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# PhD course: Topics in Mathematical Physics

**ECTS**: 1

**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
(cornean@math.aau.dk).

**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.

**References**:

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.

**References**:

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.