# Dr Gandalf Lechner

Senior Lecturer

*Email:*- lechnerg@cardiff.ac.uk
*Telephone:*- +44(0)29 208 75553
*Location:*- 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
- Deputy Director of Postgraduate Studies

August 2018 - January 2019: Acting Director of Postgraduate Studies - Member of School Research Committee
- Member of School Learning and Teaching Committee
- Member of School Student Staff Panel

### 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

### 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(1), pp. 159-202. (10.1007/s00220-017-2942-6)

### 2017

- 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)
- Lechner, G., Pennig, U. and Wood, S. 2017. Yang-Baxter representations of the infinite symmetric group. arXiv
- 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)

### 2016

- Lechner, G. and Sanders, K. 2016. Modular nuclearity: a generally covariant perspective. Axioms 5(1), article number: 5. (10.3390/axioms5010005)
- 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)
- Abdesselam, A.et al. 2016. Recent mathematical developments in quantum field theory. Oberwolfach Reports 13(3), pp. 2069-2122. (10.4171/OWR/2016/36)

### 2015

- 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 Verch, R. 2015. Linear hyperbolic PDEs with non-commutative time. Journal of Noncommutative Geometry 9(3), pp. 999-1040. (10.4171/JNCG/214)

### 2014

- Lechner, G. and Longo, R. 2014. Localization in nets of standard spaces. Communications in Mathematical Physics 336(1), pp. 27-61. (10.1007/s00220-014-2199-2)
- 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

I teach/taught the following modules:

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

### MMATH projects

- 2016/17: Charley Scotford (MMATH project, Cardiff)

Project "Supersymmetric quantum mechanics" - 2016/17: Nathaniel Fox (MMATH project, Cardiff)

Project "Transparent potentials for 1d quantum scattering" - 2015/16: Peter Sutherns (MMATH project, Cardiff)

Project: "The Radon Transform"

#### Research students, Erasmus students, Diploma/Laurea students:

- Vidal Agniel (Internship student from Rennes, France)

worked on composition operators on Hilbert spaces of analytic functions (May-July 2016) - Giovanni Cattarina (Erasmus student from Bologna, Italy)

Laurea thesis: "Link Invariants, Braid Groups and Classification of Enhanced Yang-Baxter Operators in d=2" (2015/16) - Albert Huber - Thermal Equilibrium States in Deformed Quantum Field Theory, Diploma 2012 (Vienna)
- Christian Schuetzenhofer - Multi-particle S-matrix models in 1+1 dimensions and associated QFTs, Diploma 2011 (Vienna)

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

*Operator Algebras and Quantum Physics. Programme and workshop at the Simons Center for Geometry and Physics, May/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)

**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

### 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