Mathematical Physics Research

The study of Mathematical Physics at Maynooth has a long and distinguished tradition. Modern theoretical physics and applied mathematics are exciting and dynamic and that excitement is reflected in the research projects which are pursued in the Department. Several of the permanent member of the Department holds a Research Associateship at the School of Theoretical Physics of the Dublin Institute of Advanced Studies, in whose scientific work they participate actively. In addition, the Department maintains strong, international contacts with major academic centres abroad.

Some of the research interests of the staff are detailed below. The individual staff pages may give further information about past and current research.

The main areas of research are nonlinear physics, at both the macroscopic and microscopic level, condensed matter physics, physics of information and the study of the fundamental forces of nature. Detailed studies are undertaken to understand the dynamical behaviour of fundamental physical systems far from equilibrium, in quantum computing and information processing, nonlinear dynamics, condensed matter physics, classical and quantum chaos and the topological phases of matter.  A major research effort within the Department is to develop a better understanding of the fundamental forces of nature.  There are four basic forces: Gravity, Electromagnetism and the two nuclear forces, the Strong and the Weak nuclear forces.  The last three are believed to be well described by the mathematical framework of Relativistic Quantum Field Theory, while Gravity is so far understood only at the level of classical physics.  Both the gravitational force and the other three forces are described in terms of Gauge Field Models.  Most prominent amongst these is the Yang-Mills model with its rich geometrical and topological features, and plays a central role in current understanding of these forces. Detailed information on the Department's research activities is given below.

RESEARCH INTERESTS OF STAFF

Professor Brian Dolan

Research interests: relativistic quantum field theory, general relativity, quantum gravity and condensed matter.

From a theoretical point of view geometry has always provided an immensely fertile bridge between physics and mathematics, many mathematical concepts being motivated by physical considerations and vice versa. All four of the fundamental forces of Nature (Gravity, Electro-magnetism and two nuclear forces-the Strong and Weak nuclear forces) fit nicely into the mathematical framework of the geometry of curved spaces and gauge field theories. A quantum mechanical analysis of these theories however reveals profound differences, in particular between gravity and the three forces studied in modern particle physics-the electro-magnetic, the strong and the weak forces. The aim of my research programme is to develop a deeper understanding of these fundamental physical theories and of the unifying role played by geometry and symmetry, both in classical and in quantum physics but especially in the latter. The physical and mathematical techniques involved are applicable to many other areas of physics too, such as solid state, low temperature physics and the theory of condensed matter.

Current research includes:

  1. Duality symmetries. Physical phenomena change under variations in scale (the renormalisation group), for example the parameters that determine the strength of the fundamental forces change with the energy scale at which they are measured. Discrete symmetries, sometimes called duality, can play a very important role here, for example they can be used to obtain information on the different possible phases of supersymmetric YangMills theories and in the quantum Hall effect - two apparently very different phenomena which show a remarkable similarity in terms of the structure of their different phases and their renormalisation group flow. Symmetries can be used to gain deep insights into the way in which parameters change under changes in energy scale and this is one aspect of research.
  2. Non-commutative geometry. A new technique for analysing quantum field theories is under investigation, that of non-commutative geometry in which the space-time continuum is replaced by a discrete set of points in an abstract 'fuzzy' space-in a way specially designed to preserve all of the symmetries of the theory. This has applications in renormalisation group theory and to the problem of quantum gravity.
  3. Chaotic renormalisation group flow. In some circumstances the parameters of a physical theory change in a chaotic manner when the length scale or the energy scale is changed and this phenomenon is also currently under investigation.

Dr. Mikael Fremling

My research has so far been focused on strongly correlated electron systems, especially those that arise in two dimensional election gases subject to strong magnetic fields, giving rise to the fractional quantum Hall effect. These strongly correlated systems are interesting in their own right as the fundamental excitations are neither fermions nor bosons, but something in between - anyons. On top of that, these systems offer a possibility to realize quantum computing devices where one utilized the power of quantum mechanics to make the quantum computer faster that an ordinary computer.

I study the above mentioned systems using a wide range of numerical and analytical techniques.

Dr. Masud Haque

My research area is condensed matter theory - the study of coherence and correlations in many-particle systems.

Matter, although composed of the same constituents, can exist in a vast number of possible states. This diversity of states of matter is already manifested in the familiar differences between solids, liquids and gases, and is further revealed in measurements on low-temperature materials. The emergence of new states and collective properties in assemblies of matter is the subject of condensed matter physics. From superconductivity to fractionalized charges, many-particle physics continues to confront us with fascinating new behaviours that appear due to purely collective effects.

My current interest is the dynamics of quantum systems out of equilibrium, which is widely appreciated as a fundamental challenge facing theoretical physics today. Using a variety of methods and approximation tools, I examine (by myself or with various combinations of collaborators) the time evolution of many-body systems that are kicked away from equilibrium or are driven in some way. Many fascinating questions pose themselves in the study of non-equilibrium systems. For example, do segments of a time-evolving system approach thermal equilibrium in some sense, even in the absence of thermal baths? How well does the notion of an adiabatic change describe reality, when changes occur at finite rather than infinitesimal speeds? How does transport look like in the microscopic world in truly quantum (unitary, non-dissipative) situations? How does the spectrum of the many-particle system affect dynamics, and are high-energy eigenstates really very different from low-energy eigenstates? Are there quantum dynamical phenomena that mean field theories are fundamentally incapable of describing? The questions emerging are numerous, some of them are being addressed in completely novel contexts, and many of these issues are of experimental as well as of fundamental interest. Perhaps not surprisingly, non-equilibrium quantum dynamics is currently one of the fastest growing fields in theoretical physics.

Professor emeritus Daniel M. Heffernan

The main area of research is nonlinear science.  Detailed work is being undertaken to quantify and characterize the fractal structures that occur in nature and in the phase space of dynamical systems.  The problem is a fundamental and universal one underlying many areas of physics, such as diffusion limited aggregation, percolating clusters, neural networks and turbulence in fluids.  The question of the existence of a quantum analogue of classical chaos is fundamental and is a major preoccupation of our research group.  Computational and theoretical techniques have been developed and applied to obtain an understanding of the physics of these systems.  Some of the projects currently under study by the group are:

  1. The development and utilization of generalised dimensional and f(α) spectral techniques for the study of the dynamics of physical systems.  Detailed studies were completed of the formation and evolution of multifractal structures in some simple nonlinear dynamical systems.  This is part of a major programme to develop a statistical-thermodynamic approach to the characterization and elucidation of the structural and dynamical properties of nonlinear systems, particularly spatio-temporal chaos.
  2. The detailed study of the classical and quantum chaos. In particular a fundamental and detailed programme is underway to study the nonlinear dynamics of low dimensional mesoscopic systems and the study of chaos in atomic spectra and structure.  The application of chaos theory for nonlinear control and pattern recognition was systematically investigated.
  3. The physics, both classical and quantum, of external cavity lasers and low dimensional systems.

Dr. Graham Kells

The main focus of my most recent research has been on topological superconductivity which is related efforts to isolate and manipulate Majorana fermions in proximity coupled semiconductor nanowires. Previously I have explored topological spin lattice models and their connections to topological superconductivity and stabiliser codes. I also have an interest in designing exact diagonalization algorithms for distributed high performance computing architectures and in  quantum resonance effects in driven chaotic systems.

Professor emeritus Charles Nash

Professor Nash's research is in relativistic quantum mechanical systems known as quantum field theories using both analytical and topological techniques.

It is a feature of the past twenty five years of research in quantum field theory that the way forward in many problems is considerably illuminated if, in addition to analysis, one uses topology.

The famous Yang-Mills equations, invented by theoretical physicists, have also been the source of many new results in mathematics these being mainly in three and four dimensional differential geometry and topology. Thus this is a rich field with many open problems and interconnections with other disciplines.

The analytical and topological aspects can be combined in a geometric study of the renormalisation group and this is also under study.

A key recent topic of mine is an investigation of quantum field theory models on the lattice in order to shed insight into suitable approximation methods which may be tractable for basic calculations. This has resulted so far in results relating the topology of the continuum quantum field situation with the lattice combinatorial one. This rather like the relation between the combinatorial formulation of cohomology and the continuum de Rham formulation using calculus (i.e. differential forms). There are interesting new, and calculable, features of the energy momentum tensors of such models and links with the modular invariance of string theory.  

Dr. Jon-Ivar Skullerud

My research is concentrated on understanding the strong interactions, the interactions of quarks and gluons. In ordinary conditions, quarks and gluons can never exist as free particles, but are instead confined within composite particles such as protons and neutrons. At extremely high temperatures or densities, which existed in the early universe and may exist in the cores of neutron stars, it is predicted that quarks and gluons can be liberated, forming new states of matter called the quark-gluon plasma or quark matter.

It is only at extremely high energies that standard perturbative methods can be applied to the strong interactions. At low and intermediate energies, the theory can be defined by discretising space and time to form a lattice, and studied by computer simulations.

Confinement of quarks and gluons is still far from understood. One way of approaching this issue is by studying the properties of the quarks and gluons themselves and how they couple to each other at different energy and momentum scales. Signatures of confinement will show up in the infrared régime or quark and gluon correlators. The same forces that produce confinement also give rise to about 98% of the mass of everything we see around us.

Experiments at Brookhaven and CERN are currently trying to produce the quark-gluon plasma and study its properties. Major theoretical issues surrounding this include: at what temperature does the deconfinement transition happen, what is the energy density, pressure and viscosity of the plasma, which bound states survive, and what are the properties of the deconfined quarks and gluons.

At high density, quark matter may form a number of exotic superconducting phases. Lattice simulations encounter serious problems in this r&eamp;gime, but insight may be gained from similar theories which do not suffer these problems, such as Quantum Chromodynamics with 2 colours instead of 3. Results from such simulations may also be used as input or controls for model calculations.

Dr. Joost Slingerland

My interests lie in condensed matter physics, especially collective phenomena in systems with many particles which are often strongly interacting. In particular, I work on so-called topological phases of matter and on their application to quantum computation, an approach to computing which makes full use of the power of quantum mechanics. Topological phases occur in quasi two-dimensional systems that can harbor anyons - quasiparticles with exotic exchange properties, different from those of the well known matter and messenger particles (fermions and bosons). I study the general properties of topological phases (using topological field theory) as well as models of systems with anyons and experimentally accessible systems such as the two-dimensional electron liquids of the fractional quantum Hall effect.

Dr. Jiri Vala

My research interests lie in low-dimensional quantum theory, and include specifically investigation of topological, conformal and supersymmetric theories and their mutual relations, and exploration of their microscopic formulations and physical applications.

Our particular attention is dedicated to realization of topological quantum field theories as topological phases in two-dimensional quantum lattice and condensed matter systems. The lattice models provide important microscopic platforms to study the structure of topological and other exotic phases and their properties and stability under various effects, including disorder and decoherence, and also to investigate various phenomena, like for example bulk-edge correspondence. The condense matter systems in addition provide topological materials and devices with important applications in topological quantum computation and information processing.

Dr. Paul Watts

Most of my research in mathematical physics has dealt with the application of the theory of groups and algebras to theoretical particle physics.

Basic to the understanding of modern particle physics is the realisation that if we require the universe to obey any symmetry principle then the fundamental particles that form the building blocks of the universe - quarks, electrons, photons, etc, - must live in particular representations of the associated symmetry groups. Put another way, if we expect the universe to look the same if we perform a "rotation" or "reflection" on it, then the fundamental particles must also "rotate" or "reflect" in very specific ways. I have looked into several areas where such symmetries are known to exist, and have tried to find ways to extend them consistently to larger objects in the hope that doing so will prove useful or illuminating (hopefully both, obviously!) These areas are:

  1. Using the language of Hopf algebras to extend a theory with a given "classical" symmetry group to a more general one which is symmetric under the action of a "quantum group" depending on a parameter q.
  2. Examining whether bosonic W-algebras (which arise as the symmetry algebras of certain conformal field theories) may, by the introduction of fermionic generators, be consistently extended to "super-W-algebras".
  3. Requiring that the bulk gravitational action for a braneworld respect the T-symmetry of the underlying metric, solving the equations of motion for such actions, and comparing the cosmological behaviour with that actually observed.

More recently, I have also been interested in answering a question arising out of linguistics: given a document, how could you determine if it was randomly-generated or not, and if the latter, is there any way of determining what topic(s) it might be about? Although this seems like a very specific question, it can be generalised to a set-theoretic problem where set elements may have differing frequencies, and so choosing any W objects at random will not, in general, give the same probability of producing any given set of size W. This more abstract formulation can then be useful not only for the original linguistic question, but similar problems in genetics, seismology, Internet traffic or any case in which the objects of interest have a power-law distribution (Zipf's law, in the case of mathematical linguistics).