Derivation of the Maxwell--Schrödinger Equations from the Pauli--Fierz Hamiltonian

Abstract

We consider the Maxwell-Schrödinger equations in the Coulomb gauge describing the interaction of extended fermions with their self-generated electromagnetic field. They heuristically emerge as mean-field equations from non-relativistic quantum electrodynamics in a mean-field limit of many fermions. In the semiclassical regime, we establish the convergence of the Maxwell-Schrödinger equations for extended charges towards the nonrelativistic Vlasov-Maxwell dynamics and provide explicit estimates on the accuracy of the approximation. To this end, we build a well-posedness and regularity theory for the Maxwell-Schrödinger equations and for the Vlasov-Maxwell system for extended charges.

Derivation of the Landau–Pekar Equations in a Many-Body Mean-Field Limit

“…The functional can consequently be used to monitor whether the condensate of the particles and the coherent state of the phonons is stable during the time evolution. Its definition is motivated by a previous work on the derivation of the Maxwell–Schrödinger equations [ 23 ]. In addition it is necessary to include the Gross transform in the definition of and .…”

“…A possible solution to this problem is to use a combination of the estimates from [ 23 , Chapter VIII.1] with an operator bound that is motivated by [ 10 , Lemma 10] (see Section 6 for the detailed argument). In short, we use the symmetry of the wave function and an estimate that is similar in spirit to the commutator method of Lieb and Yamazaki to obtain As shown in Lemma 5.1 , the new quantity can be bounded by and errors proportional to and .…”

“…The most difficult part is to control the interaction between the ultraviolet modes of the phonon field and the fraction of particles not in the condensate. To this end, we restrict our consideration to a subclass of the initial states which have small fluctuations in the energy per particle observable and combine estimates similar to [ 23 , Sect. VIII.1] with an operator bound that is motivated by [ 10 , Lemma 10].…”

“…VIII.1] with an operator bound that is motivated by [ 10 , Lemma 10]. The idea of using this restriction in order to treat the singular interaction between quantum fields and particles in the mean field regime was already used in [ 23 ].…”

“…In the many-particle mean-field limit considered in this work, the creation of coherent radiation happens for a different reason than in the strong coupling regime, namely because there are many particles in the same quantum state that simultaneously create the phonons. In this regard, the present work is related to [1,7,[22][23][24], where many-body mean-field limits of the renormalized Nelson model, the Nelson model with ultraviolet cutoff and the (bosonic) Pauli-Fierz model are considered. In particular, we mention [1] where the Schrödinger-Klein-Gordon equations were derived by the Wigner measure approach as a limit of the renormalized Nelson model.…”