If you are fascinated by the mathematical beauty of physical theories, we invite you to apply to join our group as a graduate student and share our excitement.

If you would like to know how to apply for admission as a graduate student you should check out the main Mathematics PhD page where you can find a link to the online application form. There are various possibilities of funding including EPSRC, PPARC and University scholarships.

For you, more important than these formal arrangements will be the personal interactions within our group. You will have plenty of opportunity for informal conversations during our morning or afternoon **coffee breaks**. These take place at 11:00 and 16:00 in G/109, a room with **Physics Seminar**comfortable chairs but also with a blackboard if you feel the urge to discuss research.

Twice a week, on Tuesday lunchtimes (14:00) and Thursday afternoons (16:15), we hold our very relaxed **seminars** in the cofffee room. We often have outside speakers on Thursdays but there remain plenty of free slots for local speakers and everyone can make suggestions for topics. The seminars are in the "Russian style", which means that the audience can interrupt the speaker with questions and often even longer discussions. It can happen that people from the audience come to the blackboard themselves to discuss a point. And the seminars are open ended; people simply leave when they have heard enough.

# PhD projects

## Integrable field theory

### Quantum and classical integrable field theory (*Ed Corrigan*)

Ed Corrigan has been investigating the properties of defects and boundaries in integrable (but not conformal) field theories with a view to classifying all possibilities. This is an extensive project with many facets. Some of the ideas and techniques are algebraic in flavour, others analytical, but most are novel and there are a number of areas suitable for development by a PhD student.

### Quantum integrability (*Niall MacKay*)

Niall MacKay works principally on algebraic aspects of quantum integrability, and is happy to supervise students on the hidden symmetry algebras (typically Yangians and other 'quantum groups') of integrable models, their representations and particle content. Most projects are suitable for students with an MSc-level background in theoretical physics (including strings, supersymmetry, QFT etc.), but some are more algebraic and may be suitable for pure mathematicians with an interest in mathematical physics. Past and current students have worked on: boundary scattering in AdS/CFT; the Yang-Baxter equation and invariant tensors of exceptional Lie groups; twist-deformed manifolds and cosmology; conserved charges in supergroup sigma models; boundary scattering in principal chiral models; quantum affine Toda solitons.

Niall has also recently developed an interest in the mathematical modelling of warfare, and has published papers on Lanchester theory, especially in its historical context. Possible research projects would be in this area and in the mathematical modelling of counter-insurgent warfare.

## Quantum field theory and quantum gravitation

### Rigorous quantum field theory *(Henning Bostelmann)*

Henning Bostelmann is interested in the mathematical foundations of physics; specifically, of relativistic quantum physics (quantum field theory). While quantum field theory is a well established topic since more than 50 years, its mathematically rigorous description, in particular of its non-perturbative aspects, is still largely incomplete. He analyses these questions using advanced methods of functional analysis (operator algebras).

Topics that he is working on, and has worked on, include: the short-distance scaling limit of quantum field theories; the relation between pointlike observables (quantum fields) and bounded observables in open regions; the covariant formulation of these for quantum theories on curved spacetime, i.e., in a gravitational background.

### Quantum Field theory in curved spacetimes *(Chris Fewster)*

Chris Fewster works on the mathematically rigorous formulation of quantum field theory in curved spacetimes (QFT in CST). In this theory, the propagation of the quantum fields is affected by the curvature of spacetime, but (usually) one neglects the "back-reaction" effect of the quantum fields on the spacetime. He has supervised seven PhD students in this area, with two main foci:

**Quantum energy inequalities** Unlike most classical forms of matter, quantum fields can have local energy densities that are negative. However, quantum field theory contains mechanisms (deeply connected to the uncertainty principle) that result in the energy density not being too negative on average. These mechanisms are expressed by results called Quantum Energy Inequalities (QEIs). Chris Fewster has supervised a number of PhD students on these questions. Simon Dawson (PhD 2006) investigated numerical QEIs and also produced the first closed-form expression for QEIs for the Dirac field in CST. Calvin Smith (PhD 2006) obtained the first "absolute" QEIs in CST, while Lutz Osterbrink (PhD 2007) found the first QEIs applicable to the nonminimally coupled scalar field. An introduction to QEIs and some of the related techniques can be found in Chris Fewster's lecture notes.

**Locally covariant QFT **Quantum field theory in Minkowski space depends in many ways on the high degree of spacetime symmetry. General curved spacetimes lack any symmetry at all, which makes it hard to prove general statements about general quantum field theories (as opposed to specific models). A major development was the introduction by Brunetti, Fredenhagen and Verch of a locally covariant formulation of QFT in CST based on techniques of category theory. Chris Fewster's former students Ko Sanders (PhD 2008), Matthew Ferguson (PhD 2013) and Benjamin Lang (PhD 2015) have worked on various aspects of locally covariant QFT, including the proof of a Reeh-Schlieder theorem and the formulation of the Dirac field (Sanders), the dynamical locality of the extension of scalar field theory to incorporate Wick polynomials (Ferguson), and an investigation of dynamical locality and related issues for electromagnetism (Lang).

**Hadamard states** Quantum field theory admits a very large space of states, most of which are unphysical. The class of Hadamard states is a well-studied set of physical states, which play an important role in QFT in CST. With Benjamin Lang, Chris Fewster has recently given a new construction of a class of Hadamard states for the Dirac field in certain spacetimes. A number of open questions remain in this area and are being pursued by current students Francis Wingham and Michael Kiss (both co-supervised with Kasia Rejzner).

In addition, Chris Fewster is interested in other aspects of QFT in flat and curved spacetime. Recently, with David Hunt (PhD 2012) he studied the quantization of linearized gravity on background spacetimes solving the Einstein equations with cosmological constant, thus making contact with Atsushi Higuchi's work (see elsewhere on this page).

### Quantum Field theory and Quantum Gravity *(Kasia Rejzner)*

Kasia Rejzner is interested in mathematical structures relevant for quantum field theory (QFT) and quantum gravity. In particular, she works on problems involving QFT on curved spacetimes, using the locally covariant setting proposed by Brunetti, Fredenhagen and Verch (CMP 2003). This setting can also be extended to include effective quantum gravity models. Kasia is also interested in abstract algebraic and analytical structures appearing in renormalization. Possible project topics could focus on:

*Mathematical structures in renormalization* Renormalization is a powerful set of tools used in Quantum Field Theory to construct interacting models (perturbatively). Usually, it is associated with the idea of "removing the divergences," but there are also more rigorous formulations of the renormalization problem that allow to see renormalization as a mathematically well-defined procedure. One of such approaches is the Epstein-Glaser renormalization scheme, which recently has been successfully applied in QFT on curved spacetimes. Potential PhD projects would aim at investigating abstract mathematical structures appearing in this renormalization scheme, or at applying this scheme in interesting physical examples.

*Quantum gravity and cosmology* Quantizing gravity is one of the greatest challenges of modern theoretical physics. At the moment, the full theory is not known and there are several competing approaches to finding it. In a recent paper of R. Brunetti, K. Fredenhagen and K. Rejzner (2013) it was shown that a perturbative model of effective quantum gravity can be constructed using the framework of locally covariant QFT (formerly used only for QFT on curved spacetimes). This opens a way for applications in cosmology and black hole physics. Potential PhD projects would involve constructing models that allow to compute quantum gravity corrections to known physical processes.

### Quantization *(Eli Hawkins)*

Eli Hawkins has interests ranging from mathematical physics to pure mathematics, and would be happy to supervise PhD projects on any of these topics. The main theme of his research is the relationship between classical (geometric) structures and quantum (noncommutative algebraic) structures. His recent interests include algebraic quantum field theory, operads, and Kontsevich formality. His previous student, James Waldron (PhD 2015) studied Lie algebroids over differentiable stacks. Some specific possible project topics are:

*Kaluza-Klein without extra dimensions* A Kaluza-Klein model is a type of unified field theory that seeks to unify gravitation with electromagnetism and Yang-Mills interactions by supposing that spacetime is more than 4-dimensional. In the mathematical framework of Lie algebroids, it is possible to formulate a similar type of model without resorting to extra dimensions. The project would be to investigate this class of models.

*Obstructions to quantization* When does there exist a continuously parametrized strict deformation quantization of a Poisson manifold? The project would be to extend results for the 2-dimensional sphere to arbitrary Poisson manifolds. This will require studying the stability of Poisson maps and Lie algebroid homomorphisms.

*Geometric quantization of symplectic orbifolds* An orbifold is a type of "singular manifold" that looks locally like the quotient of a manifold by a finite group. Standard geometric quantization provides a recipe for constructing a Hilbert space from a manifold with some additional structure; this can be used to construct example of strict deformation quantization. The project would be to extend geometric quantization to orbifolds.

*Discrete models of quantum field theory* Quantum field theory is difficult – in part because it involves infinitely many infinite-dimensional algebras. If spacetime didn't have so many points, then it might be easier. Discrete versions of spacetime (such as causal sets) have been seriously considered as models for the universe. However, the idea of this project is to use discrete spacetime as a setting for "toy" models of classical and quantum field theory.

### Quantum field theory: semi-classical limit *(Atsushi Higuchi)*

Classical and quantum electrodynamics are very old subjects and have been studied extensively. However, it has not been clear how the former is obtained as a classical limit of the latter in radiation processes. Working with a student, Giles Martin (PhD, 2008), he established that the classical back-reaction force on a radiating charged particle can be obtained as the classical limit from quantum electrodynamics (QED). He and Phil Walker (PhD, 2011) have calculated a quantum correction to the classical Larmor formula for the energy emitted from an accelerated charged particle. He is planning to extend his work on the semi-classical approximation in quantum field theory to radiation processes in curved spacetime and also apply it to gravitational radiation.

### Quantum field theory in de Sitter spacetime *(Atsushi Higuchi)*

Quantum field theory in de Sitter spacetime, which is relevant to inflationary cosmology, is believed to solve various puzzles in the big-bang cosmology. Atsushi Higuchi has supervised four PhD students in this area. With Spyros Kouris (PhD, 2003) he worked on constructing the two-parameter family of covariant propagators for gravitational waves, which will be useful in studying the gravitational fluctuation in this spacetime. With Richard Weeks (PhD, 2005) he found that the long-distance quantum correlation of quantum gravitational fluctuation is weaker than some physicists have suspected. With Yen Cheong Lee (PhD, 2010) he demonstrated how the retarded Green’s function can be used in de Sitter spacetime – there had been some confusion about this in the literature – and started investigating interacting field theory in this spacetime. With Mir Faizal (PhD, 2011) he worked on some aspects of interacting quantum gauge theory and perturbative quantum gravity. He is planning to work further on various aspects of interacting quantum field theory in de Sitter spacetime in the next few years.

## Quantum Information and Foundations

### Quantum mechanics and quantum information *(Stefan Weigert)*

You will have the opportunity to investigate topics in non-relativistic quantum mechanics with Stefan Weigert. The problems he is interested in are often motivated by questions currently addressed in quantum information.

In recent years, the properties of so-called mutually unbiased bases have attracted considerable interest in the quantum information community. Expressing complementarity of variables in quantum mechanics, they naturally give rise to fundamental questions about finite quantum mechanical systems, i.e. quantum systems accomodating a finite number of states. Recent PhD projects in this field have been based on analytic, numerical and computer-aided calculations.

### Foundations of quantum theory and quantum information *(Paul Busch)*

Paul Busch's main research interest is the development of the operational tools of quantum measurement theory and their application to the solution of conceptual problems and the modeling of practical measurement schemes. One strand of current work is concerned with structural aspects of finite-dimensional quantum systems (qudits), specifically the search for measures of approximation and degrees of unsharpness of quantum observables. Possible topics for PhD projects include the question of the connection between approximate joint measurability and approximate quantum cloning and the implications of these and other quantum measurement limitations for quantum information processing. Another possible project area is that of relativistic quantum measurements (covariant collapse, localisation vs. causality).

### Quantum information theory and the connection to quantum foundations *(Roger Colbeck)*

Roger Colbeck works on quantum information theory and quantum foundations. Within quantum information theory, he works on quantum cryptography and quantum random number generation and the use of quantum and relativistic protocols for secure computations. He also works on new uncertainty relations formulated in terms of entropy. Much of his work connects quantum information and the foundations of quantum theory, for example, he has done work on the question of whether the world is random, and whether there could be theories that improve on the predictions of quantum mechanics. There are many exciting directions this work could go in, and prospective PhD students should indicate any preferences they may have on their application form.

This year there are several additional opportunities in York created by the York centre for Quantum Technologies and the Quantum Communications hub. See here for more information.