Titlebook: Effective Evolution Equations from Quantum Dynamics; Niels Benedikter,Marcello Porta,Benjamin Schlein Book 2016 The Author(s) 2016 Bosonic

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https://doi.org/10.1007/978-3-662-39835-7in the Hilbert space anymore. However, the Araki-Wyss representation allows us to describe it by a vector in a ‘doubled’ Hilbert space. We use the Araki-Wyss method to derive the time-dependent Hartree-Fock equation for mixed states.

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Martin Aigner,Günter M. ZieglerEffective theories provide important approximations in many areas of physics. We review two examples where effective theories can be rigorously derived from the microscopic theory.

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https://doi.org/10.1007/978-3-211-69922-5The mean-field regime of bosonic systems provides one of the simplest time-dependent effective theory, the Hartree equation. We introduce the Hartree equation, notions of convergence based on reduced density matrices and review the literature. We review in some more detail proofs of the Hartree equation based on the BBGKY hierarchy.

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Hochfrequente Sonographie des BasaliomsWe introduce Bogoliubov transformations. These allow us to describe the quantum fluctuations around the mean-field limit. We also discuss the probabilistic interpretation in terms of a central limit theorem.

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Mean-Field Regime for Bosonic Systems,The mean-field regime of bosonic systems provides one of the simplest time-dependent effective theory, the Hartree equation. We introduce the Hartree equation, notions of convergence based on reduced density matrices and review the literature. We review in some more detail proofs of the Hartree equation based on the BBGKY hierarchy.

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Coherent States Approach,We present a simple proof of the Hartree equation with quantitative estimates, using the method of coherent states. Along the way, we introduce second quantization for bosonic many-body systems.