Oberseminar Mathematische Physik

I will introduce a modification of the Ghirardi-Rimini-Weber (GRW) model in which the flashes (or collapse space-time events) source a classical gravitational field. The resulting semi-classical theory of Newtonian gravity preserves the statistical interpretation of quantum states of matter in contrast with mean field approaches. It can be seen as a discrete version of recent proposals of consistent hybrid quantum classical theories. The model is in agreement with known experimental data and introduces new falsifiable predictions: (1) particles do not attract themselves, (2) the 1/r gravitational potential of Newtonian gravity is cut-off at short (. 10−7m) distances, and (3) gravity makes spatial superpositions decohere at a rate inversely proportional to that coming from the vanilla GRW model. Together, the last two predictions make the model experimentally falsifiable for all values of its parameters.

In absence of a speaker, we will consider a celebratory model for pre-christmas dynamics of many interacting group members. The effect of cookies and the consumption of warm liquids on the social behaviour of mathematicians will be empirically tested. All group members are invited!

There are a number of reasons to think that the electron cannot truly be spinning. Given how small the electron is generally taken to be, it would have to rotate superluminally to have the right angular momentum and magnetic moment. Also, the electron’s gyromagnetic ratio is twice the value one would expect for an ordinary classical rotating charged body. These obstacles can be overcome by examining the flow of mass and charge in the Dirac field (which gives the classical state of the electron). Superluminal velocities are avoided because the electron’s mass and charge are spread over sufficiently large distances that neither the velocity of mass flow nor the velocity of charge flow need to exceed the speed of light. The electron’s gyromagnetic ratio is twice the expected value because its charge rotates twice as fast as its mass.

The tunnelling time problem is almost as old as quantum mechanics itself and is a highly debated subject. Within "conventional" quantum mechanics, many conflicting theories have been developed over the decades. Here we review some of those definitions, namely the Buttiker-Landauer, Pollak-Miller, Eisenbud-Wigner, and Larmor times. While based on very different physical arguments, all of these definitions can be presented as averages using the tunneling time probability amplitudes constructed with the Feynman path integral (FPI) approach. In addition, I show how the FPI approach can be used to calculate tunneling time probability amplitudes. Bohmian mechanics, offers an alternative definition, namely Bohmian time. We show how this time can be calculated as well in the context of tunnel ionization and what it might mean. All of these theoretical tunneling times are discussed in the context of the attoclock experimental measurements and compared to the experimentally defined "attoclock tunneling time".

Vlasov equations are non-linear transport equations that describe self-consistent dynamics of densities in various physical systems. They arise as the continuum limit of many particle point mass systems with specified interaction laws. For example, Vlasov-Poisson is the limit equation of the classical Newtonian many body problem. Although the equations are the continuum limit of Hamiltonian systems, Hamiltonian structure has not yet been described in a manner fruitful from the PDE side. In the first part, we derive a simple Hamiltonian structure for general Vlasov systems, that was firstly noted by Fröhlich, Knowles, and Schwarz for the example of non-relativistic two-body interaction systems. In the second part, we sketch the proof of wellposed-ness for the Hamiltonian equations of the Vlasov-Poisson problem. At last, we shortly discuss open problems related to the equation, such as periodic solutions and mean field limit.

Phase transitions for lattice systems such as the Ising model are well-understood. For particles in R^d, the situation is much less satisfying and the existence of phase transitions has been proven only for some models, notably the Widom-Rowlinson model and Kac interactions (Lebowitz, Mazel, Presutti) model. I will present some results on the low-temperature behavior of classical Gibbs measures motivated by the search for phase transitions, and for a concrete model sketch some results on continuum Glauber dynamics at low temperature in finite volume, specifically Arrhenius law and nucleation barriers. Based on work with F. den Hollander, W. König, B. Metzger.