Abstract
John Stuart Dowker was born in Sheffield, Yorkshire, on 18 March 1937. His life, therefore, was very much influenced by the Second World War. This is evident as his father died on active service in 1945, after being called up in 1941. His grandfather also died shortly afterwards, so he did not know either of them very well. Nevertheless, it seems that he picked up a positive attitude towards natural sciences as both were technically minded. His mother later provided, often from borrowed money, all the necessary intellectual food in forms of chemistry sets, slide rules and other things that a boy needed to develop his interests. Stuart scored excellently in the 11-plus exam, which was used to decide the type of school a pupil should attend after primary school. Although Stuart was generally allowed to do what he wanted, his mother insisted that he chose King Edward VII Grammar School (KES), the top school in Sheffield at the time. KES allowed Stuart to fully develop his intellectual abilities, and after the S-level exam he received a prestigious state scholarship which allowed him to study at any university in the country.
He picked Nottingham over other possibilities, mainly because of his interest in electronics and because of the relative proximity to his family. In Nottingham, where he stayed from 1955 to 1958, his research concentration turned out to be mostly solid state physics. But with time on his hands, Stuart raided the library and taught himself things like complex analysis and quantum mechanics, with de Broglie's La mécanique ondulatoire [1] as one of his favorites. Remarkably, this book already contains a discussion of quantization on curved configuration spaces, a setting so relevant in Stuart's later career.
Stuart wanted to investigate quantum field theory for his doctoral thesis. So he wrote, among others, to Rudolph Peierls in Birmingham, and, after being interviewed by Peierls himself and J G Valatin, he received an offer of a PhD position. He went to Birmingham in 1958, where his supervisor was Leonardo Castillejo, best known from the Castillejo–Dalitz–Dyson-ambiguity; see [2]. His thesis involved stripping theory, and on this topic he wrote his first ever publication [3]. Takeaways from that time were that sometimes one simply has to do what one is told, and even more importantly, sometimes one has to get on with a calculation and 'just roll through it'. It was also around that period that Stuart started to learn quantum field theory properly following the developments of Feynman.
After his PhD, it was Peierls who helped him to find a position at the University of Pennsylvania in Philadelphia in 1961, which he accepted before he received an offer from CERN. In retrospect, this was probably fortunate as it is where he met his wife, Pwu Yih, to whom he has been married for nearly 50 years. Scientifically he remained somewhat isolated there and spent much of his time learning more about quantum field theory, for example by reading Bogoliubov and Shirkov [4]. His entry point for general relativity was Eddington's The Mathematical Theory of Relativity [5], which together with the works of Julian Schwinger and Bryce de Witt were the most influential ones at that time. A shift of focus to questions in quantum field theory in curved space time took place.
After his stay in Philadelphia, in 1963, helped by connections between Abe Klein and Brian Flowers, he went to the University of Manchester, where he remains to this day. This period was only briefly interrupted by a sabbatical in 1978 to 1979 at Austin following an invitation by Bryce de Witt.
Looking back, it seems there were two major driving forces or principles that determined much of Stuart's selection of research projects. One of them is the method of images, about which he says: '...it intrigued me that one problem (charge + plane) could be got from another (just charge) by geometrical reasoning plus uniqueness. Thomson's book [6] took this further. Chapter 5 is devoted to the image and inversion methods and I must have read this closely as there are lots of marginal notes...' The second principle can be read off from the following quote: 'After reading Eddington circa 1960 it was clear to me (and others of course) there is a strong analogy (at least) between gravitation and electromagnetism ... (His work has very strongly influenced me.) So I played a game of asking for the gravitational analogues of existing electromagnetic concepts. The basic analogue is between field strength/charge and curvature/spin... [in that] ...spin, in general relativity, plays the passive role that charge plays in electromagnetism in the sense that it is the spin–curvature coupling that knocks a particle off a geodesic.' It is quite amazing how much of Stuart's work can be traced back to these principles. This is briefly explained in the following by describing some of his most important works.
His most cited work [7] fits this bill and can be seen as doing what Julian Schwinger did in [8], for a constant electromagnetic field in the gravitational setting. In more detail he noticed Schulman's work on propagators on the three sphere [9], which he extended to Lie groups; see [10, 11]. He then noticed that having exact propagators, work like Schwinger's could be done and de Sitter space was a natural 'curved' candidate. The paper is best known for the mathematical technique introduced, namely for the zeta function method much used since for the computation of singular quantum field theoretic quantities like effective actions and the Casimir energy. The motivation for introducing this scheme goes back to reading an article of I M Gel'fand about some number theory problem involving zeta functions, image sums, propagators etc. How could an object occurring with other objects physicists were using all the time not be useful? It turned out to be very useful, although in this paper the method was only introduced but actually not used! This is probably the reason that his paper did not receive the same recognition as the one by Hawking [12], where zeta function regularization was mentioned in the title and where it was actually applied to examples in the article. Of course, Stuart's and Raymond's paper hit the pulse of the time in that after Stephen Hawking's announcement about Hawking radiation [13], the quantum field theory in curved space-time frenzy began. He had been interested in that subject for a while, the most important influence being de Witt's Les Houches lectures and also Chris Isham, who introduced him to Dennis Sciama's Oxford group, including Philip Candelas and Derek Raine, in 1973. The project about de Sitter space got somewhat delayed by work on path integrals and the ordering problem until Ray Critchley came by looking for a PhD topic.
Several more of his best known papers are in the context of quantum field theory in curved spacetime. In the article [14], conformal transformations play a fundamental role. They are used to transform static manifolds to ultra-static ones where a high-temperature expansion of the effective action can be done. The result has since been rediscovered many times for special cases. In the process they also showed results about the conformal transformation of heat kernel coefficients claimed later on by mathematicians.
The paper by Kennedy, Critchley and Dowker [15] belongs in the same context. Deutsch and Candelas had shown the occurrence of non-integrable singularities near boundaries of the Casimir energy densities, which made it impossible to obtain global energies by naively integrating local quantities. To resolve that problem, the needed surface counterterms for an arbitrarily shaped smooth boundary in curved space were computed.
How does the paper [16] fit into the general scheme? The seed to considering this topic was probably planted when Yakir Aharonov was visiting Birmingham in 1958. In the Aharonov–Bohm effect we have an electromagnetic field with finite extension that impacts particles never entering that region. What is the gravitational analogue for that situation? The analogue concerns the impact a localized curvature has, and the cone is an excellent example to shed light on that question.
Related to the method of images, Stuart has done an enormous amount of work on the influence of topology and curvature on quantum field theory. An example is [17], where the vacuum stress–energy tensor for Clifford–Klein forms of the flat or spherical type were computed.
Another strand we would like to mention is Stuart's interest in higher spin equations. In [18], Steven Weinberg wrote down a set of higher spin equations that took his fancy. They involved angular momentum theory, which has always pleased Stuart, and the description was an alternative to Roger Penrose's use of two-spinors. Investigating the inconsistencies that arose on coupling to gauge theories, Stuart extended the classic results in [19], from electromagnetism to gravity in accordance with his general philosophy; see, e.g., [20, 21, 22].
Lately, Stuart is best known for his many applications in the context of zeta function regularization and its applications to quantum field theory under external conditions and spectral theory. He can be considered the world expert on particular case calculations with a knowledge of the literature, old and recent, that is not seen very often and which originated in the many hours spent at different (mostly British) libraries. His attitude towards explicit computations is nicely summarized by himself: 'I have always been interested in exact solutions, even if unphysical, so long as they are pretty. They seem to be working mechanisms that fit together, complete in themselves, like a watch.'
The following issue in honour of Stuart's 75th birthday contains contributions that touch upon the various topics he has worked on.
References
[1] de Broglie L 1928 La mécanique ondulatoire (Paris: Gauthier-Villars)
[2] Castillejo L, Dalitz R H and Dyson F J 1956 Low's scattering equation for the charged and neutral scalar theories Phys. Rev. 101 453
[3] Dowker J S 1961 Application of the Chew and Low extrapolation procedure to K- + d → Y + N + π absorption reactions Il Nuovo Cimento 10 182
[4] Bogoliubov N N and Shirkov D V 1959 Introduction to the Theory of Quantized Fields (New York: Interscience)
[5] Eddington A S 1923 The Mathematical Theory of Relativity (Cambridge: Cambridge University Press)
[6] Thomson J J 1909 Elements of Electricity and Magnetism 4th edn (Cambridge: Cambridge University Press)
[7] Dowker J S and Critchley R 1976 Effective Lagrangian and energy momentum tensor in de Sitter space Phys. Rev. D 13 3224
[8] Schwinger J 1951 On gauge invariance and vacuum polarization Phys. Rev. 82 664
[9] Schulman L S 1968 A path integral for spin Phys. Rev. 176 1558
[10] Dowker J S 1970 When is the sum over classical paths exact? J. Phys. A: Math. Gen. 3 451
[11] Dowker J S 1971 Quantum mechanics on group space and Huygens' principle Ann. Phys. 62 361
[12] Hawking S W 1977 Zeta function regularization of path integrals in curved space-time Comm. Math. Phys. 55133
[13] Hawking S W 1974 Black hole explosions Nature 248 30
[14] Dowker J S and Kennedy G 1978 Finite temperature and boundary effects in static space-times J. Phys. A: Math. Gen. 11 895
[15] Kennedy G, Critchley R and Dowker J S 1980 Finite temperature field theory with boundaries: stress tensor and surface action renormalization Ann. Phys. 125 346
[16] Dowker J S 1977 Quantum field theory on a cone J. Phys. A: Math. Gen. 10 115
[17] Dowker J S and Banach R 1978 Quantum field theory on Clifford–Klein space-times. The effective Lagrangian and vacuum stress-energy tensor J. Phys. A: Math. Gen. 11 2255
[18] Weinberg S 1964 Feynman rules for any spin Phys. Rev. 133 B1318
[19] Fierz M and Pauli W 1939 On relativistic wave equations for particles of arbitrary spin in an electromagnetic field Proc. Roy. Soc. A 173 221
[20] Dowker J S and Dowker Y P 1966 Particles of arbitrary spin in curved spaces Proc. Phys. Soc. (Lond.) 87 65
[21] Dowker J S and Dowker Y P 1966 Interactions of massless particles of arbitrary spin Proc. Roy. Soc. A 294 175
[22] Dowker J S 1972 Propagators for arbitrary spin in an Einstein universe Ann. Phys. 71 577