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Dr Gandalf Lechner

Dr Gandalf Lechner


School of Mathematics

+44(0)29 208 70943
M/2.18, Maths and Education Building , Senghennydd Road, Cardiff, CF24 4AG
Available for postgraduate supervision


Research interests

As a mathematical physicist, I work on conceptual and mathematical problems arising in quantum field theory. I am particularly interested in new approaches to the problem of rigorously constructing quantum field theory models. A novel line of research addresses this question via operator-algebraic deformations of quantum field theories, a topic also connected to non-commutative geometry and quantum field theory on non-commutative spaces.

Further interests include integrable models in quantum field theory and related structures, such as the Yang-Baxter equation, braid group representations and questions in Operator Theory that arise in the study of spaces of holomorphic functions.

For more details about my research, you can have a look at my publications and research pages.

Administrative Roles

  • Head of the GAPT Research Group, School of Mathematics
  • Member of School Research Committee

External Positions

  • Member of the Committee of the Mathematical Physics Group of the IoP
  • INI-ICMS correspondent for the IoP

Professional Memberships

  • IAMP - International Association for Mathematical Physics
  • LMS - London Mathematical Society
  • IoP - Institute of Physics
  • DPG - Deutsche Physikalische Gesellschaft
  • Fellow of the Higher Education Academy HEA

Research group

GAPT (Geometry, Algebra, Mathematical Physics and Topology) Research Group

GAPT Seminar

Personal website

Dr Gandalf Lechner's Personal website



  • 2002 Dipl. Phys. (Georg-August-Universitaet Goettingen)
  • 2006 Dr. rer. nat. (Georg-August-Universitaet Goettingen)

Previous positions

  • 2018-2020 - Senior Lecturer, School of Mathematics, Cardiff University
  • 2015-2018 - Lecturer, School of Mathematics, Cardiff University
  • 2011-2015 - Assistant in the Quantum Field Theory and Gravity group at the University of Leipzig
  • 2008-2011 - Assistant in the Mathematical Physics Group at the University of Vienna
  • 2007 - "Experienced Researcher" in the Marie Curie Research Training network “Quantum Spaces – Noncommutative Geometry” at its Vienna node
  • 2007 - Junior Research Fellow at the Erwin Schroedinger Institute for Mathematical Physics, Vienna
  • 2006 - PostDoc in the DFG research project "Modular Structure and Particle Aspects of Quantum Field Theory" at Goettingen University




















Undergraduate modules

In the academic year 2020/21 I am teaching the following modules. Details on on/offline teaching will be communicated in due course.

Previously taught modules:

Supervision of undergraduate projects

(see tab "Supervision" for PhD projects)

  • Alexander Curis (MMATH project, Cardiff 2020/21)
    Project: "Quantum Entanglement and Operator Algebras"
  • Timothy Weaving (MMATH project, Cardiff 2019/20)
    Project: "On Quantifying the Quantum Entanglement Present Within Shor's Algorithm"
  • Owen Tanner (CUROP research student, Cardiff, July-Sept. 2019, supervised jointly with Ulrich Pennig)
    Project "Von Neumann Algebras and Applications"
  • Charley Scotford (MMATH project, Cardiff, 2016/17) 
    Project "Supersymmetric quantum mechanics"
  • Nathaniel Fox (MMATH project, Cardiff, 2016/17) 
    Project "Transparent potentials for 1d quantum scattering"
  • Vidal Agniel (Internship student from Rennes, France, May-July 2016)
    worked on composition operators on Hilbert spaces of analytic functions
  • Giovanni Cattarina (Erasmus student from Bologna, Italy, 2015/16)
    Laurea thesis: "Link Invariants, Braid Groups and Classification of Enhanced Yang-Baxter Operators in d=2"
  • Peter Sutherns (MMATH project, Cardiff, 2015/16)
    Project: "The Radon Transform"
  • Albert Huber (Diploma 2012, Vienna)
    Project "Thermal Equilibrium States in Deformed Quantum Field Theory"
  • Christian Schützenhofer (Diploma 2011, Vienna)
    Project "Multi-particle S-matrix models in 1+1 dimensions and associated QFTs"

Outreach material

Some recent research of mine is:

Yang-Baxter Representations (with Ulrich Pennig and Simon Wood)
The Yang-Baxter equation  appears in a remarkable number of different fields, such as statistical mechanics, knot theory, quantum groups. quantum field theory, quantum mechanics, quantum information theory, subfactors, … . This equation is very hard to solve directly, and the structure of the set of its solutions is insufficiently understood. We considered the case of involutive solutions R, i.e. those R-matrices that square to the identity. In this case, we found a complete classification of all solutions up to a natural equivalence suggested by applications in quantum field theory.

Quantum-mechanical Backflow (with Henning Bostelmann and Daniela Cadamuro)One incarnation of the uncertainty relation is the so-called backflow effect: A quantum-mechanical particle in one spatial dimension that has positive momentum (with probability 1) may still have negative probability flux, i.e. the probability can flow in the direction opposite to the momentum. We investigated the backflow effect for interacting systems governed by an arbitrary short-range potential. It turns out that backflow is a universal effect that exists in any such system, and its magnitude can be estimated by analytic methods.

Constructive Algebraic Quantum Field TheoryIn 2015, I wrote a book chapter reviewing the status of constructive algebraic quantum field theory.

Conferences organised

Grant income (since 2010)

  • 2018 LMS Research in Pairs Grant for a research visit to Rome, £ 941
  • 2018 IoP Grant for funding a workshop (ICFT22) at Cardiff, June 2018, £ 750 (with S. Wood)
  • 2018 LMS Grant for supporting a workshop (ICFT22) at Cardiff, June 2018, £ 2,200 (with S. Wood)
  • 2017 Grant from the Simons Foundation for a programme and workshop at the Simons Center in May/June 2019, ~$80,000 (with S. Hollands, V. Jones, R. Longo)
  • 2017 LMS grant for a conference in Cardiff, December 2017, £ 5,000 (with M. Pugh and S. Wood)
  • 2017 IoP grant for supporting a conference in Cardiff, December 2017, £ 750 (with M. Pugh and S. Wood)
  • 2017 IoP grant for supporting a conference in York, April 2017, £ 1,000 (with D. Bahns and C. Fewster)
  • 2017 LMS research in pairs grant for a research visit to York, £ 410
  • 2016 IoP grant for a workshop at Cardiff, £ 1,200 (with U. Pennig and S. Wood)
  • 2016 Oberwolfach Grant for running a (full-size) MFO workshop in July 2016 (with A. Abdesselam, S. Hollands, C. Kopper)
  • 2011-2014 FWF-Project “Deformations of Quantum Field Theories”, together with J. Yngvason (PI), € 360,000
  • 2011 ESI Grant for organizing a conference at the Erwin Schrödinger Institute in Vienna, € 24,000, with C. Kopper and C. Jäkel

Talks (since 2010)



  • High-energy bounds on wave operators
    Mathematical Physics Seminar, Leipzig University, December 2019
  • An operator-algebraic perspective on unitary R-matrices
    Mathematical Physics Seminar, University of York, November 2019
  • Yang-Baxter endomorphisms
    Subfactors and Applications, Oberwolfach, October 2019
  • From the construction of integrable QFTs to the classification of unitary R-matrices
    Mathematics of interacting QFT models, York, July 2019
  • Deformations of quantum field theories, the construction of models, and TT-deformations
    TT- and other solvable deformations of quantum field theories , Simons Center for Geometry and Physics, Stony Brook, April 2019
  • R-matrices and localised endomorphisms
    Mathematical Physics Seminar, Nottingham University, March 2019
  • Quantum-mechanical backflow and scattering
    Mathematical Physics Seminar , Northumbria University, Newcastle, March 2019
  • Braid group characters and localised endomorphisms
    43rd LQP workshop "Foundations and Constructive Aspects of QFT" , Galieo Galilei Institute Firenze, February 2019


  • The Yang-Baxter equation and extremal characters of the infinite braid group
    Mathematical Physics Seminar, Leipzig University, December 2018
  • TBA
    Conference Physics and Mathematics of Quantum Field Theory, Banff Research Station, Canada, July/August 2018
  • Quantum-mechanical backflow and scattering theory
    Conference AQFT: Where operator algebras meet microlocal analysis, Cortona, Italy, June 2018
  • Yang-Baxter characters of the infinite symmetric group and subfactors
    Deformation Quantization Seminar, Würzburg, Germany, March 2018
  • All involutive solutions of the Yang-Baxter equation
    Seminaire de Physique Mathématique, Lyon, France, March 2018


  • Modular Localization and Constructive Algebraic QFT
    Workshop on Reflection Positivity, Oberwolfach Research Centre, November 2017
  • All involutive solutions of the Yang-Baxter equation
    Advances in Mathematics and Theoretical Physics, Accademia Nazionale dei Lincei, Rome, September 2017
  • Yang-Baxter Representations of the Infinite Symmetric Group
    Modern Mathematics of Quantum Theory, University of York, September 2017
  • Yang-Baxter Representations of the Infinite Symmetric Group
    Foundations and Constructive Aspects of QFT, Leipzig University, June 2017
  • Yang-Baxter Representations of the Infinite Symmetric Group
    Subfactors, K-theory and conformal field theory, Newton Institute Cambridge, June 2017
  • Quantum Backflow and Scattering Theory
    Mathematical Physics Seminar, Imperial College, May 2017
  • Quantum Backflow and Scattering
    Mathematical Physics seminar, University of York, February 2017
  • Integrable quantum field theories and von Neumann algebras
    Theoretical Physics Seminar, King's College, London, February 2017
  • Representations of Rieffel-deformed C*-algebras and deformations of Borchers triples
    Analysis Seminar, Glasgow University, January 2017


  • Quantum Backflow and Scattering Theory
    Physics Seminar, Cardiff University , November 2016
  • Sigma models in algebraic QFT
    Local Quantum Physics and Beyond: In Memoriam Rudolf Haag , Hamburg, September 2016
  • SO(d,1)SO(d,1)-invariant Yang-Baxter operators and a dS/CFT correspondence
    Operator Algebras and Quantum Field Theory , Frascati, June 2016
  • Modular Nuclearity
    Math. Phys. Seminar, York University , March 2016
  • The inverse scattering problem for integrable QFTs in 2d and its operator-algebraic solution
    QFT Seminar, Oxford University , February 2016
  • Constructing QFTs with modular theory and R-matrices
    Graduiertenkolleg Mathematik+Physik, Universität Göttingen , February 2016


  • Singular values of Hardy space operators and relative commutants of von Neumann algebras
    Geometry and Analysis Seminar, Queen Mary, London , December 2015
  • R-matrices and integrable QFTs
    Lie Groups: Representations and Geometry , Erlangen, September 2015
  • O(d,1)-symmetric R-matrices and QFT models
    Operator Algebras and Quantum Physics , Sao Paulo, July 2015
  • Wave Equations with Noncommutative Time
    Mathematical Physics Seminar, University of York , June 2015
  • Noncommutative Wave Equations
    Mathematical Physics Seminar, Bristol University , May 2015
  • Integrable Quantum Field Theories from an Operator-Algebraic Perspective
    Wales Mathematics Colloquium , Gregynog, May 2015
  • Representations of Rieffel-deformed C*-algebras and deformations of inclusions
    Workshop on Deformation Quantization: Between Formal and Strict , Oberwolfach, February 2015
  • Modular Structure of Generalized Free Fields and Nets of Standard Subspaces
    AQFT 2015 - Workshop on New Trends in Algebraic Quantum Field Theory , Frascati, February 2015


  • Reflection Positivity and Operator Algebras in Quantum Field Theory
    Oberwolfach Workshop: Reflection Positivity in Representation Theory, Stochastics, and Physics , Oberwolfach, December 2014
  • The structure of the field algebra in non-commutative QFT and uniqueness of its KMS states
    Quantum Mathematical Physics , Regensburg, September 2014
    Hyperbolic PDEs with non-commutative time
    Frontiers in Fundamental Physics , Marseille, July 2014
  • Operator algebras and the construction of models in quantum field theory
    FJIM 2014 , Bilbao, July 2014
  • Construction of quantum field theories by deformation techniques
    DPG Spring Conference , Berlin, March 2014


  • KMS functionals and KMS states for algebras of deformed quantum fields
    Deformation Quantization Seminar, Uni Würzburg , November 2013
  • Non-local perturbations of hyperbolic PDEs and QFT models on non-commutative spacetimes
    GEP-Seminar , Leipzig, May 2013
  • Non-Local Perturbations of Hyperbolic PDEs and QFT on Noncommutative Spacetimes
    Mathematics and Quantum Physics , Accademia dei Lincei, Rome, July 2013
  • Hyperbolic PDEs with star product potentials
    GEOQUANT 2013 , ESI Vienna, August 2013


  • Equivalence of two deformation schemes in QFT
    Essays in low-dimensional quantum field theory , Universita Rom II Tor Vergata, October 2012
  • Borchers triples and the construction of models in quantum field theory
    Memorial Symposium Hans-Juergen Borchers "A Life for Relativistic Quantum Field Theory" , Goettingen, October 2012
  • KMS states of deformed quantum field theories
    Algebraic Quantum Field Theory and Local Symmetries , Hausdorff Research Institute for Mathematics, Sept. 2012
  • Thermal Equilibrium States for QFTs on Moyal Minkowski Spacetime
    ICMP 2012 , Aalborg, August 2012
  • Constructive algebraic quantum field theory at the example of integrable models
    Mathematical Aspects of Quantum Field Theory and Quantum Statistical Mechanics , DESY, Hamburg, July 2012
  • Quantum field theory on locally noncommutative spacetimes
    DPG Spring Conference 2012 , Goettingen, February 2012
  • Locally Noncommutative Spacetimes and Quantum Field Theory
    Planckland , SISSA, Trieste, February 2012


  • QFT on locally noncommutative spacetimes
    Vienna Math. Phys. Seminar , Vienna University, October 2011
  • Algebraic quantum field theory: From structural analysis to the construction of models
    Quantum Gravity Colloquium 6 , University of Erlangen, October 2011
  • Oscillatory integrals for vector-valued symbols and locally noncommutative spacetimes
    Modern Trends in Algebraic Quantum Field Theory , University of Pavia, September 2011
  • Oscillatory integrals for vector-valued symbols and locally noncommutative spacetimes
    Harmonic Analysis, Deformation Quantization, Noncommutative Geometry , Scalea, Italy, September 2011
  • Multiplicative deformations of the Borchers-Uhlmann algebra
    Local Quantum Physics workshop 28 , Goettingen, July 2011
  • Local and covariant deformations of quantum field theories
    Quantum Theory and Gravitation , Zurich, June 2011
  • Multiplicative deformations of the Borchers-Uhlmann algebra
    WIMCS workshop on quantum field theory , University Cardiff, April 2011


  • Deformations of Quantum Field Theories
    Vienna Lunch Seminar , Vienna University, December 2010
  • Local and covariant deformations of observable algebras in QFT
    Deformation Methods in Mathematics and Physics , Oberwolfach, September 2010
  • Deformations of Quantum Field Theories
    NITheP Workshop - Quantum Field Theory on Curved Spacetime , University of KwaZulu-Natal, Durban, August 2010
  • Deformations of Quantum Field Theories
    Leipzig University , June 2010
  • Deformation techniques in quantization and quantum field theory
    Hamburg University , June 2010
  • Deformations of quantum field theories
    York University , June 2010
  • Star products on Hilbert space
    Bayrischzell workshop , Bayrischzell, May 2010
  • Deformation Quantization and Deformations of Quantum Field Theories
    Math. Phys. Seminar , Vienna University, March 2010
  • Warped Convolutions und Rieffel-Deformationen
    Freiburg University , February 2010


I am supervising PhD students in areas related to my own research, including

  • mathematical physics, specifically quantum field theory
  • von Neumann algebras
  • Yang-Baxter equations and the infinite braid/symmetric groups

Current supervision

Charley Scotford

Charley Scotford

Research student

Tasarla Deadman

Research student

Past projects

  • Sabina Alazzawi (joint supervision with J Yngvason, University of Vienna) - Deformations of Quantum Field Theories and the Construction of Interacting Models, PhD 2014
  • Thomas Ludwig (joint supervision with R Verch, University of Leipzig) - Wick Rotation for Quantum Field Theories on Degenerate Moyal Space, PhD 2013
  • Eric Morfa-Morales (joint supervision with J Yngvason, University of Vienna) - Deformations of Quantum Field Theories on Curved Spacetimes, PhD 2012