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Quantum Coulomb systems : screening, recombination and van der Waals forces

The study of quantum Coulomb systems at equilibrium is important for understanding properties of matter in many physical situations. Screening, recombination and van der Waals forces are basic phenomena which result from the interplay of Coulomb interactions, collective effects and quantum mechanics. Those phenomena are introduced in the first part of this lecture, through various physical examples. Their treatment within mean-field theories and phenomenological approaches is also exposed, while related predictions are discussed. This sheds light on fundamental issues, which must be analyzed without any \textsl{a priori} approximations or modelizations. The second part of this lecture is precisely devoted to the presentation of various exact results for the quantum proton-electron hydrogen plasma. Such results are derived within the Screened Cluster Representation, which is constructed by combining the path integral representation of the Coulomb gas with Mayer-like diagrammatical techniques. They illustrate the breakdown of Debye exponential screening by quantum fluctuations, as well as the emergence of familiar chemical species in suitable low-temperature and low-density limits. Also, the amplitude of van der Waals forces is shown to be reduced by free charges.