Scattering and Irreversibility in Quantum Field Theory

• Applicant: Professor Wojciech de Roeck, Ph.D.
• Subject Area: Mathematics
• Term: from 2012 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 222227888

We consider simple quantum field theories, for example: one electron interacting with the photon field. For such theories, there is a well-known conjecture 'Asymptotic Completeness' which states the following: The dynamics of the electron and the photon field is well-described by the concept of a quasiparticle or a 'dressed electron', which has a different (renormalized) mass and dispersion relation than the original 'bare' electron. As time tends to infinity, one can describe the total system as a wavepacket of 'dressed electrons' evolving freely and a few free photons traveling off to infinity. Apart from their fundamental significance, such conjectures are very important for field theories in condensed matter: the (approximatively) free charge carriers in metals are in fact not electrons but dressed electrons, with significantly larger mass.One aim of our project is to prove asymptotic completeness in the setup described above and to prove an analogous claim in the case where the electron is bound to an atom. Up to now, and despite sustained efforts, such claims have only been proven in a few special cases, for example for massive photons.Moreover, we want to focus on irreversible phenomena taking place in these simple quantum field theories.For example, if an atom is initially prepared in an exited state, it will relax to its ground state while emitting the excess energy as radiation. This is closely connected to the question of asymptotic completeness, both on the conceptual and mathematical level.Another, probably more technically demanding, phenomenon occurs when the quantum field (where one should now preferably replace the photons by phonons) is at positive temperature. By emitting and absorbing phonons, the motion of the electron gets effectively randomized and for large times, it is well described by the diffusion equation. This often goes under the name 'Quantum Brownian Motion', but the phenomenon is universal and by no means specific to quantum field theory.In the past years, we made substantial progress on these phenomena, and we obtained a rigorous proof that diffusion occurs in some special cases. We would like to continue and extend this work.

• DFG Programme: Research Grants
• International Connection: Belgium