The Erwin Schrodinger¨ International Boltzmanngasse 9/2 Institute for Mathematical Physics A-1090 Vienna, Austria

ESI NEWS Volume 2, Issue 1, Spring 2007

Walter Thirring and the elected Thirring as its president and took Contents the formal decision to set up a research in- Erwin Schrodinger¨ Walter Thirring and the Erwin Schrodinger¨ stitute under the legal framework of the so- Institute 1 Institute ciety. Following Walter by Jakob Yngvason 1 The Erwin Schrodinger¨ International In August of 1990 Alexander Vinogradov Institute for Mathematical Physics (ESI) Walter Thirring – A Short Biographical Sketch by Wolfgang L. Reiter 2 sent a letter to Peter Michor in Vienna with started its operation in January 1993 under the suggestion to set up a research insti- the presidency and scientific directorship of Walter Thirring’s Arrival in Vienna by Her- bert Pietschmann 5 tute devoted to mathematics and physics Walter Thirring with Peter Michor as ex- in Vienna. Setting up such an institution ecutive director, and was opened officially Walter Thirring and Austria’s accession to CERN by Wolfgang Kummer 6 was seen as a potentially valuable contri- on April 20, 1993, under the auspices of bution at this time of crisis for the Eastern Vice Chancellor and Minister for Science Walter Thirring’s Path into Mathematical Physics by Heide Narnhofer 6 European scientific community: based on and Research, Erhard Busek. the cultural and scientific tradition in Vi- Unmathematical and Mathematical Physics In 1998 Walter Thirring retired as Pres- by Bernhard Baumgartner 8 enna, especially in the field of mathemat- ident of the ESI and became its Honorary ical physics, a new institute based in Vi- Gravitational Matters by Helmuth Urbantke 9 President. This did not in any way dimin- enna could provide a focal point for both ish his immensely valuable contributions: Guidelines for Students of Theoretical Eastern and Western science and an inter- Physics by Walter Thirring 10 even now, at the age of 80, Walter Thirring national platform for research in the field continues to suggest new initiatives and di- Walter Thirring: Probing for Stability by of mathematical physics. Harald A. Posch 11 rections for the Institute and its scientific This initiative was warmly welcomed programme. Almost 50 Years of the Thirring Model by Harald Grosse 12 by Walter Thirring. In a letter to the Minis- ter of Science and Research, Erhard Busek, This special issue of A Crossword Puzzle Without Clues by Peter ESI NEWS devoted to C. Aichelburg 13 dated October 18, 1990, Thirring proposed to establish the ‘Erwin Schrodinger¨ Insti- Walter Thirring on the Doing Physics with Walter by Elliott Lieb 14 tute for Mathematical Physics’ in Vienna. occasion of his 80th My Contacts with Walter Thirring by Angas Thirring’s proposal immediately won the birthday is a small to- Hurst 15 support of eminent scientists all over the ken of our gratitude Birthday Greetings to Walter Thirring by world, and Busek responded favourably in for almost 17 years of Ernest M. Henley 16 December 1990. work, support and ad- ESI News 16 The society (‘Verein’) ‘Internationales vice for the Institute, and for many years of personal friendship and scientific stimula- Current and Future Activities of the ESI 17 Erwin Schrodinger¨ Institut fur¨ Mathema- tion. Thirringfest 18 tische Physik’ was officially founded in April 1992, and on May 27 the consti- Klaus Schmidt, Joachim Schwermer and tutional general assembly of this society Jakob Yngvason

Following Walter retical physics at the University of Vienna. of the kinetic and Coulomb energies, com- Walter’s style, both in his profession and paring the result also with the analogous Jakob Yngvason his life in general is so unique that it would calculations for Bosons and also for gravi- be preposterous even to try to imitate it. But tating particles to bring out the differences. My first encounter with his papers, books and lectures continue to He then commented on the justly famous Walter was in 1976. This be a standard of reference for the assess- work of Dyson and Lenard from the year was not a personal meet- ment of any work in mathematical physics 1967 where stability of matter was proved ing though, I was simply and have been a constant stimulus for my for the first time but with an absurdly large 14 in the auditorium of the own work. constant of the order 10 in the lower Winter School in Schlad- Coming back to the Schladming lec- bound to the energy. As Walter remarked, ming where he lectured tures of 1976 I recall how I liked the way the large constant was solely due to the tour on Stability of Matter. Walter presented the case for Stability of de force method of proof that consisted of Little did I know then that 20 years later Matter. He did not jump right into the math- a long sequence of estimates, about 45 in I would be in Vienna as his successor. ematics but explained first the importance all, loosing a factor of 2 on average each Of course, it is not possible to succeed of the question and then started off with the time. If you do this 45 times, Walter ex- 14 Walter except in the trivial sense of a tem- ‘private room’ argument for the electrons plained, then you are close to 10 . After poral order in a list of professors of theo- and did the ‘back of an envelope’ estimate these explanations he proceeded with his 2 Volume 2, Issue 1, Spring 2007 ESI NEWS and Elliott Lieb’s new and beautiful proof this is one of the differences in style I men- on problems to which these methods can based on Thomas-Fermi theory that goes tioned before. be applied. In disciplines that rely heavily right to the heart of the matter and produces There is one aspect of Walter’s work- on expensive experimental equipment this a reasonable constant. This work is a true ing style that I sometimes wish I could im- strategy may even be perfectly justified for masterpiece of 20th century mathematical itate: his extreme discipline and organiza- a limited amount of time. But in physics the physics and its extensions and refinements tion. I understand that when he had admin- true aim of the game is to understand how continue to be active research topics. istrative duties at the University he man- nature ticks at its fundamental level and for Walter’s textbooks on Mathematical aged them very effectively and kept them in this task the methods have to be chosen to Physics are marvellous works and the time slots well separated from other things. fit the questions attacked. denseness of information and insights they In this way he had plenty of time for the provide is truly remarkable. One reason for things he likes most: doing physics, play- Walter has worked in Quantum Field this is Walter’s awesome skill in distill- ing the organ and composing music. Theory, Elementary Particle Physics, Sta- ing a complicated mathematical argument Characteristic for Walter’s scientific tistical Physics and General Relativity and into a few lines. Sometimes the result has work is the breath of topics he has been in- made seminal contributions to all these the character of ‘spiritus concentratus’ that terested in and worked on. In this respect fields. He is one of the pioneers of mod- must be diluted again to bring it back to he is for me the model case of a mathe- ern mathematical physics, to which I would drinking strength; this is in fact what I have matical physicist. One often distinguishes also count people like Rudolf Haag, Arthur done on several occasion when lecturing to between ‘method oriented’ and ‘problem Wightman, Hans Borchers, Ludwig Fad- students. Such elaborations have pedagog- oriented’ approaches in science and some- deev, Elliott Lieb, Joel Lebowitz and David ical merits but I sometimes wonder if I am times mathematical physics is regarded as Ruelle, among others, who in the 1960’s not depriving the students of the experience being as a discipline solely defined by its realized that there are deep questions in of being exposed directly to the hard stuff. methods. I regard this is a misunderstand- physics that require deep mathematics for Also, Walter’s method takes less time and I ing. In any field there are practitioners their proper understanding and were pre- have never in my courses managed to cover that have acquired some skills in a spe- pared to learn what it takes whenever nec- a comparable amount of material as Wal- cific method and spend most of their pro- essary. Carrying on with this tradition is the ter has apparently done in his courses. But fessional life in searching for and working legacy for Walter’s followers.

Walter Thirring – A Short elitist and socially oriented Catholic ‘Neu- now Walter’s turn to close the gap in the landschule’. The adolescent boy was soon family’s chain of physicists. Harald did not Biographical Sketch confronted with a radical change of society return from the Russian battle field. Wolfgang L. Reiter ‘which can be arbitrarily deformed in the Walter had had in mind a quite differ- hands of a demagogue’.2 A few days after ent professional career: he wanted to dedi- the occupation of Austria by Nazi Germany cate his life to music, playing the piano and ‘I never wanted to in March 1938 his father Hans Thirring composing. But the brother’s letter from write an autobiogra- (1888-1976) was dismissed from his aca- the eastern front overruled Walter’s love for phy. How despicable demic post. The Gestapo was since then a music. Responsibility for the family and its those nostalgic war frequent visitor of the Thirring’s flat. tradition, ‘Verantwortungsethik’ according stories of old men ap- ‘My brother three years older in age to Max Weber’s analysis, was the driving peared to me prov- had always been considered a great genius force — or was it the father’s order uncon- ing what hell-raisers while I was just the small appendage’.3 In sciously executed by the brother (here we they had been in their a letter to Walter written from the Russian are in genuine Freudian waters) setting the youth’.1 With these words Walter Thirring front his brother Harald (1924-1945?) ex- stage for Walter becoming a physicist? opens a biographical sketch of his early pressed no hope of surviving the war. He During the last days of World War II days overshadowed by the invasion of Nazi charged his younger brother to keep the sci- the Thirrings left Vienna for their sec- barbarism. He was born on April 29th, entific tradition of the family.4 Without be- ond home at Kitzbuhel¨ not far from Inns- 1927, in Vienna into a Protestant fam- ing able to attend school classes, because bruck, where Walter began his studies ily originating from Thuringia (the family in the meantime he had already been con- with the theoretical physicist Arthur March name Thuringer¨ finally became Thirring). scripted as a ‘Flakhelfer’ (there he exer- (1891-1957) lacking a proper final exam His ancestors came to Austria in 1623 as cised his mathematical abilities by misdi- (Matura). In 1949 he got his PhD from religious refugees during the times of the recting the guns until noticing that this was the University of Vienna with distinction. Thirty Years’ War and later settled in the to no avail) he was provided by his fa- The topic of his thesis was the Dirac small Hungarian town of Sopron. Hun- ther with the then standard textbook for equation (‘Zur kraftefreien¨ Bewegung nach gary was then and later a safe haven for theoretical physics, the ‘Joos’, a superb der Dirac-Gleichung’). Paul Urban (1905- Protestants, since the Transleithanian part choice as Thirring remarks.5 The 17 year 1995), assistant at the Institute for The- of the Habsburg empire did not submit old boy started off to follow the family’s oretical Physics at the University of Vi- to the cuius regio eius religio rule and orders issued by his brother. Following enna, raised Thirrings interests in meson the Counter Reformation. His grandfa- his grandfather Ludwig Julius who stud- theory, at that time the talk of the town ther Ludwig Julius Thirring (1858-1941) ied with Charles Hermite (1822-1901) in among physicists, by engaging him in bor- moved to Vienna in 1878 to study mathe- Paris and his father Hans, student of Boltz- ing perturbation theory calculations of var- matics and physics. mann’s pupil Fritz Hasenohrl¨ (1874-1916) ious cross sections. When he asked if there While being of Protestant denomina- and since 1927 full professor of theoretical were any experimental data available to tion Walter attended the progressive, non- physics at the University of Vienna, it was check and verify his calculations he had to

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 20073 be disappointed by his tutor. a Hitler refugee in 1938 and years spent called for new instruments for research in After having received his PhD Thirring in Mexico and later in the US (Columbia Europe: ‘The real problem of science in Eu- was granted a scholarship at the Dublin In- University, Brookhaven, Florida), she and rope seems to me to be how we shall be able stitute of Advanced Studies, where he met Thirring formed a group of young stu- to cope with these challenging fields which Erwin Schrodinger¨ (1887-1961), a close dent researchers to analyse bubble chamber require large-scale collaboration. The in- friend of his father from the times of their photographs from CERN. This initiative dividual countries are too small to deal studies with Fritz Hasenohrl.¨ The follow- was a very first bridge between experimen- effectively with these branches of science ing year he spent as a Fellow at Glas- talists at CERN and physicists in Vienna and it will be difficult to co-ordinate the gow University and during the academic and thus the starting point of high energy intentions and desires of various countries year 1950/51 he held an assistantship at the physics in Austria. Subsequently, the Insti- and various scientific disciplines. Without Max-Planck-Institut in Gottingen,¨ where tute of High Energy Physics of the Austrian something like a European Research Coun- he worked with Werner Heisenberg (1901- Academy of Sciences was founded in 1966 cil [my emphasis] which plans and super- 1976). Until his first position as a profes- and became the home base for Austria’s vises science as well as directs the spend- sor of theoretical physics at the Univer- activities at CERN, actively supported by ing of the research money on a European sity of Bern he was ‘on the road’: 1951/52 Thirring as long term chairman of the board level, I can hardly see that we will get any- 7 as UNESCO Fellow at the ETH Zurich,¨ of the Institute. In 1966 Thirring was made where’. It took quite some time until, in where he met the caustic Austrian Wolf- corresponding and in 1967 full member of 2006, Thirring’s proposal became reality gang Pauli (1900-1958), 1952/53 assis- the Austrian Academy of Sciences. (albeit in a different form) with the foun- tant at the University of Bern. In 1953/54 Thirring’s close ties to high energy dation of the European Research Coun- he worked at the Institute for Advanced physics found a new stage when he be- cil within the framework of the European Study in Princeton and met Albert Ein- came member of the Directorate of CERN Community. The incisiveness of his intel- stein (1879-1955). In 1952 Thirring mar- as head of the Theoretical Department for lect had seen a pressing need long before, ried Helga Georgiades. The couple has two the period 1968 – 1971, a time of impor- but the realisation of his proposal needed a sons, Klaus and Peter, and four grandchil- tant decisions to be made. The main tasks at further step of unification and collaboration dren have since enlarged the family. that time were to get the intersecting stor- in Europe transgressing the political model of CERN. In 1954 he returned to Europe as a lec- age ring ISR, based on a concept of the turer at the University of Bern. The fol- Austrian-born physicist Bruno Touschek Thirring started his academic career by lowing years 1956/57 he was Visiting Pro- (1921-1978), operational and to negotiate and large as a theoretical elementary parti- fessor at MIT and 1957/58 at the Uni- the siting of the next generation accelera- cle physicist. Interestingly enough by the versity of Washington in Seattle. During tor, the 300 GeV SPS, which had already mid 1960’s he shifted his style from tra- this time of academic tramping Thirring been decided on in 1964 under the direc- ditional theoretical physics to a stronger was intensively engaged with problems in torship of another Austrian-born physicist, mathematically oriented mode and this quantum field theory, becoming one of its V. F. Weisskopf (1908-2002). Thanks to the shift goes together with a shift of his re- pioneers. Among his co-authors at that interventions of the then Director General search interests. As soon as 1967 he pub- time were Stanley Deser, Marwin L. Gold- of CERN, Bernard Gregory,´ the member lished in Communications in Mathemati- 8 berger and Murray Gell-Mann. Several pa- states finally took the wise decision to build cal Physics. His first encounter with El- pers on renormalization and dispersion re- the SPS at Geneva, against strong lobbying liott Lieb happened during their stay at the lation in particle physics were the result of of several countries to build the machine on ICTP in Trieste in 1968, and subsequently these fruitful co-operations.6 During this a different site somewhere in Europe. Aus- when Thirring invited Lieb to give a talk on ten year period of work at different re- tria took part in this competition and made joint work with J.L. Lebowitz on the exis- search centres Thirring made a pivotal ex- a site proposal for Gopfritz¨ in the Waldvier- tence and convexity properties of the ther- perience, transnationality of science, an in- tel, a place surrounded by lovely woods and modynamical functions of Coulomb sys- 9 sight he brought back to his, at that time of perfect geological conditions some hun- tems. Thirring found that this result was provincial, native Austria. After one year dred kilometres north of Vienna! so much part of the general physics culture that even particle physicists ought to know as full professor at the University of Bern Interestingly enough Thirring has writ- about it.10 in 1959 he accepted a call of the University ten only occasionally on science policy of Vienna. Until his retirement in 1995 he or exposed his opinion on matters be- A period of intense collaboration on the was professor at the Institute of Theoretical yond pure science in public. On the oc- improvement of the Dyson-Lenard proof Physics in Vienna. casion of the opening of the 10th Schlad- of the stability of matter between Thirring From an early stage onward, Thirring ming Winter School for Nuclear Physics and Lieb started in 1974 when Lieb was was engaged with the first steps of Aus- in 1971 Thirring addressed the audience Schrodinger¨ Guest Professor at Thirrings tria’s membership of CERN which was with an analysis of the situation of high en- institute in Vienna. In the following sum- finalized in 1959, opening doors for ex- ergy physics in Europe stressing the impor- mer Lieb again came to Vienna and this perimental and theoretical research on a tance of fundamental science for the sci- time they succeeded in directly proving the 11 truly international basis. Since the 1950’s entific development in Europe to counter- stability of matter. During the following Thirring concentrated his research mainly act the influence of the leading powers, years this topic and other problems of the on theoretical elementary particle physics USA and the Soviet Union. Big science Thomas-Fermi-Theory was further investi- with some excursions into other fields of had become the characteristic of research gated by both, together with other collabo- physics. When Marietta Blau (1894-1970), in fields like particle physics, and Thirring rators. who pioneered the photographic method stressed the fact that it is most important During the years 1976 – 1978 Thirring in particle physics, came back to Austria for small countries to participate in this served as first president of the Interna- in 1960 after having left the country as game. Thirring went one step further and tional Association of Mathematical Physics http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 4 Volume 2, Issue 1, Spring 2007 ESI NEWS (IAMP) and succeeded in developing this much like an ex negativo formulation of Thirring to promote peace in this world.22 organisation into an international platform. Thirring’s research programme as a mathe- Let me conclude by citing an observa- Naturally, opinions are split which of matical physicist. tion Thirring made on Schrodinger:¨ ‘For Thirring’s achievements should be counted Thirring’s autonomy of choice of his him [Schrodinger¨ ] the so-called exact sci- research topics is one of the most aston- ences are just a little stone in the greater as the most important. He himself leans to- 23 wards the work on the stability of matter. In ishing observations to make regarding his edifice of a philosophical world view’. trying to do justice to the enormous breath scientific oeuvre. Till today he is working Considering Thirring’s dedication and love of his oeuvre an assessment is confronted on the forefront of mathematical physics for music, playing the organ and compos- with a wide range of subjects. Among his and at the same time he is a very inde- ing, together with his religion-based insight early work we find a paper in the field of pendent observer of the scene. His view on into the great cosmic fabric we may be applied statical mechanics, solving a prob- physics is void of fashions. Is there — one well justified to take Thirring’s words on 12 may ask — a specific ‘philosophy’ guid- Schrodinger¨ as a sort of testimony of his lem in dentistry, next in his publication 24 list to his important paper on a quantum ing Thirring’s physical insight, his intuition own world view. field theoretical model of a self-interacting and choice of subject? He never expressed Notes Dirac field which bears his name as a toy himself on this delicate point of motivation 1 ‘Ich wollte nie eine Autobiographie schreiben model stimulating research to this day.13 of his scientific work, as far as I am aware. I . . . Wie jammerlich¨ schienen mir doch die nostal- Work on gravitational theory is followed will try to formulate it my way: the central gischen Kriegsgeschichten alter Herren, mit denen 14 epistemological and methodological motor sie beweisen wollten, was sie in ihrer Jugend fur¨ by a precursor of the quark model. The Teufelskerle gewesen waren’¨ . Walter Thirring, Bi- next paper I want to mention here briefly of his style of doing physics is based on a ographical & Bibliographical Series of Classics of because Thirring frequently refers to it not deep intuition concerning the physical rele- World Science. S.S. Moskaliuk, ed., Kyiv 1997, p. 12. only in his popular writings: in 1970 the vance of a specific problem, combined with 2 ‘. . . in der Hand eines Demagogen . . . beliebig de- the aim of uncompromising mathematical formiert werden kann’, Ibid., p. 57. scientific community received with consid- 3 ‘Mein Bruder war drei Jahre alter¨ als ich und galt erable scepticism his claim, that under cer- rigour. schon immer als das große Genie, wahrend¨ ich nur tain circumstances a system, e.g. our sun Communication between scientists das kleine Anhangsel¨ war’, Ibid., p. 47. 4 or a supernova, will exhibit negative spe- from both fields, physics and mathemat- ‘die wissenschaftliche Tradition unserer Familie weiterzupflegen’, Ibid., p. 47. cific heat, i.e. the seemingly paradoxical ics, and the use of the most recent math- 5 ‘. . . eine treffliche Wahl’, Ibid., p. 47. Georg fact that a system when cooled gets hotter ematical tools are integral parts of this Joos, Lehrbuch der theoretischen Physik. Leipzig: and cooler when heated.15 The later period mode of operation and has manifested it- Akademische Verlagsgesellschaft 1932. 6 of his work is marked by dealing with ques- self on an international scale at the Erwin Cf. e.g. W. Thirring, M. Gell-Mann and M. Gold- berger, Use of Causality Conditions in Quantum The- tions on the ergodicity of quantum systems Schrodinger¨ Institute under Thirring’s ini- ory, Phys. Rev. 95 (1954) 1612-1627. and quantum chaos. An excellent overview tial directorship. The credo of Thirring — 7 W. Thirring, High Energy Physics and Big Sci- of what he has achieved can be found in and of the ESI — is the following: the ulti- ence, Acta Phys. Austr., Suppl. VIII (1971) 12-20. 8 Thirring’s ‘Selecta’ with contributions to mate criterium for the validity of a physical W. Thirring, A. Wehrl, On the Mathematical Structure of the BCS-Model. Comm. Math. Phys, 4 Mathematical Physics, Statistical Physics, argument lies in a rigorous mathematical (1967), 181-189. Quantum Field Theory, General Relativity, proof based on clearly stated premises. 9 J.L. Lebowitz, E.H. Lieb, Phys. Rev. Lett. 22 and Elementary Particle Physics.16 This does not mean that a good deal of (1969) 631-634. 10 physical inspiration and a strong feeling Walter Thirring, Biographical & Bibliographical As an author of textbooks his influ- Series of Classics of World Science. S.S. Moskaliuk, ence extended far beyond the inner circle for the right way to formulate a physical ed., Kyiv 1997, p. 221. of students attending his lecture courses. problem are not necessary conditions to 11 E.H. Lieb, W. Thirring, Bound for the Kinetic En- Thirring published advanced introductions find one’s feet in physics. ergy of Fermions Which Proves the Stability of Matter. 17 Phys. Rev. Lett. 35 (1975) 687-689. Errata 35 (1975) to quantum electrodynamics and quan- This edition of the ESI NEWS would 1116. tum field theory.18 His four volume text- not have appeared and we would not be W. Thirring, Stabilitat¨ der Materie. Naturwis- book on mathematical physics, resulting able to congratulate Walter Thirring on senschaften 73 (1986) 705. 12 ¨ from a four term lecture course during these pages had he not lent all his sup- W. Thirring, T. Hromatka, Uber die Statik fed- ernd abgestutzter¨ Freisattel¨ . Deutsche Zahnarztliche¨ the 1970’s, is considered as the standard port, enthusiasm and last, but not least, his Zeitschr. 7 (1952) 209. text in the field, known at that time to his reputation as a scholar to the initiative by 13 Walter E. Thirring, A Soluble Relativistic Field pupils as typed and mimeographed lecture Austrian and Russian physicists and math- Theory. Annals of Physics 3 (1958) 91-112. 14 notes, the green booklets (‘grune¨ Hefte’).19 ematicians to found an institute dedicated W. Thirring, Three-Field Theory of Strong Inter- actions. Nuclear Physics 14 (1959/60) 565-577. Thirring’s argument for the selection of the to mathematical physics in Vienna. With- 15 Walter Thirring, Systems with Negative Specific material presented in the textbooks is clear out Thirring as an outstanding leader and Heat, Z. f. Phys. 235 (1970) 339-352. and brief, a trade mark of his somewhat scientist this dream would not have become 16 Walter E. Thirring, Selected Papers of Walter terse style: ‘I decided to cover only those reality. Since 1993 his spirit of weaving to- E. Thirring. Providence, Rhode Island: Amer. Math. Soc., 1998. subjects in which one can work from the gether the threads of physics and mathe- 17 W. Thirring, Einfuhrung¨ in die Quantenelektrody- basic laws to derive physically relevant re- matics into one texture is manifesting it- namik. Wien: Deuticke 1955. Principles of Quantum sults with full mathematical rigour’.20 It self in the Erwin Schrodinger¨ Institute, per- Electrodynamics. New York: Academic Press 1958, follows, Thirring tells us succinctly, that haps Thirring’s most visible achievement 2nd ed. 1962. Russian Edition. 18 Ernest M. Henley und W. Thirring, Elemen- relativistic quantum theory and other im- as a scientific manager. There is a further tary Quantum Field Theory. New York: MacGraw- portant branches of physics are not taken aspect of the ESI which goes beyond its Hill, 1962. [German transl.: Manfred Breitenencker], up because they ‘have not yet matured from scientific work: as a meeting place for re- Elementare Quantenfeldtheorie. Mannheim, Wien, the stage of rules for calculations to math- searchers not only from East and West of Zurich:¨ Bibliographisches Institut 1975. Russian und 21 Japanese editions. ematically well understood disciplines’. Europe but on the global scale it reflects as- 19 W. Thirring, Lehrbuch der Mathematischen This statement, in a sense, sounds very pirations and hopes of Walter’s father Hans Physik. Wien, New York: Springer Verlag 1977

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 20075

– 1980. Band 1: Klassische Dynamische Systeme. Springer-Verlag, 1978, p. xi. Thirring, Einleitung in Erwin Schrodinger,¨ Gesam- Band 2: Klassische Feldtheorie. Band 3: Quanten- 21 Ibid. melte Abhandlungen, Band 4, Allgemeine wis- mechanik von Atomen und Molekulen¨ . Band 4: Quan- 22 Brigitte Zimmel, Gabriele Kerber (Hrsg.), Hans senschaftliche und populare¨ Aufsatze¨ . Wien: Verlag tenmechanik großer Systeme. Thirring. Ein Leben fur¨ Physik und Frieden. Wien, der Osterr.¨ Akad. d. Wiss. 1984, S. xii. 20 W. Thirring, Lehrbuch der Mathematischen Koln,¨ Weimar: Bohlau¨ 1992. 24 Walter Thirring, Kosmische Impressionen. Gottes Physik. Wien, New York: Springer Verlag 1977 – 23 ‘Fur¨ ihn waren die sogenannten exakten Natur- Spuren in den Naturgesetzen. Wien: Molden 2004. 1980. Band 1: A Course in Mathematical Physics. wissenschaften nur ein Steinchen in dem großeren¨ Classical Dynamical Systems. New York, Wien: Gebaude¨ einer philosophischen Weltsicht’. W.

Walter Thirring’s Arrival with whom I published the first papers on pearance of Zachariasons paper, he pub- Current Algebra.) lished a paper called ‘Lagrangian Formula- in Vienna Due to Walter’s US grants, we were tion of the Zachariason Model’. Today we Herbert Pietschmann also able to report our research results at all know that he was right, for the Standard international conferences or — for exam- Model of Elementary Particles is of course ple — travel to the United States to lecture based on a huge Lagrangian! But in those In 1959 a number of about our work. (In this way, I could report days it was a minority opinion and not at all graduate students at our results on Current Algebra in 1963 at obvious. (My own Habilitation was based the Institute of Theo- six American Institutes.) on the ‘Zachariason-Thirring-Model’) retical Physics of the We younger physicists greatly admired In 1966 I worked with Walter on the University of Vienna Walter’s physical intuition and ability to quark model of elementary particles. Dur- was eagerly waiting do calculations in the most simple and ing these happy days I learned a lot and for the arrival of Wal- straightforward manner. Indeed, when he again I could admire Walter’s physical intu- ter Thirring as new wrote his textbook with Henley, Ranninger ition. For the benefit of younger colleagues head of the Institute. Erwin Schrodinger¨ and I had to check the equations. At one let me just tell one story about the old days: had retired the year before and the only point, we were unable to get from one line We had to work out a three-particle phase other (associate) professor of theoretical to the next in spite of prolonged and great space integral, but in those days there were physics, Theodor Sexl, refused to teach effort. Of course we were quite ashamed no computers! So we shared the work: I Quantum Mechanics for personal reasons, when we came to Walter to confess. He was drew the integral on a then often used graph although he was expert in Nuclear Physics. not the slightest annoyed but simply went paper and counted the squares. Walter had We graduate students had to organize pri- to the blackboard, wrote down an Ansatz a slide-rule of one meter length and did a vate Seminars in order to learn Quantum from general symmetry arguments, did the rough computation. At the end, we used Mechanics and Field Theory from text- remaining algebra in his head and said with an electric calculator which was able to books. (For Quantum Mechanics we had satisfaction: ‘Don’t worry, its OK!’. I have perform the four basic algebraic manipu- chosen Dirac’s book — not exactly the eas- to add that we were relieved when a few lations. Walter, working on his slide-rule, iest way into this chapter! Field Theory we days later a letter came from Jacobson and said the numbers aloud and I typed them learned mostly from Walter’s first book on Henley asking Walter to add one line for into the calculator. In this way — after an Quantum Electrodynamics.) they also had to work for some time before hour of work — we did the integral numeri- When Walter Thirring arrived, many they found the right way to check this equa- cally. And this was just about 40 years ago! things changed spontaneously — the Insti- tion. Times have changed rapidly. tute was transformed from a historic place In the early sixtieth there was a fierce Since 1968 I had my own chair in The- to a modern research establishment. Need- fight among particle theorists as to what oretical Physics, so I was now Walter’s col- less to say, that he taught all the mod- was the future theoretical basis for the de- league. I am very grateful to Walter for the ern subjects in his course on theoretical scription of elementary particles. Double fact, that we always had a warm and per- physics; but at that time it was almost dispersion relations were the fad of a large sonal climate at the Institute; the usual ri- equally important (as well as surprising) group of theorists and many believed, that valries and quarrels among colleagues were that the Institute now had its own secre- Lagrangian field theory was old-fashioned totally absent during all these decades to tary! With research grants from the United and had to be overcome. Indeed, some re- the very day! To me, this is one of the finest States, Walter could open an experimen- spectable Universities cancelled all con- aspects of Walter’s personality, for I could tal appendix to the Institute of Theoretical ventional field theory courses and replaced not have lived in an atmosphere of tension, Physics: a group evaluating bubble cham- them by ‘S-matrix theory’. A culmination not to mention fight. (Our common interest ber photographs from CERN which later point in this fight was the publication of a in music may have been helpful.) grew into the Institute for High Energy paper by Zachariason in which he claimed Around this time, Walter became direc- Physics of the Austrian Academy of Sci- ‘we shall construct a model field theory tor of the theory division at CERN for three ence. (After returning from CERN, my first which is ... not defined in terms of a La- years. It is coincidental, that just then he job with Walter was to support this group grangian’. If this were possible, it would changed his field of interest from particle with theoretical advice.) be quite a victory for the believers in ‘S- physics to mathematical physics. One may Many prominent guests came to Vi- matrix theory’. deplore the loss of such an intuitive physi- enna to lecture and stimulate our research. Walter’s intuition told him that this cist for particle physics, on the other hand, Some of them came for a longer period and ‘was not the true Jacob’ (to use a for- one may applaud the gain for mathemati- created an international atmosphere which mulation of Einstein). For Walter it was cal physics of a person with such a great also attracted young postdocs. (One of quite obvious that Lagrangian field theory physical intuition. Without it, mathematical them was A.P. Balachandran from Madras, could not be abandoned! Soon after the ap- physics may too easily become esoteric.

http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 6 Volume 2, Issue 1, Spring 2007 ESI NEWS Walter Thirring and at the University Vienna. A fortuitous com- own fields. Powerful arguments in favour, bination of supports from quite different however, usually come from the political Austria’s Accession to (yes, also opposite political) camps created side (‘European solidarity’), but especially CERN the necessary boundary conditions. from the lobbying by industry which al- ways found CERN and its top require- Wolfgang Kummer How Walter immediately made his in- stitute a focus of international importance ments for technological products an impor- tant competitive meeting place. Clearly the In the 1950’s the in the fields of quantum field theory and mathematical physics will not be described last point did not apply to Austria at that physics institutes in time. As a young nobody I, of course, never Austria in general and here. Instead due credit must be given to his successful activities regarding Austria’s knew any details about the mechanisms of especially at the Uni- the political interventions of Walter and versity and the Tech- accession to the European laboratory for High Energy Physics CERN. This organi- his congenial partner Fritz Regler, an ex- nische Hochschule Vi- perimental physicist from the Technische enna were still in an zation had been created in 1952 as an ex- emplary European effort to collaborate in Hochschule, who politically covered the extremely poor shape: ‘right’ wing. After some ups and downs the laboratories still showed the marks of the field of fundamental physics at the sub- nuclear level. For Walter is was inconceiv- fortunate consequence was the formal ac- bombings, obsolete equipment — if avail- cession of Austria to CERN in 1959. How- able at all — from pre-war times, among able to enter the map of top level physics research without being a member in the ever, for Walter Thirring is was evident that the students many former soldiers of the this membership without proper activity in German army with large deficits in sec- new and — as it turned out soon — ex- tremely successful enterprise. Austria was useless. Therefore, with the ondary schooling and a very inhomoge- help of financial support from a US founda- neous body of teaching personal, because Especially in small countries like Aus- tion he created the ‘Plattengruppe’, an ini- only in rare cases well qualified professors tria the decision to join CERN to this day tially small group of doctoral students, who had somehow survived the nazi regime. usually faces considerable resistance from evaluated the tracks of elementary particles Also the motivation for senior Austrian different opponents: finance ministers hes- in photographic plates which had been ex- physicists abroad to return into such an itate to pay large membership fees to in- posed to the new high energy accelerator environment (moreover at ridiculously low ternational organizations, most of whom at CERN. This group represented the core salaries) was not pronounced for under- do not exhibit a glamarous reputation for of the Institute for High Energy Physics at standable reasons. Under these circum- effectiveness. Even more importantly, the the Austrian Academy of Sciences which stances it appeared somewhat of a miracle dear collegues from other fields of physics started its work officially in 1966 — again that it became possible to recruit Walter and science in general deplore the amount thanks to the joint efforts of Thirring and Thirring as a successor to his father Hans of money which, in their opinion, could Regler. The rest ist history . . . for the professorship of theoretical physics be invested much more fruitful in their

Walter Thirring’s Path into edge of the Greek alphabet.’ But already language. But already here Walter realized in 1950 Walter was giving a seminar on that it is worthwhile to deepen the mathe- Mathematical Physics the scattering matrix in elementary parti- matical background. Heide Narnhofer cle physics and Bruno Touschek, a physi- An important ingredient of this deep- cist with strong interests in experimental ening of his mathematical understanding physics and less in the mathematical beauty When looking at Wal- was the good contact with the mathemati- of the theory, raised the question whether ter Thirring’s publi- cal department. With Leopold Schmetterer Walter could control the error in the pertur- cation list in order he started to have joint seminars with bative calculation of the scattering matrix. to determine a point the purpose not so much to teach stu- As a consequence Walter tried to control of transition from dents but out of the desire that he himself this error and had to observe the divergence elementary particle wanted to learn the mathematical concepts. of perturbation theory in quantum field the- physics to mathemat- Schmetterer and Thirring studied Pontrya- ory (1952). ical physics one soon gin’s book on continuous groups. I remem- realizes that there was no turning point: Walter also in daily life wants things ber a seminar on ergodic theory based on there was a continuous evolution in the to be clear and straight and is disgusted a course by Konrad Jacobs. In addition sense that he always wanted to make rig- by any escapes into ambiguity. Therefore Walter followed an advanced course on orous statements. Only the mathematical it was clear that he was not willing to ig- the theory of manifolds given by Edmund tools he used got sharpened in the pro- nore this fact of divergence and so he was Hlawka. I remember him sitting among the cess. This process reflected a general ten- looking for statements where he was not students two rows in front of me. He would dency in the second half of the last century: forced to trust arguments without a pro- have enjoyed a joint research project with physicists realized that understanding some found basis. In quantum electrodynamics Hlawka, clarifying how general relativity is parts of physics could benefit enormously renormalization seems to offer such a ba- based on the covariance of general mani- from applying more advanced mathemati- sis. The Thirring model (1958) allowed an folds while maintaining the evidence of a cal tools. explicit solution without referring to any Riemannian metric. But for some reason or I remember that Walter told me a joke: perturbation expansion. The search for the other this cooperation did not work out. ‘In the first part of the last century it was group structure in the family of observa- In the meantime Walter covered all of said that the only mathematical tools nec- tions in particle physics (1958) could also theoretical physics in a course lasting seven essary for physics were a good knowl- be formulated in a profound mathematical semesters. This required going into de-

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 20077 tails in various disciplines, and as a con- trated on other things. He started to study vent equations, some trace estimates and sequence he got attracted to various sub- the book of J. Dieudonne´ on Functional with the appropriate mixture of these argu- jects of theoretical physics that were close Analysis. He was so much attracted by this ments he got explicit numbers. Of course to his taste of combining rigour with intu- first volume that he offered to Dieudonne´ to these numbers were not optimal. The opti- ition. But he was also attracted by the fact read the corrections of the following vol- mal estimate was carried through by Mark that some problems were asking for more umes with the motivation that this would Fannes on the basis of my idea and has delicate mathematics. force him to study all the theorems care- since found its way into the literature. We can compare Walter’s approach fully and in detail. As I mentioned earlier, Walter’s re- to more mathematically minded problems But there followed another experience search happened at the time when more ad- with the evolution of mathematical physics where the power of mathematics was im- vanced mathematics entered into physics. as a new discipline. In 1960 the Jour- portant. Whereas in disproving the conver- The necessity to control convergence was nal of Mathematical Physics was founded, gence of perturbation theory in quantum of course always present. Now it became where the term ‘Mathematical Physics’ field theory he had to realize how mathe- apparent that one has to be careful in indicated a shift of emphasis towards a matics can falsify intuition, now he expe- choosing the right topology. Puzzled by more mathematical outlook on ‘Theoreti- rienced how mathematics justifies good in- scattering theory, Walter realized that the cal Physics’. In 1965 the Communication tuition even if this intuition is against com- norm topology is useful if you concen- in Mathematical Physics with the chief ed- mon knowledge. He had to argue heavily trate on the invariance of the group (here itor Rudolf Haag followed. Both journals with the referee that his result on negative time translation) but one has to revert to started with articles on quantum field the- specific heat was correct, but even more, the strong topology to understand the time ory and statistical mechanics. In the first this struggle convinced him that this result limit and to the weak topology to incorpo- volume of CMP you can find the descrip- was not only correct but also relevant: re- rate the existence of bound states. When tion what we should understand as mathe- search on the effect of gravitation in stars I was a student, in the textbooks you matical physics, unfortunately this part dis- followed. Finally he was happy that the dy- just found an explanation on the basis of appeared in later volumes: namical part of the theory could also be jus- differential equations. In his course Wal- It is one of the goals of this jour- tified by computer simulations. ter taught me how to switch between the nal to generate among mathematicians an In the meantime important progress in viewpoints and apply both ideas of differ- increased awareness and appreciation of mathematical rigour in quantum field the- ential equations and of operator algebras current problems in physics, just as we are ory was made as well. Back in Vienna he and, here especially, the power of different hoping to acquaint a growing number of picked up this project in a seminar. But dif- topologies. physicists with methods and results of mod- ferently to previous seminars, these studies Another viewpoint close to Walter’s ern mathematics. This should not be inter- did not inspire him to do his own research. I heart is the inspiration coming from clas- preted as an encouragement of physicists to can only imagine what was the reason: this sical mechanics. What can be inherited and confine themselves to rigorous mathemat- was mathematics that was against his atti- what becomes different? Here the formula- ical argumentation. Conjectures, intuitive tude. Mathematically rigorous perturbation tion of classical mechanics in phase space judgement and plausibility arguments in- theory in quantum field is a hard job. It is is tailor-made, and quantum theory enters deed have their place, so long as it is made as if you had to find your way in a jungle by adding correction terms and by control- clear that they are not regarded as proofs. with the vague hope that after heavy strug- ling them by estimates in weighted Sobolev Evidently Walter was attracted by this gles some clearing might appear where you spaces. It is still a problem that puzzles spirit. In 1967 he published for the first would find some insight into what is im- Walter and where he has not given up, that time in CMP, together with Alfred Wehrl, portant and what not and, at last, some in- inequalities so far do not give the numeri- a paper on the BCS-model where they had tuition. We now know that this hope was cal bound on stability of matter that his in- to combine ideas of convergence with ana- not quite in vain and that some new math- tuition from classical phase space offered. lytic considerations. ematical structures really turned out to be Here we observe another element in his at- But as proposed in CMP, he also en- relevant. But this is not the way how Wal- titude to mathematical physics: existence joyed picking up new methods. I remem- ter enjoys research. He wants to start with proofs are of course important, but you can ber my thesis on various conductivity prob- intuition and then to hunt for the appropri- only be sure to understand the essential lems. The task was to control in different ate rigorous arguments that would justify facts if you can produce numbers that are settings the time evolution and this was his intuition. The more tricks are necessary close to the numbers observed in experi- possible by calculating the integral kernel the more fun. ment. and extracting from this kernel the desired Somehow typical for this attitude is a Stability of matter is a result that he ob- information. However, in a joint publica- result on the continuity of entropy that was tained jointly with Elliott Lieb. This collab- tion Walter added an appendix that was not needed for the definition of dynamical en- oration was extremely fruitful and inspir- really necessary but offered the chance to tropy. I offered a characterization with the ing. Walter and Elliott argue on the same study in an explicit example the difference necessary dependence on the dimension, footing and are interested in the same kind between self-adjoint and hermitian opera- based on the idea that first you convince of problems. In their joint work it is impos- tors, a fact that nowadays is standard for yourself that the largest variation occurs in sible to assign the various ideas to one or more mathematically minded physicists, the abelian case and then you estimate this the other. This also becomes evident in a but in those days had just become apparent. abelian situation. I was too lazy to calcu- fictitious dialogue that Walter wrote on the In 1968 Walter moved to CERN in late the explicit numbers on which Wal- occasion of Elliott’s sixtieth birthday. It is Geneva, the European center to do elemen- ter insisted. On the other hand, he consid- not complementarity but harmony that in- tary particle physics. But Walter, though al- ered my arguments too boring. Instead he spires and is the source of success. ways very interested in this field, concen- used some integral equations, some resol- What were the important achievements http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 8 Volume 2, Issue 1, Spring 2007 ESI NEWS of mathematics that inspired physics in this ered more or less independently by math- served. The macroscopic world is the world half century? There was the role played ematicians and physicists, makes it possi- with which we are familiar, and from this by different topologies. Another example ble to make statements on ergodic proper- macroscopic world he looks for his inspi- was the realization that for thermodynam- ties in the thermodynamic limit also in the ration for the quantum world that he wants ics and quantum field theory one needs the quantum setting, quite in the spirit of Boltz- to build on exact mathematical statements. initially unfamiliar type III algebras. I re- mann. Mathematics developed further in This note presents of course a rather member that as usual I was reading the this direction. Alain Connes inspired the personal view, resulting from many dis- handwritten manuscript of his book before connection between various disciplines of cussions and anecdotes that Walter told it went to Franzi Wagner to be typed. There mathematics and founded the concept of me, and observations in our collaboration. I found the sentence: ‘Unfortunately in the noncommutative manifolds. Together with Yet memory can be deceptive and I apolo- thermodynamic limit the algebras turn out topology this opened the possibility to find gize for possible errors and misinterpreta- to be type-III.’ This raised a discussion that connections between the microscopic and tions. In any case I do want to thank Wal- made Walter remove the word ‘unfortu- macroscopic world. Walter was always in- ter for the profound education and guide nately’, though he did not go as far as to terested in these developments; however, to relevant problems, and even more for a write ‘fortunately’ as I had suggested: this they did not really inspire his intuition. He long collaboration in which I explored the is really the source that enables us to find is not a mathematician who is looking for beauty and the fun in guessing, substantiat- qualitatively different behaviour in the mi- beautiful structures, but he uses physical ing the wishful thinking and finally hunting croscopic and macroscopic world without models as inspiration. Walter is a physi- for the justification. This is one of the ways the necessity to draw a sharp border. cist with all his heart whose imagination is to attack physical problems and Walter has The Tomita-Takesaki theory, discov- very much influenced by what can be ob- always been a master in it.

Unmathematical and system is thought to represent some spe- vent evaporation. Considering the orders of cific aspects of nature. To be able to con- magnitude one finds that this evaporation is Mathematical Physics centrate on those elementary details of nat- ‘slow’ in comparison with both the move- Bernhard Baumgartner ural laws which are essential to what is go- ments of particles and the approach to lo- ing on, in order to highlight the principles, cal thermal equilibrium, the error in dis- My first impression of one has to discard the other details. In the carding it for the moment is small. Then Walter Thirring was at model for stars mentioned above one disre- comes an important mathematical ingre- a talk on the thermo- gards relativity, the multitude of elements, dient: the Virial Theorem. It implies that dynamics of stars. He and — hard to swallow — the nuclear re- lowering the total energy leads to contrac- was at that time direc- actions. On the other hand one has to in- tion, to the rise of kinetic energy and to the tor of the theory divi- vent some unnatural hypotheses: contain- rise of temperature. These changes appear sion at CERN, visiting ers for the stars and a limit of particle num- in mathematical investigations as functions his home university in bers going to infinity, coupled with a limit of the theoretical input, whereas in reality Vienna. I was a student of physics at the of the volume going to zero, keeping the they happen in the course of the birth of time. Having fallen in love with mathe- proper relations between mass, size, energy a star, followed by the ignition of nuclear matics at the beginnings of my studies, and temperature. This is a kind of Thermo- fusion. Again these changes of the con- but being mainly interested in physics, I dynamic Limit adapted to Thomas-Fermi stituents happen much more slowly than was happy to hear about the existence of Theory. the local processes establishing the thermo- ‘Mathematical Physics’. And this branch The fact that such a procedure is sound, dynamic equilibrium. Of course it would of physics was led in Vienna by Walter not crazy, is realized by making obser- now be interesting to discuss the history of Thirring, well known in the physics com- vations in the realm of Unmathematical the theory of stars: how Lord Kelvin esti- munity (and also beyond). Now I saw this Physics, also designated by Walter Thirring mated that chemical reactions, as for ex- man of high reputation, short and sturdy, as Großenordnungsphysik¨ (physics of or- ample the burning of coal, give less energy talking about such a big object — a star! ders of magnitude). In the case of a star than gravitational contraction; that this is – without much ado, but with certitude: with 1057 electrons and positrons in a vol- still not enough, enabling a shining of the ‘The star gives away energy, contracts, and ume with diameter 1.4·109m it tells us that sun for only some millions of years; also becomes hotter.’ For me as a student, this the system behaves not really differently how Houtermans and Gamow found out was breath-taking — wasn’t this contrary from its description by Thomas Fermi the- that nuclear fusion was the main source of to text-book physics? A ‘Negative Specific ory. Only to reach the goal of indisputable energy. It was also of great importance that Heat’! But Walter Thirring together with mathematical exactness one has to consider the mathematically exact discussions of the Peter Hertel had proved this with math- the particles as confined and has to enact model describe an unusual kind of a phase ematical rigour,1 followed by investiga- the limiting procedures. transition, related to the collapse of a star tions including more details.2 So I got ac- For the readers coming from other after the nuclear fuel has been consumed. quainted with Mathematical Physics, ap- fields I would like to point out some of Use of ‘Großenordnungsphysik’¨ far- preciating its exactness. the ideas: in the model which has to repre- sightedly reveals that throughout history all Only in later years I realized that this sent the thermodynamic properties of stars, the ‘giants of science’ (on whose shoul- was just one of the extremal points of sci- one discusses a gas of Fermions. In reality ders we are standing) had the right intu- ence. Another one was actually the begin- each star radiates away energy and mass, ition to create models, followed by rigor- ning of any investigation: it is the art of no container exists. But in mathematical ous investigations only afterwards. How- model building, using intuition and ‘Un- investigations of statistical mechanics one ever, the final presentation is often the other mathematical Physics’. Each model of a has to incorporate a vessel in order to pre- way round: first comes the law, then the

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 20079 unmathematical popular explanation. For Sommerfeld theory and its relativistic for prospective teachers and Principles of example Newton’s Third Axiom, ‘To any corrections; Modern Physics (a lecture course initiated action there is always an equal and op- • Compressibility of atoms and its use in by Herbert Pietschmann). Walter Thirring posite reaction’ is followed by ‘If anyone finding the changeover from planets to is still a good source of ideas in this spirit.4 presses a stone with a finger, the finger is stars; His influence has been and will always be also pressed by the stone’.3 For the ‘aver- • Structures of atoms; of immeasurable value. age’ scientist, however, the danger of mak- • The role of the Pauli principle in atoms ing fundamental errors in following mis- and stars; Notes conceptions is not to be ignored. So, up • Chemical binding; 1 W. Thirring, Systems with Negative Specific Heat, • to the middle of the twentieth century, lit- The Periodic System of the elements; Z. Phys. 235 (1970), 339–352. • The question concerning existence of P. Hertel, W. Thirring, A soluble Model for a tle Unmathematical Physics can be found negative ions (‘H−− zerbricht beim An- System with Negative Specific Heat, Ann. Phys. 63 in textbooks. But nowadays the enormous schauen’ (‘H−− breaks when looked (1971), 520–533. P. Hertel, W. Thirring, Free Energy of Gravitating amount of detail in modern physics ne- at’). and ‘There are three wrong proofs cessitates its use in order to get a gen- Fermions, Commun. Math Phys. 24 (1971), 22–36. for its non-existence’); P. Hertel, W. Thirring, Thermodynamic instability eral picture. Walter Thirring gave a course • Oscillations and rotations of molecules; of a system of gravitating Fermions, in Drr H.P. (Ed.), on such themes, perhaps relaxing somehow • Aspects of solid state physics; Quanten und Felder. Braunschweig: Vieweg, 1971. 2 after having finished the first draft of his • Black body radiation, its use to deter- P. Hertel, H. Narnhofer, W. Thirring, Thermody- enormous four-volume work on mathemat- namic Functions for Fermions with Gravostatic and mine the temperature of the solar surface Electrostatic Interactions, Commun. Math Phys. 28 ical physics. At the start he stated its motto by ‘Daumenpeilung’ (taking a bearing by (1972) 159–176. ‘Es wird nie eine Gleichung aus der an- rule of thumb), and some aspects of a B. Baumgartner, Thermodynamic Limit of Corre- deren folgen (‘Never will one equation fol- light bulb; lation Functions in a System of Gravitating Fermions, low from the other one’). • Spectral lines (semiclassical aspects of Commun. Math Phys. 48 (1976), 207–213. their formation and line width); B. Baumgartner, H. Narnhofer, W. Thirring, Thomas-Fermi Limit of Bose Jellium, Ann. Phys. 150 Here is the list of contents of Thirring’s • Entropy (orders of magnitude, Gibbs course on Unmathematical Physics: (1983), 373–391. paradox, photons, phonons, Fermions) 3 Isaac Newton, The Principia, A new Translation • Gravitation (astronomical data in or- (My thanks go to Helmuth Urbantke for keeping by I. Bernard Cohen and Anne Whitman. Berkeley, ders of magnitude for earth, moon, sun, Los Angeles, London: University of California Press, the records.) Jupiter, milky way and the universe; 1999. 4 the virial theorem and its applications; The spirit of this course provided me W. Thirring, Kosmische Impressionen. Gottes Spuren in den Naturgesetzen, Wien: Molden Verlag, clouds of gas); with a firm foundation for my lectur- 2004. • Electromagnetic phenomena in atoms ing on such diverse themes as solid state C. Faustmann, W. Thirring, Einstein entformelt, in a semiclassical way a` la Bohr- physics (together with Gero Vogl), physics Wien: Seifert Verlag, 2007.

Gravitational Matters Schmetterer, on I.E. Segal’s (then) recent quently published by Springer Verlag (sev- book Mathematical Problems of Relativis- eral editions, from 1977 on). The basic Helmuth Urbantke tic Physics.3 On visiting the London con- idea was to include only subjects where ference on GR that year, Thirring became it is now possible to go from fundamental In 1959 W. Thirring impressed by the power of the geomet- laws to physically relevant results without was appointed as full ric arguments of Roger Penrose in fight- mathematical gaps, thus eliminating what professor at the Uni- ing his battle with the Landau-Lifshitz- W. Pauli called ‘wishful mathematics’ and versity of Vienna and Khalatnikov school concerning singulari- W. Thirring calls ‘Darwinian physics’. In immediately started ties in GR. While the buildup of a group compensation, each of these topics is in- his 7-semester lecture working on GR in the following years was troduced by heuristic order-of-magnitude course on Theoretical left largely to R.U. Sexl, Thirrings main remarks. Actually, many of these, and Physics. This lecture contribution to gravitational physics is non- many other heuristic considerations (in part course was accom- relativistic: his quantum statistical analysis due to V. Weisskopf) were presented by panied by a set of mimeographed notes of stellar equilibrium from first principles.4 Thirring in a one-semester course entitled Guidelines for Students of Theoretical In the relativistic regime, i.e. in GR, a con- ‘Unmathematical Physics’ (cf. p. 8). It is Physics which are reproduced on p. 10. tribution was to point out advantages of the a great pity — not only for students! — In his Classical Electrodynamics use of differential forms, in particular in the that this course was given only once and course he included, as an appendix, his context of the variational formulation, con- has never been written up. In contrast, alternative special-relativistic approach to servation laws and energy ‘pseudo-tensors’ courses along the lines laid down in the General Relativity (GR), based on two of (which were formerly based only on coor- four volumes were given by Thirring regu- 1 his papers. This is how many of us were dinate bases). This got published in the sec- larly till the nineties, and many members of introduced to the subject. In particular, Ro- ond volume5 of his four-volume textbook the Vienna Institute of Theoretical Physics man U. Sexl took up this approach and on Mathematical Physics. had the opportunity to give lecture courses used it to classify rival theories of GR in based on (various parts of) Thirring’s 2 their linear approximations. This textbook, referred to as ‘mon- course at the then recently founded SISSA The year 1965 constituted a turning umental’ by Nobel Laureate S. Chan- (Scuola Internazionale di Studi Avanzati) point in Thirring’s interests, directing them drasekhar upon a visit to Vienna, first in Trieste during the 1980’s. In fact, this set towards mathematical physics — perhaps appeared in the form of mimeographed of courses was included in the SISSA cur- also triggered by a seminar, organized lecture notes of a four-semester course riculum. jointly with the mathematician Leopold on mathematical physics and was subse- http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 10 Volume 2, Issue 1, Spring 2007 ESI NEWS

Returning to Gravity, it is perhaps domain of the linear approximation, were Notes needless to say that there is a continuing unknown or unnoticed at the time (like J. 1 W. Thirring, Fortschr. Phys. 7 (1959), 79. interest on his part in ‘Machian’ matters, Synge’s 1950 analysis7 of the global struc- W.Thirring, Ann. Phys. (N.Y.) 16 (1961), 96. in particular in the experimental results ture of Schwarzschild spacetime). Thirring 2 R. Sexl, Fortschr. Phys. 15 (1967), 269. 3 I.E. Segal, Mathematical Problems of Relativistic concerning the (H.) Thirring-Lense effect. once told us that he had even discussed par- Physics, Providence, R.I.: Amer. Math. Soc., 1963 Machian questions were touched upon in ticle creation by deep gravitational poten- 4 P. Hertel, H. Narnhofer, W. Thirring, Commun. the dissertation of F. Embacher6 under tial wells with A. Einstein — who charac- Math. Phys. 28 (1972), 159. 5 Thirring’s supervision. An early question, teristically did not like the idea. W. Thirring, Classical Field Theory. A course in Mathematical Physics, vol. 2, second edition. New dating from the fifties, was the discussion York, Vienna: Springer Verlag, 1978. of quantum fields in external gravitational 6 F. Embacher, Machian Effects and Methods of So- fields; actually, the question came too early lution for Einsteins Field Equations in the Case of in the sense that the relevant global as- Cylindrically Symmetric Stationarily Rotating Matter, University of Vienna, 1980 [in German]. pects of gravitation, which lie beyond the 7 J.L. Synge, Proc. R. Irish Acad. A 53 (1950), 83.

Guidelines for Students of written in English, knowledge of the latter is to be invested into ascertaining the correctness likewise needed urgently. To people who want of the result and discussing its significance. On Theoretical Physics to engage in Theoretical Physics as a profes- the one hand, one should deduce the result by Walter Thirring sion, active collaboration in experimental work different methods, pondering about the most di- is strongly recommended. All this knowledge is rect one. On the other, one must inquire about taken for granted in Theoretical Physics, and in the physical significance and order of magni- The following notes lectures one cannot go into details about it. tude of the result; by plausibility considerations, by Walter Thirring 2. General Advice one should see the approximate validity of the were handed to stu- result. A frequent mistake is to write down a re- dents of his lecture While studying Theoretical Physics, everybody sult with many decimal places without making course on Theoreti- will go through a few growing pains which are sure that the very first decimal place is correct. not to be taken too seriously; but it will be well cal Physics in 1959. to be informed about them. In the following, we c) Lectures The English transla- will give a few words of advices in this respect. One recommended way — but not the only one tion is due to Helmuth To the beginner, combination of mathematical — is to attend lectures; one should be careful, Urbantke. formalism and physical ideas is alien, and it will however, not to attend too many of them, since be important to be clear about significance and Theoretical Physics has become an extremely it is impossible to digest fully more than two order of magnitude of the symbols at each step broad field whose mastery presents steadily in- or three per day. Nevertheless, it is recommend- of calculation. One should believe to have un- creasing difficulties. It is, therefore, essential to able to work through a subject following a lec- derstood a result only if one is able to reproduce take the most economic path to avoid unneces- ture course even if one knows it already. In gen- from memory all the steps involved in its deduc- sary waste of time. For this reason, in the fol- eral, one gets an overview of a subject only if tion. It is not sufficient to be able to follow these lowing I shall give some guidelines to help stu- light is shed upon it from different angles, i.e., if steps by reading them. To counter these difficul- dents find a suitable path through the jungle of one has encountered it both in lectures and text- ties, the following are useful: modern Theoretical Physics. books. Just taking notes from a lecture course is a) Discussions not sufficient; one must be able to reproduce the 1. Tools and Aids An essential means to convince oneself of hav- material from memory. For the understanding of Theoretical Physics, ing reached understanding is discussing it with d) Literature others. It is only if one has an overview of ob- some aids from other parts of science are For most parts of physics, there are excellent needed. Since the material is presented in a jections and suggestions of different kinds that one may judge the soundness of one’s own ap- textbooks that enable one to work through the mathematically deductive manner, knowledge material. However, these books cannot be read of mathematical language is indispensable; but proach. In discussions, all shyness and vanity must be overcome, and one must not be con- like newspapers to be useful, but only with pa- even knowledge does not suffice — it is mas- per and pen, so that one may perform all cal- tery of mathematical techniques that enables in- tent before the objective state of affairs has been honestly ascertained. . . . It is also strongly rec- culations oneself. There are many books whose dependent work. Mainly the following branches reading is dangerous for beginners. In part, they of mathematics are needed: ommended to physics students to attend semi- nars, because there will be more room for dis- are out of age, so that notation, presentation and a) Linear algebra (vectors and tensors) cussions and questions than during the lecture content do not any more correspond to contem- b) Differential and integral calculus courses. In particular, PhD students of Theoret- porary knowledge. Also, many books by famous c) Differential and integral equations ical Physics should, besides their work on their scientists contain suggestions which later turned c) Complex analysis theses, participate in those seminars on princi- out to be useless: studying them means waste of time without much gain . . . This material has to be acquired by the student ple. The notes end with a list of recommended from lectures and suitable textbooks. b) Exercises and not recommended textbooks. Among the Furthermore, in Theoretical Physics education The execution of typical exercises and problems in experimental and general physics is required. is as important for the physicist as is exercis- textbooks which were not recommended was This will be, in the first place, knowledge of ing for the pianist. This is why lectures are ac- Arthur Stanley Eddington’s ‘Fundamental The- phenomena and of related terminology. There companied by problem sessions. It is not suf- ory’ and Arnold Sommerfeld’s ‘Atombau und are lecture courses on these as well as techni- ficient to calculate and calculate until some re- Spektrallinien’ (out of date) and, generally, all cal literature. Since a large part of the latter is sult is produced: a major part of the labour has textbooks without new editions after 1930.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 2007 11 Walter Thirring: Probing vexity of E(S) (thermodynamic stability). kind. The coarse grained entropy increased According to a theorem by Walter Thirring, both during the expansion and compres- for Stability the three stability conditions are intimately sion, but only when it was of the Boltz- Harald A. Posch connected: each pair of conditions implies mann type. Other entropies with coarse the third.2 graining solely in the momentum or con- The dynamics of thermally unstable figuration space failed the test. Using the Almost 40 years ago, systems has also provided the background concept of passivity, Walter also demon- Walter Thirring, then for my collaboration with Walter up to the strated, for an ensemble of harmonic os- member of the board of present day. Earlier than most theoretical cillators with periodically-switched force directors at CERN in physicists, he accepted computer simula- constants, that active states, which violate Geneva, faced a para- tions of complex dynamical events as a le- the Second Law, are located on a fractal set doxical problem: as- gitimate tool to check and guide the intu- in phase space with a measure which van- tronomers had known ition and to study systems still outside of ishes in the many-particle limit. for almost 100 years present-day theory. In 1989, in collabora- But a star is threatened not only by the that extracting energy from a star, for ex- tion with H. Narnhofer, we conducted a se- gravothermal effect, which would cause it ample by radiation, would make it col- ries of computer simulations with a sim- to implode and heat up in the process. It lapse and heat it up. V.A. Antonov even plified model of 400 particles in a box in- is also subjected to a second gigantic in- showed in 1962 that no equilibrium state teracting with a short-ranged attractive po- stability, the thermonuclear heating in it’s with a global maximum of the Boltzmann tential.3 We established the unusual ther- core. The later which would cause it to ex- entropy exists for a sufficiently large sys- mal properties of negative specific heat in plode if unchecked. Both instabilities, the tem of point particles interacting with a a certain range of energies and studied the gravothermal effect and thermonuclear en- 1/r potential. This means that the parti- particle-number dependence. As an exam- ergy input, are intimately linked together cles form a cluster and continue to col- ple, in Figure 1 a typical particle cluster and neutralize each other for billions of lapse for ever with their temperature di- is shown, which forms after the simulation years. The negative heat capacity changes verging to infinity, even if the total en- has been initiated with a homogeneous gas. the bifurcation point due to exponential ergy is constant. The British astronomer D. nuclear energy input and radiative energy Lynden-Bell realized that an explanation losses into a fixed point. But beware, if of this so-called gravothermal catastrophe eventually one of the gigantic instabilities in thermodynamic terms requires the spe- gets weaker than the other! Recently, we cific heat (or more correctly the heat ca- exhibited this delicate balance with two pacity) to become negative. The paradox simple examples.6 The first is the many- was that physicists “knew” that the specific body system of Figure 1. If energy is added heat of ordinary matter could be expressed to the equilibrated particles in the main in terms of the square of the energy fluc- cluster for times t > 3000, the tempera- tuations and, as a consequence, had to be ture of the cluster particles goes down as positive. Walter, at about the same time and depicted by curve B in Figure 2. independent of Lynden-Bell, realized the significance of the negative specific heat and “risked his reputation” (in the words of the editor), when he published an article in 19701 in which he explained the differ- FIGURE 1: 400 ATTRACTING PARTICLES IN BOX. ence and resolved the paradox: physicists CONFIGURATION AT A TIME t0 = 3000.THE PARTICLES were used to specifying the state of matter AREREPRESENTEDBYCIRCLESWITHDIAMETEREQUAL by the temperature in contact with a heat TO THE INFLECTION POINT OF THE NEGATIVE GAUSS bath and, hence, within the framework of POTENTIAL. the canonical ensemble. There, the specific By coupling to a heat bath, we also ex- heat is indeed always positive. Stars and plicitly demonstrated that the negative heat star clusters, however, may be considered capacity disappears and is replaced by a FIGURE 2: TIMEEVOLUTIONFORTHETEMPERATURE Tc as isolated objects with constant total en- first-order clustering phase transition when OFTHELARGESTCLUSTER.LINE A BELONGSTOTHE ergy, for which the microcanonical ensem- the simulations are carried out within the EQUILIBRATION STEP, LINE B TO THE HEATING STEP, AND ble provides the proper setting. And there 4 canonical ensemble. LINE C TO THE FINAL HEATING/COOLINGSTEP. the specific heat may become negative, de- The next logical step with such an un- pending on the particle interaction and the familiar matter was to check for the Sec- If, in addition to heating the cluster par- energy. This provided the first significant ond Law. We performed a computer exper- ticles, the system is also subjected to a example in statistical mechanics for the re- iment of an adiabatic cyclic process, where cooling process at the boundary supposed sults of different ensembles to disagree. the volume of the box was periodically ex- to mimic radiation loss off the star, the tem- The problem of negative heat capac- panded and contracted, reducing and rais- perature goes up again until it reaches a sta- ity has continued to interest Walter ever ing the energy in the process.5 During the tionary state. This is shown by curve C in since. Most notably, he showed that neg- expansion the cluster formed and disap- Figure 2. This provides a beautiful example ative specific heat is related to other no- peared again during compression. The net for the mutual stabilizing effects of the two tions of stability, namely i) extensivity (sta- effect always was an increase of the sys- otherwise destructive phenomena. bility against implosion), ii) subadditivity tem’s energy, which excludes the possibil- The second model emphasizes another (stability against explosion), and iii) con- ity of a perpetuum mobile of the second interesting observation:7 negative specific http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 12 Volume 2, Issue 1, Spring 2007 ESI NEWS heat means that the volume of the energy This simple model is also the first me- sical three-body problem.12 The work on shell expands more than exponentially with chanical device with negative specific heat, these problems always brought to light the energy. In gravity-dominated systems the which could be built in the laboratory, at deep intuitive insight Walter has of the formation of clusters always causes heat- least in principle. physical world, which is rivalled only by ing, that is expansion in momentum space. The gravitational collapse of a few par- his extraordinary mathematical skills he If, however, more than exponential expan- ticles in a box due to the gravothermal ef- brings up when solving a problem. As be- sion is provided in configuration space, the fect may also be prevented, if the particles fits a great physicist, most recently Wal- system may even cool down and still show are subjected to external constraints, which ter Thirring has turned also to another out- negative specific heat. The model Walter reduces the allowed phase space such that standing problem, the study of evolutionary proposed is shown in Figure 3. the system asymptotically remains stable. processes leading to the formation of struc- We studied, for example, what effect mo- tures, to the selection of species. Again, the mentum and/or angular momentum conser- computer turns out to be an invaluable tool vation has on such a system. Three particles and provides the opportunity for me to par- in a square box in two dimensions will col- ticipate in this adventure.13 For this, and lapse for ever, in a circular box the collapse for his generosity in all scientific and pri- comes to an end in a finite time due to an- vate matters I am forever grateful. gular momentum conservation present only in this case.8 Notes Recently, the problem of negative heat 1 W. Thirring, Z. Physik 235 (1970), 339. capacity was front-page news again, when P. Hertel and W. Thirring, Ann. Phys. 63 (1971), FIGURE 3: JUMPING BOARD MODEL AND SHORT CHAOTIC 520. laboratory experiments on nuclear frag- 2 W. Thirring, Quantum Mathematical Physics, TRAJECTORY FOR A PARTICLE.THE GRAVITATIONAL mentation and atomic clusters hinted to 2nd edition, Springer Verlag: Berlin, 2001. FORCEPOINTSINTOTHENEGATIVE y DIRECTION.AT the existence of this phenomenon also 3 H.A. Posch, H. Narnhofer, and W. Thirring, Phys. THEBOTTOM, THE PARTICLE IS ELASTICALLY for ordinary Coulombic matter, for which Rev. A 42 (1990), 1880. REFLECTED.TO AVOID NEGATIVE x, THE y AXISACTSAS H.A. Posch, H. Narnhofer, and W. Thirring, gravitation could not be the culprit. This Microscopic Simulations of Complex Systems, M. ANELASTICMIRROR. seemed to contradict a general result by Mareschal ed., New York: Plenum Press, 1990, p. 241. 4 A particle, or an ensemble of weakly inter- J. L. Lebowitz and E. H. Lieb according H.A. Posch, H. Narnhofer, and W. Thirring, Phys- ica A 194 (1993), 481. acting particles, is subjected to a gravita- to which a quantum-mechanical Coulomb 5 H.A. Posch, H. Narnhofer, and W. Thirring, J. tional field parallel to the negative y axis system consisting of electrons and nu- Stat. Phys. 65 (1991), 555. and confined to a valley, which opens up clei always has positive specific heat, 6 H.A. Posch and W. Thirring, Phys. Rev. Lett. 95 very quickly such that the accessible con- even in the microcanonical ensemble. Of (2005), 251101. 7 H.A. Posch and W. Thirring, Phys. Rev. Lett. 95 figuration space increases faster than ex- course, this theorem presupposes ergodic- (2005), 251101. ponentially with the energy (respective the ity and the thermodynamic limit. We could H.A. Posch and W. Thirring, Phys. Rev. E 74 maximum height the particle may jump to). demonstrate by computer simulations and (2006), 051103. 8 The farther the particle gets to the right simple model calculations that the failure Lj. Milanovic,´ H. A. Posch, and W. Thirring, J. Stat. Phys. 124 (2006), 843. in this way, the more potential energy is of any of the two assumptions may be suf- 9 W. Thirring, H. Narnhofer and H.A. Posch, Phys. used up and the colder the particle gets, ficient to render the specific heat negative Rev. Lett. 91 (2003), 130601. a clear sign of negative specific heat. The in some restricted range of energy.9 10 Lj. Milanovic,´ H. A. Posch, and W. Thirring, non-equilibrium version of this model (the I had the good fortune to study many Phys. Rev. E 57 (1998), 2763. Lj. Milanovic,´ H. A. Posch, and W. Thirring, J. particles farthest to the right are allowed other problems with Walter. I would like to Stat. Phys. 124 (2006), 843. to fall out of the potential well) may also mention the problem of ergodicity and the 11 H.A. Posch and W. Thirring, Phys. Rev. E 48 be looked at as the most-efficient form of Lyapunov instability of one-dimensional (1993), 4333. 12 evaporation cooling, a process extensively gravitational systems,10 the unpredictabil- H.A. Posch and W. Thirring, J. Math. Phys. 41 (2000), 3430. studied to generate ultra-cold gases in con- ity of symmetry breaking in phase tran- 13 Ch. Marx, H.A. Posch and W. Thirring, Phys. nection with Bose-Einstein condensation. sitions,11 and some aspects of the clas- Rev. E, in press (2007).

Almost 50 Years of the the construction of a solvable field theory the Bethe-Ansatz he was able to solve by Lee many people tried to obtain a simi- the appropriate many particle system in Thirring Model lar relativistic model too. The well known a Hilbert Space of fixed particle number. Harald Grosse Bethe Ansatz, successfully applied in the The next step to change the representation thirties to the Heisenberg chain model, for which the spectral condition is fulfilled served as a guide. needs renormalization and is tricky. Within The formulation of Heisenberg proposed in the fifties the physical correct representation Thirring renormalized perturba- was able to calculate the two particle cor- tive quantum field the- a Four-Fermi Interaction Model (‘Ur- formel’), which, as was noticed immedi- relation function with one particle on shell, ory by Dyson, Feyn- which is called the form factor. man, Schwinger and ately by Pauli, led to difficulties, since it is Tomonaga predicted not renormalizable. Soon after this work Glaser proposed observable corrections The analogous two-dimensional model, an operator solution to the model and ob- due to quantum fluc- formulated already by Tomonaga in the tained surprisingly for the form factor an tuations, which are confirmed by experi- early fifties, was studied by Walter Thirring answer which differed from Thirring’s an- ments to an extremely high precision. After and published 1958. With the help of swer by a change of coupling constant.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 2007 13 This puzzling situation was analyzed by a unitarity, crossing and no particle creation are nowadays well established: conductiv- number of people like Johnson, Klaiber, it was possible to deduce a possible S- ity along one-dimensional quantum wires Wightman and others from various points matrix. Again it was possible to obtain the is well described by the Luttinger liquid. of view. Nowadays, of course, it is easy form factor, but no higher correlation func- It is interesting to note that the two to state that only renormalized quantities tions have been obtained in closed form. main proponents of these developments make sense and the step from bare to renor- On the other hand from constructive meth- in two different fields, in two-dimensional malized ones depends on the renormaliza- ods the existence of these higher correla- quantum field theory and two-dimensional tion scheme. Klaiber wrote down a gener- tion functions has been derived. statistical physics, joined and obtained a alized solution of the massless model. Coleman’s duality allows to map the substantial improvement of the stability of In 1962 Schwinger extended the set massive model to the Sine-Gordon model matter proof of Dyson-Lenard. Although of solvable models. Two-dimensional QED with all its solitons, antisolitons and Thirring’s calculation of the distribution of shows mass generation and appearance of breathers. This generalized the Kramers- pressure of teeth might be of more prac- a theta vacuum. It was a common effort Wannier duality of the Ising model sub- tical importance, the first analysis of the of many physicists and mathematicians, es- stantially, and was extended through work two-dimensional spinor model gave a great pecially Australians around Carey, to es- of Olive-Montonen and many others to impulse and led to many generalizations. tablish Wightman axioms for these mod- the fascinating geometric Langlands dual- The still puzzling situation of the four- els. Two-dimensional conformal field the- ity concept, which became popular recently dimensional local or even nonlocal quan- ory ideas can be tested again on the mass- through work of Kapustin and Witten. Such tum field theory models forces us even less model and allow for many generaliza- unforeseen developments show the power nowadays to study toy models in lower di- tions. of toy models which seem to be unphysi- mensions. Of course, nowadays, an infinite num- cal at first sight, but allow to study physical During the almost 50 years which have ber of solvable, respectively almost solv- effects in a nutshell. passed since the first study of the mass- able integrable models are known and stud- The completely independent develop- less Thirring model, it has served as a ied in detail. Especially the algebraic struc- ments (almost at the same time) in sta- toy model and a testing ground for gen- ture behind the Yang-Baxter equation led tistical physics are worth mentioning. The eral ideas and phenomena. A similar four- to the invention of quantum groups which well-known Luttinger model represents a dimensional model is still to be discovered; generalize the Lie algebra structure. kind of lattice-regularized Thirring model, the best candidates are N = 4 or even Despite many attempts (also by my- were renormalization effects can be stud- N = 8 supersymmetric models. self), the massive Thirring model resists a ied easily. Its first correct solution is due complete solution. Assuming factorization, to Mattis and Lieb. Its physics implications

A Crossword Puzzle A number of challenging ideas are on the sistency i.e. that the letters at the intersec- scientific market, especially modern string tions of words must agree. Moreover, the 1 Without Clues theory which tries to unify all fundamen- puzzle should admit only a single correct Can one find a ‘Theory of Everything’ tal forces. But string theory has recently solution. by pure thinking? come under severe criticism, even from ex- If this argument were to be applied to perts in the field. The problem is experi- Peter C. Aichelburg the physical description of the word we mental verification: although most of these could do away with the sophisticated ex- I dedicate this article to my teacher Wal- proposed theories may be falsified in prin- periments in particle physics and costly ter Thirring who not only introduced me to ciple in the sense of Popper, any of their large telescopes for astronomers. All that modern theoretical physics, but also guided expected consequences are far too small for would be needed are mental efforts of the- my scientific interest towards Relativity. today’s methods of detection. oreticians. Modern theoretical Revolutions in physics were almost In fact, never before have so many sci- physics is confronted always triggered by observations which entists dedicated themselves to the con- with a dilemma. The were either in conflict with, or could not struction of theories which are far beyond standard model of be explained by, current theories. For ex- any observational verification, the only cri- particle physics is in ample, the emission of electromagnetic terion being internal consistency and con- excellent agreement waves with definite frequencies by atoms vergence to existing theories when taking with experiments and was in contradiction with classical electro- suitable limits. However, in pursuing the Einstein’s theory of magnetic theory and led to the formula- analogy with a crossword puzzle a little gravitation is confirmed every day when tion of quantum theory, forcing physics to further, an essential feature of the problem applied to the satellites of the global po- abandon a deterministic description of the is missing. Even if the solution is unique sition system. However, the decisive step world. it can only be found if the number of towards a complete theory describing our Today, however, the situation is differ- ‘squares’ in the puzzle is finite. An infi- physical world is missing: the unification ent. Since there are no observations which nite puzzle can not only not be solved com- of quantum theory and gravitation. The are in contradiction with the established pletely, but unfortunately also not partially. dilemma is that nature does not seem to theories, an essential guiding principle is It may happen that one has found a con- give us any clue how to achieve this. absent. The problem may be compared to sistent solution of, say, the upper left cor- It is fair to say that, over the past a crossword puzzle without clues, where it ner, but that one runs into contradictions at decades, the brightest scientists in the field is difficult, but not impossible to solve the a later stage. This forces one to go back and have devoted their efforts to this question. puzzle. The only guideline is internal con- abandon what has already been obtained. If http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 14 Volume 2, Issue 1, Spring 2007 ESI NEWS the puzzle is unbounded, this may happen place that traditionally was provided by re- light atomic nuclei in the universe. again and again at every stage. ligious genesis. Religious faith is replaced These calculations are based on laws Without no generally accepted the- by scientific faith. But only few are aware for subatomic matter tested in exper- ory at hand it is not too astonishing that that the scientific basis of these statements iments at accelerators and reactors. sometimes the fantasies of scientists go is rather meagre. The theory for the primordial nuclear astray. Every now and then the public is Not trying to make a case for intel- synthesis turned out to be in excel- confronted with new and surprising ideas ligent design, it is fair to say that up to lent agreement with observations. about the origin of the universe. For exam- now there does not exist a theory which ple, that our cosmos is the result of a vac- would allow us to model the beginning of On the other hand there is the seri- uum fluctuation and that these processes the universe. Einstein’s theory of General ous and so far unsolved problem of dark take place continuously and our world is Relativity gives a dynamical description of matter. As the name says, dark matter can therefore just one in an infinite ‘multi- the cosmos as a whole, but shows at the not be seen directly and its presence is verse’. Or that we live on a kind of mem- same time its limits of applicability. The solely noticed through its gravitational ef- brane embedded in a higher–dimensional big bang, where the density of the substrate fects, e.g. the bending of light rays which space, where the extra dimensions mani- that fills the cosmos increases unbound- reach us from distant objects. Dark mat- fest themselves purely through the force edly, thereby driving the geometry into a ter must have completely different prop- of gravity. If one follows the well known singularity, lies outside the scope of the erties from ordinary matter, otherwise it physicist Lee Smolin in his book The Life theory. The theory cannot make statements would have influenced the primordial dis- of the Cosmos2 one learns that for the uni- about the big bang itself. tribution of light nuclei. Scientists expect verse a cosmological natural selection acts To overcome this problem one would to obtain clues about the strange properties similarly to biological evolution. need a theory which unites Einstein’s grav- of dark matter when the Large Hadron Col- ity with quantum theory, the theory that lider (LHC) at CERN in Geneva goes into What should we think of such ideas? governs interactions at the microscopic Without doubt, there is today not a sin- operation this year. Still more mysterious level. Such a ‘theory of everything’ is is what is called the dark energy: obser- gle observation which supports these spec- the dream of many physicists. Although it ulations. In contrast, the Nobel Price 2006 vations of exploding stars in distant galax- might be possible to find a unique consis- ies indicate that the cosmic expansion is in Physics was awarded to the American tent theory of nature by pure thought, it is scientists John Mather and George Smoot accelerating, while ordinary matter should very unlikely. Knowledge about our uni- slow down the expansion rate. These find- for the detection of the structure and exact verse has to be based and oriented on ob- form of the cosmic microwave background. ings are also supported by the observed fine servation. structure in the microwave background. In- These observations give strong support to What then is observation–based knowl- the big bang theory. They show a detailed vestigations indicate the existence of a yet edge in modern cosmology? The main as- unknown substrate. picture of the universe of more than 13 bil- sertion is that the universe undergoes a con- It may well be that the search for this lions year ago and made cosmology a pre- tinuous change and that it has emerged dark energy could give a decisive clue to cise science. from a dense state about 14 billion years a connection between the quantum world How do these observations compare to ago. This big bang theory is supported by of the microcosmos and the largest dimen- claims made about the emergence of our three key observations: sion of the universe dominated by gravity. cosmos? We are confronted with a ques- (i) The shift in frequency of light com- This could bring the long–lasting search for tionable embellishment of observation– ing from distant galaxies which indi- a theory of quantum gravity back into the supported knowledge on the one side, and cates that the cosmos as a whole is state of an observation–based science. with imaginative speculations on the other. subjected to a general expansion. Such speculations reach the public through Notes popular and semi-popular books and are (ii) The cosmic microwave background 1 apparently well received. In our secular- radiation confirms that the early uni- A German version of this article was published in verse was dense and hot. the Austrian newspaper ‘Die Presse’ on February 16, ized world, where an increasing number of 2007. people find it difficult to unite ‘rationality’ (iii) Physicists were able to calculate the 2 Lee Smolin, The Life of the Cosmos, Oxford Uni- and ‘religious faith’, such concepts take the formation of the first generation of versity Press USA, 1998.

Doing Physics with Walter my memory goes back to a turning point led to lots of fruitful scientific discussion in the early seventies when we were sitting (including collaboration with Heide Narn- Elliott Lieb around a lunch counter somewhere and hofer) nothing dramatic happened except Walter asked me if I ever thought about that I lost a key to the Institute. Fortunately, My scientific life has the Dyson-Lenard proof of the stability of and this must be recorded for posterity, the natural dividing lines, matter. No, I hadn’t really, but he had, and mainstay of Walter’s group, apart, natu- like new chapters in he realized that, while correct, it needed rally, from his wife Helga, was his assis- a book, the most im- some new mathematical insight to make it tant Franziska (Franzi) Wagner. She could portant of which is physically understandable as well as math- do everything; not only type up the stuff we the day I started to ematically correct. generated but also figure out how to deal work with Walter. Walter invited me to be a ‘Schrodinger¨ with a missing, priceless, irreplaceable key Walter says we met guest professor’ at the University of Vienna that was official government propoerty and in 1968, and that is undoubtedly true, but in the summer of 1974, and while this visit must never, under any conditions be lost or

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 2007 15 duplicated. or molecules, which drops off like the neg- would announce that it was time to stop But to return to the story, I was a ative sixth power of the distance between the science and play some baroque sonatas, ‘Schrodinger¨ guest professor’ again in the them (in the absence of electromagnetic and he would very patiently overlook my summer of 1975 and this time we solved propagation corrections), was a universal missed entrances and other mistakes. It was the problem. The main new idea was the consequence of Schrodinger’s¨ equation. fun and I learned a lot, but I also realized realization that the kinetic energy of elec- There were enjoyable visits before and that I would never be organized enough to trons ( and other particles satisfying Pauli’s after 1986. I would visit Walter in his hide- do all the many things, in many fields, that exclusion principle) is always greater than away in Zweiersdorf, where Helga was a Walter is capable of doing. the integral over all space of the 5/3 power superb hostess, and he would often visit of the particle density. This, in turn, meant my wife and me in Princeton. Walter is Walter continues his work in science that the old theory of E. Fermi and L.H. one of the most organized people I know. with the energy of a newcomer. Not only Thomas gave a lower bound to the total en- The visits followed a rigorous time sched- is he interested in everything but he con- ergy of matter, and this bound was known ule starting around 6 am and ending around tinues to do original research in his inim- (by earlier work with Simon) to satisfy the 9 pm that dictated when we would wake itable style – which is to do what you think desired stability condition. up, take walks, play music, exercise and do is interesting and important without paying We had several collaborations after science. Walter is an accomplished piano much attention to the fashion of the day. that, but our last published work together and organ player, and composer, while I Welcome to the 80’s, Walter, and continue was the proof in 1986 that the attractive was a fifth rate recorder (Blockflote)¨ player. to inspire us youngsters by doing great sci- van der Waals force between pairs of atoms Following an internal alarm clock Walter ence!

Encounters with Walter bation expansion. Apparently he had been us in Adelaide for two weeks. During that instructed by Pauli to check it out as it time we took them to visit the property Thirring made some rather strong assertions. Not of a former mathematical physics student, Angas Hurst only did he check it out, but he rederived which occupied 200 square miles, and was the results by much neater methods and complete with shearing sheds and kanga- completed the outstanding gap of includ- roos. As a result they became great lovers My first contact with ing renormalisation effects. As a result the of Australian television series. Walter was when I quantum field used was called, by Bogoli- Two areas in which our interests have saw a letter in Phys- ubov and Shirkov, the Hurst-Thirring field, ical Review in 1950 overlapped have been in physical systems and so our names have always since then C∗ written by him from in which -algebras have been a potent been coupled together, although our paths method. Dublin. It was prob- have diverged widely. ably his first publi- The first was a paper with Heide Narn- cation and dealt with Walter became one of the stalwarts of hofer on the construction of covariant QED ‘Regularization as a Consequence of European physics, whilst I led a somewhat without an indefinite metric. This thinking Higher Order Equations’. This was a neat quieter life bringing mathematical physics closely parallels work that I did in collabo- connection between higher order wave to Australia, carrying on a tradition started ration with Alan Carey, Janice Gaffney and equations and the Pauli-Villars regularisa- in Adelaide University by William Bragg, Hendrik Grundling, which took off from tion. This was particularly helpful for me and it was not until I went to a Summer Re- an old paper I wrote in 1960. Our final because shortly before I had heard Pauli search Institute at Hercegnovi in 1961that conclusion was that for gauge field theo- deliver the Rouse Ball lecture to the math- we actually met. I went as someone who ries, or more generally theories with non- ematics faculty in Cambridge, and it was wanted to catch up after nearly a decade of integrable constraints, a very wide range of an example of how not to do things. Only isolation in Adelaide, and Walter was there representations may be employed so long someone with Pauli’s reputation could have as one of the lecturers. It was a very dis- as the observable part is treated by a regu- got away with it. He introduced a method tinguished gathering, both for the standing lar Hilbert space representation. What one for controlling divergences in quantum of the lecturers and for the collection of does with the non-observable part does not field theory to a general audience of mathe- students, many of whom became world fa- matter, so indefinite metric or nonseparable maticians who almost certainly knew little mous. Apart from the lectures, Walter spent representations or other pathologies can be about quantum field theory and less about most of his time drifting around the lovely employed without prejudice. For this C∗- the problems of divergences. His delivery harbour using a snorkel, and I brought algebraic methods are crucial, which Wal- consisted of walking backwards and for- some order in the Proceedings by correct- ter and Heide spelt out very thoroughly for wards in front of the audience, with his ing much of the distorted European English their system. head sunk on his chest, accompanied by provided by the lecturers. The final distinc- The second area is in a very deep study a regular squeak of a loose floorboard in tion of this meeting was that Walter invited of the description of entropy in operator al- the middle of his to and fro path. As a me to come to Vienna, where I gave a sem- gebras, and it touches on a question, which barely beginning research student I was inar on work I had done with Green on the has bothered me for years, which is how to completely lost. So Walter’s paper was a Ising model. It was also a marvellous op- understand the way in which information light in the darkness. portunity to see Vienna, and to enjoy its is parcelled between physical theories and Later on we had a more personal en- wonderful music and art galleries. raw data. I am bothered by a remark I once counter when he wrote to me about the Since then I have been to Vienna on saw Paul Davies makes about complexity content of my PhD thesis on the conver- numerous occasions and Walter and Helga of systems, and how theoretical science has gence of the quantum field theory pertur- came to Australia in 1987, staying with been able to succeed because it has been http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 16 Volume 2, Issue 1, Spring 2007 ESI NEWS possible to shunt the complexity into the has made us, and it need not have been so. privileged to have had so valuable a friend- data, whilst still having manageable theo- I shall continue to admire the depth and ship. ries. This seems to be a gift which Nature diversity of Walter’s work, and I am very

Birthday Greetings to tures were excellent. Walter conceived the of Mt.Dickerman. The mountain was en- idea of publishing these lectures and asked veloped in fog, but we thought that it would Walter Thirring me whether I would be willing to collabo- clear or that the top would be out of the Ernest M. Henley rate. I was eager to do so. After we wrote fog. Instead, the fog got worse and worse the lectures up, we asked students to find the higher we went. Fortunately, there is errors and offered monetary rewards. Still, a well known blueberry area about 3/4 of I have known Walter after the book was published as Introduc- the way up, where we gorged ourselves and Thirring for over 50 tion to Quantum Field Theory by McGraw- had lunch. On top, you could barely see years. Walter used to Hill we continued to find errors which had your own feet. Nevertheless, I pointed out come to the Univer- eluded both the students and us. I noted all all the wonderful peaks that you could see sity of Washington dur- of these errors in my copy. When McGraw- (on a clear day): Glacier Peak, Sloan peak, ing summers, which is Hill asked to borrow the annotated copy for Big Four, Vesper, and Monte Cristo, among the nicest time of year a translation into Japanese, I agreed, with others. It is such a beautiful sight that I here. We had Theoreti- the proviso that the book would be returned was deeply sorry that Walter and Helga had cal Summer Institutes, organized by a col- to me. Unfortunately, I never saw the book missed out! We never managed to repeat league (Boris Jacobsohn) and me, with the again, and McGraw-Hill could not locate this hike. help of members of the Physics Depart- it! By then, the book was out of print. Since We have remained good friends over ment. These Institutes attracted quite a few this was before Xerox was popular, I, un- the years. Whenever I can get to Vienna I well known physicists, as well as younger fortunately, had not copied the corrections. do so and always visit the Thirrings. ones, to the University. During one of the The book is only one of many pleas- summers in the late 1950’s, Walter gave a ant memories of the Thirrings’ visits. We My best wishes to Walter on his 80th series of lectures on quantum field theory went on many hikes together. I particu- birthday. I wish I could attend the sympo- and exactly soluble models in it. The lec- larly remember one notable one to the top sium and the celebration.

ESI News figures yet, the funding level of the pro- Autumn/Winter Term 2007/08 gramme for 2007 – 2009 is expected to be • Christos Likos, Heinrich-Heine Uni- at least e 200.000/year. The ESI Junior Research Fellows Pro- versity, Dusseldorf:¨ Theory of soft Currently there are 7 post-docs at the ESI, 6 gramme (ESI-JRF) has been extended by matter. a further three-year period (2007 – 2009). of whom are funded by the Junior Research Fellows programme. • Radoslav Rashkov, Sofia University: Funded by the Austrian Ministry of Sci- The most recent call for applications ended Dualities between gauge theories ence and Research, the programme pro- on April 30, 2007, and attracted 37 appli- and strings. vides support for PhD students and young cations. post-docs to participate in the scientific ac- For further information concerning the tivities of the Institute and to collaborate The deadline for the next round of appli- contents of these lecture courses and rel- with its visitors and members of the local cations for ESI Junior Research Fellow- evant literature we refer to the web page scientific community for periods between ships will be November 10, 2007. http://www.esi.ac.at/activities/courses.html. 2 and 6 months. During the initial three-year period (2004 – In the ESI Senior Research Fellows Pro- The European Post-Doctoral Institute 2006) the ESI-JRF programme established gramme the following courses will be of- (EPDI): The European Post-Doctoral In- itself as a highly successful component of fered during the Summer Term 2007 and stitute for Mathematical Sciences was the Institute’s scientific activities. Both the the Winter Term 2007/08. founded in October 1995 with the ambi- number and quality of the applicants ex- tion of facilitating the mobility of young Summer Term 2007 ceeded all expectations, and only 66 out scientists within Europe. The Institut des of 282 applications could be funded during ´ • Vadim Kaimanovich, International Hautes Etudes Scientifiques (Paris), the the years 2004 – 2006, although about 70 University Bremen: Boundaries of Isaac Newton Institute for Mathemati- % of the applicants were judged to be of groups: geometric and probabilistic cal Sciences (Cambridge) and the Max- ‘supportable’ standard. aspects. Planck-Institut fur¨ Mathematik (Bonn) as In 2004, the level of funding of the pro- founding members were later joined by • Miroslav Englis, Academy of Sci- gramme had been set at e 150.000/year. several other institutes, among them the ences, Prague: Analysis on complex In 2005 funding was increased to Erwin-Schrodinger¨ Institute. This initiative symmetric spaces. e 200.000/year, with the additional turned out to be a very successful scien- e 50.000 earmarked to help increase the • Thomas Mohaupt, University of Liv- tific enterprise which enabled young re- percentage of women engaged in mathe- erpool: Black holes, supersymmetry searchers to pursue their work at the partic- matical research. and strings (Part II). ipating institutions. This January, the 12th Although we have not been given precise call came to a conclusion in a meeting

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 1, Spring 2007 17 of the Scientific Committee at the Mittag- PIMS 10th Anniversary Distinguished Lec- trian Embassy in Dublin and the National Leffler Institute Stockholm, where the win- ture with the title ‘On some of the differ- Bank of Austria, commemorates the fa- ners of the two-year research fellowships ences between Z and Z2 in dynamics.’ mous lectures ‘What is life’ given by Erwin for the period 2007-2009 were nominated. Schrodinger¨ at Trinity College in 1943. It is worth noting the number of women In December 2006, Jakob Yngvason held Jakob Yngvason also gave the Andrejew- elected is very high compared to the num- the Erwin Schrodinger¨ Lectures at the Uni- ski Lectures at the University of Leipzig in ber within the scientific community. versity Colleges in Cork and Limerick and January 2007. at Trinity College, Dublin, Ireland. This an- In January 2007, Klaus Schmidt gave a nual lecture series, supported by the Aus-

Current and Future Activities of the ESI

Thematic Programmes Summer School, July 7 – July 18, 2008

Amenability, February 26 – July 31, 2007. Metastability and Rare Events in Complex Systems, February 1 Organizers: A. Erschler, V. Kaimanovich, K. Schmidt – April 30, 2008 Workshop on amenability beyond groups, February 26 – March Organizers: P.G. Bolhuis, C. Dellago, E. van den Eijnden 17, 2007 Workshop, February 17 – February 23, 2008 Workshop on algebraic aspects of amenability, June 18 – June 30, 2007 Hyberbolic Dynamical Systems, May 12 – July 5, 2008 Workshop on geometric and probabilistic aspects of Organisers: H. Posch, D. Szasz, L.-S. Young amenability, July 2 – July 14, 2007 Workshop, June 15 – June 29, 2008 Poisson Sigma Models, Lie Algebroids, Deformations and Higher Analogues, August 1 – September 30, 2007 Operator Algebras and Conformal Field Theory, August 25 – Organizers: H. Bursztyn, H. Grosse, T. Strobl December 15, 2008

Applications of the Renormalization group, October 15 – Organisers: Y. Kawahigashi, R. Longo, K.-H. Rehren, J. Yngvason November 25, 2007 Organizers: G. Gentile, H. Grosse, G. Huisken, V. Mastropietro Other Scientific Activities Workshop on the Renormalization Group Flow and Ricci Flow, October 22 – 26, 2007 First European Young Scientists Conference on Quantum Information, August 27 - 31, 2007 Workshop on Renormalization in Dynamical Systems, October 29 – November 3, 2007 Organizers: Simon Groblacher¨ and Robert Prevedel

Workshop on Renormalization in Quantum Field Theory, Central European joint Programme of Doctoral Studies in Statistical Mechanics and Condensed Matter, November 12 – 17, Theoretical Physics, September 24 - 28, 2007 2007 Organizer: Helmuth Huffel¨ The programme will be followed by an ESF-Workshop on Non-commutative Quantum Field Theory, November 26 – 30, There will be two lecture courses of 15 lectures each: 2007. Introductory courses to this subject will be given by J. Barrett Harald Grosse: Noncommutative Quantum Field Theory and R. Szabo in the week November 19 – 23, 2007. Jakob Yngvason: Local Quantum Physics

Combinatorics and Statistical Physics, February 1 – June 15, 2008 Fourth Vienna Central European Seminar on Particle Physics Organisers: M. Bousquet-Melou, M. Drmota, C. Krattenthaler, B. and Quantum Field Theory, November 30 – December 2, 2007 Nienhuis Theme: Commutative and Noncommutative Quantum Field theory Workshop, May 25 – June 7, 2008 Organizer: Helmuth Huffel¨

http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 18 Volume 2, Issue 1, Spring 2007 ESI NEWS

Thirringfest Tuesday, May 15, 2007

Lecture Hall of the Faculty of Physics, Strudlhofgasse 4, 1090 Vienna

10:00 – 10:15 Welcome addresses

GEORG WINCKLER, Rektor, University of Vienna ANTON ZEILINGER, Dean, Faculty of Physics, University of Vienna KLAUS SCHMIDT, President, ESI

10:15 – 11:15 Wolfgang Rindler (Dallas): Vienna, Hans Thirring, and Gravity Probe B 11:15 – 11:45 Coffee Break 11:45 – 12:45 Julius Wess (Munich): Deformed Theory of Gravity 14:30 – 15:30 Elliott H. Lieb (Princeton): Remarks on Density Functional Theory

17:30 – 18:30 Eine kleine Hausmusik: Chamber music composed by Walter Thirring Performed by friends at Pfarrsaal, Nussdorf, Pfarrplatz 3, 1190 Wien

19:00 – Heuriger – Maier am Pfarrplatz

Editors: Klaus Schmidt, Joachim Schwermer, Jakob Yngvason ESI Contact List: Contributors: Administration Peter C. Aichelburg: [email protected] Isabella Miedl: [email protected] Bernhard Baumgartner: [email protected] Scientific Directors Harald Grosse: [email protected] Joachim Schwermer: [email protected] Ernest M. Henley: [email protected] Jakob Yngvason: [email protected] Angas Hurst: [email protected] President Wolfgang Kummer: [email protected] Elliott Lieb: [email protected] Klaus Schmidt: [email protected] Heide Narnhofer: [email protected] Herbert Pietschmann: [email protected] Harald A. Posch: [email protected] Wolfgang L. Reiter: [email protected] Walter Thirring: [email protected] Helmuth Urbantke: [email protected]

The ESI is supported by the Austrian Federal Ministry for Science and Research ( ). The ESI Senior Research Fellows Programme is additionally supported by the University of Vienna and the Vienna University of Technology.

This newsletter is available on the web at: ftp://ftp.esi.ac.at/pub/ESI-News/ESI-News2.1.pdf

IMPRESSUM: Herausgeber, Eigentumer¨ und Verleger:INTERNATIONALES ERWIN SCHRODINGER¨ INSTITUTFUR¨ MATHEMATISCHE PHYSIK, Boltzmanngasse 9/2, A-1090 Wien. Redaktion und Sekretariat: Telefon: +43-1-4277-28282, Fax: +43-1-4277-28299, Email: [email protected] Zweck der Publikation: Information der Mitglieder des Vereins Erwin Schrodinger¨ Institut und der Offentlichkeit¨ in wissenschaftlichen und organisatorischen Belangen. Forderung¨ der Kenntnisse uber¨ die mathematischen Wissenschaften und deren kultureller und gesellschaftlicher Relevanz.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/