Mathematical study of non-relativistic QED: The Link between Resonances and Poles of the Scattering Matrix
Final Report Abstract
One of the striking natural phenomena that let to the discovery of quantum mechanics was the observation that the energy spectrum of light emitted by ionized hydrogen gas appears as divided into discrete energy ‘quanta‘. The first attempt of a quantitative description goes back to the renowned work of Plank in 1900. Einstein shortly after in 1905, in his famous article on the photoelectric effect, proposed that all electromagnetic radiation may be comprised of discrete quanta, the so-called photons. These entities were thought to be quantum particles whose momenta behave inversely proportional to their corresponding wave length – the first invocation of the so-called particle-wave duality. De Broglie generalized this duality further to all particles and proposed that matter waves propagate along with their particle counterparts. The electron of a H-atom was said to form standing waves that would only allow for certain discrete rotational frequencies. Based on the assumption that the electron’s angular momentum was quantized, Bohr provided a first model of the H-atom derived from first principles. While this model was supplied further with relativistic corrections by Sommerfeld, Debeye proposed that, if the electron behaves wave-like, there should also be a wave equation describing its motion. This equation was found in 1926 by Schrödinger in the non-relativistic regime. His model reproduced well the predictions of the Bohr model although the ontological interpretation of its main actor, the wave function, is still pending to this day. Assuming further, that electrons can be excited or may decay between energetically higher and lower orbits by absorption or emission of photons which ought to have wave-lengths that correspond to the energy difference of the respective orbits, a first fundamental explanation of the spectral lines of the H-atom was given. Even in nowadays textbooks, references to this assumption can be found in the rule-of-thumb notions of so-called allowed and forbidden transitions. Despite this success, Schrödinger’s model only describes the static orbits of the electron subject to the Coulomb potential of the proton, the so-called bound states, but it does not describe the dynamic process of decay of an electron from a higher orbit to a lower one through the emission of radiation. Missing was the coupling to a description of light by means of photons as postulated by Eistein. This gap was supposed to be filled by the invention of quantum electrodynamics (QED) by Dirac, Heisenberg, Pauli, Wigner and many others and culminated in the Nobel prizes of Dyson, Feynman, Schwinger, and Tomonaga. However, to this day, QED is plagued by its ill-defined equations of motion which provoke divergences in all its formally computed perturbative corrections and that have to be mended manually by a so-called renormalization procedure. We therefore studied a simplified model of QED, the so-called Spin-Boson model, in which the H-atom is replaced by a quantum two-level system and the photons by a scalar bosons in a regime for which a mathematical rigoros definition of the model is possible. Incoming bosons may excite the two-level system from its ground to excited state by absorption of bosons, and vice versa, the excited state may decay back to the ground state by emission of outgoing bosons. Despite these simplifications, this model serves as an important fundamental model of lightmatter interactions in quantum optics. In scattering experiments, e.g., when irradiating bosons on such a two-level system, one observes so-called resonances as peaks in the intensity of light at certain energy values in the measured scattering cross-sections per solid angle. The energetic locations of these peaks form the analogue of the above mentioned spectral lines of the H-atom. These are caused by the mentioned emission and absorption processes, which, contrary to the discreet notions of allowed and forbidden transitions above, occur with high probability around those peaks but are strongly suppressed at other energies. The published works funded by this grant comprise a mathematically rigorous description of resonances in the Spin-Boson model. We provide a non-perturbative formula for the respective scattering cross-sections that allows to identify the location of the resonances to arbitrary precision and makes its textbook rule-of-thumb notions of allowed and forbidden transitions precise. While similar results were obtained in the literature in the realm of N-particle quantum mechanics, to our knowledge, it is the first time such an analysis was successfully carried out in a model of quantum field theory. A possible generalization to a model of a H-atom coupled to the quantized electromagnetic field shall be explored in future endeavors.
Publications
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Analyticity of Resonances and Eigenvalues and Spectral Properties of the massless Spin- Boson Model, Journal of Functional Analysis, 8:2524-2581, 57 pages, 2019
M. Ballesteros, D.-A. Deckert, F. Hänle
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Relation between the Resonance and the Scattering Matrix in the massless Spin-Boson Model, Communications in Mathematical Physics, 370:249–290, 41, pages, 2019
M. Ballesteros, D.-A. Deckert, F. Hänle
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One-Boson Scattering Processes in the massive Spin-Boson Model, Journal of Mathematical Analysis and Applications, 489:1, 44 pages, 2020
M. Ballesteros, D.- A. Deckert, J. Faupin, F. Hänle
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One-boson scattering processes in the massless Spin-Boson model – A non-perturbative formula, Advances in Mathematics, 371:107248, 27 pages, 2020
M. Ballesteros, D.-A. Deckert, J. F. Hänle
