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Mathematical Physics – abstracts
Project Title Research Group
Quasi-classical approximation for BEC/BCS crossover phenomena in ultra-cold atom gases
Perturbative renormalization of non-linear sigma models

Non-linear sigma models in two dimensions share many features with gauge theories in four dimensions. Among these are dimensionless coupling, perturbative renormalizability, asymptotic freedom, and dynamical mass generation. Because of these similarities, non-linear sigma models are often studied as a two-dimensional laboratory for what might be happening with the gauge interactions of our four-dimensional world.
The present research internship focuses on a class of supersymmetric non-linear sigma models which play an important role as effective field theories for random matrix problems and disordered quantum systems. In the course of looking at these models as concrete and tractable examples, the research student will be introduced to Wilson's idea of the renormalization group - one of the corner stones and major tools of modern theoretical physics. By definition, non-linear sigma models are functional integrals of maps into a Riemannian symmetric target space. Thus they constitute a subclass of the more general class of "non-linear models" where the target space is any Riemannian manifold. For the broad class of non-linear models, the renormalization group flow (i.e., the change of coupling with decreasing momentum cutoff) has been worked out in a seminal paper of Friedan by perturbation theory up to two-loop order. Friedan's work gives an expansion for the renormalization group beta function in terms of the Riemann curvature tensor of the Riemannian target manifold. An analogous expansion exists in the case of the supersymmetric models describing disordered systems.
The sequence of tasks for this research internship is outlined as follows:
- To get started, do some reading in order to get acquainted with the field-theoretic notions of regularization and renormalization. In particular, get familiar with Friedan's general result for the loop expansion of the renormalization group beta function of non-linear sigma models [1].
- To prepare for the treatment of supersymmetric non-linear sigma models, study the introductory section of the foundational article on classification of Lie superalgebras by Kac [2]. In particular, learn about the basic concept of roots and root vectors of Lie algebras and Lie superalgebras.
- Friedan's formula expresses the two-loop beta function by curvature data of the Riemannian manifold. In the case of Riemannian symmetric spaces there exists a general expression for the Riemann curvature tensor in terms of the system of roots associated with the Cartan involution underlying the symmetric space. Read about this connection in Helgason's textbook [3] on the differential geometry of symmetric spaces. This finishes the reading part of the research internship.
- To begin the research part of the project, compute the root space decomposition for each of the ten large families of Riemannian symmetric superspaces. Then, use the information about the root system to compute the curvature data - in particular: the so-called Ricci curvature - for the ten large families. Finally, use the curvature data to compute the two-loop beta function. (The supervisor of the research internship will provide additional documentation to help you with performing these calculations.)
- Find out what these results mean in the context of disordered quantum systems. In particular, relate the results for the perturbative renormalization group beta function to the weak localization corrections and universal conductance fluctuations for the ten symmetry classes which are known to exist in disordered fermion systems [4].
- Write up the main facts and results which you learned/obtained in the course of doing this four-week research internship.
[1] Friedan, D.H.: Nonlinear models in 2+epsilon dimensions, Ann. Phys. 163 (1985) 318
[2] Kac, V.G.: Sketch of Lie super-algebra theory, Commun. Math. Phys. 53 (1977) 8
[3] Helgason, S.: Differential geometry and symmetric spaces (Academic Press, New York, 1962), Chapter IV, Theorem 4.2
[4] Heinzner, P., Huckleberry, A., Zirnbauer, M.R.: Symmetry classes of disordered fermions, Commun. Math. Phys. 257 (2005) 725

Statistical interpretation of black-hole entropy

According to Einstein's theory of general relativity, black holes are objects for which gravity is so strong that nothing - not even light - can escape. If, however, quantum theory is taken into account, black holes can emit thermal radiation, as discovered by Stephen Hawking in 1974. Since black holes thus possess a temperature, one can also attribute an entropy to them. While a definite expression for this entropy can be obtained from thermodynamic considerations, a statistical (microscopic) derivation in the sense of Boltzmann is only partly available. Such a derivation can only be done on the basis of a quantum theory of gravity, which so far is elusive. The task in this internship is to review the major attempts for the statistical interpretation and to present a critical examination of one particular derivation.

The problem of time in quantum gravity

Einstein's theory of general relativity and quantum theory contain drastically different concepts of time. Whereas time in quantum theory is absolute (non-dynamical), time in general relativity is part of spacetime describing gravity and is thus dynamical. A unification of quantum theory and general relativity into a quantum theory of gravity must thus have profound consequences for our concept of time - this is the "problem of time in quantum gravity". Although a full theory of quantum gravity does not yet exist in complete form, various approaches are available.
The task in this internship is to review the problem of time and to examine its consequences in detail for one particular approach to quantum gravity.