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

ESI NEWS Volume 2, Issue 2, Autumn 2007

Editorial lation. Klaus Schmidt was a key participant Contents in the workshop Interfaces between Math- Editorial 1 ematics and Physics in Vienna in May This summer saw the 1991 which laid the foundation for the Er- Wolfgang Kummer 1935 – 2007 1 deaths of two eminent win Schrodinger¨ Institute, both scientifi- Julius Wess 1934 – 2007 2 physicists who had cally and politically. He helped to impress had close links with on the Minister for Science at that time, Reminiscences of old Friendships 2 the ESI over many Erhard Busek, the desirability and, indeed, In memoriam Julius Wess 3 years and to whom necessity of creating a research institute the ESI remains grate- to provide a meeting place where scien- Entanglement in many-body quan- ful for their friendship tists from Eastern Europe could interact tum physics 4 over many years. with the international scientific community Kazhdan’s Property (T) 8 WOLFGANG KUMMER, Professor for at a period of great political and financial Theoretical Physics at the Vienna Univer- uncertainty in the post-communist world. Perturbative Quantum Field The- sity of Technology (VUT), was a mem- Julius Wess helped the ESI on a second oc- ory 9 ber of the ‘Vorstand’ (Governing Board) casion, when fell seriously Interaction of Mathematics and of the Erwin Schrodinger¨ Institute from ill in early 1992 and Julius Wess chaired Physics 11 1993 until 2005 and was elected Honorary a second workshop on Interfaces between Member of the Institute when he resigned Mathematics and Physics in March 1992 ESI News 11 from the board in 2005. He was instrumen- and took charge of the negotiations with Current and future activities of the tal in encouraging and maintaining scien- the Austrian Ministry of Science which had ESI 12 tific links between the ESI and the VUT reached a crucial point at that stage. It is by co-organizing scientific programmes at fair to say that Julius Wess made a signif- the ESI and co-hosting ESI Senior Re- icant contribution both to the foundation search Fellows on several occasions. The and the scientific development of the ESI. ESI owes him many valuable suggestions, Julius Wess was elected Honorary constructive criticism and scientific stimu- Member of the ESI in 2005.

Wolfgang Kummer nology (VUT) on a more regular basis. topics of high-energy physics and quantum Sudden heart failure prevented his plans field theory was, in an ironic twist, partly 1935 – 2007 from coming true. motivated as deliberate contrast program to the dominant research activity of his doc- Joachim Burgdorfer¨ We mourn the loss of a great scien- toral supervisor and chair of the Theoreti- tist, of an academic teacher and researcher cal Physics at the VUT at that time, Walter of highest calibre, and of a key figure The Austrian physics Glaser, which was firmly rooted in classi- that helped shaping the scientific profile community received cal dynamics. Glaser, who died already in and reputation of high energy physics in unexpectedly the sad 1960 at the age of 54, did not live to see the Austria. He was appointed full professor news of Wolfgang recognition his joint research with his ex- of Theoretical Physics in 1968 as one of Kummer’s passing perimental collaborator and winner of the the youngest full professors in Austria and away on July 15, 2007 1988 Nobel prize Ernst Ruska would even- served in this capacity for 36 years un- after a long and coura- tually receive. Glaser’s theoretical contri- til reaching emeritus status in 2004. His geous battle with can- butions to electron beam optics played a teaching career began even earlier when cer. Just a few days earlier, Wolfgang had crucial role in the developments of the high he became university assistant (or assistant given in a phone conversation with the au- resolution electron microscope, as Ruska professor) in 1958. Over a period of almost thor of these lines an up-beat assessment noted in his Nobel lecture. half a century, interrupted by several re- of a new treatment he was receiving and search visits abroad, Wolfgang taught theo- to which he responded well. He was look- Wolfgang Kummer received strong retical physics at the VUT. He received his ing forward to a further improvement of support and tutelage from Walter Thirring, diploma in 1958 and his Ph.D. in theoreti- his condition and he made already plans , who secured for him cal physics in 1960 from VUT. to return to ‘his’ Institute for Theoretical a Ford Fellowship and brought him in con- Physics at the Vienna University of Tech- His desire to specialize in the ‘modern’ tact with the high-energy physics com- 2 Volume 2, Issue 2, Autumn 2007 ESI NEWS gang became victim of a terrorist attack was unceasingly productive well beyond munity. Kummer joined Victor Weisskopf, on Vienna airport when he suffered mul- his official retirement in 2004. then director-general of CERN, as a Ford tiple injuries from hand grenade splinters His work received many accolades Fellow from October 1961 to March 1962. and shrapnels. Even though his injuries and signs of recognition. Among others Weisskopf invited him to come back as a were life threatening and he spent eleven he was elected full member of the Aus- CERN fellow and his scientific assistant days in intensive care, he quickly recov- trian Academy of Sciences in 1985. Wolf- from 1963 to 1964. Returning to Vienna, ered and resumed his duty as council presi- gang was awarded the Schrodinger¨ Prize Kummer completed his habilitation (‘ve- dent within weeks. Kummer also served in in 1988, the Walter Thirring Prize 2000 nia legendi’) from VUT in 1965. He be- numerous international organizations that (together with L. Faddeev). He received came the founding director of the Institute set science policies. Among others, he was an Honorary Doctorate from the National for High Energy Physics of the Austrian member of the Austrian UNESCO com- Academy of Sciences of the Ukraine in Academy of Sciences in 1966, which he mittee for science from 1975 to 1992 and 2005. led until 1971. His appointment to the sec- Austrian representative at the General As- ond chair of Theoretical Physics at VUT sembly of the International Union of pure Kummer’s contributions to the scien- took place during this period, as a result of and Applied Sciences (IUPAP) from 1996 tific community at large were by no means which his center of activity gravitated to- to 2002. He also was member of the limited to science. He exemplified the role wards his Alma Mater. Among the many ‘Vorstand’ (Governing Board) of the Er- of an enthusiastic teacher, of an unselfish administrative duties he took on during win Schrodinger¨ International Institute for and supporting mentor of his younger col- his tenure were the chairmanship of the Mathematical Physics, Vienna, founded in leagues, and of a colleague of impeccable physics department from 1981 to 1987 and 1993. Kummer’s foremost achievement is integrity. He will be remembered for his the directorship of the Institute for Theoret- undoubtedly the build-up of a strong theo- unfailing dedication to the cause of science, ical Physics from 1995 until 2003. retical high-energy physics group covering displayed even under adverse conditions of Over many years Kummer represented a broad range of topics in quantum field deteriorating health. the Austrian High-Energy Community at theory and (mainly 2D) quantum gravity. His many friends and close colleagues CERN. From 1966 to 1971 he was Austrian Kummer made fundamental contributions are grateful for the time we were privileged representative at the Council of CERN. to quantum gauge field theory, in particu- to share with him. Our thoughts are with his He served as its vice president from 1980 lar by using ghost-free non-covariant gauge widow, Dr. Lore Kummer who was his sup- to 1983 and as its president from 1985 fixing. Since the early 1990’s he mainly portive companion for almost half a cen- to 1987. On December 26, 1985 Wolf- worked on two-dimensional gravity and he tury.

Julius Wess 1934 – 2007 ray Gell-Mann and Yuval Ne’eman, but ries and require fewer renormalization con- not with the same representation as in the stants. It is possible to formulate SUSY ex- Walter Thirring and “eightfold way”. Instead, it included only tensions of the Standard Model (SM) of the A-particle in the same representation particle physics that do not have the diffi- Julius Wess was an as the nucleons; at that time it seemed too culties of the conventional SM. These ex- imaginative, techni- daring to include the other five particles as tensions imply new particles and fields, cally strong and in- well, as their properties seemed to be un- which could be found at LHC, fluential theoretical duly different. Working with Tom Fulton, which is due to start up in 2008. Some of physicist. He died Julius went on to formulate an SU(6) the- these theories predict particles (e.g. neu- suddenly in Ham- ory in an attempt to unify spin and isospin tralinos) that are candidates for the dark burg on 8 August at in agreement with special relativity. matter of the universe. the age of 72. Julius also worked in collaboration Julius’s path took him to many places Julius, an assistant to Walter Thirring with Bruno Zumino on the mathemati- around the world, and through his lov- in Vienna, went on to be a professor first cal structure of anomalies in non-Abelian able, unassuming manner and his conta- at Karlsruhe University and then at the gauge quantum field theory. This work gious zest for life, he rapidly made many University of Munich, later becoming a showed that anomalies must satisfy a friends. We shall all miss him greatly. director of the Max Planck Institute for consistency condition and that they give (Reprinted from CERN Courier, November Physics in Munich. He was an excellent rise to interaction terms (usually called 2007.) and friendly teacher and taught many stu- Wess-Zumino terms) which have interest- dents who now have positions in univer- ing topological properties. This pioneering sities and research institutes. He was also work has had numerous ramifications for Reminiscences of old awarded several honorary doctorates, as both physics and mathematics. Friendships well as physics prizes and medals. Julius also wrote a number of papers on Julius’s scientific work was influenced supersymmetry (SUSY) and supergravity Email of Bruno Zumino to Walter Thirring strongly by the recognition that the dynam- in collaboration with Zumino. This work ics of quantum field theories is dictated shows that there exist 4D, local, relativis- Dear Walter, largely by symmetries. His first pioneering tic quantum field theories that admit a sym- As you can imagine, Julius’ death was work was on the consequences of confor- metry between Bose and Fermi fields and a very heavy blow for me. At the ceremony mal invariance for quantum fields. He then that are renormalizable in the conventional where we were both awarded the Wigner studied the representations of SU(3) for the sense. SUSY implies that these theories prize and medal, Lochlainn O’Raifeartaigh classification of hadrons. This work was are more convergent (for instance, have no introduced us as the “terrible twins” be- done with Thirring two years before Mur- quadratic divergencies) than generic theo- cause of our numerous joint successful pa-

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 20073 at your invitation. Our friendship and our and very impressed by the breadth and pers. Twins is not the right word, Julius collaboration started there and continued the depth of his insights. His influence in was twelve years younger than me; I sug- in many different places, mostly in Karl- physics and biology has been fantastic. He gested “loving brothers” as more appropri- sruhe, which I visited numerous times and was very pleasant and gentle with me dur- ate. Besides the Wigner prize we shared the at CERN where I was a staff member and ing my visit with him. Heineman prize and we were both awarded Julius a frequent visitor. I hope that you are well and send you the Max-Planck-Medal although in differ- my very best wishes. You have always been ent years. We also received together the Of my time in Vienna, I remember with a very good friend to me, I have not forgot- Humboldt Research Award. great pleasure having the opportunity of ten it. As you know, Julius and I met first meeting Erwin Schroedinger at his house. I in Vienna during the time I spent there am a great admirer of Schrodinger’s¨ work Bruno Zumino

In memoriam Julius Wess symmetry properties of elementary par- per on Supergauge Transformations in ticles, cooperating mainly with Markus Four Dimensions, which influenced parti- 5.12.1934, Oberwolz¨ – 8.8.2007, Fierz. In 1959 he came back to Vienna as cle physics considerably and has been cited Harald Grosse assistant Professor, where Walter Thirring (up to now) 1348 times. had followed his father as chair of the In- Julius was always concerned with sym- It came as a shock when stitute for Theoretical Physics. Here Julius metries. They restrict the dynamics and we learnt that Julius met Boris Jacobsohn and Bruno Zumino. yield conserved quantities. In all branches Wess is not with us any In 1960 Zumino invited him to take a po- of physics they help to analyse physical longer. A great scien- sition as a Research Associate at the New systems. In 1973 bosonic and fermionic tist, a very close friend York University. During a subsequent half strings were formulated. Julius and Bruno passed away and we year visit at the University of Washing- deduced from these models space time su- are left behind with so ton in Seattle a very close friendship with persymmetry: in nature we have particles many memories of sci- the physicists Grace and Lawrence Spruch with integer and half-integer spin, bosons entific discussions and co-operations and started. In the years 1962 to 1966 Julius and fermions; they behave differently un- very personal relations with this extraordi- came back to Vienna and worked with Tom der rotations and other transformations and nary, deeply human scholar. His scientific Fulton on the question of the unification of obey different statistics. Electrons, protons achievements had been acknowledged by internal and Lorentz symmetry, which was and neutrons are fermions, their statistics important awards: the Gottfried Wilhelm answered negatively by the work of Sidney implies for example the stability of matter. in 1986, the Max Planck Coleman and Jeffrey Mandula. Fermions form the building blocks of mat- medal in 1987 and the Wigner medal in In 1966 Wess became associate profes- ter, while bosons are the particles which are 1992 (together with Bruno Zumino). sor at the Courant Institute in New York. responsible for the forces between them. Julius had already left Vienna at the During his two years stay at the Courant The proposed supersymmetry allows in time of my own studies at the Institute Institute from 1966 to 1968 Kurt Symanzik a fantastic manner to map from bosons to of Theoretical Physics. I learnt about his and Wolfhard Zimmermann gave impres- fermions, a transformation which has been work on conformal symmetry (which I sive lectures on quantum field theory, and achieved by using so called Grassmann was asked at my final exam with Wal- his intensive collaboration with Bruno Zu- variables, which square to zero. This en- ter Thirring) from the literature and I mino was continued. In 1968 Wess ac- largement of variables leads to an extension was deeply impressed by his work on the cepted an offer for a full professorship at of space: besides the space and time coordi- quark model and on Chiral Effective La- the University of Karlsruhe, where he spent nate a further part is introduced, which cor- grangians. more than 20 years. In 1990 he left Karl- responds to this new variables. This simple I personally met Julius for the first sruhe to become director of the Max Planck step has enormous consequences: To each time at a conference in Frascati and at the Institute for Physics (Werner-Heisenberg- particle there corresponds an appropriate Schladming Winter Schools in the early Institute) and professor at the Ludwig Max- superpartner. Nature does not fulfil this rule 1970s. imilian University in Munich. at the energies we are fit to measure up to Born in the small Alpine village During his time in Karlsruhe he worked now. Oberwolz¨ in Styria in 1934, Wess studied on various symmetry concepts in physics: This beautiful symmetry cures defi- physics and mathematics at the University together with Callan, Coleman and Zumino ciencies of our quantum field theoretical of Vienna. Here he was deeply impressed Effective Chiral Lagrangian were devel- models. (besides his readings of Robert Musil’s oped; together with Zumino in 1971 he The masses of the superpartners are ”Mann ohne Eigenschaften”) by two of his discovered the anomalous Ward identities supposed to be high, and therefore they are academic teachers, the mathematician Jo- (cited 1735 times since then). Here for not yet observed on earth. But they are ex- hann Radon and the theoretical physicist the first time the so called Wess-Zumino pected to contribute to mass estimates of . In Vienna he also met his term appears which now plays an impor- particles in the Universe and could explain later wife Waltraud Riediger and a deep tant role in model building (Wess-Zumino- missing energy. friendship developed. His thesis supervisor Witten model) as well as in conformal field It is a bitter irony that Julius no longer was Hans Thirring and Julius received his theory. This term serves up to this day as is able to follow the search for these parti- Ph.D. in theoretical physics in 1957. a prominent example of more complicated cles at CERN when the LHC is switched on After having finished his studies, a structures like gerbes and groupoids. in 2008 and starts taking data. If Julius and one year fellowship allowed him to go An absolute highlight was created by Bruno were right with their vision of super- to CERN, where he elaborated on SU(3) Julius and Bruno with their 1973 pa- symmetry we will get a better understand- http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 4 Volume 2, Issue 2, Autumn 2007 ESI NEWS Harold Steinacker, Peter Schupp, Michael his stay in Vienna he suffered a serious ing of the universe and its composition. Wohlgenannt, Paolo Aschieri and Bruno heart attack, from which he recovered as- There are extensions of the standard Zumino. tonishingly rapidly. When I visited him in model of particle physics which are based I had the pleasure to be invited many the hospital his fist question concerned the on supersymmetry, and furthermore super- times to his institutes at LMU as well as LHC measurements. On the very next day gravity might be a first step towards a for- at MPI. My own work was strongly influ- he gave me two sheets sketching the first mulation of quantized gravity. enced by Julius; he suggested, for exam- steps towards deformed Einstein gravity, Julius’ main activity from 1973 till ple, to use the Seiberg-Witten map in order which he then elaborated further together 1989 concerned Superphysics: keywords to study the question of renormalization of with Branislav Jurco, Peter Schupp and are supergauge transformations, super- noncommutative gauge field models. Paolo Aschieri when he returned to Mu- space formulation, supergravity, supersym- The Munich group in particular devel- nich. metry breaking, on which he worked oped the deformed Standard model, and In his own words: “We are in an ideal with young collaborators such as Richard again it was Julius who wanted to con- situation: from general ideas about the Grimm, Martin Sohnius, Bert Ovrut, Jan nect these more abstract developments with structure of as fundamental a concept as Louis, Jonathan Bagger, Hermann Nicolai physical predictions: certain decay pro- space-time we are led to a physical theory and others. Karlsruhe was a lively center cesses are prohibited on classical space, but that can be tested experimentally.” and I had the pleasure to visit his Institute only occur on deformed spaces. Besides his scientific activities Julius several times for a seminar. Julius was dealing with these exten- started an East European Initiative in or- His move to Munich in 1989 coin- sions of the Standard model on deformed der to help these countries to keep up with cided with a change of subject: in 1991 spaces, in order to improve the situation the rapid developments. By chance, I was the first paper with Bruno on his newly with problems of the old one, hoping that on a German committee which supported Noncommutative Quantum favored subject certain features of quantum gravity could this initiative. A number of Workshops and Field Theory was published. The mathe- already be taken into account. Schools as well as a number of visits of matician Manin and others had elaborated The Erwin Schrodinger¨ Institute for physicists from former Jugoslavia were ini- already on quantum spaces, but Julius and Mathematical Physics, created 14 years tiated by this exchange program. Bruno first formulated a consistent defor- ago, is very grateful to Julius: He was a mation of the quantum hyperplane together member of the committee initiating the in- After his retirement in Munich Julius with a differential calculus. After these stitute, served as a Vice-president of the moved to Hamburg. I was happy that two first steps, differential calculi, deformed Board (‘Vorstand’) of the Institute and was of his collaborators (Harold Steinacker and Lorentz group, respectively Poincare´ group elected Honorary Member of the ESI in Michael Wohlgenannt) joined me in Vi- were on the agenda of the Munich 2002. enna and became my collaborators. group and studied jointly with Ogievetsky, We were very happy, when he accepted We lost an eminent physicist, I lost an Schmidke, Schlieker, Schirrmacher, Brano our offer of a Senior Research Fellow- elder friend. Jurco, Stefan Schraml, John Madore, ship at the ESI four years ago. During We will miss you, Julius.

Entanglement in many-body quantum quantum mechanics however, people were too busy with the many successful applications of quantum mechanics to really pay at- physics tention to such foundational issues. Things changed drastically Frank Verstraete in the 60’s when John Bell, working as a high energy physicist in CERN, made the discovery that many-particle quantum states can in principle exhibit correlations that are stronger than corre- 3 One of the defining events in physics during lations allowed for by local hidden variable models . Although a the last decade has been the spectacular ad- loophole-free Bell experiment has still not been performed, Bell’s vance made in the field of strongly correlated work anticipated the fascinating quest to contrast the power of quantum many body systems: the observation quantum versus classical information processing and was one of of quantum phase transitions in optical lat- the main catalysts for the exceptional progress made in experi- tices and the realization that many-body en- mental quantum optics during the last decades. As a next logical tanglement can be exploited to build quantum step, visionary people like D. Deutsch, C. Bennett and P. Shor computers are only two of the notable break- understood that entanglement can be exploited to do information throughs. In a remarkable turn of events, the tools developed in tasks such as computing and cryptography much more efficiently the context of quantum information science have been shown to than possible in a classical world. The current effort in the field shed a new light on the ones used to describe strongly corre- of quantum information science is aimed at realizing these ideas. lated quantum many-body systems as studied in a wide variety The question to contrast the power of classical to quantum infor- of fields and has opened up many exciting interdisciplinary re- mation processing, and most notably to understand the power of search avenues involving mathematical physics, condensed matter quantum computers that explicitly make use of the possibility of and atomic physics, and information and computational complex- quantum interference and the quantum superposition principle, led ity theory. to an explosion of work on entanglement theory. One motivation is The key ingredient that distinguishes the quantum from the that this might lead to the discovery of new quantum algorithms by classical world is the concept of entanglement. As a response to which quantum computers can solve computational problems that the Einstein-Podolsky-Rosen paper in 19351, Schrodinger¨ coined are believed to be intractable on classical computers; the most in- the concept of entanglement2 and recognized it as being the defin- teresting algorithms that have as of today been proposed are Shor’s ing characteristic of quantum mechanics. In the early-days of algorithm for factoring large numbers (which turns out to be very

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 20075 relevant in the construction of one-way functions in cryptography) strongly related to the notion of quantum error correction. It is this and algorithms for simulating the dynamics of many-body inter- interplay between those complementary viewpoints that makes the acting quantum systems (originally proposed by Feynman). study of entanglement such a rich subject. How do you define entanglement? In the words of Recently, there has been much interest in investigating the Schrodinger,¨ a pure quantum state is entangled if and only if the amount and type of entanglement that is naturally present in whole is more than the sum of its parts. More specifically, the strongly correlated quantum systems. On the one hand, this was Hilbert space of a many-body system (where many is to be un- motivated by the question of whether the amount and type of en- derstood as largerthan 1) is a tensor product of the ones describing tanglement needed to do quantum computation could be present in single particles (those can correspond to e.g. modes in Fock space, the ground-state wave functions of quantum spin systems. On the to localized spins, to the polorization of a photon, ...). A pure quan- other hand, the hope is that the study of entanglement in strongly tum state is called separable if and only if the global wavefunction correlated quantum systems could elucidate the underlying struc- is a product of such single-particle wavefunctions, and entangled ture of the associated wavefunctions, which on its turn might lead otherwise; a mixed quantum state is called entangled iff it cannot to new ways of simulating them. be written as a convex sum of pure separable states. Note that the Concerning the first question, a local 5-body quantum spin possibility of entanglement is nothing more than a direct conse- 1/2 (qubit) Hamiltonian on a square lattice was identified whose quence of the superposition principle. Note also that the notion ground state is a so–called cluster state and allows for any (i.e. uni- of entanglement strongly depends on the choice of local Hilbert versal) quantum computation by doing adaptive local single-qubit spaces: a Slater determinant is considered unentangled with rela- measurements on it6. This was a surprising result as it showed that tion to its normal mode decomposition, but can be highly entan- ground states of local 2-D quantum spin models contain enough gled from the local point of view if these modes are delocalized. entanglement for doing universal quantum computation (note that A lot of work has been done to quantify entanglement. In the local one-qubit operations can never create entanglement). The as- case of a bipartite (i.e. 2-particle) pure quantum system |ψABi, the sociated Hamiltonian is unusual for a quantum Hamiltonian as natural measure of entanglement is the von-Neumann entropy of it consists of a sum of local commuting terms; this means e.g. 4 the local reduced density operator ρA or ρB : that local perturbations will never spread by virtue of Hamilto- nian evolution. It turns out that these cluster states and the way S(|ψABi) = −TrρA log2(ρA) = −TrρB log2(ρB). to do quantum computation with them can be understood within the formalism of valence bond states or projected entangled pair In essence, this entanglement entropy quantifies the maximal states (PEPS)7. This class of states plays also a central role in the amount of Shannon information that A can obtain by doing a mea- context of simulation of quantum spin systems, and we will later surement on his part about the measurement outcome of part B. come back to them. An entropic criterion is desirable as the amount of entanglement Concerning the question about the nature of entanglement in becomes additive for independent copies, but there is also a deeper many-body systems, we would like to get a better understanding reason why this measure is used: it can be proven that any collec- on the nature of the wavefunctions present in ground states of tion of states with a given mean entanglement can be intercon- strongly correlated quantum systems. The study of correlations, verted into any other collection of states with the same mean en- both quantum and classical, is an very rich field and lies at the 5 tropy by only local operations and classical communication . In heart of many of the most exciting discoveries in the fields of sta- essence, this means that systems with the same amount of entan- tistical physics and quantum information theory: quantum phase glement are equally useful for distributed quantum information transitions occur due to the appearance of long-range correlations, tasks such as quantum communication, and the entanglement en- and the theory of entanglement is all about quantifying the amount tropy is therefore the unique measure that quantifies how useful of quantum correlations and might lead to a better understanding entanglement is from the local point of view. Actually, it is pos- of the emergence of collective phenomena. The natural choice to sible to make a formal analogy between the theory of pure state quantify correlations in a quantum spin system endowed with a entanglement and thermodynamics, in which local operations and metric is to look at the connected correlation functions classical operations that preserve the entanglement correspond to adiabatic processes in thermodynamics. hOAOBi − hOAihOBi Entanglement appears everywhere in quantum mechanical systems, and there are many complementary viewpoints on it. as a function of the distance between two regions A and B. Non- From the point of view of quantum information theory, it is a critical systems exhibit exponentially decaying correlation func- resource that allows for revolutionary information theoretic tasks tions, leading to the definition of a correlations length. Despite the such as quantum computation and quantum cryptography (without basic nature of this result, it has only been proven very recently8; entanglement, a quantum computer would not be more powerful the main technical ingredient for that proof was the use of the so- than a classical one). From the point of view of quantum many- called Lieb-Robinson bound on the velocity by which correlations body physics, entanglement gives rise to quantum phase transi- spread with respect to local Hamiltonian evolution9. The notion of tions and exotic new phases of matter exhibiting e.g. topological a correlation length is very fundamental and quantifies the amount quantum order (i.e. a nonlocal order parameter) such as occurring of degree of localization of the relevant degrees of freedom in the in the fractional quantum Hall effect. From the point of view of system. Intuitively, the notion of a correlation length should set the numerical simulation of strongly correlated quantum systems the length scale at which ”the whole becomes equal to the sum such as quantum spin systems and also appearing in computational of its parts”; in other words, if the distance between A and B be- quantum chemistry, entanglement is the enemy number one as it comes much longer than the correlation length, we should have makes simulation so hard. Of course, these viewpoints are mutu- ρAB ' ρA ⊗ ρB. However, just looking at 2-point correlation ally compatible: the complexity of simulating entangled quantum functions can be problematic: there exist quantum states ρAB for systems is intimately connected to the power of quantum compu- which all two-point correlation functions are arbitrarily small, and tation; the possibility of topological quantum order turns out to be nevertheless they can be proven to be arbitrarily far from product http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 6 Volume 2, Issue 2, Autumn 2007 ESI NEWS which is equivalent to states10. In the case of zero-temperature quantum systems, another ob- 1 kH∂ k I(A : B) ≤ Tr (H∂ ρA ⊗ ρB) ≤ . vious choice for quantifying the amount of correlations is to calcu- T T late the entanglement entropy of a region A of spins as a function The right hand side obviously scales like the boundary of re- of the size and shape of the region (see Figure 1 below). This turns gion A as opposed to its volume, and this proves that any clas- out to be interesting as it will lead to useful insights into the nature sical and quantum thermal state exhibits an exact area law at any of the associated wavefunctions. It has been conjectured that the non-zero temperature, even in the critical case. An intriguing open entanglement entropy for noncritical systems obeys an area law, problem is to connect this behaviour to the zero-temperature be- i.e. scales as the size of the boundary |∂A|, indicating that the only 11 haviour where logarithmic corrections arise in the case of critical correlations that are relevant are the ones around that boundary . quantum spin chains. It has been shown that this area law is violated mildly for critical Those are laws prove that correlations are mainly concentrated one-dimensional quantum spin and critical two-dimensional free around the boundary and the entanglement entropy of a block fermionic systems, for which a multiplicative logarithmic correc- 12 of spins scales like its boundary as opposed to its volume. That tion has to be added means that there is very little entanglement in ground states of lo- S(ρA) ' |∂A| log(|∂A|). cal quantum Hamiltonians: all ground states live on a small man- ifold in Hilbert space with relatively few entanglement. This can been exploited to come up with a variational class of wavefunc- B tions that captures this behaviour and is still easy to simulate, and B this has precisely been the program that has been successfully pur- w sued during the last years. wA In the case of 1-dimensional quantum spin systems, power- ful numerical renormalization group (NRG) algorithms have been A devised in the 70’s by Wilson15; those methods were later gener- alized to the density matrix renormalization group (DMRG) by S. White which allows to simulate ground state properties of any spin chain16. Both of those methods have been extremely successful [ and allow to calculate correlation functions of the related systems up to very large precision. Only recently, it has become clear that both of those renormalization algorithms can be rephrased as vari- ational methods within the class of so-called matrix product states 17 FIGURE 1. A QUANTUMSPINSYSTEMONALATTICE: THELATTICEISDIVIDEDINTO (MPS) . The class of MPS is very much related to the valence-

REGIONS A AND B WITHBORDERS ∂A AND ∂B, RESPECTIVELY.INTHECASEOF bond AKLT models put forward by Affleck, Kennedy, Lieb and 18 GROUND STATES OF LOCAL HAMILTONIANS, THEENTANGLEMENTENTROPYBETWEEN Tasaki and the generalizations thereof known as finitely corre- 19 THEREGIONS A AND B SCALESLIKETHEAREA ∂A, ASOPPOSEDTOTHEVOLUME |A|. lated states . MPS can also be generalized to so-called projected entangled pair states (PEPS) which can be defined on any lattice However, there are also critical quantum spin systems known in any dimensions20. in 2 dimensions for which a strict area law holds13, and hence it

dD is in general an open question of how to relate area laws to the D P ¦¦cijk D i j k notion of a correlation length. D 11i, j,k The situation is much clearer in the case of finite temperature systems in thermal equilibrium. In that case, correlations can be quantified by using the concept of mutual information:

I(A : B) = S(ρA) + S(ρB) − S(ρAB). D ¦ i i The mutual information is zero if and only if the state is a product i 1 state ρA ⊗ ρB and has the same operational meaning as the entan- glement entropy (actually, it is equal to twice the entanglement en- tropy for a pure state). Very recently, a bound on this mutual infor- mation, valid both for classical and quantum thermal states of local Hamiltonians of the form ρAB = exp(−βH)/Tr exp(−βH), has been derived14. The argument works equally well for spin systems, FIGURE 2. A PEPS CAN BE DEFINED ON ANY LATTICE; THESMALLCIRCLES for systems with infinite dimensional local Hilbert spaces such as CORRESPOND TO VIRTUAL D-DIMENSIONALSPINS, ANDTHEMAP P MAPSTHEMTOA bosons, and for fermionic systems. The proof is short enough to PHYSICALSPINOFDIMENSION d REPRESENTEDBYTHEBIGGERELLIPSOIDS. reproduce here. Consider two regions A and B, and write H as a sum of 3 terms HA,HB,H∂ ; HA,HB contains the terms of How can we represent those MPS and PEPS? First of all, con- the Hamiltonian that acting only on A, B, and H∂ contains the sider a graph where the local d-dimensional spins lie on the nodes terms that represent the interactions between across the boundary. of the graph, and a collection of virtual bipartite entangled EPR- Thermal states are variationally characterized by the fact that they PD pairs i=1 |ii|ii distributed along all vertices of the graph (see minimize the free energy; hence, any other state has a higher free figure 2). Next, we want to identify a local subspace of those vir- energy, and in particular ρA ⊗ ρB (here, ρA, ρB are the reduced tual spins as the space of physical spins by applying a linear map density operators of the global thermal state ρAB). We hence have: P to them that maps c spins (c being the coordination number Tr (HρAB) − TS(ρAB) ≤ Tr (HρA ⊗ ρB) − TS(ρA ⊗ ρB) of the graph) to one (physical) spin of dimension d. The class

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 20077 of PEPS is now obtained by letting those projectors P vary over spin systems, and most importantly to extend the formalism of all possible Dc × d matrices. The AKLT-state is of that form, in numerical renormalization group methods to higher dimensions. which 2 qubits are mapped to a spin 1 state in the case of the 1- These methods, especially the ones in 2 dimensions, are still in dimensional spin chain and 3 qubits to a spin 3/2 state in the case development, but it has become clear that they offer crucial new of a hexagonal lattice. The cluster state discussed earlier is also of insights into the structure of the wavefunctions to be found in that form, and the quantum computation going on when doing lo- nature. Although originally formulated on the level of quantum cal measurements can be understood by identifying the underlying spin systems, it has become clear that they can also be used in the virtual qubits as the logical qubits21. In the case of a 1-dimensional broader context of bosons, fermions and field theories. Consider structure, this family of states are called Matrix Product States e.g. fermionic lattice spin systems. This is of particular interest (MPS), and a useful aspect of them is that all correlation functions as the identification of the phase diagram for the 2-dimensional can be calculated with a computational cost that scales linearly in Hubbard model is one of the central problems in condensed mat- the number of spins and polynomially in D. This is remarkable, ter theory. Motivated by those new developments on MPS/PEPS, as the dimension of the Hilbert space scales exponentially in the it has been proven that local Hamiltonians of fermions can always number of spins, and hence the class of MPS forms a subclass for be mapped to local Hamiltonians of spins27, independent of the di- which we can calculate all properties efficiently. mension, and therefore the whole formalism of PEPS turns out to By making use of entanglement theory, it has recently be- be equally applicable to fermionic lattice systems. Those methods come clear why the numerical renormalization group methods are can equally well be used to simulate quantum lattice field theo- so successful: this is a consequence of the fact that this class of ries, non-equilibrium systems, classical spin systems, and work is MPS is rich enough to approximate any ground state of a local under way to tackle problems in quantum chemistry. gapped Hamiltonian efficiently. This implies that the manifold of In conclusion, we have argued that entanglement theory pro- ground states of all local gapped one-dimensional Hamiltonians: vides fundamental new insights into the nature of strongly corre- their ground states are well approximated by MPS, and conversely lated quantum spin systems. It turns out that the amount of en- all MPS are guaranteed to be ground states of local Hamiltoni- tanglement in ground states of quantum spin systems is surpris- ans. Similar statements hold for PEPS in higher dimensions. To ingly low. Under pretty general assumptions, this has allowed us be more precise, let’s define what we mean by good approxima- to identify the manifold of wavefunctions associated to the low- PN−1 tions. Consider a family of Hamiltonians HN = i=1 hi,i+1 energy sector of strongly correlated quantum spin systems, which parameterized by the number of spins N and nearest neigbour in- on its turn can be applied to devise powerful ab initio numerical teractions hi,i+1, and associated ground states |ψN i. The goal is variational methods for simulating them. D to approximate the ground states |ψN i by a family of MPS |ψN i such that Notes D 1A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). k|ψN i − |ψN ik ≤  2E. Schrodinger,¨ Proc. Camb. Phil. Soc. 31, 555 (1935). with  independent of N. The central question is: how does D 3J.S. Bell, Physics 1, 195 (1964). 4 1/ N The reduced density operator ρA is defined as the unique density operator de- has to scale as a function of and such that this relation is fined on the Hilbert space associated to particle A that has the property that for all fulfilled? If the scaling of D is polynomial in 1/ and N, then it observables OA it holds that hψAB |OA|ψAB i = Tr (OAρA). means that the ground state is represented efficiently by a MPS: 5C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A indeed, the previous equation implies that the expectation value 53, 2046 (1996). 6R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); F. Ver- of any observable on the exact ground state is arbitrary close to straete and J.I. Cirac, Phys. Rev. A. 70, 060302(R) (2004). the one of the MPS, that the cost of getting a better precision does 7F. Verstraete and J.I. Cirac, Phys. Rev. A. 70, 060302(R) (2004). only scales polynomial22, and that all correlation functions can be 8M.B. Hastings, Phys. Rev. Lett. 93, 140402 (2004). calculated efficiently on |ψDi. 9E.H. Lieb, D.W. Robinson, Commun. Math. Phys. 28, 251 (1972); M.B. Hast- N ings, T. Koma, Commun. Math. Phys. 265, 781 (2006). The previous requirements can be met under pretty broad as- 10P. Hayden, D. Leung, P.W. Shor, A. Winter, Commun. Math. Phys. 250, 371 sumptions. First of all, it has been proven that whenever an area (2004). law is satisfied for the exact ground state, a MPS will indeed ex- 11P. Calabrese, J. Cardy, J. Stat. Mech. P06002 (2004); M.B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005). ist that approximates it well with polynomial scaling in 1/, N 12 23 G. Vidal et al., Phys. Rev. Lett. 90, 227902 (2003); M.M. Wolf, Phys. Rev. . This argument works whenever the Renyi entropy Sα(ρ) = Lett. 96, 010404 (2006). α Tr(ρ )/(1 − α) for an α < 1 of a contiguous block of spins is 13F. Verstraete, M.M. Wolf, D. Perez-Garcia, J. I. Cirac, Phys. Rev. Lett. 96, bounded above by a constant times the logarithm of the size of the 220601 (2006). 14 block. This turns out to be true for all spin chains for which this M.M. Wolf, F. Verstraete, M.B. Hastings, J.I. Cirac, arXiv:0704.3906. 15K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975); note that Wilson mapped the quantity has been calculated exactly, including the critical Heisen- Kondo impurity problem to an effective 1-D model as only s-wave electrons are berg spin chain and the Ising Hamiltonian in a transverse magnetic scattered. field. In a related recent development, it has been proven that all 16S. White, Phys. Rev. Lett. 69, 2863 (1992); U. Schollwock,¨ Rev. Mod. Phys. 77, 259 (2005). gapped local Hamiltonians on a spin chain obey a strict area law 17 24 F. Verstraete, D. Porras and J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004). in terms of the Renyi entropies , implying the approximability 18I. Affleck et al., Commun. Math. Phys. 115, 477 (1988) by MPS. The central technical tool used in the proof is again the 19M. Fannes, B. Nachtergaele and R.F. Werner, Comm. Math. Phys. 144, 443 Lieb-Robinson bound25. This provides a clear theoretical justifi- (1992). 20 cation for the numerical renormalization group methods: they are F. Verstraete and J.I. Cirac, cond-mat/0407066; V. Murg, F. Verstraete, J.I. Cirac, cond-mat/0501493. variational methods over a class of states that is rich enough to 21F. Verstraete and J.I. Cirac, Phys. Rev. A. 70, 060302(R) (2004). provide an very good approximation to the exact ground states26. 22Note that it is trivial to simulate any quantum spin system with a cost poly- The reformulation of those methods in terms of matrix product nomial in N and exponential in 1/: divide the spin system in blocks, neglect the states have opened up many new exciting possibilities and allowed boundary terms connecting those blocks, and find the ground state for all blocks independently. for new applications such as simulating spin chains at e.g. finite 23F. Verstraete, J.I. Cirac, Phys. Rev. B 73, 094423 (2006). temperature and out of equilibrium, to calculate gaps in quantum 24M.B. Hastings, JSTAT, P08024 (2007). http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 8 Volume 2, Issue 2, Autumn 2007 ESI NEWS

be of exponential complexity) in the worst case scenario (cf. N. Schuch, J.I. Cirac, 25E.H. Lieb, D.W. Robinson, Commun. Math. Phys. 28, 251 (1972); M.B. Hast- F. Verstraete, in preparation). However, this does not seem to have strong impli- ings, T. Koma, Commun. Math. Phys. 265, 781 (2006). cations for the difficulty of simulating physical quantum spin chains, as nature itself does not seem to be able to relax to exact ground states in the case of e.g. 26There is, however, one caveat: it is not because the ground state can always be spin glasses which are also known to be NP-hard problems: the situations in which approximated very well with a MPS, that the computational complexity of actually simulation with MPS fail seem to be exactly the ones for which nature cannot relax finding it is also polynomial. It has indeed been proven that this problem is NP- to its ground state. complete (Non-deterministic Polynomial complexity or in other words believed to 27F. Verstraete, J.I. Cirac, J. Stat. Mech., P09012 (2005).

Kazhdan’s Property (T) proved that the first Betti number of the lo- (iii) If a locally compact group G cally symmetric space K \ G/Γ is zero, has Property (T), then G is com- Pierre de la Harpe where K is a maximal compact subgroup pactly generated and G/[G, G] is of G. (Lattices in groups of real rank one compact. In particular, a countable Property (T) is a rigid- would require another discussion.) The fi- group Γ which has Property (T) is ity property first for- nite generation result holds for lattices in finitely generated and its abelianisa- mulated by D. Kazh- semi–simple groups G of higher ranks de- tion Γ/[Γ, Γ] is finite. dan in a three page fined over other local fields, for example 1967 paper whose in- for lattices in SLn(k((T ))), where n ≥ 3 Once again, it follows that lattices as de- fluence has been, and and k((T ))) is the field of Laurent series scribed above are finitely generated and still is, immense. over a finite field k. that locally symmetric spaces X = Γ \ The first success The main ingredient of Kazhdan’s ap- G/K with the appropriate condition on of Kazhdan’s ideas was to provide a very proach is the theory of unitary representa- ranks have finitely generated fundamen- smart solution to an old problem (going tions of groups (more precisely a small and tal group π1(X) and first Betti number back at least to Siegel) concerning finite soft part of the theory). Consider a topolog- dim(H1(X, R)) = 0. generation of lattices and vanishing of Betti ical group G, a Hilbert space H, the group Property (T) was later recognized to be numbers of Riemannian symmetric spaces. U(H) of its unitary operators, and a rep- equivalent to a fixed–point property (De- Recall that a lattice in a locally com- resentation π : G −→ U(H) such that lorme, Guichardet, Serre). More precisely, pact group G is a discrete subgroup Γ such the companion mapping G × H −→ H let G be a locally compact group which is that the homogeneous space G/Γ has a G– is continuous. We say that π has invariant σ–compact; then G has Property (T) if and invariant probability measure. For exam- vectors if there exists ξ 6= 0 in H such only if G has the so–called Property (FH), ple, the classical subject of positive def- that π(g)ξ = ξ for all g ∈ G, and that namely if and only if any continuous action inite quadratic forms on Rn leads to the π almost has invariant vectors if, for any of G by affine isometries of a real Hilbert lattice SLn(Z) in the simple Lie group compact subset Q of G and any  > 0, space has a fixed point. On the one hand, SLn(R), as well as to the locally sym- there exists a unit vector ξ ∈ H such that this indicates a very strong relevance of the notion for geometry; on the other hand, metric spaces SO(n) \ SL(n, R)/Γ, for supg∈Q kπ(g)ξ − ξk < . For example, appropriate finite index subgroups Γ of the reader can check that the representation the cohomological formulation of this fixed 1 SLn(Z). Other quadratic forms and other of the additive group R by translations of point property, namely H (G, π) = 0 for arithmetic subjects (such as division alge- L2(R) almost has invariant vectors with- any orthogonal representation π of G, has bras) lead to other lattices in other classi- out having invariant vectors. proved to be useful. cal Lie groups, and more generally in re- A topological group G has Kazhdan’s The rigidity contained in Property (T) ductive groups over local fields. A lattice Property (T) if any unitary representation has important applications in combina- Γ in G is uniform if G/Γ is compact; it of G which almost has invariant vectors ac- torics which go back to a short paper of is then a straightforward consequence of tually has invariant vectors. It is straight- Margulis (1973). Let Γ be a group gen- standard facts from general topology that Γ forward to check that compact groups have erated by a finite set S; assume that Γ is finitely generated. But the finite genera- this property. The example above shows has Property (T) and in addition has an tion of non–uniform lattices in semi-simple that R does not have it; similarly, locally infinite family (Γk)k of subgroups of fi- groups is a rather deep result (even if some compact groups which are abelian and nite index (for example SL3(Z), with the particular examples such as SLn(Z) can non–compact (or more generally amenable kernels of the reductions SL3(Z) −→ k k be shown to be finitely generated by sim- and non–compact) as well as non–abelian SL3(Z/p Z) modulo p ). Let Xk denote ple methods). It is also a crucial step for free groups do not have Property (T). Here the Cayley graph of the quotient Qk = later results of the theory, such as Margulis’ are remarkable results, essentially all from Γ/Γk with respect to the image Sk of S, arithmeticity theorems. Kazhdan’s original paper: namely the graph with vertex set Qk in Using methods of a completely differ- which q and q0 are connected by an edge −1 0 −1 ent nature of those which were used be- (i) The special linear groups SLn(K), whenever q q is in Sk ∪ Sk . Then fore, Kazhdan proved in an extremely short n ≥ 3, and the symplectic group (Xk)k is a sequence of expanders, namely way that any lattice in G is finitely gen- Sp2n(K), n ≥ 2 have Property (T) a sequence of regular finite graphs which erated when G = G1 × · · · × Gk is a for any local field K (for example have remarkable properties from the point product of simple Lie groups Gj of real K = R). It follows that G = G1 × of view of geometry (isoperimetric con- ranks at least 2. The real rank of Gj is · · · × Gk as above has Property (T). stants), spectral theory (uniform bounds on the maximal dimension of subgroups R` the non trivial eigenvalues of the corre- of Gj which are diagonalisable in the ad- (ii) A lattice Γ in a locally compact sponding simple random walks), and all joint representation; for example the real group G has Property (T) if and only kinds of applications in computer science rank of SLn(R) is n − 1. Kazhdan also if the group G has it. (networks of computations, data organisa-

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 20079 In fact, Kazhdan’s insight has turned groups reduced to one element (Popa). The tions, computational devices, and so on). out to be relevant in a large family of sub- notion has also found its way in random These ideas have had a very rich poster- jects. It has provided the solution to prob- walks, spectral theory, the theory of al- ity, including quite recent work concern- lems about the existence of finitely additive gorithms, . . . , and we should probably be ing graphs defined from infinite sequences invariant measures going back to Lebesgue open for more surprising applications. of finite groups, and more generally con- and Ruziewicz (Margulis, Rosenblatt, Sul- cerning the combinatorics of finite simple livan, Drinfeld). In ergodic theory, ergodic To come back to the main theme at groups (Alon, Bourgain, Gamburd, Kass- actions of groups with Property (T) have ESI during the spring of 2007, it is strik- abov, Lubotzky, Nikolov, Shalom, Wigder- been shown to be more rigid than a pri- ing that amenability and Property (T) are son, . . . ). ori expected (Schmidt, Connes, Weiss), and the two extreme poles of a whole range Techniques from Kazhdan’s paper, the notion is presently most important for of behaviours. For example, if we restrict closely dependent on the theory of the understanding of equivalence relations for simplicity the next statement to locally semisimple groups, provided only count- (Zimmer, Furman, Hjorth). Property (T) compact groups, a group with Property (T) ably many examples of countable groups has appropriate formulations in operator is amenable if and only if it is compact; with Property (T). But many more exam- algebras (Connes, Jones, Popa, Bekka), non–compact groups which have proper- ples have been later discovered, indeed un- where it has provided the key ingredient ties “in between” amenability and (T) are in countably many. In particular, Property (T) to solve several problems, one going back some sense the most mysterious, but what plays an important role in the theory of to Murray and von Neumann themselves: we know of the two extreme types of be- “generic groups” and “random groups” the existence of factors with fundamental haviour is often a good guide for a better (Gromov). understanding of the general situation.

Perturbative Quantum theory has been developed as a rigor- tum space. Kreimer’s original insight orig- ous mathematical framework in the fifties- inated from a study of number-theoretic Field Theory: sixties thanks to the work of Hepp, properties of Feynman integrals and related Still Surprises? Lehmann, Symanzik, Zimmermann, Stein- the amplitudes term by term in the per- mann, Epstein, Glaser, Stora, Bogoliubov, turbative expansion to polylogarithms and Romeo Brunetti Stuckelberg¨ and several others. These au- motivic theory as well as, ultimately, to thors found a mathematically consistent arithmetic geometry. method to construct the perturbation se- Quantum Field The- It turns out that Feynman graphs carry ries of quantum field theory at all orders, a pre-Lie algebra structure in a natural ory aims at a unify- thereby making mathematical sense of the ing description of na- manner. Antisymmetrizing this pre-Lie al- recipes for renormalizations suggested be- gebra delivers a Lie algebra, which pro- ture on the basis of the fore. principles of quantum vides a universal enveloping algebra whose More recently, we experienced a re- dual is a graded commutative Hopf algebra. physics and (classical) newed interest in the foundations of per- field theory. Its main It has a recursive coproduct which agrees turbation theory, which may come as a with the Bogoliubov recursion in renormal- success is the develop- surprise. Two independent directions were ment of a standard model for the the- ization theory. While this gives a mathe- traced. The first took place around 1996, matical framework to perturbation theory ory of elementary particles which describes due to Brunetti and Fredenhagen1, and was physics between the atomic scale and the in momentum space Feynman integrals, it centered around the problem of construct- also suggests to incorporate notions of per- highest energies which can be reached ing quantum field theories on curved space- in present experiments. It has, however, turbative quantum field theory into mathe- times. The other started around the end matics. turned out to be also very important in of the nineties and is due to Connes and Indeed, very similar Hopf algebras other branches of physics, in particular for Kreimer2 and deals with structural insights have emerged in mathematics in the study solid state physics. Its mathematical com- into the combinatorics of Feynman graphs of motivic theory and the polylogarithm plexity is enormous and has induced many via Hopf algebras. In both cases there arise through the works of Spencer Bloch, Pierre new developments in pure mathematics. In direct connections to the application of Deligne, Sasha Goncharov and Don Za- its original formulation it was plagued by quantum field theory to physics problems. gier. One ultimately hopes that a link can divergencies whose removal by renormal- The two settings gave a lot of striking re- be established between number theory and ization lead to fantastically precise predic- sults and applications that were unforeseen quantum field theory in studying the rele- tions which could be verified experimen- before. In particular, new aspects of the vant Hopf algebras and their relation in de- tally. renormalization group were uncovered. tail. A full construction of quantum field In the following we summarize the theories was possible up to now only for highlights of the two mentioned routes: A major problem here is the under- particular models and in too specific situ- standing of the quantum equations of mo- ations. For realistic models one still has to 1. Hopf algebras and renormalization tion, which are governed by the closed rely on uncontrollable approximations un- An important progress in the connec- Hochschild one-cocycles of the Hopf alge- der which perturbation theory, which con- tion to mathematics has been obtained re- bra. structs the models as formal power series in cently by Connes and Kreimer3. Their idea This Hochschild cohomology of per- the coupling constants, is the most impor- of using Hopf algebras in perturbation the- turbation theory illuminates the role of lo- tant one. ory has led to a better mathematical under- cality in momentum and coordinate space Perturbation theory in quantum field standing of the forest formula in momen- approaches. At the same time, it provides http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 10 Volume 2, Issue 2, Autumn 2007 ESI NEWS 3. Comparison and other results the generic Lorentzian case, something that a crucial input into the function theory of might be extremely useful in a direct attack At present a throughout comparison be- the polylogarithms, and certainly into a yet to the perturbative Quantum Gravity. tween the Hopf algebraic and the local ap- to be developed function theory of quan- To summarize, Perturbative Quantum proaches to perturbation is unfortunately tum field theory amplitudes. Extensions of Field Theory seems still extremely alive, lacking. The two framework use very dif- these ideas to gauge theories are under ac- full of new and appealing ideas that may ferent languages and structures and it is a tive investigation, as well as the connection well provide further physical insights and challenge to see to what extent they carry to motivic theory. unforeseen connections to many partly un- the same information on the renormaliza- At the same time, Connes and Mar- explored areas of mathematics. We are con- 4 tion procedure. An ongoing collaboration colli are incorporating the techniques of fident that even more surprises are coming. arithmetic geometry into quantum field between Bergbauer, Brunetti and Kreimer, is based on the realization that the renor- theories, which utilize again the underlying Notes malization procedure, similarly to what is Hopf structure in the context of Tannakian 1Brunetti, R., and Fredenhagen, K., “Interacting categories, intimately connected again to done in the local approach, e.g. as exten- quantum fields in curved space: Renormalization of n 4 the theory of the polylogarithm. sions of n-point distributions from M \ φ ,” in: “Operator Algebras and Quantum Field The- n ory,” ed. S. Doplicher, R. Longo, J.E. Roberts and L. ∪k=2∆k to the full space, can also be dis- 2. Epstein-Glaser perturbative ap- cussed using certain tools in algebraic ge- Szido, International Press 1997. proach. 2Connes, A. and Kreimer, D., “Hopf alge- ometry, namely Fulton-MacPherson com- bras, renormalization and Noncommutative geome- Another important direction of recent pactification. The hope is that this last may try,” Commun. Math. Phys. 199 (1998), 203-242; research has been put forward by Brunetti shed some light on the connection since “Renormalization in and the and Fredenhagen5 and refined by Hollands the procedure of compactification embod- Riemann-Hilbert problem I,” Commun. Math. Phys. 6 210 (2000); and Wald in a series of papers . The local ies the Hopf algebraic combinatorics. Once Renormalization in Quantum Field Theory and the point of view is emphasized, via a general- established this would provide a link from Riemann-Hilbert problem II, the Beta function, dif- ization of the Epstein and Glaser approach, the Hopf algebraic setting to the local one. feomorphisms and the renormalization group” Com- and allows a description of perturbation Establishing a link from the local approach mun. Math. Phys. 210 (2001). 3Connes, A. and Kreimer, D., “Hopf alge- theory on any background spacetime. Basic to the Hopf algebras may turn out to be bras, renormalization and Noncommutative geome- to this approach is the connection with the more entertaining. try,” Commun. Math. Phys. 199 (1998), 203-242; field of microlocal analysis pioneered by Another important point of contact is “Renormalization in Quantum Field Theory and the Riemann-Hilbert problem I,” Commun. Math. Phys. Radzikowski. These methods allowed the that renormalization group ideas seem to 210 (2000); cited authors to prove for the first time, that be crucial in both approaches. Other groups Renormalization in Quantum Field Theory and the up to possible additional invariant terms have pioneered different ideas, for instance Riemann-Hilbert problem II, the Beta function, dif- of the metric, the classification of (ultra- by making rigorous the work of Polchin- feomorphisms and the renormalization group” Com- 9 mun. Math. Phys. 210 (2001). violet) renormalization in a general space- ski and Wilson . However, a connection 4Connes, A. and Marcolli, M., “Quantum Fields time follows the same rules as that on between all these seemingly different per- and Motives,” Journal of Geometry and Physics 56 Minkowski spacetime. Actually the theory spectives is lacking and an important is- (2005) 55-85; suggests further possibilities, as envisaged sue would be a comparison and attempt “From Physics to Number Theory via Noncommuta- 7 tive Geometry. Part II: Renormalization, the Riemann- recently by Brunetti and Fredenhagen , the to find a possible unification. A first step Hilbert correspondence, and motivic Galois theory, ” most important of which is a conceptu- in this direction was done by Krajewski in: ”Frontiers in Number Theory, Physics, and Geom- ally new approach to quantum gravity, at and collaborators10. He showed how to etry, II” pp. 617–713, Springer Verlag, 2006. 5 least in the perturbative sense. In this di- use tree-like expansions and the universal Brunetti, R., and Fredenhagen, K., “Microlo- cal analysis and interacting quantum field theories: rection it is particularly important to cite Hopf algebra of rooted trees to reformulate Renormalization on physical backgrounds,” Commun. that in a very recent effort, Brunetti, Dutsch¨ the Wilson-Polchinski approach. In the lo- Math. Phys. 208 (2000) 623. and Fredenhagen, enlarged the mathemati- cal approach this connection has been re- 6Hollands, S. and Wald, R.M., “Local Wick Poly- cal framework a lot, by allowing also non cently discussed by Brunetti, Dutsch¨ and nomials and Time-Ordered-Products of Quantum Fields in Curved Spacetime,” Commun. Math. Phys. polynomial interactions, and better clarify- Fredenhagen. There, one is able to use 223 (2001) 289; ing the algebraic structures used in the local the Epstein-Glaser approach to discuss “Existence of Local Covariant Time-Ordered- approach. and compare different ideas of renormal- Products of Quantum Fields in Curved Spacetime,” Other interesting directions are that ization groups, namely, those related to Commun. Math. Phys. 231 (2002) 309-345; 8 “On the Renormalization Group in Curved Space- taken by Dutsch¨ and Rehren for perturba- Stuckelberg-Petermann,¨ to Gell-Mann and time,” Commun. Math. Phys. 237 (2003) 123-160. tion theory on AdS and connections with Low, and to Wilson-Polchinski, at least in 7Brunetti, R., and Fredenhagen, K., “Towards a the quantum field theory perspectives on the Minkowskian case. Here, one discov- background independent formulation of perturbative holography, and, more recently, Hollands ers that the Stuckelberg-Petermann¨ is re- quantum gravity,” in: Quantum Gravity – Mathemat- ical Models and Experimental Bounds, B. Fauser, J. has developed a new attack to the case of ally a group of analytic automorphisms of Tolksdorf, E. Zeidler Eds., Birkhauser¨ Basel, 2006, local and covariant pure gauge theory. The the local observables, that the Gell-Mann 151–159. results are rather appealing and points to- and Low approach to scaling gives only 8Dutsch,¨ M., and Rehren, K.-H., “A comment wards a better understanding of the (redun- a cocycle, which is a group only in the on the dual field in the AdS-CFT correspondence,” Lett.Math.Phys. 62 (2002) 171-184. dant) mathematical structures of gauge the- massless case, and for the first time a di- 9Salmhofer, M., “Renormalization: An Introduc- ories. rect derivation of the Flow Equation of tion,” Series: Theoretical and Mathematica Physics, One expects that using all these new ex- Polchinski in the Minkowskian case. A ma- Corr. 2nd printing, Springer-Verlag 2007. 10 citing developments will be possible to ad- jor improvement may come from the last Girelli, F., Krajewski, T. and Martinetti, P., “An algebraic Birkhoff decomposition for the continuous dress fundamental problems in cosmology, approach if one would be able to find a lo- renormalization group,” J. Math. Phys. 45 (2004) for instance, but this is still to be done. cal and covariant way to impose a cut-off in 4679-4697.

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 2007 11

etry, on Wednesday, December 12, 2007, The second lecture in this series will The Interaction of 18:00, ESI Schrodinger¨ Lecture Hall. be given by Professor Dr. Scott Wal- ter (Archives Henri Poincare,´ University Mathematics and Physics at Abstract: The mathematician Felix of Nancy) on Hermann Minkowski and the Turn of the twentieth Hausdorff, who also published literary and the Scandal of Spacetime, on Wednesday, philosophical writings under the name of January 16, 2008, 17:30, ESI Schrodinger¨ Century — a Series of Paul Mongre, was a singular figure in Lecture Hall. Lectures fin-de-sicle and early 20th century math- ematical culture. His intellectual career Abstract: The ubiquity in contemporary Joachim Schwermer brought together seemingly distant cultural physics of spacetime and related geomet- trends such as Nietzscheanism and mod- ric objects belies the near-universal rejec- The emergence of mathematical physics as ernist, ’abstract’ mathematics. In my talk tion by physicists of Hermann Minkowski’s an independent discipline at the end of the I will try to sketch some of Hausdorff’s theory from its inception in November 19th century brought with it profound dis- considerations on time, space, and geom- 1907 to 1911. In time, of course, space- cussions of the foundations of both math- etry, topics that he approached both as a time came to be synonymous with Ein- ematics and physics as well as a fruitful philosopher and writer, and as a mathe- stein’s special theory of relativity, the most cooperation between these two fields. Far- matician. powerful tool for discovery in relativistic reaching concepts of modern physics and physics, and the most effective means of It will be seen that philosophical rather new, fundamental mathematical structures presenting the new dynamics. How did this than mathematical considerations brought were constructed in this period. Since sum- change come about? Minkowski’s interpre- Felix Hausdorff to reflections on geome- mer 2005 a series of lectures, entitled “His- tation of spacetime was initially a scan- try and the nature of time and space in tory of Mathematics and Physics”, at the dal for physicists, challenging–and eventu- the late 1890’s. Rejecting all contempo- ESI drew attention to this topic area. The ally overturning–some of their most cher- rary attempts to sketch metaphysical or in- talks as given in this series in the previ- ished views of the nature of physical reality. tuitive ‘foundations’ for mathematical ge- ous years have found broad interest among By comparing the work of Henri Poincare,´ ometry, he strongly welcomed Hilbert’s ax- students, researchers and scholars and initi- Einstein, Minkowski and others, the scan- iomatic method as a tool for exploring the ated a new awareness of the historical con- dalous aspect of spacetime is brought into different possibilities to provide mathemat- text that goes along with the sciences in sharp focus, and its initial rejection more ical systems of geometrical notions which question. easily understood. I will argue that in this could then be compared with empirical ev- This is to announce two lectures in this instance, formal tools played an essential idence about ‘space’. In the talk I will out- series during the Winter Term 2007/08. role in quelling the scandal. line Hausdorff’s route to the resulting ‘con- The first lecture is by Professor Dr. sidered empiricism’ in order to compare it We hope that these two talks serve Moritz Epple (University of Frankfurt) on with certain other contemporary views on as another opportunity to bring physicists, the topic Beyond Metaphysics and Intu- the status of geometry, such as Poincare’s´ mathematicians, and historians of science ition: Felix Hausdorff’s Views on Geom- and Schlick’s. together in a single audience.

many-body systems”, 16. bis 18. Januar of quantum information”. ESI News 2008, and of the ESI-programme Entan- START Prize to Bernhard Lamel glement and correlations in many-body quantum mechanics in August – October Bernhard Lamel was awarded a START Awards and Prizes 2009. Prize by the Austrian Science Foundation Hermann Kummel¨ Award to Frank Ver- (FWF) for his groundbreaking work on straete Ignaz L. Lieben Prize to Markus As- ‘Biholomorphic Equivalence’ in the theory Frank Verstraete has received the “Her- pelmeyer of functions of several complex variables. mann Kummel¨ Early Achievement Award The Austrian Academy of Sciences has The START Prize is the highest award for in Many-Body Physics” 2007. This prize awarded the Ignaz L. Lieben Prize 2007 young scientists in Austria. is awarded by the International Advisory to Markus Aspelmeyer “for the outstand- Committe of the International Conference Wittgenstein Prize to Christian Kratten- ing achievments of the young scientist in thaler Series on Recent Progress in Many-Body quantum optics and quantum information.” Theories. Professor Verstraete receives the This prize is the oldest prize awarded by Christian Krattenthaler was awarded the award “for his pioneering work on the the Academy. Wittgenstein Prize by the Austrian Science use of quantum information and entan- Foundation (FWF) for his groundbreak- glemet theory in formulating new and pow- Isaac Newton Medal to Anton Zeilinger ing work on combinatorial problems. The erful nemerical simulation methods for use Anton Zeilinger is the first recipient of the Wittgenstein Prize is the most valuable and in strongly correlated systems, stochastic Isaac Newton Medal awarded by Institute prestigious prize for scientific research in nonequilibrium systems and strongly cou- of Physics (IOP). Zeilinger was honoured Austria. pled quantum field theories.” for “his pioneering conceptual and experi- Christian Krattenthaler will be one of the Frank Verstraete will be a co-organizer mental contributions to the foundations of organizers of the ESI-programme Combi- of an ESI-workshop on “Tensor network quantum physics, which have become the natorics and Statistical Physics in Febru- methods and entanglement in quantum cornerstone for the rapidly-evolving field ary – June 2008. http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics 12 Volume 2, Issue 2, Autumn 2007 ESI NEWS

Current and future activities of the ESI

Thematic Programmes 2008 Other Scientific Activities in 2007

Combinatorics and Statistical Physics, February 1 – June 15, 2008 ESF Workshop on Noncommutative Quantum Field Theory, Organisers: M. Bousquet-Melou, M. Drmota, C. Krattenthaler, B. November 26 – November 29, 2007 Nienhuis Organizer: Harald Grosse Workshop, May 25 – June 7, 2008 Fourth Vienna Central European Seminar on Particle Physics Summer School, July 7 – July 18, 2008 and Quantum Field Theory, November 30 – December 2, 2007 Theme: Commutative and Noncommutative Quantum Field theory Metastability and Rare Events in Complex Systems, February 1 Organizer: Helmuth Huffel¨ – April 30, 2008 Organizers: P.G. Bolhuis, C. Dellago, E. van den Eijnden EU-NCG Miniworkshop on Ergodic Theory and von Workshop, February 17 – February 23, 2008 Neumann Algebras, December 3 – December 14, 2007 Organizer: K. Schmidt Hyberbolic Dynamical Systems, May 12 – July 5, 2008 Organisers: H. Posch, D. Szasz, L.-S. Young Spectral theory and partial differential equations, December Workshop, June 15 – June 29, 2008 10 – December 21, 2007 Organizers: T. Hofmann-Ostenhof and A. Laptev Operator Algebras and Conformal Field Theory, August 25 – December 15, 2008 Ergodic theory — Limit theorems and Dimensions, December Organisers: Y. Kawahigashi, R. Longo, K.-H. Rehren, J. Yngvason 17 – December 21, 2007 Organizers: F. Hofbauer and R. Zweimuller¨ Thematic Programmes 2009 Other Scientific Activities in 2008 Representation theory of reductive groups — local and global aspects, January 2 – February 28, 2009 Tensor network methods and entanglement in quantum Organizers: G. Henniart, G. Muic and J. Schwermer many-body systems, January 16 – January 18, 2008 Organizers: F. Verstraete, G. Vidal and M. Wolf Number theory and physics, March 1 - April 18, 2009 Organizers: A. Carey, H. Grosse, D. Kreimer, S. Paycha, S. Ab-initio density-functional studies of intermetallic Rosenberg and N. Yui compounds, January 23 – January 25, 2008 Organizer: J. Hafner Selected topics in spectral theory, May 4 – July 25, 2009 Organizers: B. Helffer, T. Hoffmann-Ostenhof and A. Laptev 15th Anniversary of the ESI, April 14, 2008 Organizers: W.L. Reiter, K. Schmidt, J. Schwermer and J. Large cardinals and descriptive set theory, 2 weeks in June – Yngvason July 2009 Organizers: S. Friedman, M. Goldstern, R. Jensen, A. Kechris Frontiers in Mathematical Biology: Mathematical population and W.H. Woodin genetics, April 14 – April 18, 2008 Organizers: R. Burger¨ and J. Hermisson Entanglement and correlations in many-body quantum mechanics, August 18 – October 17, 2009 Topics in Mathematical Physics, July 21 – July 31, 2008 Organizers: B. Nachtergaele, F. Verstraete and R. Werner Organizers: C. Hainzl, R. Seiringer and J. Yngvason

Profinite Groups, December 7 – December 20, 2008 Organizers: K. Auinger, F. Grunewald, W. Herfort and P.A. Zalesski

Erwin Schrodinger¨ Institute of Mathematical Physics http://www.esi.ac.at/ ESI NEWS Volume 2, Issue 2, Autumn 2007 13

Editors: Klaus Schmidt, Joachim Schwermer, Jakob Yngvason ESI Contact List: Contributors: Administration Romeo Brunetti: [email protected] Isabella Miedl: [email protected] Joachim Burgdorfer:¨ [email protected] Scientific Directors Pierre de la Harpe: [email protected] Harald Grosse: [email protected] Joachim Schwermer: [email protected] Walter Thirring: [email protected] Jakob Yngvason: [email protected] Frank Verstraete: [email protected] President Bruno Zumino: [email protected] Klaus Schmidt: [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.

This newsletter is available on the web at: ftp://ftp.esi.ac.at/pub/ESI-News/ESI-News2.2.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. http://www.esi.ac.at/ Erwin Schrodinger¨ Institute of Mathematical Physics