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Leopold, Nikolai; Pickl, Peter (2020): DERIVATION OF THE MAXWELL-SCHRODINGER EQUATIONS FROM THE PAULI-FIERZ HAMILTONIAN. In: Siam Journal on Mathematical Analysis, Vol. 52, No. 5: pp. 4900-4936
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We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of nonrelativistic identical particles coupled to the quantized electromagnetic field. We study its time evolution in a mean-field limit where the number N of charged particles gets large while the coupling to the radiation field is rescaled by 1/root N. At time zero we assume almost all charged particles to be in the same one-body state (a Bose-Einstein condensate) and the photons to be close to a coherent state. In the limit N -> infinity we show that the time evolution preserves the condensate as well as the coherent structure and that it can be approximated by the Maxwell-Schrodinger system, which models the coupling of a nonrelativistic particle to the classical electromagnetic field. Our result is obtained by an extension of the method of counting, introduced in [P. Pickl, Lett. Math. Phys., 97 (2011), pp. 151-164], to condensates of charged particles in interaction with their radiation field.