Welcome to the Mathematical Physics Seminar

In this talk I will briefly introduce weak measurements and weak values. I will then present two recent experiments performed by our group using weak measurements. In the first experiment, we use weak measurements to reconstruct the trajectories of single photons as they traverse a double-slit interferometer. Weak measurements allow us to observe the photons average momentum at each position without destroying the interference pattern. Interestingly, our weakly measured trajectories are mathematically equivalent to the Bohmian trajectories of a double-slit interferometer. In the second experiment, the precision and disturbance of a measurement are measured. It is often thought that these quantities obey Heisenberg's uncertainty principle. However, the Heisenberg uncertainty principle discusses variances in quantum states not the disturbance induced by a measuring device. Recently, Ozawa proved a revised, universally valid, relationship between the precision and disturbance. I will present our experimental results showing a situation where the precision and disturbance appear to violate Heisenberg's uncertainty principle (when incorrectly applied) but Ozawa's relationship remains valid.

I will talk about a recent topic in the field of Atomic Collisions: Quantum Vortices. These fluid dynamical-like structures can be described and understood in a straightforward way from a Bohmian Mechanical point of view. I will show some results of my search of vortices in a positron-atom collision at low energies.

The idea of taking the Dirac Sea serious is the following: Try to understand the free Schroedinger or Dirac Equation for a test particle as an effective (in some sense mean field) description of an underlying microscopic many body system composed of the sea and the test particle. This approximation is widely used in nonrelativistic and relativistic quantum physics but, in my opinion, still lacks a clear physical understanding as well as a rigorous justification. The main difference compared to the derivation of other well known mean field equations is that one expects the fluctuations in the sea to determine the behaviour of the test particle. A nonrelativistic model will be discussed in order to show how such fluctuations behave and how they might be controlled.

We consider a classical system of point particles interacting by means of a short range potential. We prove that, in the low-density (Boltzmann-Grad) limit, the system behaves, for short times, as predicted by the associated Boltzmann equation. This is an extension of the celebrated result of Lanford of 1975 about the hard sphere system. Our analysis applies to any stable and smooth interaction, including forces with attractive parts.

In physics, one is used to be able to describe subsystems like a single H atom autonomously. However, in orthodox quantum mechanics, a proper definition of subsystems in terms of the theory of a larger system is not easily possible. While in spinless Bohmian mechanics (BM) the description of subsystems is straightforward and satisfactory, the situation with spin is more complicated. In the literature, a notion of conditional density matrix has been suggested to replace the notion of a conditional wave function. This opens up the question if BM should generally be reformulated in terms of density matrices. In this talk, I briefly review the analysis of subsystems in BM and critically assess the idea of a conditional density matrix. If time permits, the results will be generalised to relativistic hypersurface Bohm-Dirac models.