Oberseminar Mathematische Physik

I will show that, given a region of overwhelming phase space measure (an equilibrium region in Boltzmann's sense) and a stationary measure, a system behaves qualitatively ergodic with respect to the equilibrium state, that is, almost all trajectories spend almost all of their time in equilibrium.

An explicit formula for the one-boson scattering processes is derived in the massive Spin-Boson model which describes the interaction between a two-level atom coupling to a massive scalar field. The proof requires good control of the resolvent close to the real line which can be shown using Mourre theory.

In order to describe a Quantum-N-body system in the Schrodinger picture in a manifestly covariant way, one needs to introduce a separate time coordinate t_k for each of the particles k = {1,...,N}. The wave function therefore depends on N time coordinates (\Multi-Time wave function") and the Schrodinger equation becomes a system of N Partial Dierential Equations (PDEs). Multi-Time wave functions were already introduced in the 1930's and have become a topic of active research again throughout the past years. So far, a series of papers has been published, which investigate the existence and uniqueness of solutions to the Multi-Time Schrodinger equation systems appearing in various quantum mechanical models. However, most of the considerations in these papers were done on a non-rigorous level, for example by checking a certain consistency condition. In my master thesis, I could establish a rigorous proof of existence and uniqueness of a solution to the equations of motion for a Quantum Field Theory (QFT) toy model. This talk will concern about Multi-Time formalism in general, how to construct a solution to the equations of motion and how to show that this PDE system is actually solved.

Starting from the lift condition of unitary operators on Fock Space, I will derive an explicit formula for the lift of the one-particle time evolution of Dirac particles subject to an external potential.

I discuss the recent no-go theorem of Frauchiger and Renner based on an ``extended Wigner's friend'' thought experiment which is supposed to show that any single-world interpretation of quantum mechanics leads to inconsistent predictions if it is applicable on all scales. I show that no such inconsistency occurs if one considers a complete description of the physical situation. I then discuss implications of the thought experiment that have not been clearly addressed in the original paper, including a tension between relativity and nonlocal effects predicted by quantum mechanics. My analysis applies in particular to Bohmian mechanics, showing that it provides a perfectly consistent description of the thought experiment.

Here I explain the idea that direct interactions along light cones, similar to the Wheeler-Feynman formulation of electrodynamics, can be implemented on the quantum level using integral equations for multi-time wave functions. Multi-time wave functions are wave functions psi(x_1,...,x_N) with N spacetime arguments x_i for N particles. The crucial point is that the N time variables of the x_i make it possible to express interactions with time delay, as relativity requires. Starting from the integral formulation of the non-relativistic Schrödinger equation, I derive a covariant integral equation as an evolution equation for psi, and discuss its mathematical structure. It is shown that the equation correctly reduces to the Schrödinger equation with a Coulomb potential when time delay effects are neglected. The main mathematical results are existence and uniqueness theorems for a simplified version of the equation. This talk is partly about joint work with Roderich Tumulka.

Interest in the mathematical treatment of two-body Dirac operators $H_{2BD}$, that describe two interacting electrons, arose from relativistic quantum chemistry. There, the existence of square-integrable eigenfunctions of such operators is assumed without further justification. In this talk, we discuss a self-adjoint extension that is uniquely distinguished by means of finite potential energy. The difficulties we encounter are the unboundedness of $H_{2BD}$ from below, and that the interaction potential is not relatively bounded by the free Hamiltonian. If time allows, we also investigate the domain of $H_{2BD}$ and the possibility of infinite single particle kinetic energy states in this domain.

We consider a quantum dot in a semi-conductor device P and a train of probes sent to it. A projective measurement is applied to each probe, just after it interacts with P. This produces a discrete density-matrices valued stochastic process rho_n. We prove that the density matrices, rho_n, purify as n tends to infinity. Our results can be used to mathematically understand situations in the spirit of experiments of Haroche and Wineland (which led to the Nobel Prize in Physics 2012).

The cloud chamber problem is the problem of the quantum mechanical description of the elementary particles' position detection in the Wilson cloud chamber. In my talk I will quickly recapitulate what exactly the problem is, what has been done using standard quantum mechanics and why it is interesting and important to obtain the Bohmian description. Then I will talk about the recent progress we have made on the way towards it.