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*by Roberto Lalli***Chen Ning Yang****Nobel Prize in Physics 1957 together with Tsung-Dao Lee****"for their penetrating investigation of the so-called parity laws which has led to important discoveries regarding the elementary particles".**

Chen Ning Yang, along with his collaborator Tsung-Dao (T. D.) Lee, was the first Nobel Prize laureate of Han Chinese ethnicity. Yang made fundamental contributions to several branches of theoretical physics, notably in elementary particle physics and statistical mechanics. The work he made with Lee on the breaking of the parity (P) invariance in weak interactions, which gained them the 1957 Nobel Prize in Physics, was just one of his fundamental contributions to the development of theoretical physics. Among Yang’s major achievements, the formulation of the non-Abelian Yang-Mills gauge theory stands out. This laid the foundations for the development of the Standard Model – the set of quantum field theories that make accurate predictions concerning all the fundamental interactions but gravity. Yang’s ability to explore topics and produce analyses in advance of his contemporaries made him one of the most influential theoretical physicists of the second half of the 20th century. His uncommon skill was, in turn, related to his deep knowledge of group theory in a period when the majority of physicists were not relying on group theoretical methods to resolve physical problems.

**From China to the United States**

Yang was born on the 1st of October 1922 in Hefei, the capital of the Anhui Province in Eastern China. His father, Ko Chuen Yang, had been appointed Professor of mathematics at the prestigious Tsing Hua University in Beijing after having earned his PhD in Chicago under the famous American algebraist Leonard Dickson. Thanks to his father, Yang became acquainted with representation theory during high school, and he has been developing a special interest toward group theoretical views and symmetry principles ever since.

His university studies just increased this precocious appreciation of group theory and its applications.

Between 1938 and 1942, he attended the eminent Southwest Associated University in Kunming (Yunnan Province of China), where he obtained his bachelor’s degree under the guidance of theoretical physicist Ta-You Wu with a thesis on the application of group theory to molecular spectra. Later, he studied at the Tsing Hua University, temporarily at Kunming because of the Second Sino-Japanese of 1937-45. There, in 1944, he completed his Master’s thesis on statistical mechanics under the supervision of J. S. Wang. Soon after his graduation, Yang won a fellowship that allowed him to pursue his PhD in the United States. His desire to study with Enrico Fermi led Yang to enrol in 1946 at the University of Chicago where Fermi had become Distinguished Professor of Physics in July 1945.

During the Second World War, Fermi had been one of the leading figures in the war-related American nuclear researches. Shortly after the war, the military restrictions on these kinds of activities did not allow a foreign student, such as Yang, to work at his original experimental project with Fermi who was conducting classified research at the Argonne National Laboratories. Yang then changed his research topic and wrote a dissertation in theoretical physics under the supervision of the Hungarian-born nuclear physicist Edward Teller, receiving his PhD in 1948. In retrospect, all these events can be considered essential in the development of Yang’s style, which combined powerful physical intuition with deep understanding of group theory, both of which played a strong role in the achievements he was soon to accomplish.

**The Context: New Particles, New Problems **

Yang began to be interested in the physics of particle interactions in a period of great theoretical and experimental transformations. Thanks to fundamental improvements in experimental apparatus, since 1947 cosmic-ray observations and (later) particle accelerators disclosed the existence of several new particles that did not fit the theoretical framework then used to interpret fundamental forces, essentially based on the Yukawa meson theory for nucleon interactions and on the Fermi model for beta-radioactivity. Experimental physicists discovered that the number of particles were much greater than they had supposed. The mass of these newly discovered particles varied from about 500 MeV to about 1300 MeV. The peculiar behaviours of these particles led physicists to call them “strange particles” and soon various attempts were made to classify them. The most successful was proposed in 1953: The particles whose mass was intermediate between the π-meson and the proton were called *K*

-mesons (or kaons), while the particles whose mass exceeded the neutron’s were called hyperons. This classification did not help, however, to clear the confusion about how to interpret these particles and their experimental features (mass, charge, life-time, decay properties, etc.). In this chaotic period, Yang played a fundamental and ground-breaking role in some crucial aspects that later led physicists to formulate the satisfactory gauge field theories of quantum chromodynamics and the electroweak interaction, which constitute the Standard Model of particle physics.

**The Precursor of the Standard Model: The Yang-Mills Theory (1954)**

Ever since Yang was a graduate student, he had been impressed by the fact that charge conservation was derived by the gauge invariance of electromagnetic theory; namely, the invariance of the electromagnetic field under variations of the underlying potentials. In 1947-48 various researchers - including Sin-Itiro Tomonaga, Julian Schwinger, Richard Feynman (who shared the 1965 Nobel Prize in Physics “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles”) and Freeman Dyson (whose exclusion from the 1965 Nobel Prize sparked some controversies, but the prize can only be shared by a maximum of three) - were successful in formulating a field theory of quantum electrodynamics (QED) that was perturbatively renormalizable.

Renormalized QED solved the problems of infinities affecting the quantum field theory of electromagnetic interactions by subtracting the divergent terms and, then, allowed finite calculation of the measurable quantities. The theory showed great precision in predicting effects depending on high orders in the fine-structure constant. Following this success, many hoped that all the fundamental interactions might be understood as renormalizable quantum field theories. At the end of the 1940s, however, this confidence dramatically waned. The other two quantum field theories then employed (Fermi theory of weak interactions and Yukawa theory of strong interactions) were not renormalizable. Put simply, this meant that infinities could not be eliminated by subtraction and there was no way to use perturbation theory to make reliable predictions.

The work Yang made with the American doctoral student Robert Mills in 1954 was the necessary step to extend the quantum field theory approach to other forces in the following decades. While in Chicago, Yang had already tried to apply this procedure to the isotopic spin interactions governing the strong nuclear force, but had interrupted his research without having found any solution. In 1949, Yang moved to the Institute of Advanced Study in Princeton, then directed by J. Robert Oppenheimer, where he remained for 17 years, apart from the period 1953-54 when he worked at the Brookhaven National Laboratory. There, he resumed the gauge theory problems with his officemate R. Mills. By using the analogy between the conservation of charge in electromagnetism and the conservation of isospin in the strong nuclear force, Yang and Mills extended the gauge invariance principle of QED to include nuclear interactions, and hypothesized the existence of a field underlying these interactions, which they called B field. There was a great difference, however. The gauge transformations of the electromagnetic potentials form an Abelian group; namely, the transformations are commutative. This does not hold for the non-linear equations derived by Yang and Mills. This meant that they generalized gauge-invariance to the non-Abelian case, in which the equations were much more complicated than in the electromagnetic case. The Yang-Mills theory raised mixed reactions, but the majority expressed worries about the mass of the quanta of this new field. It seemed that the B field quanta were necessarily massless. That comported several problems, because they should have been already detected if that was the case. It took several years to resolve this issue. From 1961 to 1979 several theoreticians employed the Yang-Mills theory of non-Abelian fields as the basis of quantum field theories of strong and weak interactions. Since then, physicists have been referring to the Yang-Mills theory as one of the greatest intellectual achievements of contemporary physics.

**The Breaking of Discrete Symmetries in Weak Interactions: ****The Yang-Lee Collaboration**

The collaboration between Yang and Lee was perhaps one of the most fruitful in theoretical physics. It lasted several years and reached numerous achievements, including the work on parity violation awarded with the Nobel Price in 1957. Yang had got acquainted with Lee in Chicago, where they both were pursing their PhD.

Their first collaboration dates back to 1948. Along with M. Rosenbluth, they made a phenomenological analysis of muon (*µ*)-decay and *µ*-capture. Their analysis, along with the contemporaneous investigations of other physicists, proved that these interactions had the same strength as *β *-decay, thus leading to the understanding that weak interactions were universal and, consequently, that there were four different fundamental forces of nature.

In the same period, Yang made several researches concerning the mesons. The most original was maybe contained in a paper he wrote with Fermi in 1949. To explain the recently discovered different kinds of mesons, they proposed that the *π*-meson might be a composite particle made up of a nucleon and an anti-nucleon. The Fermi-Yang theory was the first one to propose that what were considered elementary particles were not elementary at all. Physicists did not show much enthusiasm for this model, but later it served as a basis for the development the successful quark model of strong interactions.

Despite the relevance of his previous work, his contemporaries considered Yang’s most important achievement to be the work he did with Lee in the 1950s on the non-conservation of parity and other discrete symmetry laws. The central endeavour was built on the bold proposal that parity might not be conserved in weak interactions.

Until 1956 physicists were sure that right-left symmetry - mirror symmetry - was a fundamental symmetry of nature. Very few doubted that natural phenomena should look the same if they were seen from opposite spatial perspectives, that physical laws were symmetrical with respect to spatial reflections. Parity was a name introduced in 1937 by the Physics Nobel Laureate Eugene Wigner to treat complete space inversion in theoretical group terminology. In the quantum theoretical formalism, the assumption that physical laws are invariant under space inversion implies that quantum states have a definite parity: either they are symmetric under parity transformations (even parity) or they invert their sign (odd parity). Furthermore, one could demonstrate that P had to be conserved during interactions if physical laws were invariant under space inversion.

Starting in the spring of 1956, Yang and Lee suggested that this might not be the case for weak interactions and proposed concrete experiments to test the breaking of this symmetry. Although their proposal solved one of the most disturbing problems of particle observations, the so-called *τ*-*θ **puzzle*, not many were disposed to believe that “God was a weak left-hander,” as Pauli wrote after reading the Lee-Yang analysis. However, the Chinese-born American physicist Chen-Shiung Wu soon decided to perform one of the experiments proposed by Yang and Lee, with the collaboration of a group of experts of the National Bureau of Standards. The result was astonishing. It turned out that weak interactions are not invariant with respect to parity transformations. In a few weeks, other experiments confirmed the result of Wu’s group for different kinds of weak interactions. The revolutionary discovery was considered so important that Lee and Yang were awarded the Nobel Prize in Physics “for their penetrating investigation of the so-called parity laws which has led to important discoveries regarding the elementary particles” the very same year of the experimental confirmation of parity non-conservation. The recognition of a scientific achievement with a Nobel Prize in the immediate aftermath of its proposal is a very rare case in theoretical physics. This exceptional event gives a hint of the disruptive importance that the breaking of P symmetry had in those days.

This was, however, only a part of the achievements Lee and Yang reached in the same period. The non-conservation of parity led some physicists to wonder what happened with the other two discrete symmetry operators: charge-conjugation C – the transformations indicating the substitution of particles with their anti-particles - and time reversal T – the reflections with respect to the sign of the time t. Before the Lee-Yang analysis on P non-conservation, the invariance of C and T were not brought into question.

After reading Lee and Yang’s bold proposal, Reinhard Oheme sent them a letter asking about the relationships between the invariance of P, and those of C and T. This letter prompted the two physicists to re-analyse the issue and, later, to write a paper with Oehme putting forward the CPT theorem: “if one of the three operators P, C, and T, is not conserved, at least one other must also be not conserved.” Their analysis showed that there was no proof that T and C were conserved in weak interactions and that Wu’s experiment (whose result was still unpublished) demonstrated that charge-conjugation symmetry was also violated in weak interactions. Following the proposals put forward by Yang and Lee, and, independently, by other physicists, the independent P and C symmetries were substituted with a weaker symmetry principle: the CP symmetry, called combined inversion (the simultaneous transformations of mirror inversion and charge conjugation). If P is inverted during a weak interaction, then also C is. There was much confidence about the CP symmetry principle till 1964, when James Cronin and Val Fitch showed that some weak interactions violated CP symmetry.

The collaboration between Lee and Yang lasted several years. Until 1961, they put forward fundamental analyses, including the study of symmetry principles in hypothetical high-energy neutrinos.

**Other Works**

Yang continued to be interested in particle physics and symmetry principles in the following years. After the observation of CP non-conservation in weak interactions, Yang, along with T. T. Wu, made a phenomenological analysis of the K-decay that had led to the discovery of CP symmetry violation. By extending the previous Lee-Yang analysis on the CPT theorem, Yang and Wu isolated the parameters of experimental interest and defined the equations concerning them. The concepts and parameters Yang and Wu introduced served as the basis for further analysis of the CP violation in weak interactions.

Following Fermi’s advice, Yang did not focus on a single topic, but tried to solve several problems. This approach led him to make momentous contributions to other branches of theoretical physics, more numerous than it is possible to quote here. The fields in which he significantly contributed were statistical mechanics and condensed matter physics. To mention just a few of his most important contributions it is worth recalling the works concerning spontaneous magnetization, the state transition in the quantum lattice model, the introduction of the off-diagonal long-range order by analysing the concept of Bose condensation, and the development of Bethe’s hypothesis, which, in turn, led to the formulation of the Yang-Baxter equation in 1967. Since the early 1980s, the Yang-Baxter equation has proved to be of relevance in various branches of physics and mathematics, and it is considered by some to be one of the most fundamental mathematical equations.

**A Bridge Between China and the United States **

In 1964, just in the middle of his productive career, Yang became a naturalized American citizen. The Second Sino-Japanese War was followed by the 4-year Chinese revolution that ended with the establishment of the People’s Republic of China in 1949. The tense political relationships between United States and communist China during the Cold War did not make it easy for Yang to maintain contact with his motherland. Indeed, after he had left China for the Unites States, Yang never went to China until 1971. That year, thanks to the thaw in political relations between the two countries he was the first well-known Chinese-American scholar to visit China. From then onward, he has done a lot for establishing diplomatic relationships between United States and China and for strengthening Chinese physics. After he retired from the State University of New York at Stony Brook, where he had directed what is now called the Yang Institute for Theoretical Physics, he became the honorary director of the Tsing Hua University in Beijing - the very same university where he graduated 45 years before. Yang now resides in China, where he obtained permanent residence in 2005.

**Bibliography **

Brown, L. M., Dresden, M., & Hoddeson, L. (Eds.) (2009) Pions to Quarks: Particle Physics in the 1950s. Cambridge University Press, Cambridge.

Hoddeson, L. Brown, L. M., Riordan, M. & Dresden, M., (1995) The rise of the Standard Model: Particle Physics in the 1960s and the 1970s. Cambridge University Press, Cambridge.

Li B. A., and Deng Y. (1992) Chen Ning Yang. Available at http://insti.physics.sunysb.edu/~yang/yangbio.pdf (accessed 31 October 2013)

Pais, A. (1986) Inward Bound Of Matter And Forces In The Physical World. Clarendon Press, Oxford.

Yang, C. N. (2005) Selected papers (1945-1980), with commentary. Freeman & Company, San Francisco.

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