Bohmian mechanics is a quantum theory which is free from (quantum) philosophical considerations which are needed in the conventional approach to quantum phenomena to cope with the notorious measurement problem, exemplified by Schrödinger's cat. Part of our research is the mathematical analysis of questions arising in quantum physics, for example the justification of the randomness intrinsic in quantum physics. This is known as the Born interpretation of the wave function and which appears in Bohmian mechanics as quantum equilibrium hypothesis. The quest is analogous to the justification of the use of the canonical ensembles in statistical physics. Further questions relate to the justification of the scattering formalism and the measurement of time. The extension of Bohmian mechanics to relativistic physics as well as relieving the tension between nonlocality of nature (established by Bell's inequalities and experiments) and relativity are research perspectives with great potential. The virtue of the Bohmian approach is its mathematical precision describing physics on all scales. We are therefore encouraged to look for a description of nature which is free of mathematical singularities. A famous example is the fully relativistic and interacting, yet mathematically consistent theory of electromagnetism of Wheeler and Feynman. This theory is mathematically challenging as it describes nature fundamentally in a non Markovian way, i.e., it cannot be formulated as an ordinary initial value problem. Quantum versions of that theory are still to be found.
A second pillar of our research concerns many body classical and quantum physics. A very old and yet timely and active field of research is the justification of macroscopic effective descriptions emerging from microscopic dynamics of many particles. Famous examples for classical systems are the Boltzmann equation and Vlasov equations of various types; for quantum systems the Hartree, or Hartree Fock as well as the Gross Pitaevskii equation. Our main focus is on mean field descriptions of classical and quantum systems employing the new alpha-method for proving mean field results. In addition to the justification of the effective description we are also interested in describing physical effects in the mean field situation, for example Cherenkov radiation of fast particles entering a Bose-Einstein condensate (in collaboration with Dirk-André Deckert).
Our research is also directed towards a better understanding of the quantum field vacuum in QED, where we try to understand its effective action and behaviour as it emerges from its mathematically equivalent description of the so called Dirac sea. One focus is here on experimentally controllable effects like adiabatic pair creation and annihilation which can be described with mathematical rigor. Our approach to QED is shared by many other groups and we have common research projects with the group of Felix Finster in Regensburg. Our perspectives are also directed towards applications, concerning mean field descriptions in biological systems (in collaboration with Martin Kolb).