# Dr Gandalf Lechner

Senior Lecturer

- lechnerg@cardiff.ac.uk
- +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

### Personal website

### Qualifications

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

### Previous positions

- 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

### 2020

- Lechner, G. and Scotford, C. 2020. Fock representations of ZF algebras and R-matrices. Letters in Mathematical Physics 110, pp. 1623-1643. (10.1007/s11005-020-01271-3)

### 2019

- Bostelmann, H., Cadamuro, D. and Lechner, G. 2019. High energy bounds on wave operators. arXiv, pp. -.
- Lechner, G., Pennig, U. and Wood, S. 2019. Yang-Baxter representations of the infinite symmetric group. Advances in Mathematics 355, article number: 106769. (10.1016/j.aim.2019.106769)
- Lechner, G. and Scotford, C. 2019. Fock representations of Zamolodchikov algebras and R-matrices. arXiv
- Conti, R. and Lechner, G. 2019. Yang-Baxter endomorphisms. arXiv
- Lechner, G. 2019. Subfactors and unitary R-matrices. Presented at: Subfactors and Applications, Oberwolfach, Germany, 27 october - 2 November 2019. pp. -.

### 2018

- Lechner, G.et al. 2018. Approximation numbers of weighted composition operators. Journal of Functional Analysis 274(7), pp. 1928-1958. (10.1016/j.jfa.2018.01.010)
- Hollands, S. and Lechner, G. 2018. SO(d,1)-invariant Yang-Baxter operators and the dS/CFT correspondence. Communications in Mathematical Physics 357, pp. 159-202. (10.1007/s00220-017-2942-6)

### 2017

- Alazzawi, S. and Lechner, G. 2017. Inverse scattering and locality in integrable quantum field theories. Communications in Mathematical Physics 354(3), pp. 913-956. (10.1007/s00220-017-2891-0)
- Bostelmann, H., Cadamuro, D. and Lechner, G. 2017. Quantum backflow and scattering. Physical Review a 96(1), article number: 12112. (10.1103/PhysRevA.96.012112)

### 2016

- Abdesselam, A.et al. 2016. Recent mathematical developments in quantum field theory. Oberwolfach Reports 13(3), pp. 2069-2122. (10.4171/OWR/2016/36)
- Lechner, G. and Waldmann, S. 2016. Strict deformation quantization of locally convex algebras and modules. Journal of Geometry and Physics 99, pp. 111-144. (10.1016/j.geomphys.2015.09.013)
- Lechner, G. and Sanders, K. 2016. Modular nuclearity: a generally covariant perspective. Axioms 5(1), article number: 5. (10.3390/axioms5010005)
- Abdesselam, A.et al. 2016. Recent mathematical developments in quantum field theory. European Mathematical Society.

### 2015

- Lechner, G. and Verch, R. 2015. Linear hyperbolic PDEs with non-commutative time. Journal of Noncommutative Geometry 9(3), pp. 999-1040. (10.4171/JNCG/214)
- Lechner, G. 2015. Algebraic constructive quantum field theory: Integrable models and deformation techniques. In: Brunetti, R. et al. eds. Advances in Algebraic Quantum Field Theory. Springer, pp. 397-448.
- Lechner, G. and Longo, R. 2015. Localization in nets of standard spaces. Communications in Mathematical Physics 336, pp. 27-61. (10.1007/s00220-014-2199-2)

### 2014

- Lechner, G. and Schützenhofer, C. 2014. Towards an operator-algebraic construction of integrable global gauge theories. Annales Henri Poincare 15(4), pp. 645-678. (10.1007/s00023-013-0260-x)
- Lechner, G. and Schlemmer, J. 2014. Thermal equilibrium states for quantum fields on non-commutative spacetimes. Presented at: Quantum Mathematical Physics, Regensburg, Germany, 29 September - 2 October 2014 Presented at Finster, F. et al. eds.Quantum Mathematical Physics: A Bridge between Mathematics and Physics. Springer, (10.1007/978-3-319-26902-3)

### 2013

- Lechner, G., Schlemmer, J. and Tanimoto, Y. 2013. On the equivalence of two deformation schemes in quantum field theory. Letters in Mathematical Physics 103(4), pp. 421-437. (10.1007/s11005-012-0599-9)
- Grosse, H.et al. 2013. Wick rotation for quantum field theories on degenerate Moyal space(-time). Journal of Mathematical Physics 54(2), article number: 22307. (10.1063/1.4790886)

### 2012

- Lechner, G. 2012. Deformations of quantum field theories and integrable models. Communications in Mathematical Physics 312(1), pp. 265-302. (10.1007/s00220-011-1390-y)

### 2011

- Dappiaggi, C., Lechner, G. and Morfa-Morales, E. 2011. Deformations of quantum field theories on spacetimes with killing vector fields. Communications in Mathematical Physics 305(1), pp. 99-130. (10.1007/s00220-011-1210-4)
- Bostelmann, H., Lechner, G. and Morsella, G. 2011. Scaling limits of integrable quantum field theories. Reviews in Mathematical Physics 23(10), article number: 1115. (10.1142/S0129055X11004539)

### 2010

- Buchholz, D., Lechner, G. and Summers, S. J. 2010. Warped convolutions, rieffel deformations and the construction of quantum field theories. Communications in Mathematical Physics 304(1), pp. 95-123. (10.1007/s00220-010-1137-1)

### 2009

- Lechner, G. 2009. Deformations of operator algebras and the construction of quantum field theories. Presented at: XVI International Congress on Mathematical Physics 2009, Prague, 3 - 8 September 2009.

### 2008

- Lechner, G. 2008. Construction of quantum field theories with factorizing S-matrices. Communications in Mathematical Physics 277(3), pp. 821-860. (10.1007/s00220-007-0381-5)
- Grosse, H. and Lechner, G. 2008. Noncommutative deformations of Wightman quantum field theories. Journal of High Energy Physics 2008(9), article number: 131. (10.1088/1126-6708/2008/09/131)

### 2007

- Lechner, G. 2007. Towards the construction of quantum field theories from a factorizing S-matrix. Progress in Mathematics -Boston- 251, pp. 175-197. (10.1007/978-3-7643-7434-1_13)
- Grosse, H. and Lechner, G. 2007. Wedge-local quantum fields and noncommutative minkowski space. Journal of High Energy Physics 2007(11), article number: 12. (10.1088/1126-6708/2007/11/012)

### 2005

- Lechner, G. 2005. On the existence of local observables in theories with a factorizing S-matrix. Journal of Physics A: Mathematical and General 38(13), pp. 3045. (10.1088/0305-4470/38/13/015)

### 2004

- Buchholz, D. and Lechner, G. 2004. Modular Nuclearity and Localization. Annales Henri Poincaré 5(6), pp. 1065-1080. (10.1007/s00023-004-0190-8)

### 2003

- Lechner, G. 2003. Polarization-free quantum fields and interaction. Letters in Mathematical Physics 64, pp. 137-154.

### Undergraduate modules

In the academic year 2020/21 I am teaching the following modules:

- MA2008 Linear Algebra II
- MA0414 Mathematical Foundations of Quantum Physics

Previously taught modules:

- MA01005 Foundations of Mathematics I (2017/18, 2018/19, 2019/20)
- MA0322 Knots (2015/16)
- MA0414 Mathematical Foundations of Quantum Physics (2016/17, 2017/18, 2018/19, 2019/20)

### Supervision of undergraduate projects

(see tab "Supervision" for PhD projects)

- 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

- Zopfgruppen, die Yang-Baxter Gleichung, und Unterfaktoren (Braid groups, the Yang-Baxter equation, and subfactors) February 2020

This is a contribution to the series Snapshots of Modern Mathematics published by the Oberwolfach Research Institute for Mathematics which aims to explain mathematical problems and ideas of present relevance in an accessible and understandable way to a wide audience. My article is currently available as a preprint in German, a final version and an English translation will follow. - Visual Mathematics: Braids and Knots

A short video for prospective mathematics students, produced for our virtual open day in July 2020

braids and knots video

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 Theory**In 2015, I wrote a book chapter reviewing the status of constructive algebraic quantum field theory.

### Conferences organised

- LQP Colloquium, since Sept. 2020 (with S. Hollands, Y. Kawahigashi, R. Longo, K.-H. Rehren)
- Virtual LQP1 / LQP45 online workshop, June 2020 (with P. Naaijkens)
- Operator Algebras and Quantum Physics. Programme and workshop at the Simons Center for Geometry and Physics, June 2019 (with S. Hollands, V. Jones, R. Longo)
- ICFT 22 -- 22nd UK Meeting on Integrable Models, Conformal Field Theory and Related Topics, Cardiff, June 2018 (with S. Wood)
- Algebraic Structures & Quantum Physics, LMS Regional Meeting for South West and South Wales, Cardiff, December 2017 (with M. Pugh and S. Wood)
- Quantum Field Theory: Concepts, Constructions, and Curved Spacetimes, York, April 2017 (with D. Bahns and C. Fewster)
- Yang-Baxter Equations: A Tangle of Physics & Mathematics, Workshop in Cardiff, December 2016 (with U. Pennig and S. Wood)
- Recent Mathematical Developments in Quantum Field Theory, Oberwolfach, 2016 (with A. Abdesselam S. Hollands, C. Kopper)
- Session "Operator Algebras and Quantum Field Theory" at the 14th Marcel Grossmann meeting, Universita La Sapienza, July 2015 (with G. Ruzzi)
- LQP 31 – Workshop “Foundations and Constructive Aspects of QFT”, Leipzig University, November 2012 (with R. Verch)
- Rigorous quantum field theory in the LHC era, ESI Vienna, September 2011 (with C. Jaekel and C. Kopper)

### 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)

**2019**

- 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

**2018**

- 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

**2017**

- 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

**2016**

- 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

**2015**

- 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

**2014**

- 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

**2013**

- 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

**2012**

- 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

**2011**

- 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

**2010**

- 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

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