PhD course: Topics in Mathematical Physics


Time: Wednesday May 2 and Thursday May 3, 2012.

Place: auditorium G5-109, Department of Mathematical Sciences, Aalborg University, Frederik Bajers Vej 7G, 9220 Aalborg, Denmark.

Contact person: Horia Cornean (

Lecturers: Jan Derezinski (Warsaw), Søren Fournais (Aarhus), Klaus Mølmer (Aarhus), Thomas Garm Pedersen (Aalborg) and Baptiste Savoie (Bucharest). PhD students must register here. Postdocs and/or other senior researchers are also more than welcomed. A short description of each lecture together with a tentative schedule can be found below.


Jan Derezinski (Wednesday and Thursday, both days from 10:30 to 12:00).

Title: Mathematical introduction to Quantum Electrodynamics

Abstract: QED is a very successful theory. However, it has its problems: full QED most probably exists only as a perturbative theory. In my lectures I will describe theories that are incomplete versions of QED. I will discuss photons in an external current and charged particles in an external electromagnetic potentials. These theories allow for a fully non-perturbative rigorous treatment. Many difficult ideas of full QED, such as the infrared problem, renormalization of the vacuum polarization, freedom of choice of the photon propagator, Ward identities, the wave function renormalization and the non-implementability of dynamics can be understood non-perturbatively on this level. Therefore, in my opinion, a discussion of these theories serves as a good introduction to the full QED. Here is the plan of the lectures:

1. Relativistic quantum physics (Minkowski space, Haag-Kastler and Wightman axioms as reasonable guiding principles, not as a dogma)

2. Neutral scalar bosons (Klein-Gordon equation and its quantization, linear and quadratic perturbations)

3. Massive vector bosons (quantization of the Proca equation in the presence of an external current)

4. Massless vector bosons (quantization of the Maxwell equation in the presence of an external current, infrared problem)

5.Charged scalar bosons (quantization of complex solutions of the Klein-Gordon equation in the presence of external potentials)

6. Dirac fermions (quantization of the Dirac equation in the presence of external potentials).


Thomas Garm Pedersen (Wednesday, 13:00-14:30)

Title: Optical properties of graphene

Abstract: Graphene is an exceptional two-dimensional material holding great technological promise. Moreover, it is a dream material for theoretical physics because of its simplicity. In the lecture, I will discuss the simple tight-binding model of graphene. This leads directly to the famous "2D massless Dirac fermion" picture for low-energy excitations describing electrons and holes as relativistic particles in a 2D world with effective speed of light c/300. Based on these results, I will discuss the optical properties with and without magnetic field. Finally, mechanisms of opening energy gaps in graphene will be mentioned. In the Dirac picture, this amounts to providing mass to the carriers. The implications for the optical response will be described.


1. T. G. Pedersen, C. Flindt, J. Pedersen, A-P. Jauho, N.A. Mortensen and K. Pedersen: Graphene antidot lattices - designed defects and spin qubits, Phys. Rev. Lett. 100, 136804 (2008)

2. T. G. Pedersen, C. Flindt, J. Pedersen, A-P. Jauho, N.A. Mortensen and K. Pedersen: Optical properties of graphene antidot lattices, Phys. Rev. B. 77, 245431 (2008)

3. T. G. Pedersen, A-P. Jauho, and K. Pedersen: Optical response and excitons in gapped graphene, Phys. Rev. B. 79, 113406 (2009)

4. J. G. Pedersen and T. G. Pedersen: Tight-binding study of the magneto-optical properties of gapped graphene, Phys. Rev. B. 84, 115424 (2011)


Baptiste Savoie (Wednesday 14:30-15:15)

Title: Some rigorous aspects on the orbital magnetism of Bloch electrons in solids

Abstract:We present a rigorous mathematical treatment of the zero-field orbital magnetic susceptibility of non-interacting Bloch electrons in crystalline ordered solids at zero-temperature and fixed density of particles. In the metallic case, we notably investigate the validity of the Landau-Peierls approximation in the low density regime. The semiconducting situation also is treated; in this instance, we mainly focus on the particular case of gapped graphene-like solids.


Søren Fournais (Wednesday 15:30-17:00)

Title: The mathematics of superconductivity

Abstract: We will give an introduction to some recent mathematical result on the Ginzburg-Landau model of superconductivity. The minimizers of the Ginzburg-Landau functional solve a system of coupled elliptic non-linear differential equations with a very rich structure. We will focus on a certain regime where methods from semiclassical spectral theory can be applied. In particular this gives a precise characterization of the strength of the external magnetic field for which the superconductor looses its superconducting properties.


1. F. Bethuel, H. Brezis, and F. Helein. Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications 13. Birkhaauser (1994)

2. S. Fournais, B. Helffer. Spectral Methods in Surface Superconducitivity. Progress in Nonlinear Differential Equations and Their Applications, Vol. 77 Birkhäuser Boston (2010).

3. S. Sandier, S. Serfaty. Vortices in the magnetic Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, 70. Birkhäuser Boston (2007).


Klaus Mølmer (Thursday 13:00-14:30)

Title: Quantum state control by quantum measurements

Abstract: With different examples, I will explicitly discuss the connection between the unpredictable character of measurements in quantum theory, the stochastic master equation formalism and realistic experimental situations. I will then present a number of examples, where we can use measurements as a tool to prepare states of quantum systems for which we have no other means of preparation, e.g., non-classical states of light from conventional light sources, entangled states of remote particles, squeezed states of many non-interacting particles, ... . These examples also lead us to consider cases where the observed system Hilbert space is too large to permit a straightforward numerical implementation of the stochastic wave function or density matrix formalism. Some cases may be exactly dealt with by the (multi-mode) Gaussian covariance matrix formalism, while for others we suggest to implement a stochastic version of the Matrix Product state formalism.