Dirac first made it clear that he was using the principle of mathematical beauty in the late 1940s, when he dismissed the renormalised theory of photons and electrons on the grounds that it was too ugly. He was, however, unable to use his principle constructively, to build new theories. It could therefore be argued that Dirac’s passion for beauty was to some extent destructive, but he knew no other way: he was temperamentally unable to focus on any other subject in particle physics until he had found a truly beautiful theory of electrons and photons, without the disfiguring infinities.
A way out of this alleged flaw in quantum field theory arrived, tragically, just too late for him: a particularly promising, infinity-free theory of electrons and photons began to circulate among theoreticians in the autumn of 1984, as he lay dying. Michael Green, of the University of London, and John Schwarz, of Caltech, had written a crucial paper showing that string theory might be able to form the basis of a unified theory of fundamental interactions.22 Previously, the theory appeared to say that the weak interactions must have perfect left–right mirror symmetry, contrary to experimental evidence. By proving that the theory can naturally describe the breaking of this symmetry, and by resolving other embarrassing anomalies in the theory, Green and Schwarz began a revolution. Within weeks, string theory was the hottest topic in theoretical physics. Although the theory was far from complete – it was really a collection of inchoate concepts, all in need of development – there were strong signs that it contained the seeds of an exciting new framework for giving a unified account of all the fundamental interactions, encompassing the Standard Model and Einstein’s general relativity.
The new theory describes nature not in terms of point-like particles but of pieces of string, so small that if they could be aligned end to end, it would take a billion billion of them to span a single atomic nucleus. In this picture of the fundamental constituents of the universe, there is only one fundamental entity – the string – and every type of particle, including the electron and the photon, is simply an excitation of the string, analogous to a mode of vibration of a tuning fork.23 The mathematics of the theory is fearsome, but underneath the complexities is a modern version of John Stuart Mill’s desideratum of fundamental physics: a unified description of all the fundamental interactions.
What would surely have impressed Dirac is that modern string theory has none of the infinities he abhorred. He would have revelled in the mathematical beauty of the theory, which delights not only the physicists who use it but also many mathematicians who have mined it for new concepts. It has turned out that string theory, much like the Dirac equation, is a fertile source of purely mathematical ideas that have a value for their own sake, not just as tools to understand nature. Dirac often said that he was interested in theories only as ways of accounting for nature, but he would probably have been intrigued to see, at the heart of string theory, mathematics known as complex projective geometry, a generalisation of his favourite branch of geometry.24
No one has done more to shed light on string theory than the mathematical physicist Edward Witten, at the Institute of Advanced Study. In 1981, when he was a lecturer at the Erice summer school and thirty years old, he met Dirac briefly and heard his familiar condemnation of renormalisation but chose not to follow his advice. Dirac followed Witten’s work and, in 1982, wrote – in his trembling hand – to the Papal Academy, supporting Witten’s nomination for a special award and describing his mathematical work as ‘brilliant’.25 From the early 1980s, Witten’s reputation among string theorists has been comparable to Dirac’s among quantum theorists half a century before.
Witten believes that string theory seems to be the kind of theory that Dirac had in mind when he argued that a revolution was needed to produce a new theory free of infinities so that renormalisation was not needed:
In some ways Dirac’s reaction to renormalization was vindicated because the better theories he said he wanted were eventually developed, with the advent of string theory. But by far the most progress towards the new theory was made by physicists who used and studied renormalization. So you’d have to look at the outcome for Dirac as bittersweet: he was partly right, but his approach was not entirely pragmatic.26
It is hard to disagree with this tactfully expressed judgement about Dirac’s principled but counterproductive attitude to renormalisation. If he could have shed some of the insistence on rigour that he learned as a student of pure mathematics and been able to retain some of the pragmatism he learned when training to be an engineer, his achievement would, in all likelihood, have been even greater. Perhaps, if he had been more active in quantum field theory, it would have advanced more quickly, and modern string theory would have arrived sooner.
Although string theory is the only strong candidate for a unified theory of the fundamental interactions, by no means all theoreticians are convinced of its value. A substantial number of physicists worry that the theory makes sense only in more than four dimensions of space-time (it is easiest to formulate in ten or even eleven dimensions). More worrying, it has received little support from experiment: string theory has yet to make a clear-cut prediction that experimenters have been able to test. These are among the key signals, several physicists have argued, that the theory is absurdly overvalued and that it would be better to pursue other avenues. One of the most vocal sceptics is the Standard-Model pioneer Martin Veltman: ‘String theory is mumbo jumbo. It has nothing to do with experiment.’27
But it is clear from the comments Dirac repeatedly made in his lectures on the way theoretical physics should be done that he would have disagreed with these criticisms: he would have counselled string theorists to let the theory’s beauty lead them by the hand, not to worry about the lack of experimental support and not to be deterred if a few observations appear to refute it. But he would have cautioned string theorists to be modest, to keep an open mind and never to assume that they are within sight of the end of fundamental physics. If past experience is anything to go by, another revolution will follow eventually.
Such was the advice this extraordinarily unemotional man offered to his colleagues: be guided, above all, by your emotions.
Notes - Chapter thirty-one
Chapter thirty-one
1 Weinberg wrote these words for me to read aloud at the Centenary meeting. Text checked by Weinberg, 22 July 2007 (e-mail).
2 Interview with Freeman Dyson, 27 June 2005.
3 Quoted in Charap (1972: 332).
4 E-mail from Sir Michael Atiyah, 15 July 2007.
5 Woolf (1980: 502).
6 Letter from Dirac to Abdus Salam, 11 November 1981, reproduced in Craigie et al. (1983: iii).
7 ’t Hooft (1997: Chapter 14).
8 Stephen Hawking appeared in an episode of Star Trek first broadcast on 21 June 1993, and in episodes of The Simpsons first broadcast on 9 May 1999 and 1 May 2005.
9 Letter from Nicolas Kurti to New Scientist, 65 (1975), p. 533; letter from E. C. Stern (1975) to Science, 189, p. 251. See also the comments by Dalitz in ‘Another Side to Paul Dirac’, in Kursunoglu and Wigner (1987: 87–8).
10 Freimund et al. (2001). The Kaptiza–Dirac effect had been observed for atoms, but not for electrons, in 1986 (Gould et al. 1986). I thank Herman Betelaan for his advice on modern experiments on the effect.
11 Deser (2003: 102).
12 Interview with Nathan Seiberg, 26 July 2007, and e-mail, 20 August 2007.
13 In his interviews, Leopold Halpern often stressed the importance to Dirac of the large numbers hypothesis (interview with Halpern, 26 February 2006).
14 By conventional measure, the gravitational force is a millionth of a billionth of a billionth of a billionth the strength of the next strongest fundamental force, the weak interaction.
15 Rees (2003). I thank Martin Rees for his advice on the status of Dirac’s large numbers hypothesis.
16 E-mails from James Overduin, 20–2 July 2006.
17 Overduin and Plendl (2007).
18 I thank Rolf
Landua of CERN for his expert help on the current state of experimental research into anti-matter.
19 See Yang (1980: 39).
20 These words, written on 27 November 1975, seem to have been special to Dirac. He wrote them on a single sheet of paper and filed them among his lecture notes: Dirac Papers 2/29/17 (FSU). The words replaced by [this happened] are ‘I have felt the mathematics lead me by the hand.’
21 The first reference to beauty in Dirac’s papers appears to be in the paper he co-wrote with Kapitza in 1933, ‘The Reflection of Electrons from Standing Light Waves’, where they refer to the beauty of the colour photography introduced by Gabriel Lippmann.
22 Green and Schwarz’s paper was received on 10 September 1984 by the academic journal Physics Letters B, which published it on 13 December.
23 For a popular account of modern string theory, see Greene (1999).
24 Dirac told his student Harish-Chandra, ‘I am not interested in proofs but only in what nature does’: Dalitz and Peierls (1986: 156).
25 Dirac’s notes commend Witten’s ‘brilliant solutions to a number of problems in mathematical physics’, Dirac Papers, 2/14/9 (FSU).
26 Interview with Edward Witten, 8 July 2005, and e-mail, 30 August 2006.
27 E-mail from Veltman, 20 January 2008. For a sceptical assessment of string theory, see Woit (2006), especially Chapters 13–19.
Dirac family, 3 September 1907
Paul Dirac, 17 August 1907
Left to right: Felix, Betty and Paul Dirac c.1909. A French grammar book rests on Paul’s lap.
Technical drawing by Paul Dirac at Bishop Road School, Bristol, 9 December 1913
Bristol University Engineering Society’s visit to Messrs Douglas’ Works, Kingswood, 11 March 1919. Dirac is in the front row, fourth from the right.
Charles Dirac, c.1933
Felix Dirac, 1921
6 Julius Road, Bristol, here Dirac lived with his family from April 1913 until he left for Cambridge in 1923. He regularly returned home and began his work on quantum mechanics in his bedroom here.
Max Born (seated, central) with everal younger colleagues at his home in Göttingen, spring 1926. Dirac is, as usual, diverted. Oppenheimer is in the back row, fourth from the left.
Some members of the Kapitza Club, after a meeting c.1925, in the room of Peter Kapitza, Trinity College, Cambridge. Kapitza is directly beneath the drawing of a crocodile on the easel.
Patrick Blackett and Paul Ehrenfest, c.1925
Isabel Whitehead with her husband Henry, and their son Henry, 1922.
Dirac (standing close to the doorway) at a meeting in Kazan, Russia, 12 October 1928
Left to right: Heisenberg’s mother, Schrödinger’s wife, Flo Dirac, Dirac, Heisenberg and Schrödinger. They have just arrived at Stockholm railway station, 9 December 1933, for the Nobel celebrations.
Extract from a letter from Dirac to his friend Manci Balazs, 9 May 1935
Dirac and Manci on their honeymoon, Brighton, January 1937
The Dirac family in the garden of their Cambridge home, c.1946. Left to right: Dirac, Monica, Manci, Gabriel, Mary and Judy.
Dirac and Manci (on the far left) with a party during a crossing of the Atlantic on the SS America, 2 April 1963
Dirac and Richard Feynman at a conference on relativity, Warsaw, July 1962
Dirac at the Institute for Advanced Study, Princeton, c.1958
The Diracs’ home in Tallahasse, 223 Chapel Drive
Kapitza and Dirac at the Hotel Bad Schachen, Lindau, summer 1982
One of the last photographs taken of Dirac, Tallahassee, c.1983
Abbreviations in Notes
AHQP Archives for the History of Quantum Physics, multiple locations, provided by Niels Bohr Library & Archives, American Institute of Physics, College Park, Maryland., USA (http://www.amphilsoc.org/library/guides/ahqp/).
AIP American Institute of Physics, Center for the History of Physics, Niels Bohr Library, Maryland, USA.
APS Archive of the American Philosophical Society, Philadelphia, USA.
BOD Bodleian Library, University of Oxford, UK.
BRISTU Bristol University archive, UK.
BRISTRO Bristol Records Office, UK.
CALTECH California Institute of Technology, archive, USA.
CHRIST’S Old Library, Christ’s College, Cambridge University, UK.
CHURCHILL Churchill Archives Centre, Churchill College, Cambridge University, UK.
DDOCS Dirac letters and papers, property of Monica Dirac.
EANGLIA Tots and Quots archive, University of East Anglia, Norwich, UK.
FSU Paul A. M. Dirac Papers, Florida State University Libraries, Tallahassee, Florida, USA. All of the letters Dirac’s mother wrote to him are in this archive.
IAS Institute for Advanced Study, archive, USA.
KING’S King’s College, Cambridge; unpublished writings of J. M. Keynes.
LC Library of Congress, Collections of the Manuscript Division.
LINDAU Archive of Lindau meetings, Germany.
NBA Niels Bohr Archive, at the Niels Bohr Institute, Copenhagen.
PRINCETON Eugene Wigner Papers, Manuscripts Division, Department of Rare Books and Special Collections, Princeton University Library, USA.
ROYSOC Archives of the Royal Society, London, UK.
RSAS Royal Swedish Academy of Sciences, Center for History of Science, Stockholm.
SOLVAY Archives of the Solvay Conferences, Free University of Brussels, Belgium.
STJOHN St John’s College archive, Cambridge, UK.
SUSSEX Crowther archive, Special Collections at the University of Sussex, UK (the university holds the copyright of the archive).
TALLA Dirac archive at the Dirac Library, Florida State University, USA, http://pepper.cpb.fsu.edu/dirac/diracFA (HTML)_011. htm.
UCAM University of Cambridge archive, UK.
UKNATARCHI National Archives of the UK, Kew.
WISC University of Madison, Wisconsin, archives, USA.
1851COMM Archives of the Royal Commission of 1851, Imperial College, London, UK.
Notes
Prologue
1 A version of the ‘more people who prefer to speak than to listen’ remark, one of Dirac’s favourites, is cited by Eugene Wigner in Mehra (1973: 819).
2 Dirac made the ‘God is a mathematician’ remark in his Scientific American article in May 1963.
3 The quote from Darwin is taken from Part VII of his autobiography. The words were written on 1 May 1881.
4 The author of the quote relating to Shakespeare was the late Joe Lannutti, a leading member of the Physics Department at Florida State University when Dirac arrived. The source of the quote is Peggy Lannutti, interview 25 February 2004. Lannutti also tells the story in J. Lannutti (1987) ‘Eulogy of Paul A. M. Dirac’ in Taylor (1987: 44–5).
5 This account is taken from interviews with Kurt Hofer on 21 February 2004 and 25 February 2006, and many subsequent e-mails. The account was checked in detail via e-mails on 22 September 2007. Hofer’s recollections are consistent in every detail with the account given by Dirac in Salaman and Salaman (1986), in his interview, AHQP, 1 April 1962 (pp. 5–6), and in the account he gave of his early life to his friends Leopold Halpern and Nandor Balázs. I spoke to these former colleagues of Dirac on 18 February 2003 and 24 July 2002, respectively. Dirac’s wife gives her recollections of his experiences at the dining table in her letter to Rudolf Peierls, 8 July 1986, Peierls archive, additional papers, D23 (BOD).
Chapter one
1 Letter from André Mercier to Dirac and his wife, 27 August 1963, Dirac Papers 2/5/10 (FSU).
2 Interview with Dirac, AHQP, 1 April 1962, p. 5.
3 Dirac Papers 1/1/5 (FSU), see also the records of the Merchant Venturers’ School in BRISTRO.
4 See, for example Jones (2000:
Chapter 5).
5 Pratten (1991: 8–14).
6 Although Flo lived in Cornwall only briefly, she would later insist that she was not English but Cornish. Source: interview with Christine Teszler, 22 January 2004.
7 Flo Dirac mentions this in an undated letter to Manci Dirac, written in early February 1940 (DDOCS). By 1889, when Richard Holten was fifty, he was captain of the 547-ton Augusta.
8 Richard Holten was aware that official documents often name his wife as the head of the family. His sailing record is in ‘They Sailed Out of the “Mouth”’ by Ken and Megan Edwards, microfiche 2001, BRISTRO, FCI/CL/2/3. See also Holten’s Master’s certificates, stored in the archives at the National Maritime Museum, Greenwich, London, UK.
9 The details of Charles and Flo’s early life together are in Charles’s documents in Dirac Papers 1/1/8 (FSU).
The Strangest Man Page 64