Abstract
The focus of this broad historical overview of “the steady evolution of theoretical ideas” from Shelter Island I in 1947 to Shelter Island II in 1983 is some of the developments in “fundamental” physics after the establishment of the standard model, in particular, the adoption of the view that all present day field theories are “effective field theories” based on the gauge concept; taking seriously big bang cosmology, grand unified field theories (GUTs), and inflation; and the emergence of a new symbiosis of physics and mathematics.
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Notes
It consisted of a complex scalar field ϕ with the Lagrangian density
$$\begin{aligned} L & = \partial_{\mu } \phi^{*} \partial^{\mu } \phi - V(\phi ) \\ V(\phi ) & = m^{2} \phi^{*} \phi + \frac{1}{2}\lambda (\phi^{*} \phi )^{2}, \\ \end{aligned}$$where m and λ are the mass and self-interaction coupling constant of the scalar field. The model is invariant under a global change of phase ϕ(x)→ e iα ϕ(x), transformations which define the abelian symmetry group U(1).
Expressed in term of Weyl spinors, the charges obey an anticommutation algebra \(\{ Q_{\alpha }^{i} ,Q_{\beta }^{j} \} = 2(\sigma_{\mu } )_{\alpha \beta } \delta_{ij} P^{\mu } .\) In writing the anti-commutation rules an additional index i= 1,2,…,N has been inserted to label the possible different supersymmetry charges. Since renormalizable gauge theories are restricted to spin ≤ 1 at most four changes of spin 1/2 by supersymmetry, charges Q i α are allowed \(S = + \,1\,{\kern 1pt} \,\mathop \to \limits_{Q} \,\, + \frac{1}{2}\,\,\mathop \to \limits_{Q} \,\,0\,\,\mathop {\mathop \to \limits_{Q} \,}\limits_{{}} \,{\kern 1pt} \mathop { - \frac{1}{2}}\limits_{{}} \,\mathop {\, \to }\limits_{Q} \,{\kern 1pt} \, - 1\), hence N ≤ 4 for gauge theories, and N ≤ 8 for supergravity theories (the graviton has S=2).
The topological charge is given by \(k = - (\frac{1}{{8\pi^{2} }})\int {d^{4} } x\,{\text{Tr}}{\kern 1pt} \, (\varepsilon^{\mu \nu \alpha \beta } \,F_{{^{\mu \nu } \,}} F_{{^{\alpha \beta } \,}} )\).
References
Steven Weinberg, “Conceptual Foundations of the Unified Theory of Weak and Electromagnetic Interactions,” Reviews of Modern Physics 52 (1980), 515–23.
My use of “collective” resonates with the definition given by Fleck of a “thought collective” as “a community of persons mutually exchanging ideas or maintaining intellectual interactions, [in which] we will find by implication that it also provides the special ‘carrier’ for the historical development of any field of thought, as well as for the given stock of knowledge and level of culture. This we have designated thought style.” Ludwik Fleck, Genesis and Development of a Scientific Fact, ed. Thaddeus J. Trenn and Robert K. Merton, trans. Fred Bradley and Thaddeus J. Trenn (Chicago: The University of Chicago Press, 1979), 39. I would translate “thought style” as style of reasoning as expounded in Silvan S. Schweber, “Hacking the Quantum Revolution,” European Journal of Physics H 40 (2015), 53–149. See also the informative and perceptive Introduction by Ilana Löwy, Nathalie Jas, and Johannes Fehr, “De l’originalité et de la richesse de la pensée de Ludwik Fleck,” in Penser aver Fleck: Investigating a Life Studying Life Sciences, ed. Johaness Fehr, Nathalie Jas, and Ilana Löwy (Zurich: Collegium Helveticum, 2009).
By the late 1960s and early 1970s, as estimated by attendance at the Rochester conferences, the size of the theoretical high energy physics community worldwide was over 350.
Chen Ning Yang, “Symmetry and Physics,” in The Oskar Klein Memorial Lectures, vol. 1, Lectures by C. N. Yang and S. Weinberg, ed. G. Ekspong (Stockholm: Stockholm University, 1991), 27; Chen Ning Yang, “My Experience as Student and Researcher,” International Journal of Modern Physics A 27(9) (2012), 1230009-1–19, on 13.
There is by now a huge literature on the standard model at the level of Scientific American articles. For an accessible introduction to the subject together with an insightful historical overview see Gerard ’t Hooft, “Gauge Theory of the Forces between Elementary Particles,” Scientific American 242 (June 1980), 104–36; Under the Spell of the Gauge Principle (Singapore: World Scientific, 1994); In Search of the Ultimate Building Blocks (Cambridge: Cambridge University Press, 1996).
See for example Robert P. Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in Twentieth-Century Physics (New York: Macmillan, 1986); Tony Hey and Patrick Walters, The Quantum Universe (Cambridge, UK: Cambridge University Press, 1987).
In the field theoretical description, the left and right handed fields describing the leptons and the quarks belong to different representations of SU(2) × U(1).
In the simplest model, the Higgs scalar field is assumed to be a doublet of weak interaction SU(2), so that it can generate the masses of the W and Z bosons.
To be more precise, to the charged leptons. In the early 1980s, it was believed that the neutrinos were massless.
The photon, the W ±, and the Z only have flavor attributes.
See for example Gerard ’t Hooft et al., eds., Recent Developments in Gauge Theories (New York: Plenum Press, 1980).
See Martinus J. G. Veltman, “The Higgs Boson,” Scientific American 255 (November 1986), 88–94.
For the details of the count see for example John Ellis, “Unification and Symmetry,” in The Constants of Physics: Proceedings of a Royal Society Discussion Meeting held on 25 and 26 May 1983, ed. William H. McCrea and Martin J. Rees (London: The Royal Society, 1983) and Chris Quigg, Gauge Theory of the Strong, Weak and Electromagnetic Interactions (Reading MA: Benjamin Cummings Publishing Inc.,1983).
See Christopher H. Llewellyn Smith, “The Strong, Electromagnetic, Weak Couplings,” in McCrea and Rees, Constants (ref. 13), 43–48; Paul Langacker, The Standard Model and Beyond (Boca Raton, FL: Taylor and Francis, 2010).
See Edward Witten, “Reflections on the Fate of Spacetime,” Physics Today 49(4) (1996), 24–28.
The characterization of QFT as an adequate mathematical language is due to Israel M. Gelfand, “Mathematics as an Adequate Language,” in The Unity of Mathematics: In Honor of the Ninetieth Birthday of I. M. Gelfand, Progress in Mathematics, vol. 244, ed. Pavel Etingof, Vladimir Retakh, and Isadore M. Singer (Boston: Birkhäuser 2006), x–16.
See for example Polyakov, who writes: “String theory is a beautiful and dangerous subject. On one hand it is a top achievement of theoretical physics exploiting the most advanced and daring methods. On the other—without a guidance from the experiment it can easily degenerate into a collection of baroque curiosities, some kind of modern alchemy looking for philosopher’s stone.” Alexander Polyakov, “String Theory as a Universal Language,” ArXiv, http://arxiv.org/abs/hep-th/0006132, June 2000. Or Ellis and Silk who write: “In our view, the issue boils down to clarifying one question: what potential observational or experimental evidence is there that would persuade you that [string] theory is wrong and lead you to abandoning it? If there is none, it is not a scientific theory.” George Ellis and Joe Silk, “Scientific Method: Defend the Integrity of Physics,” Nature 516 (2014), 321–23.
For example, Schwinger, Feynman, Tomonaga, Bethe, Weisskopf, Landau, Bogoliubov, Pauli, Marshak, Pais, Dyson, Salam, Ward, Källen, Dalitz, Goldberger, Treiman, Gell-Mann, Fubini, Chew, Low, T. D. Lee, C. N. Yang, T. T. Wu, Michel, Takahashi, Cabbibo, Amati, S. Weinberg, Feinberg, Sudarshan, Mills, Schnitzer, Grisaru, Deser, Ne’eman, Zweig, Sakharov, Zel’dovich, Drell, Bjorken, B. Lee, Mandelstam, Wolfenstein, Bludman, Ruderman, K. Wilson, Sakata, Fubini, Wess, Zumino, Nambu, Nishijima, Goldstone, Skyrme, Pomeranchuk, Coleman, Gottfried, Glashow, Veltman, Higgs, Englert, Brout, Kibble, Brezin, Zinn-Justin, Itzikson, Gaillard, Gervais, Neveu, Penrose, Hawking, Dashen, Georgi, Adler, Weisberger, Callan, Faddeev, Gribov, Popov, Kirzhnits, Kadanoff, Veltman, ’t Hooft, Jackiw, Bell, Adler, Ramond, B. Lee, Veneziano, Virasoro, Nielsen, M. Green, Dimopoulos, Susskind, Quinn, E. Weinberg, Dolan, Politzer, Gross, Polyakov, Preskill, Rebbi, Wilczek, Abbott, Zee, Scherk, Schwarz, and Witten. Many more names could easily be added to this list of theorists, including “mathematical” physicists such as Jost, Haag, Lehmann, Symanzik, Wightman, Zimmerman, Ruelle, Khuri, Jaffe, Froissart, Martin, Hepp, Stora, and Fröhlich; condensed matter theorists such as Anderson, P. Martin, Pines, Thouless, Schrieffer, Langer, and Parisi; or nuclear theorists such as Villars, Baranger, Kerman, G. E. Brown, Negele, Scalapino, Walecka, and Wilets, who also contributed importantly to the development of the field.
See David Kaiser, “Epilogue: Textbooks and the Emergence of a Conceptual Trajectory,” in Research and Pedagogy: A History of Quantum Physics through its Textbooks, ed. Massimiliano Badino and Jaume Navarro (Berlin: Max Planck Institute for the History of Science, 2013), 285–89; “Booms, Busts, and the World of Ideas: Enrollment Pressures and the Challenge of Specialization,” Osiris 27 (2012), 276–302.
I have only mentioned those that were initiated from the late 1950s until the 1970s and operated on a regular basis. “Summer” schools have proliferated since then, and Cargèse and les Houches hold one to three weeks long “schools” on a variety of subjects almost year round.
Borrelli’s article illustrates this in the case of the 2013 award of the Nobel prize to Peter Higgs and François Englert “for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN’s Large Hadron Collider.” Arianna Borrelli, “The Story of the Higgs Boson: The Origin of Mass in Early Particle Physics,” The European Physical Journal H 40 (2015), 1–52.
Except for Germany. See Christa Jungnickel and Russell McCormmach, Intellectual Mastery of Nature (Chicago: The University of Chicago Press, 1986).
Furthermore, there was no sharp difference between mathematical physicists and theoretical physicists. Thus, besides being outstanding mathematicians, Poincaré and Hilbert could be considered mathematical or theoretical physicists. See Jeremy Gray, “Mathematics and Natural Science in the Nineteenth Century: The Classical Approaches of Poincaré, Volterra and Hadamard,” in Changing Images in Mathematics: From the French Revolution to the New Millenium, ed. Umberto Bottazzini and Amy Dahan Dalmedico (London: Routledge, 2001), 113–36. For an insight into the Göttingen mathematics Denkstil and Hilbert’s metaphysics, see Skuli Sigurdsson’s review of David Hilbert: Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Gottingen, Lecture manuscript prepared by Paul Bernays, edited by David E. Rowe (Basel: Birkhauser Verlag, 1992) in Isis 84 (1993), 600–602. Furthermore, Skuli Sigurdsson pointed out to me that in the early 1930s Hilbert was nominated several times for the Nobel Prize in physics (by mathematicians!). See Elisabeth T. Crawford, John L. Heilbron, and Rebecca Ullrich, The Nobel Population 1901–1937: A Census of the Nominators and Nominees for the Prizes in Physics and Chemistry (Berkeley, CA: Office for History of Science and Technology, University of California, 1987). Concerning the difference between theoretical physicists and mathematical physicists, John von Neumann made the following incisive observations. Answering questions posed to him by R. O. Fornaguera regarding the nature of mathematical physics and theoretical physics, von Neumann stated that: “I have always drawn a somewhat vague line of demarcation between the two subjects, but it was really more a difference in distribution of emphasis. I think that in theoretical physics the main emphasis is on the connection with experimental physics and those methodological processes which lead to new theories and new formulations, whereas mathematical physics deals with the actual solution and mathematical execution of a theory which assumed to be correct per se, or assumed to be correct for the sake of the discussion. In other words, I would say that theoretical physics deals rather with the formation and mathematical physics rather with the exploitation of physical theories. However, when a new theory has to be evaluated and compared with experience, both aspects mix.” John von Neumann, letter to R. O. Fornaguera, December 10, 1947, in John von Neumann: Selected Letters, ed. Miklós Rédei (Providence, RI: American Mathematical Society, 2005), 118–19.
It takes its name from verse 25:29 in the Gospel of Matthew: “For unto every one that hath shall be given, and he shall have abundance: but from him that hath not shall be taken even that which he hath.” See Robert K. Merton, “The Matthew Effect in Science,” Science 159(3810) (1968), 56–63. See also Margaret W. Rossiter, “The Matthew Matilda Effect in Science,” Social Studies of Science 23 (1993), 325–41. I thank Skuli Sigurdsson for pointing me to the Rossiter article.
There are several folders in the Bethe Nachlass at the Cornell Kroch Library archives which contain his notes on the seminar he attended. Those in Box 14 contain his notes on the seminars from 1975 to 1981. The three folders contain over 120 pages of notes on some fifty seminars, all dealing with the most recent developments in high energy physics, often given by the person responsible for the advance.
See in this connection the recollections of Sheldon Glashow and of Alan Guth. Sheldon L. Glashow, Interactions: A Journey through the Mind of a Particle Physicist and the Matter of this World, with Ben Bova (New York: Warner Books, 1998); Alan H. Guth, The Inflationary Universe: The Guest for a New Theory of Cosmic Origins (Reading, MA: Addison-Wesley Publishing, 1997).
The concept of a community of practice is due to Jean Lave and Etienne Wenger who in their book Situated Learning: Legitimate Peripheral Participation (Cambridge, UK: Cambridge University Press, 1991) offered a new viewpoint of “learning.” Classical learning theory conceived of learning as an individual process whereby a formal body of knowledge is acquired from a teacher or from an expert. Learning was a psychological process whose foci were the brains of individuals. For Lave and Wenger social participation is the essential mechanism for learning. Moreover, for them knowledge is not a stable commodity that can be transmitted. After the publication of Wenger’s books many sociologists, historians, and philosophers of science have adopted the view of considering the sciences as communities of practice committed to research enterprises that are expansive and prospective, and have placed particular emphasis on the community’s pervasive social entanglement. Helpful points of entry to that literature are Joseph Rouse, Engaging Science: How to Understand Its Practices Philosophically (Ithaca NY: Cornell University Press, 1996); How Scientific Practices Matter: Reclaiming Philosophical Naturalism (Chicago: The University of Chicago Press, 2003); Articulating the World: Conceptual Understanding and the Scientific Image (Chicago: The University of Chicago Press, 2015).
See in this connection Karin Knorr-Cetina, The Manufacture of Knowledge: An Essay on the Constructivist and Contextual Nature of Science (Oxford: Pergamon Press, 1981); The Social Process of Scientific Investigation, ed. Karin D. Knorr, Roger Krohn, and Richard Whitley (Dordrecht: D. Reidel, 1981); Epistemic Cultures: How the Sciences Make Knowledge (Cambridge, MA: Harvard University Press, 1999).
See Sharon Traweek, Beamtimes and Lifetimes: The World of High Energy Physicists (Cambridge, MA: Harvard University Press, 1988).
I am here referring to the fact that the period from 1945 to the mid-1970s did not conform to the usual pattern of increasing concentration of income and wealth in the hands of a few, as argued in Thomas Picketty, Capital in the Twenty-First Century, trans. Arthur Goldhammer (Cambridge, MA: Belknap Press, 2014).
See Silvan S. Schweber, “A Historical Perspective on the Rise of the Standard Model,” in The Rise of the Standard Model: Particle Physics in the 1960s and 1970s, ed. Lilian Hoddeson, Laurie Brown, Michael Riordan, and Max Dresden (Cambridge, UK: Cambridge University Press, 1997), 445–64; “Writing the Biography of Hans Bethe: Contextual History and Paul Forman,” Physics in Perspective 16 (2014), 179–217, where some of these elements are discussed and further references given. See also Hallam Stevens, “Fundamental Physics and its Justifications, 1945–1993,” Historical Studies in the Physical and Biological Sciences 34 (2003), 151–97. In his informative and insightful article Stevens analyzes the justifications for the huge public expenditure in the US on accelerator laboratories over the course of the Cold War, and the reasons for the disenchantment with high energy physics starting in the 1970s. See also the seminal articles by Aubin and Dalmedico wherein longue durée socio-historical analyses are presented for mathematics. David Aubin and Amy Dahan Dalmedico, “Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures,” Historia Mathematica 29 (2002), 273–339; David Aubin, “From Catastrophe to Chaos: The Modeling Practices of Applied Topologists,” in Changing Images in Mathematics: From the French Revolution to the New Millenium, ed. Umberto Bottazzini and Amy Dahan Dalmedico (London and New York: Routledge, 2001), 255–80; Amy Dahan Dalmedico, “An Image Conflict in Mathematics after 1945, in” Bottazzini and Dalmedico, Changing Images in Mathematics (ref. 31), 233–54; “Chaos, Disorder, and Mixing: A New fin-de-siècle Image of Science?,” in Growing Explanations: Historical Perspective on Recent Science, ed. M. Norton Wise (Durham: Duke University Press, 2000), 67–94.
Thus, that research in general relativity was only marginally pursued from the mid-1920s until the mid-1960s is a reflection of the fact that no new data beyond that connected with Einstein’s three tests—the deflection of light in its passage near the sun, and the gravitational red shift of light—was introduced until Pound and Rebka’s experiment, and the all important experiments by Irwin Shapiro making use of the new radar technology. Similarly, Dicke’s novel experimental approach in the mid-1960s is a crucial factor in making research in general relativity acceptable to the physics community at large.
See for example P. Söding and G. Wolf, “Experimental Evidence on QCD,” Annual Reviews of Nuclear Science 5 (1981), 231.
Christine Sutton, The Particle Connection (New York: Simon and Schuster, 1984).
See also Hoddeson et al., Rise of the Standard Model (ref. 31) and Gordon Kane, Modern Elementary Particle Physics (Reading, MA: Addison-Wesley, 1987), chs. 12 and 13.
R. E. Lanou in Proceedings of the 1982 DPF Summer Study on Elementary Particle Physics and Future Facilities, June 28-July 16, 1982, Snowmass, Colorado, ed. René Donaldson, Richard Gustafson, and Frank Paige (New York: American Physical Society, 1983), 661.
See Lanou’s and Richardson’s comments on the demographics in high energy physics in ibid., 651–54.
Snowmass is a part of the Aspen/Snowmass ski resort complex located in Snowmass Village near Aspen, Colorado.
Donaldson et al., DPF Summer Study (ref. 36), v.
Freeman Dyson, e-mail to the author, August 22, 2015.
See “Larry Abbott,” National Academy of Sciences Member Directory, accessed February 13, 2016, http://www.nasonline.org/member-directory/members/20033167.html.
See Schweber, “Hacking the Quantum Revolution” (ref. 2). Regarding effective field theories, see Weinberg’s Nobel Prize lecture and his talk at the 1996 conference on quantum field theory that Tian Yu Cao organized at Boston University. Steven Weinberg, “Conceptual Foundations of the Unified Theory of Weak and Electromagnetic Interactions,” Reviews of Modern Physics 52 (1980), 515–23; “What is Quantum Field Theory, and What Did We Think It Was?,” in Conceptual Foundations of Quantum Field Theory ed. Tian Yu Cao (New York: Cambridge University Press, 1999), 241. For a more general history and an incisive analysis of EFTs see Hartmann’s informative article: Stephan Hartmann, “Effective Field Theories, Reductionism and Scientific Explanation,” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 32 (2001), 267–304.
Ian Hacking, Representing and Intervening: Introductory Topics in the Philosophy of Natural Science (Cambridge: Cambridge University Press, 1983), ch. 12.
Ian Hacking, “Philosophers of Experiment,” PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, vol. 2, Symposia and Invited Papers (Chicago: University of Chicago Press, 1988), 147–57, on 152.
Ian Hacking, “On the Stability of Laboratory Science,” The Journal of Philosophy 85 (1988), 507–14.
Steven Weinberg, “Effective Field Theory, Past and Future,” arXiv, http://arxiv.org/abs/0908.1964, September 26, 2009.
Steven Weinberg, “Critical Phenomena for Field Theorists,” in Understanding the Fundamental Constituents of Matter, ed. Antonino Zichichi (New York: Plenum Press, 1977), 1–52.
See Lepage for a readily accessible exposition of Wilson’s approach. G. Peter Lepage, “What Is Renormalization?,” arXiv, http://arxiv.org/abs/hep-ph/0506330, June 30, 2005.
Thomas Appelquist and J. Carrazzone, “Infrared Singularities and Massive Fields,” Physical Review D 11(10) (1975), 2856–62; Steven Weinberg, “Effective Gauge Theories,” Physics Letters B 91 (1980), 51–55.
The interaction term in the effective action is given by \(S_{eff} [ {\text{fermions}} ]{\kern 1pt} = \int {d^{4} } x[ {\frac{{G_{F} }}{\sqrt 2 }( {j_{l}^{\lambda } + {\kern 1pt} J_{q}^{\lambda } } )( {j_{l\lambda } + J_{q\lambda } } )} ] \;\;{\text{with}}\; \; j_{l}^{\lambda } = \overline{e} \gamma^{\lambda } (1 - \gamma_{5} ){\kern 1pt} \nu_{e} + \mu ,\tau \;{\text{terms}} \; J_{q}^{\lambda } = \overline{u} \gamma^{\lambda } (1 - {\kern 1pt} \gamma_{5} )(d\cos {\kern 1pt} \theta_{C} + s\sin \theta_{C} ) + {\text{c}},{\text{t}},{\text{b}}\;{\text{terms}}.\)
Ludvig D. Faddeev and Victor N. Popov, “Feynman Diagrams for the Yang-Mills Field,” Physics Letters B 25 (1967), 29–30.
Pierre Ramond, Field Theory: A Modern Primer (Reading, MA: Benjamin/Cummings, 1981).
Steven L. Adler, “Einstein Gravity as a Long Wavelength Effective Field Theory,” in Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics, ed. Roman Jackiw, Nicola N. Khuri, Steven Weinberg, and Edward Witten (Cambridge, MA: MIT Press, 1985), 162–70. See also Steven L. Adler, “Einstein Gravity as a Long Wavelength Effective Field Theory,” in McCrea and Rees, Constants (ref. 13), 63–68; and “Einstein Gravity as a Symmetry-Breaking Effect in Quantum Field Theory,” Reviews of Modern Physics 54 (1983), 729–37; “Erratum,” Reviews of Modern Physics 55 (1983), 837.
Equivalently stated, the effective Lagrangian that describes gravitation at ordinary energies is a power series \(L_{eff} = \sqrt g \left[ {c_{0} M_{G}^{4} + c_{1} M_{G}^{2} R + c_{2} R^{2} + c_{2}^{\prime } R^{\mu \nu } R_{\mu \nu } + c_{3} M_{G}^{ - 2} R^{3} + \ldots } \right]\). If c 0 ,c 1 ,c 2 ,c 2 ′,c 3 were of order 1 it then would be the case that only the effects of the terms c 0 and c 1 would be experimentally observable. If M G is defined so that c 1 =1, then M G =(16πG)-½ = 1.72x1018 GeV, where G =6.674 × 10−8 cm3 g−1 sec−1 is Newton’s gravitational constant. But it turns out that c0 is not “quite” of order unity; from upper limits to the cosmological constant one concludes that c 0 < 10−120! Steven Weinberg, Cosmology (Oxford: Oxford University Press, 2008).
Although I do not refer to them, important developments in general relativity occurred in the 1970s, in particular, with regard to the singularity theorems of Penrose and of Hawking, with deep implications for cosmology. See Steven W. Hawking and Roger Penrose, The Nature of Space and Time (Princeton: Princeton University Press, 1995); Stephen W. Hawking and George F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge, UK: Cambridge University Press, 1995); Robert Geroch and Gary T. Horowitz, “Global Structure of Spacetimes,” in General Relativity: An Einstein Centenary Survey, ed. Stephen Hawking and Werner Israel (Cambridge: Cambridge University Press, 1979), 212–89; John Earman, “The Penrose-Hawking Singularity Theorems: History and Implications,” in The Expanding Worlds of General Relativity, ed. Hubert Goenner, Jürgen Renn, and Tilman Sauer (Boston: Birkhäuser, 1999), 235–67.
Tian Yu Cao, an erudite and accomplished historian and philosopher of physics, does so. Private communication.
Yuri I. Manin, Mathematics and Physics, trans. Ann and Neal Koblitz (Boston: Birkhäuser, 1982). Quoted in Freeman Dyson, “Review of Mathematics and Physics by Yuri Manin,” The Mathematical Intelligencer 5(2) (1983), 54–57, on 54.
See note 38.
See Phillip James Edwin Peebles, Physical Cosmology (Princeton, N.J.: Princeton University Press, 1971); Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley, 1972); Helge Kragh, Cosmology and Controversy: A History of Two Theories of the Universe (Princeton: Princeton University Press, 1996); Conceptions of Cosmos: From Myths to the Accelerating Universe: A History of Cosmology (Oxford: Oxford University Press, 2007); Higher Speculations: Grand Theories and Failed Revolutions in Physics and Cosmology (Oxford: Oxford University Press, 2011).
See for example Leon M. Lederman and David N. Schramm, From Quarks to the Cosmos, Tools of Discovery (New York: The Scientific American Library, 1989).
Murray Gell-Mann, “From Renormalizability to Calculability,” in Jackiw et al., Shelter Island II (ref. 53), 33. I say “final version,” i.e. the version appearing in Shelter Island II, because the notes that Bethe took at Shelter Island II do not indicate that in his oral presentation Gell-Mann pointed to the unification of particle physics and cosmology. By then the “unification” was certainly accepted. The 1982 Nuffield workshop on the very early universe cemented the symbiosis between elementary particles and cosmology and made “inflation” respectable. See John D. Barrow and Michael S. Turner, “The Inflationary Universe: Birth, Death, and Transfiguration,” Nature 298 (1982), 801–5, on 802; G. W. Gibbons, S. W. Hawking, and S. T. C. Siklos, The Very Early Universe: Proceedings of the Nuffield Workshop, Cambridge, 21 June to 9 July, 1982 (Cambridge University Press Archive, 1985).
See Schweber, “Hacking the Quantum Revolution” (ref. 2).
See e.g. Shuanggen Jin, Haghighipour Nader, and Ip Wing-Huen, eds., Planetary Exploration and Science: Recent Results and Advances (Berlin: Springer, 2015).
See e.g. Stephen Finney Mason and Stephen F. Mason, Chemical Evolution: Origin of the Elements, Molecules, and Living Systems (Oxford: Clarendon Press, 1991); Hubert Klahr and Wolfgang Brandner, eds., Planet Formation: Theory, Observations, and Experiments (Cambridge, UK: Cambridge University Press, 2006).
See David J. Stevenson, “Making the Moon,” Physics Today 67(11) (2014), 32–38.
See Kevin Zahnle et al., “Emergence of a Habitable Planet,” Space Science Reviews 129 (2007), 35–78; Helmut Lammer et al., “What Makes a Planet Habitable?” The Astronomy and Astrophysics Review 17 (2009), 181–249.
See John D. Barrow, Simon Conway Morris, and Stephen J. Freeland, eds., Fitness of the Cosmos for Life: Biochemistry and Fine-Tuning (Cambridge, UK: Cambridge University Press, 2008), and in particular George Whitesides’s foreword, “The Improbability of Life.”
Charles S. Peirce, “The Architecture of Theories,” The Monist 1(2) (1881), 161–76.
Larry F. Abbott and So-Yong Pi, Inflationary Cosmology (Singapore: World Scientific, 1986). See their introduction where they have given a beautiful exposition of the evolution of inflationary models.
The Planck scale is defined in terms of the natural constants associated with quantum mechanics (h) and special and general relativity (c and G). The Planck length is \(l_{P} = \sqrt {\frac{\hbar G}{{c^{3} }}}\) (1.616 × 10−33 cm); the Planck mass is \(m_{P} = \sqrt {\frac{\hbar c}{G}}\) (2.176 × 10−5 gr); and the Planck time is \(t_{P} = \,\frac{{l_{P} }}{c} = \frac{\hbar }{{m_{P} c^{2} }} = \sqrt {\frac{\hbar G}{{c^{5} }}}\) (5.391 × 10−44 s).
See the introduction in Larry F. Abbott and So-Yong Pi, Inflationary Cosmology (Singapore: World Scientific, 1986).
See Andrei Linde, Inflation and Quantum Cosmology (New York: Academic Press, 1990); Guth, Inflationary Universe (ref. 26); and in particular, for an account of the genesis of inflation Chris Smeenk, “Approaching the Absolute Zero of Time: Theory Development in Early Universe Cosmology” (Ph.D. diss., University of Pittsburgh, 2003); “False Vacuum: Early Universe Cosmology and the Development of Inflation,” in The Universe of General Relativity (Boston: Birkhäuser, 2005), 223–57; “Inflation and the Origins of Structure” (unpublished manuscript 2012), http://publish.uwo.ca/~csmeenk2/files/BEPaper.pdf.
For an accessible overview of monopoles see ’t Hooft, Ultimate Building Blocks (ref. 5).
Andrei Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy, and Primordial Monopole Problems,” Physics Letters B 108 (1982), 389–93; Andrei Linde, “Coleman-Weinberg Theory and the New Inflationary Universe Scenario” Physics Letters B 114(6) (1982), 431–35; Andreas Albrecht and Paul Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Physical Review Letters 48 (1982), 1220–23.
Dyson “Review of Manin” (ref. 57), 56.
See in particular the writings on Jackiw on this subject in Roman Jackiw, Diverse Topics in Theoretical and Mathematical Physics (Singapore: World Scientific, 1995). Also Sam Treiman and Roman Jackiw, Current Algebra and Anomalies (Princeton: Princeton University Press, 2014); Roman Jackiw, “Nonperturbative and Topological Methods in Quantum Field Theory,” in Jackiw et al., Shelter Island II (ref. 53), 65–78; Roman Jackiw, “My Encounters—as a Physicist—with Mathematics,” arXiv, http://arxiv.org/abs/hep-th/9410151v1, 1994; “Axial Anomaly,” Scholarpedia 3(10) (2008), 7302.
For a beautiful, readily accessible exposition of this viewpoint see Jurg Fröhlich and Bill Pedrini, “New Applications of the Chiral Anomaly,” in Mathematical Physics 2000, ed. A. Fokas, A. Grigorian, T. Kibble, and B. Zegarlinski (London: Imperial College Press, 2000), 9–47.
For an informative, lucid introduction to anomalies and their connection to Yang-Mill theories and topology, see Jackiw, “Axial Anomaly” (ref. 76).
Dependence on the dimensionality of space-time was already encountered in the analysis of anomalies: the axial vector current has special properties in four-dimensional theories. See ibid.
For a helpful presentation of the mathematical concepts and methods involved see Tohru Eguchi, Peter B. Gilkey, and Andrew J. Hanson, “Gravitation, Gauge Theories and Differential Geometry,” Physics Reports 66(6) (1980), 213–393.
Gerard ’t Hooft, “The Evolution of Quantum Field Theory, from QED to Grand Unification,” arXiv, http://arxiv.org/abs/1503.05007arXiv:1503.05007, March 17, 2015.
See Shing-Tung Yau, ed., The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer (Somerville, MA: International Press, 2003). For a simple exposition of the relation of the famous Atiyah-Singer index theorem and anomalies in Yang-Mill theories see the section entitled Physical Consequences of Axial Symmetry Anomalies in Jackiw, “Axial Anomaly” (ref. 76).
’t Hooft et al., Gauge Theories (ref. 11), v.
See the Introduction of Betozzini and Dalmonico, Changing Images (ref. 31) and the articles by Gray and Rowe therein.
See Michael Atiyah, “The Unreasonable Effectiveness of Physics in Mathematics,” in Fokas et al., Mathematical Physics (ref. 77), 25–38.
See Atiyah’s comments when awarding Witten the Fields Medal. Michael Atiyah, “The Work of Witten,” International Conference of Mathematicians. Kyoto 1990. Reprinted in Michael Atiyah Collected Works, vol. 6 (Oxford: Oxford University Press, 2004), 209–13.
See e.g. Atiyah, “Unreasonable Effectiveness” (ref. 85); Israel M. Gelfand, “Mathematics as an Adequate Language,” in The Unity of Mathematics: In Honor of the Ninetieth Birthday of I. M. Gelfand, Progress in Mathematics, vol. 244, ed. Pavel Etingof, Vladimir Retakh, and Isadore M. Singer (Basel: Birkhäuser 2006), x–16; Edward Witten, “Physics and Geometry,” in Proceedings of the International Congress of Mathematicians: August 3–11, 1986 (Providence, RI: American Mathematical Society, 1987), 267–303; Witten, “Fate of Spacetime” (ref. 15); “What Every Physicist Should Know about String Theory,” Physics Today 68(11) (2015), 38–43.
Pierre Cartier has given a (technically demanding) overview of the evolution of the concept of space and symmetry in mathematics from after World War II until the beginning of the twenty first century. Pierre Cartier, “A Mad Day’s Work: From Grothendieck to Connes and Kontsevich: The Evolution of Concepts of Space and Symmetry,” Bulletin of the American Mathematical Society 38(4) (2001), 389–408.
Leo Corry, “Mathematical Structures from Hilbert to Bourbaki: The Evolution of an Image of Mathematics,” in Bottazzini and Dalmedico, Changing Images of Mathematics (ref. 31), 167–86.
See Edward Witten, “Unravelling String Theory,” Nature 438(7071) (2005), 1085–88.
Pierre Deligne et al. Quantum Fields and Strings: A Course for Mathematicians, 2 vols. (Providence, RI: American Mathematical Society, 1999).
Ibid.
I am not competent to review either how computers have transformed theoretical physics, or how computers have transformed all of experimental physics, or how they have transformed the teaching of physics, chemistry, and biology at all levels. See e.g. Benjamin A. Stickler and Ewald Schachinger, Basic Concepts in Computational Physics (Cham: Springer International Publishing, 2014); Rolly Belfer, “Absent Knowledge: The Impact of Information-Theory on 20th Century Physics” (PhD diss., Bar-Ilan University, 2012) and the references therein.
See Kenneth G. Wilson, “Quantum Chromodynamics on a Lattice,” in New Developments in Quantum Field Theory and Statistical Mechanics Cargèse 1976 (New York: Springer, 1977), 143–72.
See e.g. Kurt Symanzik, “Cutoff Dependence in Lattice ø 44 Theory,” in ’t Hooft et al., Gauge Theories (ref. 11), 313–30.
See Michael Creutz, Quarks, Gluons and Lattices (Cambridge, UK: Cambridge University Press, 1983).
Michael Creutz, “Monte Carlo Study of Quantized SU(2) Gauge Theory,” Physical Review D 21(8) (1980), 2308; Kenneth Wilson, “Monte-Carlo Calculations for the Lattice Gauge Theory,” ’t Hooft et al., Gauge Theories (ref. 11), 363–402.
Frank Wilczek, “Particle Physics: A Weighty Mass Difference,” Nature 520 (April 16, 2015), 303–4.
See for example the research of Stuart Shapiro at http://physics.illinois.edu/people/profile.asp?slshapir and his publications listed therein.
See Margaret Morrison, “Where Have All the Theories Gone?,” Philosophy of Science 74 (2007), 195–227; “Models as Representational Structures,” in Luc Bovens, Stephan Hartmann, and Carl Hoefer, eds., The Philosophy of Nancy Cartwright (London: Routledge, 2008), 67–88; “Models, Measurement and Computer Simulation: The Changing Face of Experimentation,” Philosophical Studies 143 (2009), 33–57; “Emergent Physics and Micro-Ontology,” Philosophy of Science 79 (2012), 141–66; especially Morrison (2009).
Mathematical physics was added sometime in the late 1960s.
See Stevens, “Fundamental Physics” (ref. 31).
Frank Wilczek in an interview on October 9, 2015, stressed this point. While an undergraduate student at the University of Chicago from 1966 to 1970, he did not participate in any of the student demonstrations against the Vietnam War but keenly observed, and was deeply affected, by the upheavals occurring around him. That the student protests likewise deeply affected him was stated by Alan Guth in an interview on August 25, 2015.
See Dalmedico, “An Image Conflict” (ref. 31) for an account of the tension within the American mathematical community between “pure” and “applied” mathematicians after World War II.
McCrea and Rees, Constants (ref. 13), 211.
Ibid., 211.
Percy Bridgman, The Logic of Modern Physics (New York: Macmillan, 1927); “Operational Analysis,” Philosophy of Science 5 (1938), 114; Reflections of a Physicist (New York: Philosophical Library, 1955).
McCrea had obtained his PhD at Cambridge in 1929 with a seminal dissertation on the constitution of the sun under William Fowler. This at a time when theoretical physics was still part of the department of applied mathematics. In 1965, he became the head of the Astronomy Centre at Sussex University. Martin Rees was born in 1942, and obtained his PhD in 1967 at Cambridge University under the supervision of Dennis Sciama.
I have not made any attempt to give a detailed bibliography of theoretical contributions, nor am I competent to do so. For a general overview of QCD see Andreas S. Kronfeld and Chris Quigg, “Resource Letter QCD-1: Quantum Chromodynamics,” American Journal of Physics 78 (2010), 1081–92; for a detailed historical exposition of the development of QCD see Hoddeson, et al., Rise of the Standard Model (ref. 31); ’t Hooft, Under the Spell (ref. 5); Tian Yu Cao, From Current Algebra to Quantum Chromodynamics: A Case for Structural Realism (Cambridge University Press, 2010). For an introduction to some of the mathematical developments to Eguchi et al., “Gravitation” (ref. 80); Martin H. Krieger, Doing Mathematics: Convention, Subject, Calculation, Analogy, 2nd ed. (River Edge, N.J.: World Scientific, 2016).
Peter Galison, Image and Logic: A Material Culture of Microphysics (Chicago: The University of Chicago Press, 1997).
Lochlainn O’Raifeartaigh, The Dawning of Gauge Theory (Princeton: Princeton University Press, 1997); Lochlainn O’Raifeartaigh, “Gauge Theory: The Gentle Revolution,” in Fokas et al., Mathematical Physics (ref. 77), 183–215.
Daniel Z. Freedman and Peter van Nieuwenhuizen, “Supergravity and the Unification of the Laws of Physics,” Scientific American 238(2) (1978), 126–43.
Bruno Zumino, “Supersymmetry: A Personal View,” in Fokas et al., Mathematical Physics (ref. 77), 316–26.
John H. Schwarz, ed., Superstrings: The First 15 Years of Superstring Theory, 2 vols. (Singapore: World Scientific, 1985); John H. Schwarz and Nathan Seiberg, “String Theory, Supersymmetry, Unification, and All That,” Reviews of Modern Physics 71 (1999), S112; John H. Schwarz, “String Theory Origins of Supersymmetry,” arXiv, http://arxiv.org/abs/hep-th/0011078, November 9, 2000; “The Early Years,” arXiv, http://arxiv.org/abs/hep-th/0007118v3, July 15, 2000; “Anomaly Cancellation: A Retrospective From a Modern Perspective,” arXiv, http://arxiv.org/abs/hep-th/0107059v1, July 9, 2001; “The Early History of String Theory and Supersymmetry,” arXiv, http://arxiv.org/abs/1201.0981v1, January 4, 2012.
Andrea Cappelli, Elena Castellani, Filippo Colomo, and Paolo Di Vecchia, The Birth of String Theory (Cambridge, UK: Cambridge University Press, 2012).
Peter Galison, “Theory Bound and Unbound: Superstrings and Experiment,” in Laws of Nature: Essays on the Philosophical, Scientific, and Historical Dimensions, ed. Friedel Weinert (Berlin: Walter de Gruyter, 1995), 369–408; “Mirror Symmetry: Persons, Values, and Objects,” in Wise, Growing Explanations (ref. 31).
I thank Chris Bostock for this important communication.
’t Hooft, “Evolution of Quantum Field Theory” (ref. 81).
Silvan S. Schweber, “A Short History of Shelter Island I,” in Jackiw et al., Shelter Island II (ref. 53), 301–43; QED and the Men Who Made It (Princeton: Princeton University Press, 1994).
Bethe’s notes of Shelter Island I have been posted at https://ecommons.cornell.edu/handle/1813/14146 with the gracious permission of the estate of Hans A. Bethe and the Division of Rare Books and Manuscript Collection of the Cornell University Library. The originals can be found in The Bethe Papers at the Cornell Archives. I thank Rose Bethe for making this possible.
Hendrik Kramers, “Die Wechselwirkung zwischen geladenen Teilchen und Strahlungsfeld,” Nuovo Cimento 15 (1938), 108–14. Reprinted in Hendrik Kramers, Collected Scientific Papers (Amsterdam: North-Holland, 1956); J. Serpe, “Sur la quantification du problème de l’intéraction d’un oscillateur linéaire harmonique,” Physica 8 (1941), 226–32; W. Opechowski, “Sur la quantification du system de l’électron et du rayonnement,” Physica 8 (1941), 161–76.
See Robert E. Marshak, “Origin of the Two-Meson Theory,” in Jackiw et al., Shelter Island II (ref. 53), 355–62.
According to Breit’s notes, at the end of the third day of the conference Feynman gave a talk on his path integral formulation of quantum mechanics. Since Feynman was at Cornell at the time, Bethe knew of this work and did not take any notes.
Hans A. Bethe, “The Electromagnetic Shift of Energy Levels,” Physical Review 72 (1947), 339.
Simon Pasternack, “Note on the Fine Structure of H α and D α ,” Physical Review 54 (1938), 1113.
Victor Weisskopf, in “Panel on Shelter Island I,” in Jackiw et al., Shelter Island II (ref. 53), 344–354, on 344.
Max M. Dresden, H. A. Kramers (Berlin: Springer Verlag, 1987).
Dresden’s conclusion is reinforced by an analysis of Bethe’s contribution to the eighth Solvay Congress, which was held from September 27 to October 2, 1948. The proceedings of that meeting were never published, but Bethe’s article, entitled “Electromagnetic Shift of Energy Levels,” is available from the Institut Solvay. In section II of that article Bethe analyzed the non-relativistic limit of Schwinger’s Lorentz covariant and gauge invariant formulation of quantum electrodynamics which he had presented at the Pocono conference and at the Michigan summer school earlier that year. In fact, a preprint of Schwinger forthcoming 1948 article on QED was available at the Michigan Summer School. In Bethe’s exposition, the one reference to Kramers consists of the statement “Using the same approximation [of making the dipole approximation and neglecting the recoil of the electron], but in classical electrodynamics, Kramers had developed a theory similar to Schwinger’s in the spring of 1947.” Bethe gave as a reference to Kramers’s work, Kramers’s presentation at the Shelter Island Conference. But in fact Kramers’s work, and that of his students, had been done in the early 1940s. Furthermore, Schwinger’s approach had been based on what he had learned from Kramers’s presentation at Shelter Island.
Stated field theoretically, in hole theory one subtracted from the Hamiltonian for the electron field the expectation value of the Hamiltonian operator for the state representing the vacuum in the absence of the electromagnetic field. Thus: \(H_{0}^{matter} = \int {\left[ {\psi ^{*}(r)(\varvec{\alpha}\cdot \varvec{p}{\kern 1pt} + \beta m)\psi (r) - \left\langle {\psi^{*}(\varvec{\alpha}\cdot \varvec{p} + \beta m)\psi } \right\rangle_{vac} } \right]}\). Similarly, the charge current four vector of the quantized electron field was defined as \(\{ \varvec{j}(r){\kern 1pt} ,\rho (r)\} = e\left[ {\psi ^{*}(r)\{\varvec{\alpha},1\} \psi (r) - \left\langle {\psi ^{*}\{\varvec{\alpha},1\} \psi } \right\rangle_{vac} } \right]\). For the free electron field \(\psi (r) = \sum\limits_{m} {a_{m} e^{ip \cdot r} } u_{m};\;\psi^{*}(r){\kern 1pt} = \sum\limits_{m} {a_{m}^{*} e^{ - ip \cdot r} \overline{{u_{m} }} }\), where u m are the Dirac spinor functions, a m *, a m the creation and annihilation operators, and E m the absolute value of the energy for an electron in the state m. The symbol m denotes a definite value of the linear momentum p and a specification of the four possible spin-states corresponding to this momentum p.
All three were sponsored by the National Academy of Sciences (NAS). The second, the Pocono conference was held from March 30 to April 2, 1948, and the third, Oldstone, was held for four days from April 11 to April 14, 1949.
See John Polkinghorne, Rochester Roundabout: The Story of High Energy Physics (New York: W. H. Freeman, 1989).
See the perceptive preface Kevles wrote for the 1987 reprinting of The Physicists. He there asserted that by the 1980s in the United States “among many social groups science as such enjoys something closer to tolerance than reverence.” Daniel J. Kevles, The Physicists: The History of a Scientific Community in Modern America (Cambridge, Mass.: Harvard University Press, 1987), xi.
Khuri was born in Beirut, Lebanon, and obtained his BA from the American University of Beirut in 1952. He received his PhD from Princeton University 1957 under the supervision of Sam Trieman and Res Jost. His dissertation, a rigorous study of scattering amplitudes in non-relativistic quantum mechanics established the validity of dispersion relations in that context. At the suggestion of Robert Oppenheimer he returned to Lebanon and from 1957 to 1964 he taught at the American University of Beirut, but found himself isolated there. A visiting appointment at the Institute for Advanced Study from 1959 until 1963 allowed him to return to the United States. In 1964, he accepted a position at Rockefeller University. Khuri’s research was in mathematical physics, closely coupled to high energy experimental physics. He made important contributions to the elucidation of Feynman’s path integral. See “Faculty Publications,” Rockefeller University, https://appext.rockefeller.edu/facpub/do/query_pub_list?author1=%22Khuri+N%22&yearFrom=1900.
A function describing a property of hadrons that can be measured in high-energy scattering experiments is said to scale when its value is not determined by the energy at which the measurement is being carried out, but is determined by dimensionless kinematic quantities such as the ratio of the energy to a momentum transfer. Increasing the energy of the probing particle implies decreasing its wavelength, and thus being able to resolve more precisely the spatial features of what is being probed. Thus the property of “scaling” for a cross-section implies that the resolution scale of the scatterer does not depend on length, and hence that effectively what ever is responsible for the scattering has a point-like substructure. Scaling behavior for the structure functions of deep inelastic scattering of electrons on nucleons was first proposed by James Bjorken in 1968.
See Freedman and Nieuwenhuizen, “Supergravity” (ref. 112); Zumino, “Supersymmetry” (ref. 113).
The outstanding universities then created amalgamated teaching at the undergraduate level with a commitment to research at the graduate level, with Stanford University, Johns Hopkins University, the University of Chicago, and Duke University prime examples.
But located in Manhattan with such institutions.
These buildings were designed to make possible the fulfillment of four new functions in the development of the institute: (1) to encourage scientists from all over the world to meet and live together for brief or extended periods of time on campus, in order to exchange ideas and synthesize important sectors of the rapidly accumulating mass of scientific knowledge; (2) to transmit scientific knowledge to as wide a spectrum of students as possible, to young people, and to the general public; (3) to provide model facilities for meetings of scientific societies; and (4) to nurture the cultural life of the institute faculty, visiting scholars and students, through exhibitions of paintings, programs of music, and lectures in many fields of the creative and performing arts. John Walsh, “The Rockefeller University: Science in a Different Key,” Science 150(3704) (1965), 1692–95. These reflected Cardinal Newman’s idea of a university, which Bronk had spoken of repeatedly: “a University is a place of concourse, whither students come from every quarter for every kind of knowledge.…It is the place for seeing galleries of first-rate pictures, and for hearing wonderful voices and performers of transcendent skill. It is the place for great preachers, great orators, great nobles, great statesmen. In the nature of things, greatness and unity go together; excellence implies a centre….It is the place to which a thousand schools make contributions; in which the intellect may safely range and speculate, sure to find its equal in some antagonist activity, and its judge in the tribunal of truth. It is a place where inquiry is pushed forward, and discoveries verified and perfected, and rashness rendered innocuous, and error exposed, by the collision of mind with mind, and knowledge with knowledge. It is the place where the professor becomes eloquent, and is a missionary and a preacher, displaying his science in its most complete and most winning form, pouring it forth with the zeal of enthusiasm, and lighting up his own love of it in the breasts of his hearers.…It is a place which wins the admiration of the young by its celebrity, kindles the affections of the middle-aged by its beauty, and rivets the fidelity of the old by its associations.” John Henry Newman, “What is a University, Lecture Dublin 1852,” in A Newman Treasury, ed. Charles Frederick Harrold (London: Longmans Green, 1943), 41.
New York Times, June 17, 1961.
Ted Berlin died shortly after he came to Rockefeller and was replaced by Ezekiel G. D. Cohen in 1963.
Pais had decided in 1963 to leave the Institute of Advanced Study in Princeton. He had found himself thinking, “I am in great danger. I [am] about to become too content with myself, and to stop striving toward new goals.” Abraham Pais, A Tale of Two Continents: The Life of a Physicist in a Turbulent World (Princeton, NJ: Princeton University Press, 1997), 385. Pais was attracted to Rockefeller University both because of its location, Manhattan, and because Uhlenbeck had become a faculty member there. Uhlenbeck had been Pais’s teacher at Amsterdam University when Pais started his studies in physics in the mid-1930s. Uhlenbeck helped arrange an appointment for Pais, and Pais joined Rockefeller in 1964.
From 1943 to 1947, Kac was associated with the Radiation Lab at MIT. He there met George Uhlenbeck and began collaborating with him. This reawakened his interest in statistical mechanics and was a decisive factor in his moving to Rockefeller University. In his autobiography, Kac commented that while Detlev Bronk’s vision for the university was not fully realized either then or afterwards, nonetheless, it afforded him the opportunity to immerse himself in the statistical mechanics of phase transitions in the company of Berlin, Cohen, and Uhlenbeck, among others. Mark Kac, Enigmas of Chance: An Autobiography (New York: Harper & Row, 1985).
Bronk was given an honorary PhD degree by Rockefeller University at the first commencement Seitz presided over. It was his 58th. It indicates Bronk’s standing as a statesman for science during his long career.
Bronk was president of the NAS from 1950 to 1962. See Frank Brink, “Detlev Wulf Bronk 1897–1975. A Biographical Memoir” (Washington DC: National Academy of Sciences, 1978).
Seitz had been president of the NAS since 1962.
See Frederic Seitz, A Selection of Highlights from the History of the National Academy of Sciences, 1863–2005 (Lanham, MD: University Press of America, 2007).
Daniel S. Greenberg, “Rockefeller University: Seitz To Succeed Bronk as President,” Science 160(3824) (1968), 173–74.
M. S. Handler, New York Times, June 23, 1968.
John Walsh, “The Rockefeller University: No Time for Philosophers,” Science 195(4275) (1977), 272–75, on 272.
Davidson at Chicago; Frankfurt at Yale; Feinberg at the University of Arizona; and Kripke at Princeton.
Walsh, “The Rockefeller University” (ref. 147).
Ibid., 274.
When Shenker died in 2007, his obituary in New York Times began with the statement, “Israel Shenker, a scholar trapped in a newsman’s body who was known to readers of The New York Times for his vast erudition and sly, subversive wit.”
Shenker was referring to Hao Wang and Donald A. Martin.
For a different assessment of the events, see Seitz’s autobiography, Frederick Seitz, On the Frontier: My Life in Science (Woodbury, NY: American Institute of Physics, 1994), 325–26.
See his NAS biographical memoir, S. Gaylen Bradley, “Joshua Lederberg, 1925–2008,” National Academy of Sciences (2009), www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/lederberg-joshua.pdf.
See “Rodney W. Nichols: Consultant on Science and Technology Policy,” Rockefeller University, http://www.rockefeller.edu/graduate/files/ScienceDiplomacy/nichols.pdf.
All the letters quoted in the following, except for the letters addressed to Bethe, were found in the box in which Khuri had stored all the materials he had kept regarding Shelter Island II. The content of the box is to be deposited in the Rockefeller Archives. I thank Professor Khuri for permission to quote from the materials in the box.
The list consisted of some eighteen names, some of them starred to indicate “higher priority.” The starred names were: Michael Peskin, Michael Green, Larry Yaffe, Alan Guth, Georgio Parisi, Sergio Ferrara, and Martin Luscher. In a footnote he added, “one who I’ve never met but has written one or two impressive papers is Kawai.” Witten to Khuri, October 25, 1982.
Salam suggested three names.
Three of the people from the Soviet Union that were on Jackiw’s and were invited—Bogoliubov, Faddeev, and Polyakov—could not attend.
Samios, Lederman, and Panofsky had been on Jackiw’s list of persons to be invited.
I have in mind theorists like Philip Anderson, Leo Kadanoff, J. Robert Schrieffer. Ken Wilson, who was Jackiw’s teacher, was invited. Kadanoff at the time was doing work in regional planning. Brezin and Jinn-Justin were invited and attended, but they could be considered outstanding field theorists.
Feynman began the letter with “Dear Hans,” but signed the letter “R. P. Feynman.” One might have expected, given the number of years Feynman had known Bethe, and the closeness of their association, that he would have signed it more informally as “Dick.”
There is a letter by Jackiw to Khuri informing him that he was writing to ’t Hooft and Faddeev and would soon be speaking with Feynman and Bethe.
Tsung Tao Lee, Marshall Rosenbluth, and Chen Ning Yang, “Interaction of Mesons with Nucleons and Light Particles,” Physical Review 75(5) (1949), 905.
Tsung-Dao Lee and Chen-Ning Yang, “Question of Parity Conservation in Weak Interactions,” Physical Review 104(1956), 254, 1671–77.
See Paul Forman, “The Fall of Parity,” Physics Teacher 20(5) (1982), 281–88.
For a review of these developments see David Kaiser, “Do Feynman Diagrams Endorse a Particle Ontology? The Roles of Feynman Diagrams in S-Matrix Theory,” in Conceptual Foundations of Quantum Field Theory, ed. Tian Yu Cao (New York: Cambridge University Press, 1999), 343–56; and “Nuclear Democracy,” Isis 93 (2002), 229–68.
One of the reasons for Chew holding this view was that none of the hadrons seemed more fundamental than any of the others. He therefore rejected the idea of formulating field theories that singled out some subset of the hadrons. Echoing Heisenberg in 1925, he hoped that by focusing on physical observables, in particular the S-matrix, he would be able to formulate a theory that would determine the hadron spectrum and the hadronic scattering amplitudes.
See Hoddeson et al., Rise of the Standard Model (ref. 31); Arianna Borrelli, “The Making of an Intrinsic Property: ‘Symmetry Heuristics’ in Early Particle Physics,” Studies in History and Philosophy of Science Part A 50 (2015), 59–70.
Murray Gell-Mann, “From Renormalizability to Calulability?,” in Jackiw et al., Shelter Island II (ref. 53), 3–23; Steven Weinberg, “Calculation of Fine Structure Constants,” in Jackiw et al., Shelter Island II (ref. 53), 24–37.
See Cao, Current Algebra (ref. 109) for the historical account.
See Kurt Gottfied and Victor F. Weisskopf, Concepts of Particle Physics, 2 vols. (Oxford: The Clarendon Press, 1984); Kane, Modern Elementary Particle Physics (ref. 35); Rabindra N. Mohapatra, Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics (New York: Springer-Verlag, 1986).
For a historical account of these developments see Schweber, “Hacking the Quantum Revolution” (ref. 2).
See Gottfried and Weisskopf, Concepts (ref. 174).
Yoichiro Nambu, “Quasi-Particles and Gauge Invariance in the Theory of Superconductivity,” Physical Review 117 (1960), 648; Yoichiro Nambu and Giovanni Jona-Lasinio, “Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity,” pts. 1 and 2, Physical Review 122 (1961), 345; 124 (1961), 246. See also Yoichiro Nambu, “A Superconductor Model of Elementary Particles and its Consequences,” International Journal of Modern Physics A 23 (2008), 4063–79. Their model was based on the Lagrangian density \(L{\kern 1pt} = i{\kern 1pt} \overline{\psi } (x)\gamma^{\mu } \partial_{\mu } \psi (x) + g\left[ {\overline{(\psi } (x)\psi (x))^{2} - (\overline{\psi } (x)\gamma_{5} {\kern 1pt} \psi (x))^{2} } \right]\), which is invariant under the phase changes \(\psi (x)\, \to \,e^{i\alpha } \,\psi (x)\quad {\text{and}}\quad \psi (x) \to e^{{i\beta \gamma_{5} }} \psi (x)\). By Noether’s theorem, there would be a vector current, \(j^{\mu } = \overline{\psi } (x)\gamma^{\mu } \psi (x)\), which is conserved and an axial vector current, \(j_{5}^{\mu } = \overline{\psi } (x)\gamma^{\mu } \gamma_{5} {\kern 1pt} \psi (x)\), which is conserved. Nambu and Joan-Lasinio then stipulated that the ground state (i.e. the vacuum state) does not respect the chiral symmetry and that the symmetry is broken spontaneously by a non-zero expectation value: \(\left\langle 0 \right|{\kern 1pt} \overline{\psi } (x)\psi (x)\left| 0 \right\rangle \ne 0\).
It consisted of a complex scalar field ϕ with the Lagrangian density \(L = \partial_{\mu } \phi^{*} \partial^{\mu } \phi - V(\phi ) \;\; V(\phi ) = m^{2} \phi^{*} \phi + \frac{1}{2}\lambda (\phi^{*} \phi )^{2}\), where m and λ are the mass and self-interaction coupling constant of the scalar field. The model is invariant under a global change of phase ϕ(x)→ e iα ϕ(x), transformations which define the abelian symmetry group U(1).
Interestingly, Nambu and Jona-Lasinio noted that in the BCS theory the particle that plays the role of the Goldstone boson is not massless because of the existence of the long range Coulomb interaction.
Jeffrey Goldstone, “Field Theories with Superconductor Solutions,” Nuovo Cimento 19 (1961), 154–64.
Furthermore, the representation of the canonical commutation relations on this Hilbert space is inequivalent to the usual Fock-space representation.
Jeffrey Goldstone, Abdus Salam, and Steven Weinberg, “Broken Symmetries,” Physical Review 127(1962), 965–70.
Nambu and Jona-Lasinio, “Dynamical Model” (ref. 177); Philip W. Anderson, “Plasmons, Gauge Invariance, and Mass,” Physical Review 130(1) (1963), 439.
Peter W. Higgs, “Broken Symmetries and the Masses of Gauge Bosons,” Physical Review Letters 13 (1964), 508–9.
François Englert and Robert Brout, “Broken Symmetry and the Mass of Gauge Vector Mesons,” Physical Review Letters 13 (1964), 321–23.
Gerald S. Guralnik, Carl R. Hagen, and Thomas W. B. Kibble, “Global Conservation Laws and Massless Particles,” Physical Review Letters 13 (1964), 585–87.
For an informative contemporaneous presentation see Jeremy Bernstein, “Spontaneous Symmetry Breaking, Gauge Theories, the Higgs Mechanism and All That,” Reviews of Modern Physics 46(1) (1974), 7–48. For retrospective accounts by the theorists who made the advances see Peter W. Higgs, “Spontaneous Breaking of Symmetry and Gauge Theories,” in Hoddeson et al., Rise of the Standard Model (ref. 31), 506–10; Robert Brout, “Notes on Spontaneously Broken Symmetry,” in Hoddeson et al., Rise of the Standard Model (ref. 31); 485–500; Gerald S. Guralnik, “The History of the Guralnik, Hagen, and Kibble Development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles,” International Journal of Modern Physics A 24 (2009), 2601–27; Thomas W. B. Kibble, “Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism,” Scholarpedia 4 (2009), 6441; Thomas W. B. Kibble, “Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism (History),” International Journal of Modern Physics A 24 (2009), 6001–9.
One must take into account that the right handed component of leptons and quarks are SU(2) singlets whereas and the left hand components are doublets. This would have been apparent had we written down the full Lagragian for the standard model including the terms specifying the dynamics of the quarks and leptons.
Ludvig D. Faddeev and Victor N. Popov, “Feynman Diagrams for the Yang-Mills Field,” Physics Letters B 25 (1967), 29–30; Bryce S. DeWitt, “Quantum Theory of Gravity,” pt. 1, “The Canonical Theory,” Physical Review 160 (1967), 1113; Ludvig D. Faddeev and Andreĭ Alekseevich Slavnov, Gauge Fields: Introduction to Quantized Theory (Reading, MA: Benjamin-Cummings, 1980).
Gerard ’t Hooft, “Renormalizable Lagrangians for Massive Yang-Mills Fields,” Nuclear Physics B 35 (1971), 167–88; Gerard ’t Hooft and Martinus Veltman, “Regularization and Renormalization of Gauge Fields,” Nuclear Physics B 44 (1972), 189–213; Benjamin W. Lee and Jean Zinn-Justin, “Spontaneously Broken Gauge Symmetries,” pt. 1, “Preliminaries,” Physical Review D 5 (1972), 3121; pt. 2, “Perturbation Theory and Renormalization,” Physical Review D 5 (1972), 3137; pt. 3, “Equivalence,” Physical Review D 5 (1972), 3155; pt. 4, “General Gauge Formulation,” Physical Review D 7 (1973), 1049.
Steven Weinberg, “A Model of Leptons,” Physical Review Letters 19 (1967), 1264–66; Abdus Salam, “Weak and Electromagnetic Interactions in Elementary Particle Theory,” in Elementary Particle Physics: Relativistic Groups and Analyticity,” Eighth Nobel Symposium, ed. N. Svartholm (Stockholm: Almquvist and Wiksell, 1968), 367.
H. David Politzer, “Reliable Perturbative Results for Strong Interactions,” Physical Review Letters 30 (1973), 1346–49; “Asymptotic Freedom: An Approach to Strong Interactions,” Physics Reports 14 (1974), 129–80.
David J. Gross and Frank Wilczek, “Ultraviolet Behavior of Non-Abelian Gauge Theories,” Physical Review Letters 30 (1973), 1343; “Asymptotically Free Gauge Theories,” Physical Review D 8 (1973), 3633.
Curtis G. Callan, “Bjorken Scale Invariance in Scalar Field Theory,” Physical Review D 2 (1970), 1541–47; “Bjorken Scale Invariance and Asymptotic Behavior,” Physical Review D 5 (1972), 3202–10; Curtis G. Callan and David Gross, “Bjorken Scaling in Quantum Field Theory,” Physical Review D 8 (1973), 4383–94.
Kenneth Wilson and John Kogut, “The Renormalization Group and the ϵ Expansion,” Physics Reports 12(2) (1974), 75–199.
See the lectures David Gross gave at the 1975 Les Houches Summer School in Theoretical Physics. David Gross, “Applications of the Renomalization Group to High Energy Physics,” in Methods in Field Theory, ed. Roger Balian and Jean Zinn-Justin (Amsterdam: North Holland, 1975), 141–250.
See for example P. Söding and G. Wolf, “Experimental Evidence on QCD,” Annual Reviews of Nuclear Science 5 (1981), 231.
The same interaction that is responsible for proton decay allows an initial state of the universe which is symmetric in particle anti particle composition, to evolve into one in which in which matter is preponderant and that contains on the average 1010 photons for every nucleon of matter, as observed at present. GUTs are thus able to explain previously unsolved problems in cosmology. See Howard Georgi, “A Unified Theory of Elementary Particles and Forces,” Scientific American 244 (1981): 48–63.
See for example Maurice Goldhaber, “The Search for Proton Decay and Other Rare Phenomena,” in McCrea and Rees, Constants (ref. 13), 15–17; Maurice Goldhaber, The Search for Proton Decay: A Look Back, BNL-41757, CONF-8806204-5, Brookhaven National Laboratory, Upton, NY, 1981; J. M. LoSecco, Frederic Reines, F. Sinclair, and D. Sinclair, “The Search for Proton Decay,” Scientific American (June 1, 1985), 54.
Howard Georgi, Helen R. Quinn, and Steven Weinberg, “Hierarchy of Interactions in Unified Gauge Theories,” Physical Review Letters 33(7) (1974), 451.
For the history of the big bang theory and of its rival, the steady state theory, see Kragh, Conceptions of Cosmos (ref. 59).
A detailed history of these developments is given in Steven Weinberg, The First Three Minutes: A Modern View of the Origin of the Universe (New York: Basic Books, 1977). See also his Cosmology (ref. 54) for the detailed calculations. Bernstein and Feinberg also narrate that history, and in addition present the original papers. Jeremy Bernstein and Gerald Feinberg, Cosmological Constants: Papers in Modern Cosmology (New York: Columbia University Press, 1986). See also Guth, Inflationary Universe (ref. 26).
Penzias and Wilson had measured the cosmic background radiation at only one wavelength. By the early 1980s, further measurements indicated that at least for long wavelengths the distribution of wavelengths agreed with what was expected from black body radiation at a temperature of 2.9 K.
Neutrons would decay into protons, neutrons and protons would combine to form deuterons, deuterons would combine with neutrons to produce He3, and the latter combining with neutrons would produce He4.
See for example David N. Schramm, The Big Bang and Other Explosions in Nuclear and Oarticle Astrophysics (Singapore: World Scientific, 1996).
David A. Kirzhnits, D. A. Linde and Andrei D. Linde, “Macroscopic Consequences of the Weinberg Model,” Physics Letters B 42 (1972), 471–74.
Steven Weinberg, “Gauge and Global Symmetries at High Temperature,” Physical Review D 9 (1974), 3357–78; Louise Dolan and Roman Jackiw, “Symmetry Behavior at Finite Temperature,” Physical Review D 9 (1974), 3320–41. See also David A. Kirzhnits and Andrei D. Linde, “Symmetry Behavior in Gauge Theories,” Annals of Physics 101 (1976), 195–238.
Sidney Coleman, “Fate of the False Vacuum: Semiclassical Theory,” Physical Review D 15(10) (1977), 2929.
James S. Langer, “Theory of the Condensation Point,” Annals of Physics 41(1) (1967), 108–57.
Coleman, “Fate of the False Vacuum” (ref. 207).
Alexei Starobinsky, “Spectrum of Relic Gravitational Radiation and the Early State of the Universe,” Journal of Experimental and Theoretical Physics 30(11) (1979), 682.
Weinberg, Cosmology (ref. 54), 201.
Barrow and Turner, “Inflationary Universe” (ref. 61).
Guth, “Inflationary Universe” (ref. 26); Andrei Linde, “The New Inflationary Scenario: Problems and Perspectives,” in Jackiw et al., Shelter Island II (ref. 53), 190–217; Stephen Hawking, “The Cosmological Constant is Probably Zero,” in Jackiw et al., Shelter Island II (ref. 53), 217–19.
Weinberg, Cosmology (ref. 54), 202.
Gibbons, Hawking, and Siklos, Very Early Universe (ref. 61).
But see Frank Wilczek, “Conference Summary and Concluding Remarks” in Shelter Island II (ref. 53), 475–80.
See Barrow and Turner, Inflationary Universe (ref. 61), and Guth’s account of his participation in the workshop in Guth, “Inflationary Universe” (ref. 26).
Gibbons, Hawking, and Siklos, Very Early Universe (ref. 61), 3.
See Guth, “Inflationary Universe” (ref. 26).
The participants were: Moisey Alexandrovich Markov (born May 13, 1908); Dennis William Siahou Sciama, FRS (born November 18, 1926); Igor Dmitriyevich Novikov (born November 10, 1935); George Francis Rayner Ellis (born August 11, 1939); James Maxwell Bardeen (born May 9, 1939); James Burkett Hartle (born August 20, 1939); Martin John Rees (born June 23, 1942); Stephen William Hawking (born January 8, 1942); Alan Harvey Guth (born February 27, 1947); Andrei Dmitriyevich Linde (born March 2, 1948); Alexei Alexandrovich Starobinsky (born April 19, 1948); Alexander Vilenkin (born May 13, 1949); Michael S. Turner (born July 29, 1949); Paul Joseph Steinhardt (born December 25, 1952); Frank Anthony Wilczek (born May 15, 1951); John David Barrow (born November 29, 1952); Larry Ford (PhD, Physics, Princeton University, 1974, MA, Physics, Princeton University, 1971, BS, Physics, Michigan State University, 1970); Qaisar Shafi (PhD student of Salam in the mid-1970s). No dates are available for M. Yu Khlopov, A. G. Polnarev, Vladimir Nikolaevich Lukash, A. D. Dolgov, G. Lazarides; nor for several even younger participants: R. J. Perry a PhD student of Kenneth Wilson, Andreas Albrecht a PhD student of Steinhardt.
See Stanley J. Gates, Marc T. Grisaru, Martin Rocek, and Warren Siegel, Superspace: or One Thousand and One Lessons in Supersymmetry (Reading, MA: Benjamin/Cummings, 1983).
Julius Wess and Bruno Zumino, “Supergauge Transformations in Four Fimensions,” Nuclear Physics B 70(1) (1974), 39–50.
See Mohapatra, Unification and Supersymmetry (ref. 174); S. P. Martin, “A Supersymmetry Primer,” arXiv, http://arxiv.org/abs/hep-ph/9709356, September 6, 2011.
Peter C. West, “Supersymmeyry and Finiteness,” in Jackiw et al., Shelter Island II (ref. 53), 127–61.
See Freedman and van Nieuwenhuizen for an accessible explanation of the properties of supersymmetric quantum field theories. Also Howard E. Haber and Gordon L. Kane, “Is Nature Supersymmetric?,” Scientific American (June 1, 1986), 42–50. For a much more technical exposition see Peter C. West, Introduction to Supersymmetry and Supergravity (Singapore: World Scientific, 1986). See also Peter C. West, ed., Supersymmetry: A Decade of Development (Bristol: Adam Hilger, 1986). For the present status of supersymmetry see John Ellis, “The Physics Landscape after the Higgs Discovery at the LHC,” supplement, Nuclear Physics B Proceedings 267–269 (2015), 3–14.
See Schwarz, Superstrings (ref. 114); “String Theory Origins of Supersymmetry” (ref. 114); “String Theory” (ref 114); “Early History of String Theory and Supersymmetry” (ref. 114).
Edward Witten, “Magic, Mystery and Matrix,” in Mathematics: Frontiers and Perspectives. ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur (Providence, RI: American Mathematical Society, 2000), 243–52, on 248.
Bruno Zumino, “Supersymmetry and the Index Theorem,” in Jackiw et al., Shelter Island II (ref. 53), 79–94.
Michael J. Duff, “Superunification and the Seven-Sphere,” Jackiw et al., Shelter Island II (ref. 53), 95–126; Edward Witten, “Fermion Quantum Numbers in Kaluza-Klein Theory,” in Jackiw et al., Shelter Island II (ref. 53), 227–77.
See Schwarz’s presentation of the subject in Ernest M. Henley and Stephen D. Ellis, eds., 100 Years of Subatomic Physics (Hackensack, NJ: World Scientific, 213).
See Zvi Bern, L. J. Dixon, and David Kosower, “Loops, Trees and the Search for New Physics,” Scientific American 306 (May 21, 2013), 34–41.
Philip J. Davis and Reuben Hirsh, The Mathematical Experience (Boston: Birkhausen, 1980).
David Mumford, “The Dawning of the Age of Stochasticity,” in Arnold, Mathematics (ref. 227), 197–218. See also Dirk J. Struik, A Concise History of Mathematics, 3rd rev. ed. (New York: Dover Publications, 1967).
Ernst Breitenberger, “Gauss’s Geodesy and the Axiom of Parallels,” Archive for History of Exact Sciences 31(3) (1984), 273–89.
Witten, “Magic” (ref. 227), 347.
The proof of the theorem makes use of elliptic differential operators on a manifold to study its geometry and topology.
The notes Feynman took of some of the lectures at Shelter Island II have been posted at feynmanlectures.info with the gracious permission of the estate of Richard P. Feynman and the California Institute of Technology. The originals can be found in Box 76, Folder 2, Singer-Index Theorem, in The Feynman Papers at the Caltech Archives. I thank Michael A. Gottlieb for making this possible. I have not attempted to figure out who is the person whose portrait Feynman sketched on the first page of the notes (possibly Bethe?). Feynman’s extended notes in that folder are fascinating as they reveal his meanderings while listening to some of the lectures. It is also interesting to note that among Feynman’s papers in the Archives of Caltech there is a Folder 76.1, whose content are some 100+ pages of calculations trying to prove quark confinement in a 2+1 dimensional Yang-Mills field theory, which seem to extend some of his earlier work (See R. P. Feynman, “The Qualitative Behavior of Yang-Mills theory in 2+1 Dimensions,” Nuclear Physics B 188 (1981), 479–512). They contain some of the materials he presented in his lecture on QCD at Shelter Island II. The notes Feynman prepared for the presentation he made at Shelter Island II are to be found in Box 51, Folder 18, “How QCD works,” in the Feynman papers.
Dan Friedan and Paul Windey, “Supersymmetric Derivation of the Atiyah-Singer Index and the Chiral Anomaly,” Nuclear Physics B 235(3) (1984), 395–416. I thank Scott Axelrod for deciphering Feynman’s notes on Singer’s lecture and for pointing me to Friedan and Windey.
Roman Jackiw and J. Robert Schrieffer, “Solitons with Fermion number 1/2 in Condensed Matter and Relativistic Field Theories,” Nuclear Physics B 190 (1981), 253–61.
Atiyah, “Work of Witten” (ref. 86).
Ibid.
Witten’s work gave rise to a controversy within the mathematics community. A. Jaffe, and F. Quinn, “Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics,” Bulletin of the American Mathematical Society 21 (1993), 1–13 expressed concerned about the rigor displayed in Witten’s “conjectures” and suggested that these results be confined to a new subdiscipline branded as “theoretical mathematics” to safeguard the standards and purity of the traditional mathematical practice. For the response of leading mathematicians and mathematical physicists to Jaffe and Quinn’s proposal see Michael Atiyah et al. “Responses to ‘Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics,’ by A. Jaffe and F. Quinn,” Bulletin of the American Mathematical Society 30(2) (1994), 178–207.
After the paper had been submitted to Physics in Perspective, Leo Corry lent me his copy of Fernando Zalamea, Synthetic Philosophy of Contemporary Mathematics (New York: Synthetic Press, 2012). In it Zalamea describes “contemporary” mathematics, i.e. from 1950 until 2000, analyzes its evolution on its own terms, and contrasts his description and analysis with “contemporary” philosophy of mathematics, theory of culture. It is a wide ranging, very stimulating account of developments in mathematics during the second half of the twentieth century, which merits wide dissemination.
Yang, “My Experience” (ref. 4).
Gerard ’t Hooft, “Nobel Lecture: A Confrontation with Infinity,” Nobelprize.org, December 8, 1999, http://www.nobelprize.org/nobel_prizes/physics/laureates/1999/thooft-lecture.pdf.
Alexander Polyakov, “A View From the Island,” arXiv, http://arxiv.org/abs/hep-th/9211140, November 30, 1992; “Physics of Scales Project,” Dibner Institute, interview with Babak Ashrafi and S. S. Schweber in Princeton, NJ on February 6, 2003.
Edward Witten, “My Interactions with Atiyah and with Singer,” in Yau, Founders of Index Theory (ref. 82).
Zumino, “Supersymmetry” (ref. 113).
Yang, “My Experience” (ref. 4), 13.
Witten, “Interactions with Atiyah” (ref. 247).
Michael Atiyah, “A Personal History,” in Atiyah Collected Works (ref. 86).
Yau, Founders of Index Theory (ref. 82).
Jackiw, “My Encounters” (ref. 76); Diverse Topics (ref. 76).
But see Galison, “Mirror Symmetry” (ref. 116).
But see the previously quoted works of Aubin and Dalmedico for the Institut des Hautes Études Scientifiques in Bures-sur-Yvette (ref. 31).
But see Ludvig D. Faddeev, 40 Years in Mathematical Physics (Singapore: World Scientific, 1995); Ludvig D. Faddeev, “Modern Mathematical Physics: What it Should Be,” in Fokas et al., Mathematical Physics (ref. 77), 1–8.
Yuri I. Manin, “Mathematics as Profession and Vocation,” in Arnold, Mathematics (ref. 227), 153–160, on 155–56.
In a footnote appended to “algebra,” Arnold noted that Viète, the creator of modern algebra, was the cryptographer of Henry IV.
Vladimir Arnold, “Polymathematics: Is Mathematics a Single or Science or a Set of Arts,” in Arnold, Mathematics (ref. 227), 403–17, on 403–4.
Recall that the most serious accident that occurred in a US commercial nuclear power plant, the partial meltdown of the Three Mile Island Unit 2 (TMI-2) reactor located near Middletown, Pa., happened on March 28, 1979.
David E. DeCosse, “A Conclave of Physics,” East Hampton Star, June 8, 1983.
See Gottfried and Weisskopf, and in particular ’t Hooft’s lectures in Gauge Theories (ref. 11).
Acknowledgments
It would not have been possible to write this article without Nicola Khuri making available all the materials in his possession regarding Shelter Island II. I cannot adequately thank him. I am deeply indebted to Roman Jackiw for sharing with me his recollections of Shelter Island II. It gives me great pleasure to acknowledge the consideration and thoughtfulness with which Larry Abbott, Steven Adler, Scott Axelrod, Jeremy Butterfield, Leo Corry, Freeman Dyson, Jeffrey Goldstone, Kurt Gottfried, David Gross, Alan Guth, Matthew Headrick, Arthur Jaffe, Albion Lawrence, Peter Lepage, Roman Jackiw, Howard Schnitzer, Isadore Singer, Shlomo Sternberg, Steven Weinberg, Frank Wilczek, and Edward Witten answered my queries in writing or in interviews. My weekly luncheon meetings with George Smith were stimulating and very effective and valuable in clarifying issues. I am indebted to Chris Bostock for reading the manuscript and for his corrections and very helpful suggestions. I had the benefit of Skuli Sigurdsson carefully reading the manuscript and giving me the benefit of his wide erudition. It is a better paper by virtue of his constructive criticisms and his suggestions. And the same is true of the paper by virtue of Joseph Martin’s and Peter Pesic’s editing. I thank them. I am grateful to Evan Fay Earle, at the Kroch Library at Cornell University, and to Loma Karklinks, at the Cal Tech Archives, for their help in making the Bethe notes and the Feynman notes of the Shelter Island Conferences publically available on the web. As always, the support and encouragement of Snait Gissis were invaluable and the discussions with her regarding contexts and historiography deeply appreciated.
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A condensed version of this essay appeared as the introduction to the reissued Shelter Island II proceedings volume: Roman Jackiw, Nicola N. Khuri, Stephen Weinberg, and Edward Witten, eds., Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and Fundamental Problems of Physics (Mineola, NY: Dover, 2016).
S. S. Schweber is professor of physics and Koret Professor of the History of Ideas, emeritus, at Brandeis University and an associate in the department of the history of science at Harvard University. He is at work on a biography of Hans Bethe.
Appendices
Appendix A
If G is a symmetry group whose elements are expressed in terms of its generators θ a, then \(G = {\kern 1pt} e^{{i\varepsilon^{a} \theta^{a} }}\), where the θ as satisfy the Lie algebra of the group \([\theta^{a} ,\theta^{b} ] = if^{abc} {\kern 1pt} \theta^{c}\), with f abc the structure constants of the Lie algebra; local symmetry means that the ε a become functions of x, ε a(x).
Consider the set of spin ½ matter field described by ψ(x) with ψ(x) transforming as an irreducible representation of G as follows:
Under this transformation, the kinetic energy term in the Lagrangian transforms as follows:
To make the Lagrangian invariant, it is necessary to introduce a spin one field B µ (x) that transforms under G as follows:
If the kinetic part of the Lagrangian is modified so that it becomes L = \(\overline{\psi } \gamma^{\mu } (\partial_{\mu } + B_{\mu } )\psi\), then the matter Lagrangian will be invariant. For B µ (x) to be a full-fledged dynamical field, a G invariant kinetic energy term must appear for it in the Lagrangian. Since the function
transforms covariantly under G(x), TrF μν F μν is G invariant. The full gauge invariant Lagrangian is then
The introduced B field is massless, and thus gives rise to a long-range force, contrary to what is necessary for the description of the strong nuclear forces. How to “break” the symmetry and give the B field quanta a mass was the problem that had to be solved.
Appendix B
For the Lagrangian describing the interaction of a complex scalar field with an abelian massless vector field A μ to be invariant under local gauge transformations requires that it be written in terms of covariant derivatives
The Lagrangian is given by
The particular shape of the potential function U(ϕ) can be left open, except for its having certain minima, but for concreteness we will once again assume it is given by
with μ 2 < 0.
The gauge invariance is now local:
In the case of a local gauge invariance the transformations yield physically equivalent states and the vacuum is not degenerate. A particular choice of gauge allows us to constrain the Higgs field to be a single real field ϕ(x) = v + h(x), with h real and v the value of the minimum of the potential. Inserting this into the Lagrangian yields the result that the A field acquires a mass M A = gv—so that the forces mediated by the A bosons become short ranged.
The situation becomes slightly more complicated in the case of a non-abelian gauge group, especially when attempting to obtain non-perturbative results.263 The Higgs field in the simplest version of electroweak theory has SU(2) × U(1) quantum numbers I = ½ and Y = ½, (the hypercharge Y is related to the electric charge by Q = I 3 +Y). Thus
where both ϕ + and ϕ 0 are complex fields.
The Lagrangian for the Higgs field in the simplest version of electroweak theory is taken to be
V is invariant under the local gauge transformation
where the τ i are Pauli matrices appropriate to the I = ½ representation of the SU(2) group. For the full Lagrangian L to be invariant, ∂ μ has to be replaced by the covariant derivative
The potential has a minimum at \(\phi^{\dag } \phi = \mu^{2} /\lambda = v^{2}\). Local gauge invariance allows the vacuum state to be such that
which corresponds to an electrically neutral vacuum state. The Higgs field configuration is then taken to be of the form
corresponding to small excitations around the vacuum. This choice breaks the SU(2) symmetry. Upon inserting this representation of ϕ into
one finds upon defining
that the charged components of W μ , W 1 μ , and W 2 μ , acquire a mass M W =vg 2, the Z field acquires a mass \(M_{z} = \frac{1}{2}v\sqrt {g_{1}^{2} + g_{2}^{2} }\), but that no A μ A μ term appears in L. Hence the A field describes the electromagnetic field with its massless photons.
Appendix C: Attendees
S. Adler (IAS), D. Amati (CERN), T. Applequist (Yale), J. Bernstein (Stevens IT), H. A. Bethe (Cornell), K. M. Bitar (American U Beirut), E. Brezin (CERN), L. Brown (U Washington), N. Byers (UCLA), N. Cabbibo (Rome U), E. Case (Rockefeller), L. L. Chau (Brookhaven), L. Christ (Columbia), S. Coleman (Harvard), R. Cool (Brookhaven), J. M. Cornwall (UCLA), M. Creutz (Brookhaven), R. Dashen (Princeton), S. Deser (Brandeis), L. Dolan (Rockefeller), S. Drell (Stanford), M. Dresden (SUNY Stony Brook), M. Duff (U Texas), F. Dyson, (IAS), G. Farrar (Rutgers), N. Ferrara (CERN). H. Feshbach (MIT), R. P. Feynman (Caltech), Mary Gaillard (UC-Berkeley), M. Gell-Mann (Caltech), S. Glashow (Harvard), D. Freedman (MIT), M.L. Goldberger (Caltech), J. Goldstone (MIT), A. Goldhaber (SUNY Stony Brook), M. Goldhaber (Brookhaven), D. Gross (Princeton), G. S. Guralnick (Brown), F. Gursey (Yale), A. Guth (MIT), S. W. Hawking (Cambridge), K. Huang (MIT), R. Jackiw (MIT), A. Jaffe (Harvard), R. L. Jaffe (MIT), K. Johnson (MIT), T. Kibble (Imperial College), T. Kinoshita (Cornell), N. Khuri (Rockefeller), W. Lamb (U Arizona), L. Lederman (Fermilab), T. D. Lee (Columbia), A. D. Linde (Lebedev Physical Institute USSR), F. E. Low (MIT), Gloria Lubkin (American Institute of Physics), S. MacDowell (Yale), S. Mendelstam (UC-Berkeley). W. Marciano (Virginia Polytechnical Institute), L. Michel (IHES), J. Muzinich (Brookhaven), Y. Nambu (Chicago), Y. Ne’eman (Tel Aviv U), H. Nicolai (CERN), K. Nishijima (Tokyo), H. Pagels (Rockefeller), L. Pauling (Caltech), M. Perry (Princeton), S. Y. Pi (Boston U), D. Politzer (Caltech), H. Quinn (Stanford), I. I. Rabi (Columbia), P. Ramond (U Florida), C. Rebbi (Brookhaven), B. Sakita (CCNY), N. P. Samios (Brookhaven), S. Schweber (Brandeis), J. Schwarz (Caltech), F. Seitz (Rockefeller), R. Serber (Columbia), L. Singer (UC-Berkeley), A. Sirlin (NYU), W. Slansky (Los Alamos), C. Sommerfield (Yale), P. J. Steihardt (U Pennsylvania), Mr. Walter Sullivan (New York Times), K. Symanzik (DESY), ’t Hooft (Utrecht), T. Tamboulis (Princeton), S. Treiman (Princeton), L. T. Trueman (Brookhaven), G. Veneziano (CERN), E. Weinberg (CERN), S. Weinberg (U Texas), V. Weisskopf (MIT), P. C. West (King’s College), J. A. Wheeler (U Texas), E. P. Wigner (Princeton), P. K. Williams (DOE), E. Witten (Princeton), T. T. Wu (Harvard), A. Zee (U Washington), W. Zimmerman (Max Planck) J. Zinn-Justin (Saclay), G. Zuckerman (Yale), B. Zumino (UC-Berkeley), S. Zhou (Peiking).
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Schweber, S.S. The Shelter Island Conferences Revisited: “Fundamental” Physics in the Decade 1975–1985. Phys. Perspect. 18, 58–147 (2016). https://doi.org/10.1007/s00016-016-0180-5
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DOI: https://doi.org/10.1007/s00016-016-0180-5