Quantisation of integrable field theories and Lagrangian multiform

Summary: This project aims to study the covariant quantization of integrable field theories. Historically, the most successful approach to quantizing such theories has been through the so-called Quantum Inverse Scattering Method which is based on the notion of quantum R matrix and quantum Yang-Baxter equation. The latter appears as the canonical quantisation (in the sense of Dirac) of the classical Yang-Baxter equation, a key property of the classical r matrix. One of the conceptual drawbacks of canonical quantization is that it breaks (Lorentz) covariance of the underlying spacetime coordinates. <br /> <br /> Recently, two independent discoveries were made that open the way to investigate covariant quantization of integrable field theories from different. One is the fact that the classical r matrix has been identified as playing a key role in a covariant version of the Hamiltonian description of certain integrable field theories. The other one is the notion of Lagrangian multiforms which captures integrability in a Lagrangian framework. This offers the possibility of covariant quantization from the perspective of Feynman's path integral. What ties this project together is the beautiful connection made between the above two new approaches in the area of integrable systems - the classical r matrix appears naturally in the context of Lagrangian multiforms. This suggests that such a connection should survive quantization and relate the quantum R matrix with Feynman's path integral formalism.

<p>Detailed description: The discovery of integrable partial differential equations(PDEs) in the pioneering paper by Gardner, Greene, Kruskal and Miura, marked the birth of the modern era of research generally known as {\it integrable systems}. It triggered a myriad of mathematical discoveries (infinite dimensional Hamiltonian systems, infinite dimensional Lie algebras of symmetries, quantum groups...) which led to breakthroughs in classical and quantum physics (soliton dynamics, exact correlation functions in quantum spin chains and quantum field theories, partition functions in gauge theories...). This rich area of Theoretical/Mathematical Physics comprises a large variety of systems whose key feature is that it is possible to ``solve them exactly'', in sharp contrast with usual perturbation and approximate techniques.</p> <p>(Quantum) field theory is a dominant framework by which one models fundamental laws of nature, such as electromagnetism (Maxwell theory), gravity (general relativity) or fluid dynamics (Navier-Stokes), and tries to predict or understand the occurrence of certain phenomena. Within this vast arena, a large number of {\it integrable} (quantum) field theories have been found over the years. Given their special properties, they provide valuable "theoretical laboratories" where one can obtain explicit and exact results to guide our understanding of intricate phenomena. </p> <p>Historically, the process of quantizing these integrable models has used the idea of canonical quantization going back to Dirac: in short, one promotes the classical Poisson bracket of observables to a commutator of quantum operators. Beautiful structures have been discovered thanks to this: Poisson-Lie and quantum groups based on the classical and quantum Yang-Baxter equation (the Fields medal rewarded V.G. Drinfel'd for his work on this topic). In turn, this has allowed for tremendous success in computing the all important correlation functions. </p> <p>From a conceptual point of view, this approach has limitations which have been recognised for decades: the choice of Poisson bracket and Hamiltonian function breaks the natural (covariance) symmetry between the space-time coordinates. The most famous alternative is Feynman's path integral quantization which is based on Lagrangian as opposed to Hamiltonian description of the field theory. Another, less known idea, relies on constructing a covariant Poisson bracket and Hamiltonian at the classical level before trying to quantize canonically. </p> <p>Because of the overwhelming success of the quantum Yang-Baxter approach, the other two methods just mentioned have received essentially no attention. However, recently two new discoveries have emerged in integrable field theory: 1) a covariant classical $r$-matrix formalism [1,2], 2) a Lagrangian formalism which captures integrability: Lagrangian multiform theory [3,4]. </p> <p>This PhD project proposes to investigate this quantisation problem by taking advantage of the integrability of the model encoded in the multiform. Since multiforms apply equally well for finite or infinite dimensional systems, we expect that one should be able to describe quantum mechanical systems and quantum field theories for which the classical Lagrangian multiform is known. Current examples abound and include: the Toda chain, Gaudin models, the sine-Gordon equation, the modified Korteveg-de Vries equation, the nonlinear Schrödinger equation, Zakharov-Mikhailov models which contain<br /> the Faddeev-Reshetikhin model and recently introduced deformed sigma/Gross-Neveu models as particular cases, etc. Possible tools of investigation can include discretisation of the path integral, as in Feynman's original work, or equivariant localisation techniques which have proved powerful in the exact computation of the partition function in certain (supersymetric) gauge theories. An alternative and complementary direction would be to develop covariant canonical quantisation based on results obtained by the supervisor on covariant Poisson brackets and classical r-matrix structures. Comparison with, as well as guidance from, the well-established theory of the quantum Yang-Baxter equation and quantum groups will be important components of the project.</p> <p>With this project, you will</p> <p>1) Discover and learn the rich world of integrable field theories;</p> <p>2) Contribute to the existing theory of classical integrable field theories and develop new methods to study their covariant quantization using one or both methods described above.</p> <p>3) Apply your results to compare with existing predictions in certain quantum integrable field theories and, if time allows, investigate the possibility to tackle the very challenging issue of out-of-equilibrium physics using the tools developed in the second step. </p> <p>Applicants with a strong background in theoretical/mathematical physics are encouraged. Throughout the project, the successful candidate will interact with international research and researchers, attend international conferences, give seminars, and publish the results in international peer-reviewed journals.</p> <p>Bibliography</p> <p>[1] V. Caudrelier, M. Stoppato, <em>A connection between the classical r-matrix formalism and covariant Hamiltonian field theory</em>, J. Geom. Phys. 148 (2020), 103546.</p> <p>[2] V. Caudrelier, M. Stoppato, <em>Multiform description of the AKNS hierarchy and classical r-matrix</em>, J. Phys. A 54 (2021), 235204.</p> <p>[3] S. Lobb, F.W. Nijhoff, <em>Lagrangian multiforms and multidimensional consistency</em>, J. Phys. A42 (2009), 454013.</p> <p>[4] D.G. Sleigh, F.W. Nijhoff, V. Caudrelier, <em>Variational symmetries and Lagrangian multiforms</em>,  Lett. Math. Phys. 110 (2020), 805.</p>

<p>Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Applied Mathematics FT,</strong></em> in the research information section that the research degree you wish to be considered for is <em><strong>Quantisation of integrable field theories and lagranian multiform</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4011/dr-vincent-caudrelier">Vincent CAUDRELIER</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong></em>.</p> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p style="margin-bottom:11px"><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered after the closing date.  Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p class="MsoNoSpacing"><strong>Please note that you must provide the following documents in support of your application by the closing date of Monday 3 March 2025 if applying for Leeds Doctoral Scholarship or Tuesday 1 April 2025 for Leeds Opportunity Research Scholarship.</strong></p> <p><strong>If you are applying for the School of Mathematics Scholarship 2025/26, or with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<p style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong> – The <a href="http://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> <strong>(closing date: Monday 3 March 2025)</strong>, <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: Tuesday 1 April 2025) </strong>and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to UK applicants.</p> <p><strong>Non-UK</strong> – <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to all International applicants.  The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> <strong>(now closed for October 2025 entry)</strong> is available to nationals of China. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£320 in Session 2024/25), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Important:</strong> Please note that that the award does <em><strong>not </strong></em>cover the costs associated with moving to the UK.  All such costs (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source. </p> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p> <p>For further information about this project, please contact Dr Vincent Caudrelier by email to <a href="mailto:v.caudrelier@leeds.ac.uk">v.caudrelier@leeds.ac.uk</a></p>