Physicalia Mag. 27 (2005) 2 pp 166-171

RESEARCH ACTIVITIES IN MATHEMATICAL PHYSICS IN BELGIUM J.-P. Antoine

Institut de Physique Théorique, Université Catholique de Louvain B – 1348 Louvain-la-Neuve

Before going into details, a word of warning is in order. Mathematical Physics is an ill-defined concept, various interpretations are possible, ranging from the very narrow to the broadest sense – and the boundary with other fields is fuzzy. In this survey, we take the widest possible definition: Mathematical Physics is present whenever a physical problem is studied with mathematically rigorous methods, thus it comprises various forms of applications, to (applied) mathematics, physics, engineering, etc. We feel that these borderline themes are among the liveliest parts of research and should not be left out. We have chosen to list the various aspects of Mathematical Physics according to research themes, rather than by universities. The opposite choice could be made as well, but it would result in a much heavier presentation. The reason is that mathematical physics is extremely scattered among universities. Indeed, contrary to experimental physics or theoretical physics (particle, atomic, condensed matter physics), it is being practiced mostly by very small groups, sometimes single academics with a couple of students, and a given group is often active in several research directions. This characteristic makes mathematical physics closer in spirit to mathematics, although the importance of the physical content of a given research line makes the difference. For the convenience of the reader, we have listed at the end the web sites of the research groups quoted in the text. For greater clarity, the research themes have been regrouped into four categories, namely (1) Statistical physics; (2) Nonrelativistic quantum theories; (3) Quantum field theory and its generalizations; and (4) Non quantum aspects. 1. Statistical physics (classical and quantum) Statistical physics is a traditional domain for Belgian mathematical physicists. A more precise terminology would be “statistical physics of equilibrium and non-equilibrium systems, both classical and quantum”. This covers traditional topics like irreversibility, large deviation theorems, quantum fluctuations (general theory and applications), which are all actively pursued (ULB-1, KUL-1). Occasionally attention is given to some aspects of the foundations of classical and quantum statistical mechanics (UA-1). Besides, more specialized topics are also being investigated. For instance, the representation of canonical commutation and anti-commutation relations, models of BoseEinstein condensation and superconductivity, biophysical modeling (KUL-1), or statistical 166

mechanics of small systems (UA-1). A recurrent theme is the investigation of the microscopic foundations of thermodynamics (KUL-1). A nice example is a microscopic derivation of the Fourier law of heat conduction : starting from microscopic models of interacting anharmonic oscillators, one tries to prove the existence of stationary nonequilibrium states exhibiting the Fourier law (UCL-1). Another topic currently under investigation is related to phase transitions. Namely, it is well known that weakly coupled piecewise expanding maps have a unique SRB (SinaiRuelle-Bowen) measure. However, it is not clear that such a system can undergo phase transitions - non-uniqueness of the SRB measure - when the coupling becomes strong. It has been shown that phase transitions do occur for a given class of piecewise expanding maps, but the kind of coupling considered is clearly unnatural. The aim of the current research is to find a more general result, proving the existence of a phase transition for a more physical class of systems (UCL-1). A very active domain concerns surface and interface phenomena, more precisely, physico-chemical processes in the neighborhood of interfaces between two fluids or between a fluid and a solid surface. This is an interdisciplinary analysis involving physicists, chemists, mathematicians and computer scientists (UMH-1), working with various techniques for modification of surface properties, for experimental characterization of solid surfaces, and for analysis of fluid/fluid and fluid/solid interfaces. The research themes involved here concern surface treatment, wetting and surface phenomena, molecular modeling, biosensors, nanotechnologies, image treatment and analysis, both at the fundamental level (modeling) and in practical applications, including industrial valorization in the fields of medical imaging, computing and biotechnology. Finally, we conclude with two-dimensional conformal field theory. This yields an appropriate transition to the next item, since here statistical mechanics (equilibrium critical phenomena) meets field theory. Sandpile models, in particular, are under scrutiny (UCL-1). This is a special class of open dynamical systems that exhibit a wide variety of critical behaviors and have a faithful description in terms of the so-called logarithmic conformal field theories.

2. Classical and quantum field theory QFT nowadays takes various forms, all more or less linked with physics beyond the Standard Model of fundamental interactions. Many research directions are being investigated and most Belgian universities are involved. The research themes overlap with each other, and also with the physics of fundamental interactions (or elementary particles). Although the ultimate goal of post-Standard model physics is the unification of the gravitational interaction with the three other ones, the former is still an object of intensive study. The paradigm, of course, is Einstein’s General Relativity, but several generalizations are actively studied, for instance, the scalar-tensor theories of gravitation. The hot topic is cosmology, i.e., the study of the Universe as a whole and its time evolution. In this context, one finds several groups analyzing Black holes and their attendant paradoxes (in 3 and 4 dimensions), Belinsky-Khalatnikov-Lifshitz asymptotic analysis and cosmology, or the

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structure of the Cosmic Microwave Background. In the latter case, the crucial question is the (non)Gaussianity of fluctuations; this is being analyzed with novel techniques of image analysis such as spherical wavelets (UCL-1, ULB-2, UMH-2). Gauge theories occupy a special position, since the proof of their renormalizability by ‘t Hooft and Veltman, and accordingly many aspects are being studied, again with large overlap between them. Clearly the presence of symmetries is crucial for the construction of new models, a prime example being quantum field theory models with BRST symmetry (ULB-2, UMH-2). Other topics are being studied as well, like, for instance, algebraic aspects of renormalization and anomalies; quantization of systems with constraints; generalized duality, of the type electric-magnetic, for nonabelian gauge theories; extensions of gauge theories to non-semisimple gauge groups; classical solutions of gauge theories, in particular for Einstein-Yang-Mills equations and Einstein-non-linear-sigma-models equations; black holes and soliton type solutions; or self-dual equations in space-time of dimension larger than four (ULB-2, UMH-2, UMH-3). As said above, the unification of all four fundamental interactions is the ultimate goal of today’s physics, and the best candidate for quantizing General Relativity. This trend has given rise to a large number of new approaches, characterized in general by their heavy mathematical content (and also, unfortunately, by the lack of experimental justification). The main examples are supersymmetry (combining bosons and fermions in a unified formalism), quantum gravity, superstring theory, M-theory and the underlying supergravity theories, hidden symmetries (Lorentzian Kac-Moody algebras), noncommutative geometry, all under study in Belgian universities. This is clearly a borderline case, between mathematical physics, mathematics, and also elementary particle physics. It is also the latest manifestation of the Holy Grail of physics, the “theory of everything” (KUL-1, UCL-2, ULB-2, UMH-2, VUB-1). For the sake of completeness, one should mention here also the interface with elementary particle physics. This is more traditional and in general not counted as a part of mathematical physics. Yet in certain cases, mathematical techniques are being developed, upon motivation from such applications in physics as the classification and the interactions of fundamental particles, the application of group theory to the theory of fundamental interactions, the applications of supersymmetric algebras to the classification of algebras with spin higher than two and/or half integer spin, or the relations with supersymmetric partners of fundamental particles (UCL-1, UCL-2, UMH-3). On the other hand, good old QFT is still a domain of interest and some work is devoted to it (UA-1).

3. Nonrelativistic quantum theories Besides the relativistic theories discussed in the previous section, a number of research topics, rather disconnected from each other, pertain to the nonrelativistic realm. One of the oldest problems in quantum theories is to find a consistent quantization scheme. Besides the standard textbook approach of Schrödinger, various alternative techniques of quantization are investigated, such as star-products, Moyal quantization, or coherent states. This is again on the borderline between physics and mathematics, and indeed most of the people involved in this domain are mathematicians, essentially differential

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geometers. One should also mention the fact that few concrete situations have been amenable to these methods, while many abstract results have been obtained (UCL-3, ULB-3, UMH-3). Whereas coherent states (CS) yield a particular form of quantization (initiated by Berezin), they are interesting objects by themselves, both mathematically and physically. The former aspect has been mostly treated in the context of quantization, but several physical aspects and applications are also under scrutiny, such as CS for exactly solvable potentials, connection with quantum optics, algebraic construction of CS, CS for finite dimensional systems, etc. (UCL-1). More generally, nonrelativistic quantum mechanics still generates a number of mathematical methods that are of intrinsic interest and susceptible of far reaching generalizations. For instance, application of Lie groups and Lie algebras in microscopic physics is an old story, but it is being rejuvenated by considering instead their deformations, namely, quantum groups and algebras. Here all sorts of standard problems can be reformulated, such as classification issues, representations, CS, etc. (UG-1, ULB-4, VUB-2). Another extension of standard problems is the study of inverse problems by supersymmetric transformations for nuclear and atomic systems (ULB-5). On the other side, physical concepts also get generalized. A striking example is the current explosion of quantum information theory. Phenomena like state entanglement, teleportation, etc. are being actively explored (KUL-1, ULB-6, VUB-2). Besides group-theoretical techniques, methods of functional analysis find new applications in quantum theories. Here we are clearly at the interface between mathematics and physics, and sometimes one should rather speak of physical mathematics rather than mathematical physics! We may quote singular interactions in quantum mechanics (e.g. deltapotentials or surface potentials), the study of non-self-adjoint operators with partially real spectrum, and their relations with self-adjoint Hamiltonians, solvable and partially solvable quantum lattice solitons (UCL-1, UCL-3, UMH-3). A special case is the development and application of (partial) operator algebras. The idea is to find a suitable generalization to unbounded operators of well-known objects like C*-algebras and von Neumann algebras, and then explore the abstract algebraic structure underlying them. Applications of such techniques aim at quantum field theory or statistical physics of systems with long-range interactions (some spin systems, for instance). But it is fair to say that the momentum is almost exclusively on the mathematical side, where a considerable mount of results has been obtained in the last years (UCL-1).

4. Non-quantum aspects In addition to quantum theories, there are a number of research avenues in mathematical physics that pertain more to classical physics, sometimes with a strong engineering or mathematical flavor. A recent and spectacular example is wavelet analysis. Wavelets were originally designed as a particular case of CS, namely those associated with the similitude group of the line or the plane. As such they belong to the class of time-frequency methods used in signal and image processing, and they have become a standard tool, in particular among engineers.

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An alternative view is to consider them as the exploitation of ideas from quantum measurement in the essentially classical realm of signal processing. A characteristic aspect is that they have at the same time a close connection with (pure and applied) mathematics and with concrete (engineering) applications. One-dimensional wavelet analysis finds applications in various domains of signal processing, such as medical signal processing (NMR), statistics, finance, seismology, etc., but it also influences several topics in mathematics, such as approximation theory, resolution of PDEs, etc. As a result, many groups exploit and develop this technique (KUL-2, UG-2, UCL1, ULB-5, ULg) The 2-D version applies primarily in image analysis, including tomography and medical imaging. More recently, wavelet analysis has been extended to the 2-sphere, with applications in geophysics or cosmology (analysis of the Cosmic Microwave Background radiation, as discussed above) (UA-2, UCL-1, ULB-5, UMH-1). Inverse problems are another topic of wide interest, since very often the physicist seeks to retrieve information on a source from observed data. Examples are scattering processes, astronomy, optics, microscopy, medical imaging. Analytical or numerical methods for such data processing are being investigated in several groups (ULB-5, UMH-1, VUB-2). Another borderline case with mathematics is soliton theory and its extensions. Whole classes of nonlinear partial differential equations, or hierarchies of equations, exhibit solutions of the soliton type, that is, localized and stable solutions with a number of striking properties (nonlinear superposition principle, infinitely many conservation laws, asymptotic particle-like behavior, etc.). These equations mostly describe nonlinear wave phenomena, but the subject extends to integrable Hamiltonian systems with infinitely many degrees of freedom, and specific methods are developed for this (tau functions, cellular automata, etc.). Besides, there are also attempts to integrate Hamiltonian systems with a finite number of degrees of freedom (e.g. construction of explicit solutions to Hénon-Heiles-like systems) (UG-1, VUB-2). The game is mostly to find new equations or new systems of that type and to analyze soliton-like solutions for known equations, including the development of new mathematical tools. For instance, these questions often lead to the study of infinite dimensional Lie algebras (Kac-Moody algebras). Clearly such work involves close contact with mathematicians . From the more physical point of view, one should note also the search of exact analytical solutions to physically challenging 'partially' integrable nonlinear PDEs (such as Ginzburg-Landau and Kuramoto-Sivashinsky) (UCL-3, VUB-2). Finally, mathematical physics has found its way into financial matters. Namely, some physical theories (in particular, auto-organized models) are being applied to the time evolution of interest rates in theoretical finance. More generally, time series analysis, including with wavelets, is relevant in macroeconomy (ULB-5, UMH-3) Conclusion From this survey, one may conclude that mathematical physics is alive and well, and Belgian groups are very active on the national and the international scene. But it has a distinct flavor that sets it apart. On one hand, as indicated in the beginning, it is spread among many little groups, which tend to interact with each other. But, conversely, it is often practiced in

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close contact with other scientists, mathematicians, engineers, particle physicists, etc. Thus there is teamwork, but mostly with heterogeneous teams.

Web addresses of the research groups mentioned in the text : KUL-1 : http://itf.fys.KUL.ac.be KUL-2 : http://www.cs.kuleuven.ac.be/cwis/research/nalag/ UG-1 : http://twiserv.ugent.be/ UG-2 : http://cage.ugent.be/clg/ UA-1 : http://nat-www.uia.ac.be/DptUA/Nederlands/Onderzoek/wisnat.html UA-2 : http://nat-www.uia.ac.be/DptUA/Nederlands/Onderzoek/visielab.html UCL-1 : http://www.fyma.ucl.ac.be UCL-2 : http://www.fynu.ucl.ac.be UCL-3 : http://www.math.ucl.ac.be ULB-1 : http://www.ulb.ac.be/rech/inventaire/unites/ULB164.html ULB-2 : http://www.ulb.ac.be/rech/inventaire/unites/ULB150.html ULB-3 : http://www.ulb.ac.be/rech/inventaire/unites/ULB175.html ULB-4 : http://www.ulb.ac.be/rech/inventaire/unites/ULB186.html ULB-5 : http://www.ulb.ac.be/rech/inventaire/unites/ULB178.html ULB-6 : http://www.ulb.ac.be/rech/inventaire/chercheurs/5/CH3685.html ULg : http://www.ulg.ac.be/sectmath/ UMH-1 : http://crmm.umh.ac.be/ UMH-2 : http://w3.umh.ac.be/metg/ UMH-3 : http://w3.umh.ac.be/pmt/compo.htm VUB-1 : http://tena4.vub.ac.be/ VUB-2 : http://tena4.vub.ac.be/nld/

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