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We report on a simple strategy to treat mean-field limits of quantummechanical systems in which a large number of particles weakly couple to asecond-quantized radiation field. Extending the method of counting, introducedin [Lett. Math. Phys. 97, 151-164], with ideas inspired by[http://www.mathematik.uni-muenchen.de/%7Ebohmmech/theses/Matulevicius_Vytautas_MA.pdf]and [J. Math. Phys. 54(1), 012303] leads to a technique that can be seen as acombination of the method of counting and the coherent state approach. It issimilar to the coherent state approach but might be slightly better suited tosystems in which a fixed number of particles couple to radiation. The strategyis effective and provides explicit error bounds. As an instructional example wederive the Schr\"odinger-Klein-Gordon system of equations from the Nelson modelwith ultraviolet cutoff. Furthermore, we derive explicit bounds on the rate ofconvergence of the one-particle reduced density matrix of the non-relativisticparticles in Sobolev norm. More complicated models like the Pauli-FierzHamiltonian can be treated in a similar manner [arXiv:1609.01545].

Mean-field limits of particles in interaction with quantized radiation fields