Mathematical Physics Cluster

1. Sefydliad Gwyddorau Cyfrifiadurol a Mathemategol Cymru SGCMC WIMCS Wales Institute of Mathematical and Computational Sciences Mathematical Physics Cluster Research interests include: Symmetry and Independence in Quantum Probability Strongly coupled dynamics of quantum field theories Claus Köstler, Institute of Math and Physics, Aberystwyth Carlos Nunez, Department of Physics, Swansea • Motivation and Aim • Development of a quantized theory of distributional symmetries and invariance principles • Investigation of quantized counterparts of conditional independence in quantum probability • Identification of asymptotic models for scaling limits of interacting quantum dynamical systems Motivation and Aim To develop and exploit techniques to investigate the strongly coupled dynamics of quantum field theories (QFTs) • Provide new computational methods for exactly solvable models in quantum statistical physics • Application of the developed tools to collective quantum effects, quantum control and information • Method • To use the fact that certain QFTs are known to have a dual description involving an emergent higher dimensional gravitational theory. In particular, the strong coupling regime of the QFT maps to the classical regime of the dual gravitational theory and many observables can be calculated exactly. • To exploit the fact that the dual gravitational theory is described beyond the classical regime by a QFT in order to understand quantum gravitational effects. • To discretize the QFT by defining it on a spacetime Lattice in order that the functional integral can be calculated numerically on state-of-the-art computers by Monte Carlo techniques. • Method • Systematic quantization of classical probability using the language of operator algebras • Investigate to which extent classical de Finetti type results transfer to the quantum world • Investigate the relation of symmetry and independence in the general quantized setting  • Selected Results •  - indicate deep connections between important areas in modern analysis and its applications. • Noncommutative extended de Finetti theorem: 'exchangeability' and 'spreadability imply 'noncommutative conditional independence' in terms of 'commuting squares' • Failure of common folklore facts from classical probability: ‘exchangeability' and 'spreadability’ of an infinite sequence of random variables are no longer equivalent in the quantized setting. • Braided extended de Finetti theorem: we introduce 'braidability' as a new symmetry intermediate to 'exchangeability‘ and 'spreadability'. It applies to subfactor inclusions with small Jones index • Outcome • The group has produced new strong coupling results in QFTs with minimal and extended supersymmetry. • The results apply not only in the zero temperature theory, but also at finite temperature and density (important for heavy ion collisions and also in the early universe). • The observables which have been investigated include the spectrum of states and their interactions, the expectation values of important operators. • The group has used these results to formulate new dualities involving some QFTs which may have direct relevance in the interpretation of experimental observation at the Large Hadron Collider. • Free de Finetti theorem: ‘quantum exchangeability' yields a beautiful characterization of freeness with amalgamation. This quantum probabilistic symmetry comes from a compact quantum group. • Thoma theorem is a quantum de Finetti theorem: we give a new operator algebraic proof of the characterization of extremal characters of the infinite symmetric group. • Selected Publications •  R. Gohm; C. Köstler. Noncommutative independence from the braid group B∞. Comm. Math. Phys. 289 (2009), 435–482. • R. Gohm; C. Köstler. Noncommutative independence from characters of the infinite symmetric group S∞. arXiv:1005.5726 [math.OA](2010). • C. Köstler. A noncommutative extended de Finetti theorem. • J. Funct. Anal. 258 (2010), 1073–1120. • C. Köstler; R. Speicher. A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation. Comm. Math. Phys. 291 (2009), 473–490. Collaborators include: Juan Maldacena (IAS), Daniel Freedman (MIT)Alfonso Ramallo (Santiago de Compostela), Nick Dorey (Cambridge) Kostas Skenderis (Amsterdam), Dario Martelli (King's College) Selected Cluster Activities Large N at Swansea Swansea University, July 7-10 2009 A conference drawing together the world's experts to discuss recent advances in strong coupling dynamics in large N gauge theories. Speakers inlcuded: Thomas Appelquist (Yale), Lawrence Yaffe (Seattle), Mikhail Shifman (Minnesota), … Cardiff Semester on Noncommutative Geometry and Physics, February – July 2010 Organized by David E Evans (Cardiff) Long term visitors as well as a series of short term events, with 120+ participants in all, including: WIMCS MP Colloquium, 26 February 2010 held at Swansea Computer Science Speakers: Marcos Mariño (Geneva), Richard Thomas (Imperial) Planar Period Cardiff University, 15 February – 5 March 2010 Speakers included: S Bigelow (UCSB), J Fuchs (Karlstad), … London Mathematical Society Spitalfields Day Cardiff University, 17 May 2010. Speakers: Nigel Higson (Penn State), Terry Gannon (Alberta) London Mathematical Society Regional Meeting and Workshop Operator Algebras and Physics, Cardiff University, 21-25 June 2010. Series of lectures by Constantin Teleman and other speakers included Werner Nahm (Dublin) Annual Meeting of the EU Research Training Network in Noncommutative Geometry, Cardiff University, 28 June – 2 July 2010 Speakers included: J Mickelsson (Helsinki), Y Kawahigashi (Tokyo), A Verbeure (Leuven), J Yngvason (ESI Vienna), …. First Frontiers Lecture of the Learned Society of Wales Cardiff University, 28 June 2010 Speaker: Dan Voiculescu (Berkeley) Quantum field theory on curved space-times and curved target-spaces Erwin-Schrodinger Institute, Vienna, March+April 2010 , organized by Stefan Hollands (Cardiff) Future Cluster Activities David E Evans (Cardiff) EU Research Training Network in Noncommutative Geometry Appointments in 2011: Stefan Hollands (Cardiff) ERC Starter Research Grant 2011-16 INI-WIMCS follow up meeting to Isaac Newton Institute  2006 programme on Noncommutative Geometryto be held at Cardiff or Gregynog in 2012 - application pendingOrganizers David E Evans (Cardiff ), Nigel Higson (Penn State), Shahn Majid (Queen Mary, London) Themes of meeting include: Noncommutative Algebraic GeometryRepresentation Theory aspects of Baum-ConnesFree aspects of Noncommutative GeometryNoncommutative Geometry and Conformal Field TheoryNoncommutative Geometry and CategorificationNoncommutative Spacetime and CosmologyThe Standard Model and Beyond • Alin Galatan (Bucharest) • Antti Harju (Helsinki) • Grace Kennedy (UCSB) • Jennifer Maier (Hamburg) • Makoto Yamashita (Tokyo) Frontiers Lecture of the Learned Society of Wales Cardiff University, 17 January 2011 Speaker: Sir Michael Atiyah (Edinburgh) Dr Claus Köstler, Institute of Math and Physics, Aberystwyth Dr Carlos Nunez, Department of Physics, Swansea