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, 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

Prof. Vladimír Bužek

The research interests of Prof. V. Buzek can be divided into several mutually related groups:

(1) quantum information processing -- universal optimal manipulations with quantum information (e.g. universal quantum machines such as quantum cloners, or universal NOT gates)

(2) quantum state reconstruction -- reconstruction of states of quantum systems from incomplete data  (e.g. application of the principle of maximum entropy and the Bayesian quantum inference)

(3) dynamics of open quantum systems -- stochastic quantization, quantum decoherence, nonclassical effects in quantum optics, reconstruction of Liouvillian superoperators, description of dynamics of open systems from the point of view of quantum information theory.

(4) quantum entanglement in multi-particle systems -- generation of entanglement (non-classical correlations) in many body systems, utilization of multi-partite entanglement to communication protocols (e.g. quantum secret sharing).

(5) theoretical quantum optics -- various aspects of matter-light interactions and generation of non-classical states of light.

Dr. 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:

i) 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.

ii) 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.

iii) 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.

Professor 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(alpha) 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.

Professor Charles Nash

Professor Nash's research is in quantum field theories using both analytical and topological techniques.  Of interest on the analytical side is the renormalisation of quantum field theories and the relation of this to quark confinement and QCD.  The topological investigations centre on the properties of Yang-Mills gauge theories and the relation of these to quantum field theories in dimensions 2, 3 and 4. These involve investigations of classical solutions representing instantons, solitons and monopoles and relate to chiral anomalies, Donaldson invariants and Witten-Turaev-Viro invariants.  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.  

A development closely related to the above is that of the discovery by Witten of an Abelian monopole description of the 4-manifold polynomial invariants of Donaldson. This is also related to the new equations of Seiberg and Witten and this is under study at present. In particular some new results on the three dimensional Seiberg-Witten equations have been derived recently and are under active development.  Other separate work is a new project on non-commutative matrix models of Quantum field theories.

Dr. Jonivar 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 régime, 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.

Professor T. Tchrakian

Higher dimensional field theories, including dimensional descent. Abelian and non Abelian gauged Higgs and Skyrme field theories, including their gravitating cases. Special emphasis on higher curvature gravitational terms, as well as inclusion of (negative) cosmological constant. Specific problems tacked are listed:

Construction of a U(1) (Maxwell) gauged Skyrmion in 3+1 dimensions. This is a stable soliton when the electric potential vanishes, and is quasistable otherwise. In the second case we succeeded in finding a static spinning finite energy lump. This is the only such example that does not involve stationary fields.

Continued our study of the systems consisting of two distinct gravitating members of the Yang--Mills hierarchy, going beyond the first two members of the Yang--Mills hierarchies in dimensions d=6,..,12 extending our previous results restricted to the first two members of this hierarchy, and to dimensions d=6,..,8. Using fixed point analysis combined with numerical techniques, we established that the various solutions are characterised by three types of fixed points, at zero and infinity, Reissner-Nordström type, and what we called a 'conical', this last one being a feature exclusively in dimensions 5,9,... etc. In this study, we restricted to Einstein-Hilbert gravity, and to regular solutions only.

Again in this area, and restricting to Einstein-Hilbert gravity, we studied such systems in the presence of a negative cosmological constant. The results are qualitatively similar to those of the usual $3+1$ dimensional case, except that the peculiar accident in the latter case allowing finite mass solution with monopole like rather than sphaleron like asymptotic behaviour. Here, only the latter type exist, except in diemsnions where a Pontryagin charge is defined, when their asymptotic behaviour is instanton like.

Finally, again in this area, we studied higher diemensional gravitating Yang--Mills systems, but with the gravitational term being a member of the gravitational hierarchy (e.g. Gauss-Bonnet). It was shown that the qualitative properties of the usual 4 dimensional case repeated modulo 4p dimensions.

Constructed 'monopoles' in 4p-1 dimensions, employing the 2p-th members of the Yang--Mills hierarchy. The corresponding dyons were also discussed, and in a subclass of these models its existence is guaranteed.

This project is ongoing and at present we are gravitating these systems. It is planned also to apply these objects to brane theories.

Started on the construction of axially symmetric instantons in all even Euclidean dimensions, and of axially symmetric monopoles in all odd Euclidean dimensions. In the latter case we have also considered monopole antimonopoles, and monopole antimonopole chains.

Started the construction of axially symmetric instantons of the 4 dimensional Yang-Mills system, which unlike Witten's axially symmetric instantons are labeled by a voertex like charge n. This project is ongoing.

Dr. Jiri Vala

Dr. Vala's iterests are: topological phases of matter, topological quantum computation, theory of decoherence, quantum error correction and suppression and fault-tolerant quantum computation.