# Covariant quantisation of integrable field theories

PGR-P-1168

## Key facts

Type of research degree
PhD
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Dr Vincent Caudrelier
Schools
School of Mathematics
Research groups/institutes
Algebra, geometry and integrable systems, Applied Mathematics

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. Very recently, 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. This opens the way to investigate canonical quantization of integrable field theories in a covariant fashion, and compare with the well-established quantum Yang-Baxter approach. <br /> <br /> Another recent discovery in integrable field theories is the notion of Lagrangian multiforms which captures integrability in a Lagrangian framework. This also opens the possibility of covariant quantization but from another perspective: by applying Feynman's path integral ideas to Lagrangian multiforms. One of the beautiful connections made between the above two new approaches is that it has been shown that the classical r matrix also 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. In this project, you will contribute to the development of the existing results at the classical level in order to prepare the ground for tackling the quantization problem, using one or both of the methods mentioned above.

<p>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...). 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&#39;&#39;, 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 &quot;theoretical laboratories&quot; where one can obtain explicit and exact results to guide our understanding of intricate phenomena.&nbsp;</p> <p>Historically, the process of quantizing these integrable field theories 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: the classical and quantum Yang-Baxter equation (which led to V.G. Drinfel&#39;d winning the Fields medal for establishing the mathematical theory behind these equations). In turn, this has allowed for tremendous success in computing the all important correlation functions in a quantum field theory.&nbsp;</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&#39;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.&nbsp;</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)&nbsp;a covariant classical $r$-matrix formalism [1,2], 2)&nbsp;a Lagrangian formalism which captures integrability: Lagrangian multiform theory [3,4].&nbsp;</p> <p>Each aspect opens the way to tackle the exact quantization of an integrable field theory in a covariant way, either using ideas from covariant Hamiltonian theory to perform a canonical quantization based on the $r$-matrix or by using Feynman&#39;s path integral ideas based on the Lagrangian multiform formalism. 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.&nbsp;</p> <p>Applicants with a strong background in theoretical/mathematical physics are strongly 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]&nbsp;V. Caudrelier, M. Stoppato, <em>A connection between the classical r-matrix formalism and covariant Hamiltonian field theory</em>, J. Geom. Phys. 148&nbsp;(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&nbsp;(2021), 235204.</p> <p>[3] S. Lobb, F.W. Nijhoff, <em>Lagrangian multiforms and multidimensional consistency</em>, J. Phys. A42&nbsp;(2009), 454013.</p> <p>[4] D.G. Sleigh, F.W. Nijhoff, V. Caudrelier, <em>Variational symmetries and Lagrangian multiforms</em>, &nbsp;Lett. Math. Phys. 110&nbsp;(2020), 805.</p>

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University&#39;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> and in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Covariant quantisation of integrable field theories</strong></em>&nbsp;as well as&nbsp;<a href="https://eps.leeds.ac.uk/maths/staff/4011/dr-vincent-caudrelier">Vincent CAUDRELIER</a>&nbsp;as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;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 on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. &nbsp;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>Please note that you must provide the following documents at the point you submit your application:</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> <li>Funding information including any alternative sources of funding that you are applying for or if you are able to pay your own fees and maintenance</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.