32, which aroused his interest in algebraic geometry.

The doctoral degree that Busemann obtained in 1931 would have led him in principle to a position of lecturer, but practically it did not. He recounts:

Normally I would become an assistant but this was of course the depression time. Since my father had money and Courant knew it, he asked my father whether he wouldn't support me instead of having me become an assistant. This would make it possible for some other youngster to become a mathematician. That fact did me a lot of harm later on, in the sense that the German government after the Second World War tried to make good, but in my case they refused, because I had never had a paid position. If I had a paid position I would have had a pension.

Courant, as the director of the Mathematics Institute in Göttingen, told Busemann that they had run over their funds and asked him to make it possible to meet his father in order to get some financial support from him for the institute. The two men met. Busemann says: "How they settled this situation I don't know and I thought I shouldn't ask." Reid reports on this in her book on Hilbert ([9] p. 132):

During the increasingly hard times, a number of informal assistantships came into being at the institute in addition to the official ones funded by the government. Often duties were vague or non-existent. Courant once gave a student a stipend because he thought the young man was on the verge of a nervous breakdown and needed a skiing vacation. He also contrived to have some students work without pay. One of them was Busemann.

Busemann adds, concerning his former advisor:

Courant was rather reactionary in his mathematical outlook, and so he prevented many things which should not have been prevented. In Göttingen, he constantly tried to prevent the concept of Lebesgue integral. This has in the meantime conquered the whole world. He didn't see the importance of many things of modern mathematics. He had no relations at all with algebraic geometry. [...]

The Russians had played quite a role in Göttingen. I believe that I was the only one who was directly inspired by them. But their course was very popular. They filled a gap. They were familiar with certain modern tendencies that were not represented in Göttingen, e.g. topology.

On January 30, 1933, Hitler was named Chancellor of Germany. Busemann experienced the immediate decline of the University of Göttingen and in the month of May of the same year, he left to Denmark, where he got a friendly reception by Harald Bohr, Werner Fenchel and his wife, Jakob Nielsen, the physicist and Nobel Prize winner James Frank, and others. Busemann spent three years in Copenhagen on an unpaid position, lecturing occasionally, watching the political developments in Europe with the hope of returning to Germany after things get normal again. During these three years, Busemann does return to Germany for occasional visits, he does pay attention to what is happening there, he even intentionally strolls along streets in Göttingen, Berlin and other cities, talking to chance acquaintances to see how they think about the situation in Germany. When his hope of returning for good to Germany became unrealistic, Busemann decided to emigrate to the United States. His plans to emigrate are reflected in his correspondence with Courant; cf. [START_REF] Karbe | On the Busemann-Courant Correspondence[END_REF] and [START_REF] Karbe | Correspondence related to the paper by Busemann and Feller Zur Differentiation der Lebesgueschen Integrale (On the differentiation of Lebesgue's integrals)[END_REF]. Courant expressed some will to help Busemann and came up with various suggestions. For instance, he proposed that Busemann should come to the US on trial to get to know life there and see whether he feels that settling there is an option. However, there was no paid position within sight yet, and the correspondence with Courant gives the impression that the latter, in view of Busemann's financial health related to his family background, did not put particular emphasis to this point. In a letter to Courant, dated May 12, 1935, Busemann writes: "I once thought of Baltimore, because the book of Zariski made so much impression on me." Princeton showed interest, probably because Lefschetz had the chance to encounter Busemann at some earlier event and received a good impression, but nothing resulted in a way satisfactory to Busemann. In a letter to Courant, Busemann reports that he received a letter from Oswald Veblen in which he was invited to come to Princeton, but at the same time Veblen made clear that he had no chance to obtain any scholarship or position whatsoever (not even an "epsilon-scholarship" in the words of Busemann, letter to Courant dated November 3, 1935). The colleagues in Denmark were in favor of Baltimore; only Bohr seems to prefer Princeton: "Nobody knows Zariski." But Bohr also considers advantageously Frank's presence there. In a letter to Busemann, dated October 29, 1935, Courant writes:

I have now talked with both the people of Baltimore (but not directly with Zariski, who lives a very secluded life) and Lefschetz. My impression is the following: Baltimore does not seem so favorable because of Zariski's isolation and the difficulty of having personal contact with him. Otherwise, maybe in Baltimore, the possibility of getting a scholarship for the year 1936-37 would indeed exist. However, under the present circumstances, I would greatly prefer Princeton. Lefschetz, with whom I spoke for a long time about the case, was very friendly, expressed himself with great warmth and appreciation, and would very much like to have you there. He suggested that you should contact him directly by letter.

During his stay in Copenhagen, Busemann published several papers with William Feller, who had also fled Germany. Harald and Niels Bohr were actively helping German refugees in Denmark and Busemann recalls that, in Copenhagen, "Niels Bohr used to call up the American consulate to see how things were going for me."

Busemann's family did not leave Germany. His father stayed as one of the main directors at Krupp, until his retirement in 1943.

In 1936, upon the recommendation of Veblen, Busemann was invited on a temporary position at Princeton's Institute of Advanced Study. Veblen was one of the first Faculty members there. Busemann spent three years at the Institute, first as an assistant to Marston Morse and then as a member. After that, Busemann stayed temporarily at the Johns Hopkins University and Smith College. His first permanent position was at the Illinois Institute of Technology in Chicago, a position which he described as a "horrible permanent job." He recalls that this was a period where "everybody was looking for jobs, and one had to take whatever." He says he spent "five miserable years" in Chicago, from 1940 to 1945. He adds:

The head of the department made it difficult. He did not like foreigners in the first place. He belonged to those people who had done a couple of good things when they were quite young and he was against anyone who was too active mathematically. On the other hand the administration forced him to take good people, and he resented them.

It is interesting to note that Karl Menger, who did foundational work on topology and metric geometry, joined the Faculty of the Illinois Institute of Technology, in 1946 (Busemann had already left).

In 1945, Busemann was appointed Assistant Professor at Smith College in Northampton.

Busemann stayed in contact with Courant, and the two men had a regular correspondence, but essentially on practical matters. Talking about the institute that Courant founded later in New York, Busemann says, in his interview with Reid: "In America, Courant tried to do again what he did before in Göttingen. [...] His institute is excellent but very one-sided too. The mathematics represented there goes all, or most of it, in one direction."

In 1947, Busemann was appointed professor at the University of Southern California, and he spent there the rest of his career. In 1964, he was made distinguished professor.

At USC, Busemann worked in relative isolation, and practically his only collaborators were his PhD students. His work started to be recognized only in the 1980s, when metric geometry was revived in the West, especially by M. Gromov, and when the methods of synthetic global geometry were introduced in the study of geodesic metric spaces. W. P. Thurston, in his approach to geometry, also started from basic principles. Before that, Busemann's work was only appreciated in the Soviet Union, where A. D. Alexandrov founded an important school on the subject, with a large number of collaborators and students. Alexandrov, like Busemann was only interested in the most basic notions of geometry. Classical problems of convexity, isoperimetry and isoepiphany became the forefront of research, and classical projective geometry took its revenge upon a certain Riemannian geometry based on linear algebra and tensor calculus in tangent spaces. In some sense, it was a return to Euclid and Archimedes. In a tribute to Alexandrov's memory, S. S. Kutateladze writes [START_REF] Kutateladze | Alexandrov of Ancient Hellas[END_REF]:

Alexandroff contributed to mathematics under the slogan: "Retreat to Euclid." He remarked that "the pathos of contemporary mathematics is the return to Ancient Greece."

V. A. Zalgaller, in another paper dedicated to Alexandrov [START_REF] Zalgaller | Memoirs of A. D. Alexandroff and his Leningrad geometry seminar[END_REF], mentions Busemann:

In 1961, a well-known American geometer, H. Busemann (1905Busemann ( -1994) ) arrived at the IV All-Union Mathematical Congress; he also was a guest of the seminar. As a student, Busemann (almost the same age as A. D.) took part in preparation of the fundamental book by Bonnesen and Fenchel on convexity, and so he was an expert in A. D.'s area of research. He reviewed many publications of Alexander Danilovich's school for Mathematical Reviews. For the sake of that, he learned Russian and gave his talk at the Congress in Russian.

In 1963, Busemann was elected foreign member of the Royal Danish Academy of Arts and Sciences. But the most important recognition that he got for his work came from the Soviet Union, namely, the Lobachevsky Medal, which was awarded to him in 1985, "for his innovative book The Geometry of Geodesics" which he had written 30 years before. The first recipients of this prestigious prize were Sophus Lie in 1897, Wilhelm Killing in 1900 and David Hilbert in 1903. A. D. Alexandroff obtained it in 1951. The list of recipients also includes Weyl, Pontryagin, H. Hopf, P. S. Alexandroff, Kolmogorov, Hirzebruch, Arnol'd, Margulis, Gromov and Chern.

Busemann's work is profound. He was capable of formulating problems and working on them, without relying on the trends that were fashionable in his time. In an article that appeared in the Los Angeles Times on June 14, 1985, [START_REF] Dembart | An unsung geometer keeps his own plane[END_REF], on the occasion of the attribution of the Lobachevsky prize to Busemann, the author reports the following:

Few mathematicians ever make it into public consciousness, but Busemann has had a hard time even within his own field, in part, at least, because he never worked on trendy problems and never followed the crowd.

The journalist quotes Busemann saying: "If I have a merit, it is that I am not influenced by what other people do." He then quotes Bob Brooks, who was Busemann's colleague at USC: Tastes change a lot and interests change a lot in the space of five years. But there are people who aren't so interested in keeping up with today's fads. Busemann is very definitely in that category.

We also read in the same article:

Busemann characterizes his basic mathematical approach this way: "Any apparently difficult problem can be done with very simple methods. This is the property of many of my things. I see a simple geometric reason which others have overlooked."

Talking about the Geometry of Geodesics for which the Lobachevsky Medal was attributed, Busemann declares, in the Los Angeles Times article, that the approach is more important than the results: "The emphasis is more on the radically new approach than on the individual problem."

Busemann retired in 1970. In 1971, he received from USC a honorary degree of Doctor of Laws. He was a also a linguist. He spoke several Languages, including, besides German and English, French, Spanish, Italian, Russian, and Danish. He wrote papers or translated articles and monographs from all these languages. Busemann published in the Mathematical Reviews a large number of reports on articles written in Russian. He could also read Arabic, Latin, Greek and Swedish. In the article in the Los Angeles Times [START_REF] Dembart | An unsung geometer keeps his own plane[END_REF] mentioned above, the journalist quotes Busemann: "Every two years I read the Odyssey, I like it so much. And Plato." Busemann was also a painter. In the same article in the Los Angeles Times, we read: "Despite a lifelong desire to paint, Busemann never took it up, fearing that it would divert him from the arduous work of mathematics. But retirement freed him, and he built a studio in his home that is now chockablock with dozens of large canvases painted in vibrantly colored geometrical designs. His mathematical vision carried over into art."

Busemann got married in 1939. Little is known about his personal life. In a recent letter to the author of the present article, Peter Woo, who was a student of Busemann, writes:

Busemann had a funny way of writing on the blackboard. Some important theorem, he would write in big letters, from left edge to right edge, some 15 feet wide. Then he would scribble the proof underneath, and say, "See how easy it is," and then erase his proof. We had to beg him not to wipe away until we finished copying it. He would give us a puzzled look, as if we were wasting his time.

He wanted me to discover theorems about new geometrical spaces with the rule : AB + BC ≤ AC unless A, B, C lie on a "geodesic curve." First I found this funny. Then I began to make conjectures, and he encouraged me to prove them, first in some particular cases, then in more and more general cases. At some point he said, "You have done enough for the PhD. dissertation."

He often took us to his home. Together with three other PhD students, we used go to his house one afternoon per week. Each of us was supposed to present some theorem or unsolved problem, on a blackboard hanging on the wall at his backyard. He would make some remarks on what direction we should turn to, some easy special cases we had to study first, etc. After that, he would invite us into his house, to have a piece of pastry, and tea or coffee. He was always positive. He did not tell us much about his life. He knew a lot about history of mathematics, and this tied us with the European cultural heritage. He liked plants. He had a cactus garden where we loved a little thing about 9" tall, 4" wide, like a ridged dark green okra, with much white hair Spreading from the top all around. He called it "the Old Man." We loved it. He had no children, he loved us almost like his children. In all sense of the word, we were his disciples.

The list of Busemann's students includes John Beem, John Featherstone, Donald Glassco II, George Lewis, Flemming Pederson, Clinton Petty, Benson Russell, Bhalchandra Phadke, Jànos Szenthe, Steven Weinstein, Peter Woo and Eugene Zaustinsky.

There is not much more that can be added on Busemann's life. His work will speak for him.

3

  An English translation of Courant's report on Busemann's dissertation is contained in the Volume II of the present Selected Works.


Tian Ma
email: matian56@sina.com

Shouhong Wang
email: showang@indiana.edu

RADIATIONS AND POTENTIALS OF FOUR FUNDAMENTAL INTERACTIONS

Keywords: . 1991 Mathematics Subject Classification. 83C, 70S15 fundamental interactions, principle of general relativity, principle of equivalence, principle of gauge invariance, principle of representation invariance, principle of interaction dynamics, principle of symmetry-breaking, interaction potential, gravitational field equations, gravitational field particles, gravitational radiation, weak radiation, strong radiation, electromagnetic radiation

First we demonstrate that the principle of general relativity, the Lorentz and gauge invariance are basic symmetries, dictating the Lagrangian actions of the four fundamental interactions, and the field equations are then derived with the principle of interaction dynamics (PID) and the principle of representation invariance (PRI). Second, we identify that each individual interaction possesses two basic attributes: 1) field particles and their radiation, and 2) the interaction potential and force formulas. In particular, due to the presence of dark matter and dark energy, the Einstein general theory of relativity can be uniquely modified using PID to take into account the effect of dark energy and dark matter phenomena, and to preserve the Einstein's two fundamental principles: the principle of equivalence and the principle of general relativity. The gravitational field particle is described by the dual field {Φ µ }, which is a spin-1 massless particle. This field particle can be regarded as the dark matter, and the energy it carries is the dark energy.

There are four fundamental interactions in nature: the gravity, the electromagnetism, the weak and the strong interactions. It was to the credit of Albert Einstein, Herman Weyl and Paul Dirac that physical laws of these four interactions should be built based on fundamental first principles. There have been extensive studies in the last 100 years or so; see among many others [START_REF] Lee | Question of parity conservation in weak interactions[END_REF][START_REF] Wu | Experimental test of parity conservation in beta decay[END_REF][START_REF] Maki | Remarks on the unified model of elementary particles[END_REF][START_REF] Salam | On parity conservation and neutrino mass[END_REF][START_REF] Landau | The classical theory of fields, Translated from the Russian by Morton Hamermesh[END_REF][START_REF] Gross | Ultraviolet behavior of non-abelian gauge theories[END_REF][START_REF] Politzer | Reliable perturbative results for strong interactions?[END_REF][START_REF] Weisskopf | Physics in the Twentieth Century: Selected Essays[END_REF][START_REF] Nambu | Quasi-particles and gauge invariance in the theory of superconductivity[END_REF][START_REF] Nambu | Dynamical model of elementary particles based on an analogy with superconductivity. I[END_REF][START_REF] Griffiths | Introduction to elementary particles[END_REF][START_REF] Halzen | Quarks and leptons: an introductory course in modern particle physics[END_REF][START_REF] Englert | Broken symmetry and the mass of gauge vector mesons[END_REF][START_REF] Guralnik | Global conservation laws and massless particles[END_REF][START_REF] Higgs | Broken symmetries and the masses of gauge bosons[END_REF][START_REF] Weinberg | A model of leptons[END_REF][START_REF] Glashow | Gauge theories of vector particles[END_REF][START_REF] Yang | Conservation of isotopic spin and isotopic gauge invariance[END_REF] and the references therein.

The main objectives of this paper are to examine these first principles, and to study radiations and potentials for the four interactions. Hereafter we address the main ingredients of this study.

First, the Einstein theory of general relativity is the most profound scientific theory in the recorded human history. The Einstein theory is built on two first principles: the principle of equivalence (PE) and the principle of general relativity (PGR). In essence, PE amounts to saying that space time is a four-dimensional Riemannian manifold (M, {g µν }) with the metric being the gravitational potential.

The PGR is a symmetry principle, and says that the law of gravity is the same (covariant) under all coordinate systems. In other words, the Lagrangian action of gravity, called the Einstein-Hilbert action, is invariant under all coordinate transformations.

The Einstein-Hilbert functional (2.1) is uniquely dictated by this profound and simple looking symmetry principle, together with simplicity of laws of Nature. Indeed, in Riemannian geometry, the invariant quantities satisfying the principle of general relativity and containing the second order derivative terms of {g µν } is just the scalar curvature R, which is unique.

The presence of the dark matter and dark energy phenomena requires the inevitable need for modifying the Einstein general theory of relativity. Such modification needs to preserve the following basic physical requirements:

• conservation of energy-momentum,

• inclusion of dark matter and dark energy effect, and • preservation of Einstein's two principles: PE and PGR. We have shown that, under these basic requirements, the unique route for altering the Einstein general theory of relativity is through the principle of interaction dynamics (PID), which takes variation of the Einstein-Hilbert action subject of energy-momentum conservation constraint. This leads to the new gravitational field equations (2.7), stated here as well for clarity [START_REF]Gravitational field equations and theory of dark matter and dark energy[END_REF]:

(1.1) R µν - 1 2 g µν R = - 8πG c 4 T µν -∇ µ Φ ν .
Also, we have shown [13] that PID is the direct consequence of the presence of dark energy and dark matter, is the requirement of the presence of the Higgs field for the weak interaction, and is the consequence of the quark confinement phenomena for the strong interaction.

Second, it has been shown that the basic symmetry for the electromagnetic, the weak and the strong interactions is the gauge invariance and the Lorentz invariance. We discovered the principle of representation invariance (PRI) [START_REF] Ma | Unified field theory and principle of representation invariance[END_REF], which is a logic requirement for the gauge theory. Basically, PRI requires that the gauge theory be independent of the choices of the representation generators. These representation generators play the same role as coordinates, and in this sense, PRI is a coordinate-free invariance/covariance, reminiscent of the Einstein principle of general relativity.

PRI shows that the unification of four interactions through large symmetric group is not mathematically feasible. Consequently, PRI leads us to postulate the principle of symmetry-breaking (PSB); see [START_REF]Astrophysical dynamics and cosmology[END_REF]13].

The three principles PID, PRI and PSB offer us to establish a unique route of unification for the four fundamental interactions:

• the PGR dictates uniquely the Einstein-Hilbert action, and the Lorentz and gauge invariances dictate uniquely the actions for the electromagnetic, the weak and the strong interactions; • the PID and PRI determine the field equations. Our unification is the unique unification, which provides the unified field equations and is based on first principles. In fact, all other known attempts of unification are not able to provide proper field equations. It has successfully provided solutions to a number of challenge problems associated with the four fundamental interactions that modern physics has faced for many decades, including for example the following:

(1) what are dark energy and dark matter?

(2) what is the mechanism of active galactic nucleus jets?

(3) What are the strong and weak interaction potential formulas? (4) Why the Yukawa potential of nucleus is in disagreement with experiments in short range? (5) What is the mechanism of quark confinement and asymptotic freedom? (6) Why do leptons not participate in strong interactions? [START_REF] Higgs | Broken symmetries and the masses of gauge bosons[END_REF] Is there a first principle approach to the Higgs mechanism? [START_REF] Landau | The classical theory of fields, Translated from the Russian by Morton Hamermesh[END_REF] What is the mechanism of subatomic decays and reactions? [START_REF] Lee | Question of parity conservation in weak interactions[END_REF] What is the mechanism of bremsstrahlung? [START_REF] Ma | Unified field theory and principle of representation invariance[END_REF] What are the field equations of the strong and weak interactions? Third, our PID and PRI based models of the four interactions demonstrate the following two characteristics:

(1) Each individual interaction possesses its own field particles and its own radiations, and the field particles are massless, electric neutral bosons; (2) each interaction possesses its own potentials and force.

Fourth, the gravitational field particle is described by the dual field {Φ µ } in (1.1), and the governing radiation equations are (3.3), recalled here for convenience:

(1.2) ∇ µ ∇ µ Φ ν = - 8πG c 4 ∇ µ T µν ,
where T µν stands for the energy-momentum tensor of the visible matter, and {Φ µ } is a massless, spin-1 and electric neutral boson.

In fact, the gravitational field particle {Φ µ } represents the dark matter that we have been searching for, and the energy that {Φ µ } carries is the dark energy. Equation (1.2) is the field equations for dark matter and dark energy. The gravitational effect of the field particle {Φ µ } is manifested through the mutual coupling and interaction with the gravitational potential {g µν }, through the field equations (1.1), leading to both attractive and repulsive behavior of gravity, which is exactly the dark energy and dark matter phenomena.

Fifth, the basic symmetry for the field particles and radiations of the electromagnetic and the strong interactions are respectively the U(1) and SU(3) gauge invariances. The corresponding field particles for electromagnetism are the scalar and vector photons, and the field particles for strong interactions are the vector and scalar gluons.

The massive W ± and Z particles do not qualify for the field particles for the weak interaction as they are massive. We conjecture that the symmetry for the field particles and radiation of the weak interaction should be SU(2) gauge symmetry, and the corresponding field particles should be the ν-particles: ν 1 , ν 2 , ν 3 , whose weakton constituents are ν 1 = ν e νe , ν 2 = ν µ νµ and ν 3 = ν τ ντ .

The field equations for the field particles and radiations are then given by scalar photon:

∂ µ ∂ µ φ = 0, vector photon: ∂ µ (∂ µ A ν -∂ ν A µ ) = eJ ν , vector ν-particles: ∂ µ A k µν - g w c ε k ij g αβ A i αν A j β = g w Q k ν ? scalar ν-particles: ∂ µ ∂ µ φ k w = -g w ∂ ν Q k ν ? vector gluons: ∂ µ A a µν - g s c λ a bc g αβ A b αν A c β = g s Q a ν , scalar gluons: ∂ µ ∂ µ φ a s = -g s ∂ ν Q a ν ,
Sixth, by the modified gravitational field equations (1.1), we derive the following approximate gravitational force formula:

F (r) = mM G - 1 r 2 - k 0 r +k 1 r , k 0 = 4×10 -18 km -1 , k 1 = 10 -57 km -3 .
Seventh, the potentials for electromagnetic, the weak and the strong interactions depend on the number N of particles carrying the same charges. For an N particle system with each particle carries the same charge g, the system obeys the SU(N) gauge invariance, and the governing field equations are

∂ µ A a µν - g c λ a bc g αβ A b αν A c β -gΨγ ν τ a Ψ (1.3) = ∂ ν - 1 4 k 2 x ν + g c γA ν φ a , iγ µ D µ - mc Ψ = 0. (1.4)
By PRI, using the contraction with an SU(N) vector, we deduce the total potential (1.5)

A µ = (A 0 , A) = α a A a µ , φ = α a φ a ,
where A 0 is the interaction charge potential, and A is the corresponding interaction magnetic potential. Namely, the interaction force F and the interaction magnetic force F m are given by

F = -g∇A 0 , F m = g c v × curl A,
where v is the velocity of the particle with charge g. We can then derive the unified layered interaction potential A 0 for the electromagnetic, the weak and the strong interactions:

A 0 = qe -k 1 t 1 r - A ρ (1 -kr)e -(k-k 1 )t , q = nN ρ 0 ρ 3 g,
where n is the number of charges g, ρ 0 is the radius of the particle, and ρ is the radius of the system of N particles.

The paper is organized as follows. In Section 2, we recall the fundamental first principles for the four interactions. Section 3 is devoted to field particles and radiations, and Section 4 derives interaction potential and force formulas.

First Principles of Fundamental Interactions

Two intrinsic properties of interactions.

There are four fundamental interactions in Nature: gravitational, electromagnetic, strong, weak.

There are two basic attributes for each fundamental interaction:

(a) the field particles and their radiations, and (b) the potentials and forces.

In this paper, we demonstrate that the above two attributes of each interaction are governed by their respective physical laws which are determined by the following first principles:

(1) the principle of general relativity, which dictates the Lagrangian action of the gravity, (2) the principle of gauge invariance and the principle of Lorentz invariance, which dictate the Lagrangian actions of the electromagnetic, weak and strong interactions, and (3) the principle of interaction dynamics and the principle of representation invariance, which dictate the field equations through their Lagrangian actions.

2.2. Symmetry principles. Symmetry plays a fundamental role in understanding Nature. The Einstein principle of general relativity (PGR), the principle of gauge invariance, the principle of Lorentz invariance and the principle of representation invariance are all symmetry principles. Together with the simplicity of laws of Nature, these symmetry principles dictate the Lagrangian actions for the four fundamental interactions.

1) Gravity. Among the four interactions, the gravity is unique and different from the other three interactions. The Lagrange action of gravity is the Einstein-Hilbert functional:

(2.1)

L EH = M R + 8πG c 4 S √ -gdx,
where M is the 4-dimensional Riemann space, representing the three dimensional space and the one dimensional time, R is the scalar curvature of the space-time, S is the energy-momentum density, G is the gravitational constant, c is the speed of light, and

g = det(g µν ).
Here {g µν } is the Riemannian metric of M , representing the gravitational potential. The basic symmetry principle for gravity is the principle of general relativity (PGR), which is recalled here: Principle of General Relativity 2.1 (Einstein, 1915). The law of gravity is the same (co-variant) under all coordinate systems. In other words, the Lagrange action of gravity is invariant under any coordinate transformations.

The Einstein-Hilbert functional (2.1) is dictated by this profound and simple looking symmetry principle, together with simplicity of laws of Nature. Indeed, in Riemannian geometry, the invariant quantities satisfying the principle of general relativity and containing the second order derivative terms of {g µν } is just the scalar curvature R, which is unique.

In fact, we believe it is this very reason that Albert Einstein stated in different occasions that his gravitational theory is unique, and we believe that the uniqueness refers to the uniqueness of the action (2.1). In short, any modification of the action (2.1) is not warranted.

2). Electromagnetic, weak, and strong interactions. Modern theory and experimental evidence have suggested that the electromagnetic, the weak and the strong interactions obey the gauge symmetry, to be stated below as the principle of gauge invariance. The Lagrangian actions of the electromagnetic, weak and strong interactions are the functionals of gauge fields:

(2.2) Ψ = {ψ 1 , • • • , ψ N }, A a µ = {A 1 µ , • • • , A N 2 -1 µ },
where N ≥ 1 represents the number of particles in the associated interaction system, ψ i is the wave function corresponding to the i-th particle, A a µ (1 ≤ a ≤ N 2 -1) are the functions of SU(N) gauge fields, and for each a,

A a µ = (A a 0 , A a 1 , A a 2 , A a
3 ) is a 4-dimensional vector field defined on the space-time manifold M .

The Lagrangian action of the SU(N) gauge field (2.2) is expressed as

(2.3) L SU(N) = M - 1 4 G ab g µα g νβ A a µν A b αβ + Ψ iγ µ D µ - mc Ψ dx
where Principle of Gauge Invariance 2.3. The electromagnetic, weak and strong interactions obey gauge invariance. Namely, for an N-particle system with the interactions mentioned above, its Lagrangian action is invariant under the following SU(N) gauge transformation

{g µν } is the Minkowski metric, Ψ = Ψ † γ 0 , γ ν is the Dirac matrix, γ µ = g µν γ ν , (2.4) A a µν = ∂ µ A a ν -∂ ν A a µ + gλ a bc A b µ A c ν , D µ = ∂ µ + igA a µ τ a ,
( Ψ, A a µ τ a ) = ΩΨ, A a µ Ωτ a Ω -1 + i g (∂ µ Ω)Ω -1 , (2.5)
for any Ω = e iθ a τa ∈ SU (N ), where θ a (x)(1 ≤ a ≤ N 2 -1) are functions, {τ a } are the representation generators of SU(N), g is the charge, (Ψ, A a µ ) are the gauge fields as in (2.2).

Principle of Interaction Dynamics (PID).

The three symmetry principles stated above dictate the expressions (2.1) and (2.3) of Lagrangian actions for the four interactions. Then PID determines the field equations from the Lagrangian actions (2.1) and (2.3). PID is stated as follows.

Principle of Interaction Dynamics 2.4. Let L(u) be a Lagrangian action of an interaction system, and u be the field functions with u = g µν for the gravity, and u = (Ψ, A a µ ) for the electromagnetic, weak, strong interactions. Then the state of u is the extremum point of L(u) with the constraint of energy-momentum conservation, i.e.

(2.6)

d dλ L(u + λX) = M δL(X) • Xdx = 0, ∀ divX = 0.
Based on the PID, the field equations of the four fundamental interactions are derived as follows:

1). Gravitational field equations. For the Einstein-Hilbert functional (2.1), the equation (2.6) can be written in the form

(2.7) R µν - 1 2 g µν R = - 8πG c 4 T µν -∇ µ Φ ν
, where Φ ν is a 4-dimensional vector field, called the dual gravitational potential.

2). SU(N) gauge field equations. For the Lagrangian action (2.3) of SU(N) gauge field, the field equation (2.6) can be expressed as

G ab ∂ µ A b µν - g c λ b cd g αβ A c αν A d β -gΨγ ν τ a Ψ (2.8) = ∂ ν - 1 4 k 2 x ν + g c γA ν φ a , iγ µ D µ - mc Ψ = 0, (2.9)
where (Ψ, A a µ ) are as in (2.2), g is the interaction charge, A a µν and D µ are as in (2.4), k, γ are constants, A µ = ∂ a A a µ is the SU(N) tensor construction, representing the total interacting potential, G ab = 1 2 Tr(τ a τ + b ). 2.4. Principle of representation invariance (PRI). We note that in (2.5), the matrices Ω of gauge transformation are expressed as ) is an (N 2 -1) × (N 2 -1) complex matrix, then the gauge fields {A a µ } transform as (2.12)

A a µ = y a b A b µ , where (y a b ) = A -1
is the inverse matrix of A. In addition, the quantities λ a bc and G ab transform also as (N 2 -1)-dimensional tensors:

(2.13)

λ a bc = y a k x i b x j c λ k ij , G ab = x i a x j b G ij .
However, basic logic indicates that the gauge theory should be invariant under the transformation (2.11), i.e. the Lagrangian action of gauge field should be invariant. Consequently, we have the following principle of representation invariance (PRI):

Principle of Representation Invariance 2.5 (Ma & Wang [13]). For the SU(N) gauge theory, under the representation transformation (2.11), the Lagrangian action of gauge filed (2.2) is invariant, and the field equations are covariant.

As we have indicated, PRI is a basic logic requirement for an SU(N) gauge theory, and has profound physical implications. The quantities satisfying (2.12) and (2.13) are called the SU(N) tensors.

It is easy to see that the field equations (2.8)-(2.9) are covariant under the transformation (2.11), and (3) for A µ = (A 0 , A), the temporal component A 0 and the spatial components A = (A 1 , A 2 , A 3 ) represent respectively the interaction potential and the interaction magnetic potential. The force and the magnetic force generated by the interaction are given by

F = -g∇A 0 , F m = g c v × curl A,
where v is the velocity of the particle with charge g.

Field Particles and Radiations

3.1. Gravitational field particles and radiations. First we recall the PID gravitational field equations (2.7) for convenience:

(3.1) R µν - 1 2 g µν R = - 8πG c 4 T µν -∇ µ Φ ν .
We deduce then from (3.1) the following conclusions for the gravitational interaction, which will be explored in detail below:

(a) gravity possesses a field particle Φ ν , called the graviton, which is massless with no electric charge and spin J = 1; (b) there exists gravitational radiations, emitting gravitons at the speed of light.

1). Gravitational field particle. In the field equations (3.1), there are three groups of state functions, whose physical meaning are given as follows:

(1) {g µν } represents the gravitational potential, depicting the curved space-time; (2) {Φ µ } is the dual gravitational potential, representing the gravitational field particle and carrying the field energy; and (3) {T µν } is the energy-momentum tensor, representing the visible matter field. Physically, the dual gravitational potential {Φ µ } plays the role of the dark matter and dark energy. In fact, as a state function describing the gravitational field particle, (

3.2)

{Φ µ } represents the graviton.

In this sense, {Φ µ } is the dark matter, and the energy carried by {Φ µ } is the dark energy.

2). Gravitational radiation. By the Bianchi identity,

∇ µ (R µν - 1 2 g µν R) = 0.
Hence we infer from (3.1) that

(3.3) ∇ µ ∇ µ Φ ν = - 8πG c 4 ∇ µ T µν .
For the case where the gravity is weak, ∇ µ ∇ µ is approximated by the wave operator:

(3.4) ∇ µ ∇ µ 1 c 2 ∂ 2 ∂t 2 -∇ 2 . Consequently, (3.
2) is the graviton and (3.3) is the gravitational radiation equation, representing the field equation of gravitons.

3). Gravitational radiation wave. In vacuum, T µν = 0. Then equations (3.3) becomes

(3.5) 1 c 2 ∂ 2 ∂t 2 -∇ 2 Φ ν = 0,
which is the wave equation of gravitational radiation.

Remark 3.1. The gravitational radiation wave is different from the gravitational wave. The radiation wave is about the field particle {Φ µ } in (3.2), representing the propagating of gravitational field energy. The gravitational wave is about space metric {g µν }, representing the propagation of space deformation. In fact, because the gravitational wave is very weak, {g µν } can be written as

(3.6) g µν = δ µν + h µν ,
where δ µν is the Kronecker sign, and |h µν | 1. Then h µν satisfy the wave equation

(3.7) 1 c 2 ∂ 2 ∂t 2 -∇ 2 h µν = 0.
Therefore, (3.7) describes the wave of space-time metric (3.6), reflecting the propagation of space-time deformation.

3.2. Electromagnetic field particles and radiation. The symmetry dictating electromagnetic field particle is the U(1) gauge invariance. The particle described by ψ is (3.8) ψ particle in the U(1) gauge theory = electron.

By (2.8), the U(1) gauge field equation is written as

(3.9) ∂ µ A µν -eJ ν = ∂ ν φ,
where J ν = ψγ ν ψ is the 4-dimensional current density, and

A µν = ∂A ν ∂x µ - ∂A µ ∂x ν .
Based on (3.9), we can deduce the following conclusions:

1). Electromagnetic field particles. According to (3.9), the electromagnetic field particles are a pair of dual photons:

(3.10)

A µ = vector (spin-1) photon γ, φ = scalar (spin-0) photon γ 0 .

The weakton constituents of γ and γ 0 are

(3.11) γ = cos θw 1 w 1 -sin θw 2 w 2 ( , ), γ 0 = cos θw 1 w 1 -sin θw 2 w 2 (↑↓, ↓↑).
where w 1 , w 2 are the weaktons, and w 1 , w 2 are their antiparticles; see Ma and Wang [START_REF]Weakton model of elementary particles and decay mechanisms[END_REF]13].

2). Electromagnetic radiation. By the identity and the electric charge conservation:

∂ ν ∂ µ A µν = 0 and ∂ ν J ν = 0,
we derive from (3.9) the radiation equation for the scalar photon γ 0 :

(3.12) 1 c 2 ∂ 2 ∂t 2 -∇ 2 φ = 0.
Taking φ = 0 in (3.9), we get the radiation equation for the vector photon γ as follows

(3.13) ∂ ∂x µ ∂A ν ∂x µ - ∂A µ ∂x ν = eJ ν .
The equation (3.13) is the Maxwell equation. Equations (3.12) and (3.13) are also regarded as the field equations for the scalar photon γ 0 and the vector photon γ.

3.3.

Strong interaction field particles and radiations. Based on QCD, the symmetry dictating strong interaction field particles is the SU(3) gauge invariance. The particles described by ψ 1 , ψ 2 , ψ 3 are the quarks with three different colors: red, blue, and green:

(3.14) three particles ψ 1 , ψ 2 , ψ 3 of SU(3) = (q 1 , q 2 , q 3 ),

where q 1 = q red , q 2 = q blue , q 3 = q green . The gauge field equation (2.8) for SU(3) is written as

(3.15) ∂ µ A a µν - g s c λ a bc g αβ A b αν A c β -g s Q a ν = ∂ ν φ a
, where g s is the strong charge, and we have used the Gell-Mann matrices {τ a } as the generators of SU(3) so that

A a µν = ∂ µ A a ν -∂ ν A a µ + g s c λ a bc A b µ A c ν , Q a ν = Ψγ ν τ a Ψ, G ab = δ ab .
Here τ a = τ a , and Ψ = (ψ 1 , ψ 2 , ψ 3 ) satisfies the Dirac equation (2.9).

Based on (3.15), we have the following conclusions.

1). Strong interacting field particles. For SU(3) gauge theory of the strong interaction, the field particles are two groups of gluons, each of which consists of eight gluons with either spin J = 1 or spin J = 0:

(3.16) {A a µ } = spin-1 gluons {g a | 1 ≤ a ≤ 8}, {φ a } = spin-0 gluons {g 0 a | 1 ≤ a ≤ 8}.
The weakton constituents for g a and g 0 a are (3.17) g a = w * w * ( , ), g 0 a = w * w * (↑↓, ↓↑), where the index a corresponds to the different combinations of colors of w * and w * .

2). Gluon radiation. The equations (3.15) provide the model describing gluon radiations. As {g 0 a } and {g a } are independent to each other in strong interaction radiation, we deduce from (3.15) the field equations for {g 0 a } as:

(3.18) 1 c 2 ∂ 2 ∂t 2 -∇ 2 φ a = -g s ∂ ν Q a ν (1 ≤ a ≤ 8),
and the field equations for {g a } as:

(3.19) ∂ µ A a µν - g s c λ a bc g αβ A b αν A c β = g s Q a ν (1 ≤ a ≤ 8).
3.4. Weak interaction field particles and radiation. In the classical standard model, the weak interaction field particles are dictated by SU(2) gauge invariance, and are given as the three massive bosons:

(3.20) W ± , Z and Higgs H 0 .

However, in the weakton model by Ma and Wang [START_REF]Weakton model of elementary particles and decay mechanisms[END_REF]13], the massive bosons (3.20) are just the transition particles in decaying and scattering processes. Also, for an interaction radiation, the following basic rule should be satisfied: radiation particles must be massless and electric-neutral. Hence the W and Z particles (3.20) are not the weak interaction field particles.

Due to the limitation of physical experiments, the massless and electric neutral particles are hard to be directly detected. Therefore, the theory for the weak interaction radiation is completely open. The key point is that we don't know what the weak interaction field particles are. It is likely that the field particles are confined in the interiors of electrons and quarks.

We conjecture that the symmetry for the field particles and radiation of the weak interaction should be SU(2) gauge symmetry, and the corresponding field particles should be the ν-particles: ν 1 , ν 2 , ν 3 , whose weakton constituents are ν 1 = ν e νe , ν 2 = ν µ νµ and ν 3 = ν τ ντ . In this case, the field equations for the field particles and radiations are then given by

∂ µ A k µν - g w c ε k ij g αβ A i αν A j β = g w Q k ν , ∂ µ ∂ µ φ k w = -g w ∂ ν Q k ν , Q k ν = Ψγ ν τ k Ψ, for k = 1, 2, 3.

Interaction Potentials and Forces

4.1. Gravitational potentials. Gravity is different from the other three fundamental interactions, and the difference is reflected in four aspects as follows:

(1) the symmetry that gravity obeys is the principle of general relativity, rather than the gauge symmetry; (2) the field equations of the gravitational potential are the same as these of gravitational radiation; (3) the gravitational force is the curved effect of the space-time, and the electromagnetic, weak, strong interactions are the twisted effects of the underlying complex vector bundles M ⊗ p C N ; (4) the gravitational mass charge is continuous, and the electric, weak, strong charges are discrete.

Remark 4.1. We will see that the electromagnetic, weak and strong interaction potentials are layered. In fact, it is the difference (4) between gravity and the other three interactions that makes the gravitational potential non-layered.

Recall that for the Newtonian gravitational potential and force are given by ϕ = -M G r , (4.1)

F = -m∇ϕ = - mM G r 2 . (4.2)
In the general theory of relativity, the gravitational potentials are (4.3) {g µν } and the dual potential {Φ µ }.

In the Newtonian law, there is a single potential function ϕ in (4.1), which yields the gravitational force F in (4.2). However, in the general theory of relativity, there are many components in the gravitational potentials in (4.3). A natural question is that which component of (4.3) represents the gravitational force F , and how the potentials {g µν } and {Φ µ } are related? We address this question as follows.

1). Gravitational potentials in spherically symmetric fields.

Let M be the total mass of a centrally symmetric ball. The gravitational potentials of (4.3) are the functions of r, i.e. g µν = g µν (r), Φ µ = Φ µ (r). In this case, the time-component g 00 of g µν and the Newton potential ϕ have the following relation

g 00 = -1 + 2 c 2 ϕ .
Also the gravitational force F exerted on an object of mass m is written as

(4.4) F = -m∇ϕ = - c 2 m 2
dg 00 dr .

In addition, mathematically the Riemann metric is (4.5) ds 2 = g 00 c 2 dt 2 + g 11 dr 2 + g 22 dθ 2 + g 33 dϕ 2 , where (r, θ, ϕ) is the spherical coordinate, and g 00 = -e u(r) , g 11 = e v(r) , g 22 = r 2 , g 33 = r 2 sin θ, and the dual potential Φ µ is given by

Φ µ = ∇ µ φ(r).
Thus, the gravitational potential and dual potential become Let r 0 be the radius of the ball with mass M . Then, for r > r 0 the energy-momentum tensors is zero, i.e.

T µν = 0. Thus, the gravitational field equations (3.1) are the ordinary differential equations, written as

(4.8) v + 1 r (e v -1) = - r 2 u φ , u - 1 r (e v -1) = r(φ - 1 2 v φ ), u + ( 1 2 u + 1 r )(u -v ) = - 2 r φ ,
where u, v, φ are as in (4.6), and the prime stands for derivative in r.

By solving u, v, φ from (4.8), we deduce the Riemannian metric (4.5) and the gravitational force (4.7) in r > r 0 .

2). Gravitational potential in astrophysical fluids. In astronomy, many physical motions are governed by the astrophysical fluid dynamics equations, which are expressed in the following general form; see Ma and Wang [START_REF]Astrophysical dynamics and cosmology[END_REF]13]

∂u i ∂t + u k ∂u i ∂x k + Γ i kj u k u j = ν[div(∇u i ) + g ij R jk u k ] (4.9) - 1 ρ g ij ∂p ∂x j + mc 2 2
g ij ∂g 00 ∂x j , ∂u k ∂x k + Γ k kj u j = 0, (4.10) where g = g 00 0 0 g ij is the gravitational potential, (g ij ) represents the space metric and g 00 is the time component; these potentials satisfy the gravitational field equations

(4.11) R µν - 1 2 g µν R = - 8πG c 4 T µν -∇ µν φ, where T µν = (c 2 ρ + p)u µ u ν + pg µν , R µν is the Ricci curvature tensor, R = g µν R µν is the scalar curvature, and Γ α µν = 1 2 g αβ ∂g µβ ∂x ν + ∂g νβ ∂x µ - ∂g µν ∂x β , div(∆u i ) = g kl ∂ ∂x l ∂u i ∂x k + Γ i kj u j + Γ i lj ∂u j ∂x k + Γ j ks u s -Γ j kl ∂u i ∂x j + Γ i js u s .
We see from the model (4.9)-(4.11) that the fluid dynamical equations (4.9) and (4.10) are coupled with the gravitational field equations (4.11), to form astrophysical fluid dynamics equations. In this manner, the gravitational potentials {g µν } play an important role in astrophysical fluid dynamics. 4.2. Gravitational forces of central matter fields. For a central matter field with radius r 0 , the gravitational force in r > r 0 is given by (4.12) r) ,

F = - c 2 m 2 d dr e u(
where g 00 = e -u is the gravitational potential generated by the central matter field with mass M , and m is the mass of a body at distance r, and u = u(r) is determined by (4.8).

Based on (4.8) and (4.12), for a central gravitational field we can deduce the following conclusions; also see Ma and Wang [START_REF]Gravitational field equations and theory of dark matter and dark energy[END_REF]13].

(1) The gravitational force F given by (4.12) is asymptotically zero at large distance:

F (r) → 0 as r → ∞.
(2) There exists a sufficiently large r 1 such that the gravitational force F is repulsive for r > r 1 , i.e.

F (r) > 0 if r > r 1 .

(3) The force F has an approximate expression:

F (r) = mM G - 1 r 2 - k 0 r + k 1 r for r 0 < r < r 1 ,
where k 0 and k 1 are given by k 0 = 4 × 10 -18 km -1 , k 1 = 10 -57 km -3 .

These conclusions (1)-( 3) provide an explanation of the dark matter and dark energy phenomena. In other words, dark matter and dark energy are intrinsic properties of gravity, through the nonlinear interactions of the gravitational potentials {g µν } and {Φ ν } through the gravitational field equations (3.1). 4.3. Electromagnetic, weak and strong potentials. The electromagnetic, weak and strong interaction potentials are dictated by the SU(N) gauge invariance, where N is the number of particles in the interacting multi-particle system with the same interaction charge. If we take the standard representation basis {τ

a | 1 ≤ a ≤ N 2 -1} of SU(N) such that 1 2
Tr(τ a τ + b ) = δ ab , then the SU(N) gauge field equations (2.8) and (2.9) for the N-particle system are in the form Taking the contraction (2.14) we obtain the total interaction potentials (4.15)

∂ µ A a µν - g c λ a bc g αβ A b αν A c β -gΨγ ν τ a Ψ (4.13) = ∂ ν - 1 4 k 2 x ν + g c γA ν φ a , iγ µ D µ - mc Ψ = 0, ( 4
A µ = α a A a µ = (A 0 , A) and φ = α a φ a . The system (4.13) contains 4(N 2 -1) equations, and has 5(N 2 -1) unknown functions A a µ , φ a , 1 ≤ a ≤ N 2 -1, µ = 0, 1, 2, 3. Therefore we need to supply N 2 -1 equations for (4.13), which are called the gauge fixing equations.

In the following, we establish the field equations for the dual potentials A 0 and φ in (4.15) of the associated interaction from the fields (4.13)- (4.14).

Taking inner product of (4.13) with α a , we get where κ bc = α a λ a bc , and Q ν = α a Ψγ ν τ a Ψ,

A µν = ∂ µ A ν -∂ ν A µ + g s c κ bc A b µ A c ν .
As A µν = -A νµ , we have

∂ ν ∂ µ A µν = 0.
Then taking divergence on both side of (4.16), we get the field equation of φ as follows (4.17)

1 c 2 ∂ 2 ∂t 2 -∇ 2 φ + k 2 φ -g∂ ν Q ν = g c ∂ ν [κ ab g αβ A a αν A b β + γA ν φ].
For the weak interaction, γ = 0. By the short-range property of the weak interaction, take the following transformation in (4.16)

φ → φ + φ 0 , φ 0 = constant.

Then (4.16) become Based on the superposition property of the interaction charge forces, the dual potentials A 0 and φ should satisfy linear equations, i.e. the equation (4.17) and the time-component ν = 0 equation of (4.18) are linear. Therefore, the gauge fixing equations for the interactional potentials must contain the following three equations

∂ µ (∂ µ A ν -∂ ν A µ ) -k 2 1 A ν + 1 4 k 2 x ν φ -gQ ν (4.
∂ ν (κ ab g αβ A a αν A b β + γA ν φ) = 0, ∂ ν + gγ c A ν φ + 1 4 k 2 a 0 x ν - g c κ ab (∂ µ A a µ A b ν -g αβ A a αν A b β ) = 0, ∂ ∂t (∂ µ A µ ) = 0.
In addition, under the static conditions ∂A 0 ∂t = 0, ∂φ ∂t = 0, the equation (4.17

-∆φ + k 2 φ = g∂ µ Q µ , -∆A 0 + k 2 1 A 0 = -gQ 0 - 1 4 k 2 cτ φ,
where cτ is the interaction wave length of φ, k 1 is as in (4.19). The equation of (4.20) determine the potential A 0 in the form

A 0 = qe -k 1 t 1 r - A ρ (1 -kr)e -(k-k 1 )t , (4.21)
where A is a parameter, q is the effective charge written as q = N ρ 0 ρ 3 g, (4.22) and ρ 0 is the particle radius and ρ is the radius of the particle system. Then the interaction forces read as

  where g is interactional charge, A a µ as in (2.2), λ a bc are the structure constants of SU(N), and {τ a | 1 ≤ a ≤ N 2 -1} are the representation generators of SU(N), G ab = 1 2 T r(τ a τ + b ). The expression of L SU (N ) in (2.3) is essentially determined by the principle of Lorentz invariance and the principle of gauge invariance, which are stated as follows.Principle of Lorentz Invariance 2.2. Physical laws are invariant under the Lorentz transformation, i.e. the Lagrangian action of the electromagnetic, weak and strong interactions is invariant under the Lorentz transformation.

( 2 .

 2 [START_REF] Ma | Unified field theory and principle of representation invariance[END_REF]) Ω = e iθ a τa ∈ SU (N ), where τ a (1 ≤ a ≤ N 2 -1) are the representation generators of SU(N), which constitute a coordinate basis of the tangent space of SU(N) at the identity matrix I. In fact, if we choose another basis { τ a }, i.e. take a transformation(2.11) τ a = x b a τ b , where A = (x b a

(2. 14 )

 14 A µ = α b A b µ is invariant, where {α b } and {A b µ } are SU(N) tensors. Furthermore, PRI leads to the following physical conclusions: (1) the coupling constant g in (2.4) represents the interaction charge, playing the same role as the electric charge e in the U(1) gauge theory for quantum electrodynamics; (2) the potential A µ = α b A b µ represents the total interacting potential, and each α b represents the portion of the component A a µ contributed to the total interaction potential; and

(4. 6 )

 6 u = u(r), v = v(r) and φ = φ(r),

  .14) where Ψ = (ψ 1 , • • • , ψ N ) T represents the wave functions of the N particles, k -1 is the interaction range, and γ = 0 for the weak interaction, = 0 for the electromagnetic and strong interactions.Also the SU(N) gauge potentials {A a µ | 1 ≤ a ≤ N 2 -1} represent the interaction potentials between the N particles; seeMa and Wang [13].

Remark 4 . 2 .

 42 The parameter k in (4.13) represents the attracting range of the electromagnetic, weak and strong interactions, and γ represents the weak interaction range.

( 4 .

 4 [START_REF] Nambu | Quasi-particles and gauge invariance in the theory of superconductivity[END_REF])∂ µ A µν -g c κ bc g αβ A b αν A c β -gQ ν = ∂ ν -1 4 k 2 x ν + g c γA ν φ,

  ) and the time-component ν = 0 equation (4.18) become(4.20)

  electric charge g = e, weak force : f = -g w d dr A 0 for weak charge g = g w , strong force : f = -g s d dr A 0 for strong charge g = g s .

The work was supported in part by the US National Science Foundation (NSF), the Office of Naval Research (ONR) and by the Chinese National Science Foundation.