Effective dynamics for particles coupled to a quantized scalar field

We consider a system of N non-relativistic spinless quantum particles (``electrons'') interacting with a quantized scalar Bose field (whose excitations we call ``photons''). We examine the case when the velocity v of the electrons is small with respect to the one of the photons, denoted by c (v/c= epsilon << 1). We show that dressed particle states exist (particles surrounded by ``virtual photons''), which, up to terms of order (v/c)^3, follow Hamiltonian dynamics. The effective N-particle Hamiltonian contains the kinetic energies of the particles and Coulomb-like pair potentials at order (v/c)^0 and the velocity dependent Darwin interaction and a mass renormalization at order (v/c)^{2}. Beyond that order the effective dynamics are expected to be dissipative. The main mathematical tool we use is adiabatic perturbation theory. However, in the present case there is no eigenvalue which is separated by a gap from the rest of the spectrum, but its role is taken by the bottom of the absolutely continuous spectrum, which is not an eigenvalue. Nevertheless we construct approximate dressed electrons subspaces, which are adiabatically invariant for the dynamics up to order (v/c)\sqrt{\ln (v/c)^{-1}}. We also give an explicit expression for the non adiabatic transitions corresponding to emission of free photons. For the radiated energy we obtain the quantum analogue of the Larmor formula of classical electrodynamics.Comment: 67 pages, 2 figures, version accepted for publication in Communications in Mathematical Physic

Response Ranking with Deep Matching Networks and External Knowledge in Information-seeking Conversation Systems

Intelligent personal assistant systems with either text-based or voice-based conversational interfaces are becoming increasingly popular around the world. Retrieval-based conversation models have the advantages of returning fluent and informative responses. Most existing studies in this area are on open domain "chit-chat" conversations or task / transaction oriented conversations. More research is needed for information-seeking conversations. There is also a lack of modeling external knowledge beyond the dialog utterances among current conversational models. In this paper, we propose a learning framework on the top of deep neural matching networks that leverages external knowledge for response ranking in information-seeking conversation systems. We incorporate external knowledge into deep neural models with pseudo-relevance feedback and QA correspondence knowledge distillation. Extensive experiments with three information-seeking conversation data sets including both open benchmarks and commercial data show that, our methods outperform various baseline methods including several deep text matching models and the state-of-the-art method on response selection in multi-turn conversations. We also perform analysis over different response types, model variations and ranking examples. Our models and research findings provide new insights on how to utilize external knowledge with deep neural models for response selection and have implications for the design of the next generation of information-seeking conversation systems.Comment: Accepted by the 41th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2018), Ann Arbor, Michigan, U.S.A. July 8-12, 2018 (Full Oral Paper

Analytic Wave Front Set for Solutions to Schro&#776;dinger Equations II \u2013 Long Range Perturbations

This paper is a continuation of a previous work, where short range perturbations of the flat Euclidian metric where considered. Here, we generalize the results to long-range perturbations (in particular, we can allow potentials subquadratic at infinity). More precisely, we construct a modified quantum free evolution acting on Sjo&#776;strand\u2019s spaces, and we characterize the analytic wave front set of the solution to the Schro&#776;dinger equation, in terms of the microlocal semiclassical exponential decay of the corresponding modified quantum free evolution. The result is valid for t &lt; 0 near the forward non trapping points, and for t > 0 near the backward non trapping points

Analytic Singularities for Long Range Schr"odinger Equations

We consider the Schrödinger equation associated to long range perturbations of the flat Euclidean metric (in particular, potentials growing subquadratically at infinity are allowed). We construct a modified quantum free evolution G(s) acting on Sjöstrand’s spaces, and we characterize the analytic wave front set of the solution e?itH u_0 of the Schrödinger equation, in terms of the semiclassical exponential decay of G(?th^{?1})Tu_0, where T stands for the Bargmann-transform. The result is valid for t &lt; 0 near the forward non-trapping points, and for t > 0 near the backward non-trapping points

Propagation of analytic singularities for the Schr"odinger Equation

We present recent results concerning the regularizing effects and the propagation of analytic singularities for the solutions to the Schroedinger equation. In particular, for short-range perturbations of the laplacian, we give a complete characterization of the analytic wave front set of such solutions in terms of that of the free evolution of the initial data