SIAM Journal on Mathematical Analysis, Volume 52, Issue 5, Page 4900-4936, January 2020.
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 $1/\sqrt{N}$. 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 $N \rightarrow \infty$ we show that the time evolution preserves the condensate as well as the coherent structure and that it can be approximated by the Maxwell--Schrödinger 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.

中文翻译：

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Pauli-Fierz哈密顿量的麦克斯韦-薛定er方程的推导

SIAM数学分析杂志，第52卷，第5期，第4900-4936页，2020年1月。
我们考虑不旋转的Pauli-Fierz哈密顿量，该量描述了耦合到量子电磁场的非相对论性相同粒子的量子系统。我们研究了其在平均场范围内的时间演化，其中带电粒子的数量$ N $变大，而与辐射场的耦合按$ 1 / \ sqrt {N} $重新定标。在零时间，我们假设几乎所有带电粒子都处于同一单体状态（玻色-爱因斯坦凝聚态），光子接近相干态。在极限$ N \ rightarrow \ infty $中，我们证明了时间演化既保留了凝结水又保持了相干结构，并且可以通过麦克斯韦-薛定er系统（Maxwell-Schrödingersystem）对其进行近似，该系统模拟了非相对论粒子与经典粒子的耦合。电磁场。我们的结果是通过扩展计数方法获得的。Pickl，Lett。数学。Phys。，97（2011），pp.151--164]，以使带电粒子与其辐射场相互作用而冷凝。