Quasi-Classical Dynamics

We study quantum particles in interaction with a force-carrying field, in the quasi-classical limit. This limit is characterized by the field having a very large number of excitations (it is therefore macroscopic), while the particles retain their quantum nature. We prove that the interacting microscopic dynamics converges, in the quasi-classical limit, to an effective dynamics where the field acts as a classical environment that drives the quantum particles.


Introduction and Main Results
This paper is devoted to the study of the quasi-classical dynamics of a coupled quantum system composed of finitely many non-relativistic particles interacting with a bosonic field.The quasi-classical regime is concretely realized by taking a suitable partial semiclassical limit, introduced by the authors in [CF18,CFO19] to derive external potentials as effective interactions emerging from the particle-field coupling.The physical meaning of such limit is discussed in § 1.1.
Our analysis clarifies, both mathematically and physically, the role played by external macroscopic classical force fields on quantum systems, and in which regime such macroscopic fields provide an accurate description of the interaction between an open quantum system and its environment (bosonic field).
In order to study the dynamical quasi-classical limit, we develop a mathematical framework of infinite dimensional quasi-classical analysis, in analogy with the semiclassical scheme initially introduced and in [AN08, AN09, AN11, AN15a], and further discussed in [Fal18a,Fal18b].Such a framework allows to characterize the quasi-classical behavior of quantum states which are not factorized, i.e., in which the degrees of freedom of the quantum particles and the bosonic field are entangled.Although our mathematical scheme is more general, we are going to focus our attention on three concrete models of interaction between particles and force-carrying fields: the Nelson, Pauli-Fierz, and Fröhlich polaron models (see § 1.4).Note that partial semiclassical limits have already been studied, with somewhat different purposes, in [GNV06, AN15b, AJN17, ALN17], as well as in the context of adiabatic theories (see, e.g., [Teu02, PST03, TT08, ST13]).
1.1.Physical Motivation.The quasi-classical description, combining a quantum system with a classical force field, is often used in physics to model external macroscopic forces acting on a quantum particle system.The best known examples are atoms and electrons in a classical electromagnetic field (see, e.g., [CTDRG97]), and particles subjected to external potentials, such as systems of trapped atoms and of particles in optical lattices.Since these external force fields are macroscopic, they are heuristically taken as classical, and inserted in the particles' Hamiltonian in the same way their microscopic counterparts would appear.Note that in literature the terminology "quasi-classical" is often used as a synonymous of semiclassical, while here we use it to stress that the classical limit we consider is not complete, but applies only to a part (radiation field or environment) of the microscopic system.
In this paper we provide a detailed analysis of the quasi-classical dynamical scheme, and discuss its validity as an approximation of a more fundamental microscopic model, thus justifying and completing the above heuristic picture.The basic idea is the following: in experiments, the external force fields are considered macroscopic because they live on an energy scale much larger than the ones of the quantum particles under study: the number of field's excitations is much larger than the number of quantum particles in the system.Let us denote by N the number of particles in the system.The force field is itself a quantum object, and its excitations are created and annihilated by the interaction with the particles.Let us denote the field's number operator by dGp1q " ż dk a : pkqapkq, where G stands for the second quantization functor.Therefore, the field is macroscopic if the state Ψ of the coupled system particles+field is such that xΨ |dGp1q| Ψy " N .The number of particles N is fixed, and therefore of order 1.In other words, the quasi-classical configurations are the ones for which xΨ |dGp1q| Ψy " 1 . (1.2) We thence introduce a quasi-classical parameter ε, playing the role of a semiclassical parameter but only for the field's degrees of freedom: when ε Ñ 0, the system becomes quasi-classical.We quantify ε as follows: a quasi-classical state Ψ ε is a state such that In other words ε is proportional to the inverse of the average number of excitations of the force-carrying field.It follows that on quasi-classical states, where a # ε p ¨q :" ?εa # p ¨q.The creation and annihilation operators a # ε satisfy ε-dependent semiclassical canonical commutation relations: " a ε pkq, a ε pk 1 q ‰ " εδpk ´k1 q . (1.4) It is therefore clear that a quasi-classical state is a state that behaves semiclassically only with respect to the field's degrees of freedom.It remains to understand which microscopic dynamics would yield, in the quasi-classical limit, an external potential acting on the particles and generated by the macroscopic field.In concrete applications, the macroscopic field is not affected by the quantum system and acts as an environment.Therefore, the coupling should be such that the particles do not back-react on the environment, at least to leading order in ε and for times of order 1.In addition, we may think that the environment itself either evolves freely, or it remains constant in time.The absence of back-reaction is determined by the ε-scaling of the microscopic interaction, while the dynamical behavior of the environment is determined by the ε-scaling of the field's free part.The ε scaling that we put on the microscopic Hamiltonian H ε (see § 1.5 below) reflects the above: the interaction is weak enough to have no back-reaction to leading order, and the field's free part is such that either the environment evolves freely, or it is constant in time.Let us remark that the latter is mathematically equivalent, up to a time rescaling, to a strong coupling regime, even if the physical interpretation is rather different.
In § 1.5 we prove that indeed the quasi-classical description can be obtained from microscopic models of particle-field interaction in the limit ε Ñ 0 of very large number of average field's excitations.Since such limit is a semiclassical limit on the field only, the resulting structure of quasi-classical systems is that of an hybrid quantum/classical probability theory.The quantum system is driven by the classical environment, whose configuration is a classical probability with values in the quantum states for the particles.This mathematical structure is described in detail in next § 1.3.1.2.Notation.Since we are going to consider a tensor product Hilbert space of the form H b K ε , we will distinguish between the full trace of operators Trp ¨q on H b K ε , and the partial traces tr H p ¨q and tr Kε p ¨q w.r.t.H and K ε , respectively.
We adopt the following convenient notation: an operator acting only on the particle space H is denoted by a calligraphic capital letter (e.g., T or T ε ), whereas an operator on the full space H b K ε is identified by a roman capital letter (e.g., H ε ).Given an operator T on H , we also conveniently denote its extension to H b K ε , i.e.T " T b 1, by the roman counterpart T .
Given a Hilbert space X , we denote by L p pX q, p P r1, 8s, the p-th Schatten ideal of BpX q, the space of bounded operators on X .More in general the set LpX q identifies all linear operators on X .We also denote by L p `pX q and B `pX q the cones of positive elements, and by L p `,1 pX q the set of positive elements of norm one.The corresponding norms are denoted by keeping track of the space, except for the case of the operator norm, for which we use the short notation } ¨} :" } ¨}BpX q .
Throughout the paper, given a set S we denote by 1 S its indicator function.The symbol C also stands for a finite positive constant, whose value may vary from line to line.
1.3.Quasi-Classical System.We consider a microscopic system consisting of two parts in interaction.The first one contains the objects whose microscopic nature remains relevant, while the second is a semiclassical environment.For the sake of clarity, we focus on a specific class of systems: non-relativistic quantum particles in interaction with a semiclassical bosonic force-carrying field (electromagnetic, vibrational, etc.).It is not difficult to adapt the techniques to other coupled systems as well, consisting of a quantum and a semiclassical part.We denote by H the Hilbert space of the quantum part, and by K ε the Hilbert space of the semiclassical part, that carries an ε-dependent, semiclassical, representation of the canonical commutation relations as in (1.4).Therefore, the microscopic theory is set in the Hilbert space H b K ε .
We restrict our attention to Fock representations of the canonical commutation relations.Therefore, we assume that the symmetric Fock space constructed over a separable Hilbert space h.The space h is the space of classical fields1 .The canonical commutation relation (1.4) in K ε reads, for any z, w P h, " a ε pzq, a : ε pwq ‰ " ε xz|wy h , and the quasi-classical limit corresponds to the limit ε Ñ 0. According to the notation set above, a microscopic Fock-normal state is thus described by a density matrix The dynamics is generated by a self-adjoint and bounded from below Hamiltonian on H bK ε , that we denote by H ε .Given the unitary dynamics e ´itHε , the evolved state is Γ ε ptq :" e ´itHε Γ ε e itHε . (1.6) Let us now turn the attention to the effective quasi-classical system in the limit ε Ñ 0. This is an hybrid quantum-classical system, in which the classical part acts as an environment for the quantum part.In fact, as we will see, the classical field affects the quantum particles, but the converse is not true, the interaction is not strong enough to cause a back-reaction of the particles on the classical field.
The basic observables for the classical fields are the elements z P h, or, more precisely, the real vectors of the form z `z˚.Scalar observables in a generalized sense are functions z Þ Ñ f pzq P C semiclassically called symbols.In addition to scalar or field observables, there are more general observables involving both subsystems, which are thus represented by operatorvalued functions z Þ Ñ Fpzq, where Fpzq is a linear operator on the particle Hilbert space H .Note that one can easily associate an operator-valued function to a scalar symbol as well, by simply setting Fpzq " f pzq ¨1, where 1 P BpH q stands here for the identity operator.
A state of the classical field (environment) is a Borel probability measure µ P M phq, while a state of the quantum particles is a density matrix γ P L 1 `,1 pH q.Since in the quasi-classical regime the environment affects the behavior of the quantum particle system, a quasi-classical state is a state-valued measure m P M `h; L 1 `pH q ˘. (1.7) A state-valued measure thus takes values in L 1 `pH q, but it can also be conveniently described by its norm Radon-Nikodým decomposition (see Proposition 2.2): a pair pµ m , γ m pzqq consisting of a scalar Borel measure µ m , and a µ m -integrable, almost everywhere defined function γ m pzq P L 1 `,1 pH q taking values in normalized density matrices, i.e., dmpzq " γ m pzqdµ m pzq.
(1.8) (1.9) Note that, when integrating against the state-valued measure, it is a priori relevant to keep the order as in the above expression, since Fpzq might not commute with γ m pzq.
The quasi-classical evolution also consists of two parts: an evolution of the environment's probability measure µ m , and one of the quantum system for (almost) every configuration of the classical field.The evolution of the environment depends on the choice of a scaling parameter for the field's part in H ε , and we consider two cases: either the environment is stationary, e.g., it is at equilibrium, or it evolves freely.Concretely, the environment is evolved by a unitary, linear, flow e ´itνω : h Ñ h, t P R, of classical fields, where ω is a positive self-adjoint operator on h (typically, a multiplication operator by the dispersion relation of the field), and ν P t0, 1u, depending on the chosen scaling.This flow pushes forward the measure µ m , yielding µ m,t :" `e´itνω ˘7 µ m . (1.10) The explicit action of the pushforward, as it is well-known, is as follows: for all measurable Borel sets B Ă h, "`e ´itνω ˘7 µ where e itνω B stands for the preimage of B w.r.t. the map e ´itνω .The quantum part of the evolution is generated by a map from field configurations to two-parameter groups of unitary operators z Þ Ñ `Ut,s pzq ˘t,sPR , and it acts as γ m,t,s pzq :" U t,s pzqγ m pzqU : t,s pzq . (1.12) Let us remark that the pushforward of the measure does not affect the Radon-Nikodým derivative γ m,t,s pzq, but only the integrated functions.The quantum evolution is unitary for (almost) all configurations of the field.However, a measurement on the classical system modifies the quantum state in a non-unitary, but explicit, way.Let f pzq be a scalar field's observable.For λ P C, let us define the level set of f as B λ " tz P h, f pzq " λu , and suppose it is µ m -measurable.Then the conditional quantum state γ m,t,s f "λ P L 1 `pH q at time t P R, describing the state of the quantum system conditioned to an observed value λ of the classical observable f , is given by t,s pzq .
The conditional evolution pt, sq Þ Ñ γ m,t,s f "λ is clearly non-unitary in several respects, but it preserves positivity: on the one hand, the norm of the evolved density matrix γ m,t,s f "λ can be strictly smaller than 1, simply because µ m pe ipt´sqνω B λ q ď µ m phq " 1; furthermore, the dynamics is actually non-Markovian, in general, unless either B λ " tz λ u or µ m " δ z 0 , i.e., the group property might not be satisfied.One should indeed not expect that, for any t, s, τ ą 0, there exists some two-parameter unitary group W t,s P BpH q, such that The quantum state at time t P R, conditioned to the fact that f is observed, irrespective of its value, is denoted by γ m,t,s f , and is given by If f is everywhere defined, the conditional evolution does not actually depend on f , Furthermore, in this case the conditional evolution pt, sq Þ Ñ γ m,t,s f preserves both positivity and the trace, but it is still non-Markovian in general.It would be interesting to study the states of the environment, if any, not concentrated in a single field configuration, that make the conditional evolution Markovian, and possibly non-unitary.Such measures would yield a quasi-classical evolution on the open quantum system of Lindblad type (see, e.g., [Kos72,Lin76]).
1.4.The Concrete Models: Nelson, Pauli-Fierz, and polaron.Let us define more concretely the three models of interaction between non-relativistic particles and bosonic force carrier fields that we consider throughout the paper: the Nelson, Pauli-Fierz, and polaron models.
1.4.1.Nelson Model.The Nelson model describes quantum particles (e.g., nucleons), interacting with a force-carrying scalar field (e.g., a meson field), and was firstly rigorously studied [Nel64].In this paper, we restrict our attention to the regularized Nelson model, where the interaction is smeared by an ultraviolet cutoff.We consider N , d-dimensional, non-relativistic, spinless particles, and therefore H " L 2 pR dN q.The classical fields are usually taken to be in h " L 2 pR d q, but other choices may be possible, e.g., a cavity field, whose classical space would then be ℓ 2 pZ d q.The Hamiltonian H ε has the form where K 0 " K 0 b1, with K 0 self-adjoint and bounded from below on H , ω a positive operator on h and dG ε pωq its second quantization, i.e., the Wick quantization of the symbol κpzq :" xz |ω| zy h , (1.13) and λ P L 8 pR d ; hq is the coupling factor.
If one naively replaces the quantum canonical variables a # with their classical counterparts, i.e., z # , one can easily deduce that the quasi-classical effective potential for the model above is given by the symbol z Þ Ñ Vpzq, where (see also [CF18,Sect.
This leads to the effective potential Vpzq being the Fourier transform of an integrable function, and thus continuous and vanishing at infinity.In order to obtain more singular potentials, it is necessary to consider microscopic states whose measures are not concentrated as Radon measures in h [CF18, Sect.2.5].This would, however, make the analysis more involved.
We thus restrict our attention to states whose measures are indeed concentrated in h (see Remark 1.9 for additional details).
1.4.2.Pauli-Fierz Model.We consider the class of Pauli-Fierz models describing N nonrelativistic, spinless, extended d-dimensional charges moving in R d , d ě 2, interacting with electromagnetic radiation in the Coulomb gauge.Adding spin, adopting a different gauge, or constraining particles to an open subset of R d would not affect the results, but make the analysis more involved.The particles' Hilbert space is thus H " L 2 pR dN q, while the classical fields are in h " L 2 pR d ; C d´1 q.The Hamiltonian H ε is customarily written as `´i∇ j `Aε px j q ˘2 `W px 1 , . . ., x N q `νpεqdG ε pωq , with A ε pxq " a : ε `λpxq ˘`a ε `λpxq ˘.
In this case, we have K 0 " ´∆ `W and the effective potential can be easily seen to become [CFO19, Sect.1.2] Vpzq " 4 N ÿ j"1 " ´iRe xz|λpx j qy h ¨∇j `´Re xz|λpx j qy h

¯2
. (1.15) Notice that the interaction term in H ε is not the Wick quantization of the above symbol Vpzq, because H ε is not normal ordered and an additional term is missing, i.e., but such a contribution vanishes in the limit ε Ñ 0. Similarly to the Nelson model, the effective interaction Vpzq describes the minimal coupling of the particles with a magnetic potential that is continuous and vanishing at infinity.1.4.3.Polaron.The Fröhlich's polaron [Fro37] describes electrons moving in a quantum lattice crystal.The N d-dimensional electrons are modeled as non-relativistic spinless particles, and thus again H " L 2 pR dN q.For the phonon vibrational field, h " L 2 pR d q.The Hamiltonian H ε is formally written as H ε " ´∆ `a: ε `φpx j q ˘`a ε `φpx j q ˘`W px 1 , . . ., x N q `νpεqdG ε p1q , with the particles' potential W satisfying the same assumptions given in § 1.4.2 for the Pauli-Fierz model.In addition, φpx; kq :" α e ´ik¨x |k| d´1 2 , α P R, is the polaron's form factor and, for all x P R d , it does not belong to h.Hence, H ε as written above is only a formal expression.However, it makes sense as a closed and bounded from below quadratic form: one can find a parameter r P R `, a splitting φ " φ r `χr , with φ r px; kq :" 1 t|k|ďru pkqφpx; kq, and some λ r P L 8 pR d ; h d q, such that, as a quadratic form, H ε " ´∆ `a: ε `φr px j q ˘`a ε `φr px j q ˘`" ´i∇ j , a ε `λr px j q ˘´a : ε `λr px j q ˘‰ `W px 1 , . . ., x N q `νpεqdG ε p1q , where the commutator between two vectors of operators involves a scalar product.
In the polaron model K 0 " ´∆ `W, and the effective potential is given by [CF18, Sect.
Notice that one could formally resum the two terms above, obtaining the same expression (1.14) as in the Nelson model.In the case of the polaron, the potential Vpzq is not necessarily bounded, but still relatively form bounded w.r.t.´∆.In fact, Vpzq can be any function in 9 H d´1 2 pR d q X L 2 loc pR d q.Let us also remark that in the polaron case, the quasi-classical limit is mathematically analogous to the strong coupling limit.Strongly coupled polarons have been widely studied in the mathematical literature both from a dynamical and a variational point of view (see, e.g., [Gro76, LT97, GW13, FS14, FG17, Gri17, FS19, LS19, LRSS19, FG19]).Compared to the available dynamical results [FS14, FG17, Gri17, LRSS19], our quasi-classical approach has the advantage of being applicable to a very general class of microscopic initial states.However, we have no control on the errors and we are not able to derive the higher order corrections to the effective dynamics, i.e., the ones given by the Landau-Pekar equations.
1.5.Main Results.Before stating our main results, we provide more technical details about the general structure of the models we are considering in this paper, by specifying some assumptions that are sufficient to prove our main results, and that are satisfied in the above concrete models.We do not strive for the optimal assumptions nor for the most general setting.
First of all, we remark that all the Hamiltonians introduced in § 1.4 can be cast in the following form where: K 0 is self-adjoint and bounded from below on H , and describes the particle's system when it is isolated; νpεq is a quasi-classical scaling factor, such that ν " lim εÑ0 ενpεq P t0, 1u, (1.17) and the two relevant scalings are νpεq " 1, yielding an environment that remains constant in time, and νpεq " 1 ε , yielding an environment that evolves freely; κ is the symbol given by (1.13) for a densely defined, positive operator ω on h.Given a symbol z Þ Ñ Fpzq, we denote by Op Wick ε pFq its Wick quantization, so that in particular Op Wick ε pκq " dG ε pωq.The symbol z Þ Ñ Vpzq is operator-valued and polynomial, and it describes the interaction between the particles and the environment.The possible concrete choices of V have been presented in § 1.4.Finally, Opεq is a bounded particle operator of order ε.
To study the limit ε Ñ 0 of evolved states Γ ε ptq, we make the following very general assumption on2 Γ ε p0q " Γ ε : which is for instance satisfied if the state scales with ε as in (1.3), or if it is formed by a coherent superposition of vectors with a finite number of force carriers.Such assumption is sufficient to prove the existence of a subsequence tε n u nPN Ñ 0 such that Γ εn converges to a quasi-classical state m in the sense of the Definition 1.1 below.For the polaron and Pauli-Fierz models, an additional assumption is necessary to study the limit ε Ñ 0 of Γ ε ptq, due to the fact that such models are "more singular" than the Nelson model: Finally, in order to ensure that no loss of mass occurs along the weak limit, or, equivalently, that the quasi-classical limit point m is still normalized and }mphq} L 1 pH q " 1, we also need a control of the particle component of the state Γ ε .We thus define the reduced density matrix for the particles as γ ε :" tr Kε Γ ε P L 1 `,1 pH q, (1.18) and impose the following alternative conditions on γ ε : We are going to comment further about the above conditions in Remark 1.6 and Remark 1.7, but we point out here that the two assumptions do not seem to be strictly related and their implications besides the conservation of the norm are quite different.A simple but relevant case in which (A2 1 ) is trivially satisfied is given by product states of the form γ b ς ε with γ P L 1 `,1 pH q independent of ε.Contrarily, Assumption (A2) seems at a first glance more arbitrary, but it could be put in relation with the physics of the model (see Remark 1.8).
Let us define by p Γ ε the noncommutative Fourier transform or generating map of a state for any η P h, where W ε pηq is the Weyl operator on K ε : W ε pηq :" e ipa : ε pηq`aε pηqq . (1.20) Analogously, to any state-valued measure m P M ph; L 1 `pH qq there corresponds the Fourier transform p mpηq :" if and only if p Γ ε pηq Ñ p mpηq pointwise for all η P h in weak-˚topology in L 1 pH q, i.e., when testing against compact operators B P L 8 pH q.
The above Definition 1.1 is given in terms of the Fourier transforms in order to completely characterize the limit quasi-classical measure m.On the other hand, from the physical point of view, it is relevant to study the convergence of expectation values of quantum observables, which is discussed in § 2 and specifically in Theorem 1.14.Note that in light of Proposition 2.3, assumption (A1) guarantees that any such Γ ε admits at least one limit point in the sense of Definition 1.1.

Remark 1.2 (Reduced density matrix).
We point out that the reduced density matrix γ ε for the particle system given in (1.18) can be obtained by evaluating the noncommutative Fourier transform (1.19) in η " 0, i.e., m can be easily seen to imply that where we have denoted by w ´op the weak operator topology.

Remark 1.3 (Product states).
As a special case, we observe that, if Γ ε is a physical product state 3 , i.e., if there exist γ P L 1 `,1 pH q and µ m and γ m pzq " γ. (1.24) 3 Product states are the mathematical formulation of the fact that the two parts of the system are independent.
Since ε characterizes only the behavior of the field, it is not physically relevant to put an ε-dependence on the particle part.
The proper definition of the convergence between scalar measures was given in [CF18], but it coincides with Definition 1.1 when H " C.
Our main result (see Theorem 1.4 and Corollary 1.12 below) is that, for all t P R, Γ εn ptq also converges to the quasi-classical state m t defined by the norm Radon-Nikodým decomposition where U t,s pzq is the above mentioned quasi-classical two-parameter unitary group of evolution, that turns out to be weakly generated by the time-dependent Schrödinger operator Notice again that the pushforward in (1.25) does not affect the Radon-Nikodým derivative U t,0 pzqγ m pzqU : t,0 pzq.The interplay between the quasi-classical limit and the time-evolution can be summed up in the following commutative diagram involving the Radon-Nikodým derivatives e ´iHεt εÑ0 εÑ0 (1.25) (1.28) where we have decomposed the initial state-valued measure as dmpzq " γ m pzqdµ m pzq, with γ m P L 1 `,1 pH q and µ m P M phq, and the convergence is always along a given subsequence tε n u nPN .
We state now the first result in detail.Recall that we say that m P M `h; L 1 `,1 ˘is a probability measure whenever }mphq} L 1 pH q " 1.
Let νpεq be such that ενpεq Ñ ν P t0, 1u, when ε Ñ 0 and let Γ ε P L 1 `,1 pH b K ε q be a state satisfying assumptions (A1) and (A2) or (A2 1 ).Let also (A1 1 ) be satisfied for the polaron and Pauli-Fierz models.Then, there exist at least one subsequence tε n u nPN and one probability measure m P M `h; L 1 `,1 pH q ˘, such that and, if (1.29) holds, then for all t P R, where m t is given by (1.25).
Let us point out that, as anticipated above, the limit measure m at initial time, according to Definition 1.1, might depend on the choice of the subsequence tε n u nPN Ñ 0. However, we stress that the convergence at time t stated in (1.30) occurs along the same subsequence.
The result is proven assuming that either (A2) or (A2 1 ) holds true.The only part of the statement affected by such assumptions is that the limit measure is a probability measure.If the aforementioned assumptions are dropped, a loss of probability mass may in fact occur as k Ñ `8, and we only know that }mphq} L 1 pH q ď 1.The characterization of the quasi-classical evolved measure given by (1.30) is however still true, and since such evolution preserves the mass of the state-valued measure, we have that }m t phq} L 1 pH q " }mphq} L 1 pH q for all t P R. In other words, the commutative diagram (1.28) is true without assuming (A2) or (A2 1 ), and the only loss of mass for the quasi-classical state could happen at the initial time.As suggested by the fact that physical factorized states γ b ς ε do not lose mass, the loss of mass is due either to a "bad" correlation between the field and particle subsystems, or to a somewhat artificial dependence of the particle subsystem on the quasi-classical parameter in an uncorrelated state.
The implications and the meaning of assumptions (A2) and (A2 1 ) is quite different.For instance, (A2 1 ) provides a uniform control on the reduced density matrix γ ε but has little physical consequences.Assumption (A2) on the other hand is somehow stronger, since it implies the convergence of Γ ε to m in weak sense, i.e., when tested against bounded operators (see the proof of Proposition 2.3).Such a stronger convergence holds true however only at initial time, and its propagation along the time-evolution is typically impossible (see also next Remark 1.8).
Remark 1.8 (Topology).The convergence in (1.30) holds in the sense of Definition 1.1, i.e., in weak-˚topology for the Fourier transforms (1.19) and (1.21).Whether such a convergence could actually be lifted to weak topology, i.e., with a test against bounded operators, is an intriguing question.
We do not address it here, but, as anticipated, this is the case at initial time if (A2) is assumed.Note that the propagation in time of such a priori bound is far from obvious and therefore it is difficult to show that the convergence can be lifted also at later times.A notable exception is given by trapped particle systems, i.e., when K 0 has compact resolvent and thus one can take A " pK 0 `1q δ , for some δ ą 0, in (A2).Thus, in this case the assumption TrpΓ ε pK 0 `dG ε pωqq δ q ď C on the initial state is sufficient to strengthen the convergence at any time.
As already remarked in § 1.4, states satisfying (A1) yield effective potentials V t that are "regular".For example, no confining potential can be obtained with such quasi-classical states.It is possible to obtain more general effective potentials relaxing assumption (A1) to accommodate states whose limit are cylindrical measures [Fal18b], however the analysis becomes more complicated.In the polaron model, for coherent states, whose cylindrical measure is concentrated in a single "singular" point (a suitable tempered distribution), the analysis has been carried out in [CCFO19] to obtain an effective (time-dependent) point interaction.
Before proceeding further we discuss in some detail the scaling factor νpεq that appears in front of the free energy of the field in the Hamiltonian H ε .Physically, one should distinguish between two relevant situations: νpεq " 1, and νpεq " 1 ε ; all the other possibilities are physically less relevant, and yield the same qualitative results, up to rescaling of the parameters.Let us consider separately the two cases.Let us remark first however that, despite the fact that the two cases yield different evolutions for the classical field, the interaction is always too weak to cause a back-reaction of the particles on the field, when ε Ñ 0. Thus, the quasi-classical field can indeed be seen as an environment.
Remark 1.10 (ν " 0).When νpεq " Opε ´δ q, δ ă 1, the quasi-classical field remains constant in time.In fact, in such a case ν " lim εÑ0 ενpεq " 0, and therefore U t,s pzq " U t´s pzq is the strongly continuous group generated by the self-adjoint operator K 0 `Vpzq, with, e.g., Vpzq " ř N j"1 2Re xλpx j q| zy h for the Nelson model.Also, the measure µ t is constant: µ t " µ for all t P R. Therefore, in the scaling yielded by νpεq " Opε ´δ q the radiation field does not evolve.Let us remark that, in the case of the polaron, this is the scaling equivalent, up to suitable rescalings, to the well-known strong coupling regime.
Remark 1.11 (ν " 1).When ν " 1, e.g., if νpεq " 1 ε , the quasi-classical radiation field evolves in time in a nontrivial way, obeying a free field equation, and therefore the effective evolution operator for the particles U t,s pzq has a time-dependent generator.For the regularized Nelson model, such a free evolution is given by the Klein-Gordon-like equation where ωpDq is the pseudodifferential operator defined by the Fourier transform of the function ω; for the Pauli-Fierz model it is given by the free Maxwell equations in the Coulomb gauge, and for the polaron by the equation pB 2 t `1qA " 0. Here, for clarity, we have written such equations in the usual form, that involves the real field A and its time derivatives.Throughout the paper however, we use the complex counterpart of such real field, that we denoted by z, and which is given in terms of A, e.g., in the regularized Nelson model, by Hence, the evolution equation for z becomes iB t z " ωz.
A consequence of Theorem 1.4 is that, for any compact operator B P L 8 pH q, we have that its Heisenberg evolution satisfies There is also a counterpart of the above statement for the particle degrees of freedom alone: for any Γ ε as in Theorem 1.4, the following weak-* convergence holds in L 1 pH q, i.e., the particle state obtained by tracing out the field degrees of freedom evolves as ε Ñ 0 into the r.h.s. of the above expression.When the state is a product state, the above result can be made more explicit (see also Remark 1.3): Corollary 1.12 (Quasi-classical evolution of product states).
Let ς ε P L 1 `,1 pK ε q be a field's state such that for all γ P L 1 `,1 pH q, γ b ς ε satisfies assumption (A1), and (A1 1 ) for the polaron and Pauli-Fierz models, so that there exists µ P M phq such that ς εn ÝÝÝÝÑ nÑ`8 µ. (1.34) Then, for all B P L 8 pH q and all t P R, Remark 1.13 (Bounded operators).It would obviously be more satisfactory to extend the above result to bounded operators B P BpH q.However, this can not be done in full generality because the convergence in Definition 1.1 holds in weak-˚topology.As explained in Remark 1.8, one can lift the convergence to weak topology, and thus extend the statement above to bounded observables, if an additional regularity on the initial state is assumed and such a regularity can be propagated by the dynamics, which can be done for example whenever the particle system is trapped.
The analogue of the above Corollary 1.12 for non-product states and more complicated observables, i.e., self-adjoint operators acting on the full Hilbert space, is more involved to state and holds true only for a subclass of such operators.We indeed introduce a class of operators on H b K ε , consisting of polynomials with m creation and n annihilation normal ordered operators, with arguments possibly depending on the particle's positions: explicitly, we consider operators Op Wick εn pFq obtained as the Wick quantization of symbols F P S n,m , i.e., of the form xz |λ 1 px j q y h ¨¨¨xz |λ ℓ px j q y h xλ ℓ`1 px j q| zy h ¨¨¨xλ ℓ`m px j q| zy h , (S ℓ,m ) where λ j P L 8 pR d ; hq, j " 1, . . ., m `ℓ.
To state the result, we also need to make more restrictive assumptions on the initial state Theorem 1.14 (Quasi-classical evolution in the Heisenberg picture).
Let Γ ε P L 1 `,1 pH b K ε q be a state satisfying assumption (A δ ), so that there exists m P M `h; L 1 `pH q ˘such that Γ εn ÝÝÝÝÑ nÑ`8 m. (1.36) Then, for all F P S ℓ,m , with ℓ`m 2 ă 2δ, for all t P R and for all S, T P BpH q, such that either S or T P L 8 pH q, The constraint δ " 1 for the Pauli-Fierz model is due to some technical difficulties in propagating in time higher order regularity of the number operator, due to the fact that the number operator and the field's kinetic term are not comparable in such case, since the field carriers may be massless.
The rest of the paper is organized as follows.In § 2 we develop the main technical tools for the subsequent analysis, that we call quasi-classical analysis, in analogy with the more familiar semiclassical analysis.In fact, quasi-classical analysis is semiclassical analysis on a bipartite system, where only one part is semiclassical, and the other is quantum.In § 3 we describe the relevant features of the microscopic Nelson model, that we use as a reference to explain the strategy of the proof of Theorem 1.4.We then take the limit ε Ñ 0 of the microscopic integral equation of motion in § 4, while we discuss in § 5 the uniqueness of solutions to the quasi-classical equation obtained performing the aforementioned limit.In § 6 we put together the results obtained in § 2 to 5, and prove Theorem 1.4 for the Nelson model, and thus consequently also Corollary 1.12 and Theorem 1.14.In § 7, we provide the technical modifications needed to prove the aforementioned theorems, for the Pauli-Fierz and polaron models.Finally, in Appendix § A, we collect some results concerning state-valued measures.

Quasi-Classical Analysis
In this section we introduce the quasi-classical asymptotic analysis, needed to study the dynamical limit of quasi-classical systems.In particular, we have to develop a semiclassical theory for operator-valued symbols, since the latter are crucial to characterize the interaction part of the dynamics.The key tools presented here are ‚ the convergence of regular states to state-valued measures in the quasi-classical limit (Proposition 2.3) in the sense of Definition 1.1; ‚ the convergence of expectation values of suitable classes of operators to their classical counterparts (Proposition 2.6).
We start by clarifying the notion of state-valued measure.
An additive measure m on a measurable space pX, Σq is H -state-valued iff ‚ mpSq P L 1 `pH q for any S P Σ; ‚ mp∅q " 0; ‚ m is unconditionally σ-additive in trace norm.
We denote by M pX, Σ; H q or simply M pX; H q the space of H -state-valued measures.A H -state-valued measure is a probability measure iff mpXq L 1 " 1.
Using the Radon-Nikodým property and positivity, there is a simple characterization of state-valued measures: Proposition 2.2 (Radon-Nikodým decomposition).
For any measure m P M pX, Σ; H q, there exists a scalar measure µ m P M pX, Σq, with µ m pXq " mpXq L 1 pH q , and a µ m -a.e.defined measurable function γ m : X Ñ L 1 `,1 pH q, such that for any S P Σ, mpSq " ż S dµ m pzq γ m pzq, (2.1) with the r.h.s.meant as a Bochner integral.In addition, a measure m P M pX, Σ; H q is a probability measure iff µ m is a probability measure.We call pµ m ; γ m pzqq the Radon-Nikodým decomposition of m.
Proof.First of all we point out that the separable Schatten space of trace class operators L 1 pH q has the Schatten space of compact operators L 8 pH q as predual, and therefore it has the Radon-Nikodým property (see, e.g., [DP40,DU77]).In addition, since m takes values in positive operators, we can define its "norm" measure as mp ¨q :" mp ¨q L 1 pH q . (2.2) In fact, m is a scalar measure (see § A, Proposition A.5) such that m !m !m, i.e., m and m are absolutely continuous w.r.t. each other.The latter property can indeed be easily seen as follows: mpSq " 0, as an element of the vector space L 1 `pH q, if and only if mpSq " mpSq L 1 pH q " 0.
Moreover, the Radon-Nikodým property guarantees the existence of the Radon-Nikodým derivative dm dµ P L 1 `X, dµ; L 1 `pH q ˘, such that mpSq " for any measurable S P Σ and for any scalar measure µ such that m is absolutely continuous w.r.t.µ.
In our setting, compared to the more general case of Banach-space-valued vector measures, there is an additional notion of positivity, as discussed above.Such notion naturally singles out a given scalar measure, w.r.t. which m is absolutely continuous.Such measure is the "norm" measure m defined in (2.2).Indeed, combining the mutual absolute continuity of m and m with the existence of the Radon-Nikodým derivative, we deduce that, for any measurable and that, m-a.e., dm dm ‰ 0 .
Therefore, we can rewrite and setting dµ m pzq :" we obtain the sought Radon-Nikodým decomposition.
Let now F : X Ñ BpH q be a measurable function with respect to the weak-* topology on BpH q.It is then natural to define the (L 1 -Bochner) integrals of f with respect to m as follows: for any S P Σ, ż (2.9) Notice that one has to keep track of the order inside the integral, i.e., putting the measure on the right or on the left of the integrand is not the same, because γ m might not commute with F pzq, since both are operators on H .In Appendix § A we characterize the state-valued measures and corresponding integration in more detail and from a more general perspective, possibly useful for the investigation of quasi-classical coupled quantum field theories.
State-valued measures are important since they are the quasi-classical counterpart of quantum states (see [Fal18b] for a detailed discussion).Operator-valued symbols, such as the aforementioned F, are correspondingly the quasi-classical counterpart of quantum observables.From a general point of view, we can summarize the main objective of quasi-classical analysis as follows: let Op ε pFq be a "quantization" of F acting on H b K ε , where the space K ε carries a semiclassical representation of the canonical commutation relations corresponding to a symplectic space of test functions pV, σq, and let Γ ε be a quantum state converging to the Borel state-valued measure m on the space V 1 of suitably regular classical fields, then we would like to prove that lim Fpzq dmpzq ; (2.10) where the convergence holds in a suitable topology of L 1 pH q.
It is however difficult to obtain results such as the above for general symbols and quantum states.The most important obstruction is indeed the difficulty of define a proper quantization procedure for symbols acting on infinite dimensional spaces.However, for the theories of particle-field interaction under consideration (Nelson, polaron, Pauli-Fierz), the interaction terms in the quasi-classical Hamiltonians contain only symbols of a specific form.We can therefore restrict our analysis to such type of symbols.
Let us recall that we are considering the following concrete setting: H " L 2 pR dN q, where d is the spatial dimension on which the particles move and N is the number of quantum particles; K ε " G ε phq, the symmetric Fock space over the complex separable Hilbert space h, carrying the standard ε-dependent Fock representation of the canonical commutation relations " a ε pzq, a : ε pηq Finally, we are interested in the case V 1 " h, i.e., the space of test functions coincides with the space of classical fields.The type of symbols F is given by the class defined in (S ℓ,m ), i.e., xz |λ 1 px j q y h ¨¨¨xz |λ ℓ px j q y h xλ ℓ`1 px j q| zy h ¨¨¨xλ ℓ`m px j q| zy h , where the functions λ j P L 8 pR d ; hq for any j P t1, . . ., ℓ `mu should be considered as fixed "parameters", and Fpzq acts as a multiplication operator on L 2 pR dN q.
Since F is a polynomial symbol with respect to z and z ˚, it is natural to quantize it by the Wick quantization rule.For such simple symbols the Wick rule has a very easy form: substitute each z with a ε and each z ˚with a : ε , and then put the so obtained expression in normal order, by moving all the creation operators to the left of the annihilation operators.Therefore, we obtain ε `λ1 px j q ˘¨¨¨a : ε `λℓ px j q ˘aε `λℓ`1 px j q ˘¨¨¨a ε `λℓ`m px j q ˘, (2.11) as a densely defined operator on L 2 pR dN q b G ε phq.
In order to prove a weak convergence as in Eq. (2.10) for T Op Wick ε pFqS, with S, T P BpH q, we need suitable hypotheses on the quantum state Γ ε , and some preparatory results.The following condition ensures that all the quasi-classical Wigner measures corresponding to a state Γ ε P L 1 `,1 `H b K ε ˘are concentrated as Radon L 2 -state-valued probability measures on h.Recall the definition (1.20) of the Weyl operator W ε pηq, η P h, and the Fourier transform (1.21) of a measure m P M ph; L 1 `pH qq.
Proposition 2.3 (Convergence of quantum to classical states).
Let Γ ε P L 1 `,1 `H b K ε ˘be such that there exist δ ą 0, so that Tr `Γε pdG ε p1q `1q δ ˘ď C . (2.12) Then, there exists at least one subsequence tε n u nPN and a H -state-valued cylindrical measure m (that may depend on the sequence) such that in the sense of Definition 1.1.Furthermore, all cluster points m of Γ ε are state-valued Radon measures on h, and, for any 0 ď δ 1 ď δ, there exists In order to prove the last part of the above proposition, we need a couple of preparatory results, that will be useful as well in § 7.1.
Lemma 2.4.Let T be a densely defined self-adjoint operator on H , and 1 m pT q its spectral projection on the interval r´m, ms, m P N.Then, the set of operators K :" tB m :" 1 m pT qB 1 m pT q , B P L 8 `pH q, m P Nu , (2.16) separates points in L 1 `pH q w.r.t. the weak-˚topology.Proof.Let γ P L 1 `pH q be such that, for all B m P K, tr H pγB m q " 0 .
Let ř j λ j |ψ j yxψ j | be the decomposition of γ.Then, it follows that for all j P N, by positivity of B. Taking the limit m Ñ `8 of the last equation, one obtains that for any B P L 8 `pH q and j P N, xψ j |B| ψ j y H " 0 , (2.17) but, taking in particular B " |ψ j y xψ j |, we get ψ j " 0 for any j P N, and therefore γ " 0.
Proposition 2.5 (Convergence of general state sequences).
‚ tr H pT γ ε T q ď C for some self-adjoint T P LpH q, where γ ε is given by (1.18).

Then,
T where the latter is defined by the Radon-Nikodým decomposition pµ m , T γ m pzq T q .
Proof.Since tr H pT γ ε T q ď C, T Γ ε T is a quasi-classical family of states and thus it follows that there exists a generalized subsequence pΓ εn α q αPA of Γ εn and a cylindrical state-valued measure n such that (see [Fal18b] for additional details): ‚ T Γ εn α T converges to n when tested on the Weyl quantization of smooth cylindrical symbols, ‚ tr H `T p Γ εn α pηq T B ˘converges to tr H `γpηqB ˘for all η P h and B P L 8 pH q, where γpηq P L 1 pH q has yet to be determined.Now, let K P K.Then, T KT P L 8 pH q and therefore lim However, the set K separates points by Lemma 2.4, and therefore we can conclude that γpηq " T p mpηq T .
On the other hand, an analogous reasoning when testing with the Weyl quantization of smooth cylindrical symbols yields that n " T m T .
Therefore, we conclude that T Γ εn α T ÝÝÑ αPA T m T .Finally, let Γ εn α 1 be any generalized subsequence such that for any η P h and B P L 8 pH q Then, repeating the above reasoning it follows that γ 1 pηq " T p mpηq T .In other words, the cluster point is unique, and therefore T Γ εn T ÝÝÝÝÑ nÑ`8 T m T .
Proof of Proposition 2.3.The key result about the weak-˚convergence in the semiclassical case is proven in [AN08, Theorem 6.2].The generalization to the quasi-classical setting is trivial: for all compact operators B P L 8 pH q and all η P h, one immediately gets where B :" B b 1.Moreover, the Fourier transform p m : h Ñ L 1 pL 2 q identifies uniquely the measure m by Bochner's theorem [Fal18b].The bound (2.14) is also an immediate extension of [AN08, Theorem 6.2] to the quasi-classical case.
It remains to prove that under either Assumption (A2) or (A2 1 ) m is a probability measure.Let us start assuming (A2).Then, by Proposition 2.5, we have that for any bounded B P BpH q, and η P h: In particular, for η " 0 and B " 1, we have that It is clear that together with Proposition 2.3, all the other results that hold in semiclassical analysis for infinite dimensions can be adapted to quasi-classical analysis, considering the semiclassical symbols and corresponding quantizations in tensor product with the identity acting on H , replacing Wigner scalar measures with state-valued Wigner measures, and replacing convergence of the trace with L 1 pH q-weak-˚convergence of the partial trace, i.e., one should test the partial traces and integrals with compact operators.

Proposition 2.6 (Convergence of expectation values).
Let F P S ℓ,m , and let Γ ε P L 1 `,1 pH b K ε q.Assume that there exist δ ą n`m 2 , such that Tr ´Γε pdG ε p1q `1q δ ¯ď C . (2.21) m, for any S, T P BpH q, B P L 8 pH q and η P h, with analogous statement when the positions of Γ εn and T Op Wick εn pFqS are exchanged.
To prove Proposition 2.6, we need the following preparatory lemma, which introduces the approximation of F by simple functions.
Lemma 2.7.Let F P S ℓ,m .Then, there exists a sequence of operator-valued functions tF M u M PN , F M : h Ñ BpH q, such that ‚ for all z P h, lim }Fpzq ´FM pzq} H " 0 ; (2.23) ‚ F M pzq acts as the multiplication operator by where J : N Ñ N, ϕ j,l P h, l P t1, . . ., ℓ `mu, and 1 B j is the characteristic function of the Borel set B j Ď R d and the B j are pairwise disjoint.
Proof.It is sufficient to prove the convergence in the case N " 1, n " 1, m " 0, since the case N " 1, n " 0, m " 1 is perfectly analogous, and the general one N P N, n P N, m P N, can be obtained combining the approximation for each term of the product within each term of the sum and possibly reorder the sum.So let us restrict to the case Fpzq " xz|λpxqy h , x P R d , acting as a multiplication operator on H " L 2 pR d q.Since both Fpzq and Now, let us fix z P h and consider Fpzq " F z pxq only as a function of x P R d .We can decompose F z pxq " F R pxq `iF C pxq, and split both the real and imaginary part as F R{C pxq " F R{C,`p xq ´FR{C,´p xq.Setting K :" λ L 8 pR d ,hq , we can partition the real positive half-line as (2.26) Let us now focus on the real positive part F R,`p xq: we can introduce the measurable sets D `:" F ´1 R,`p Aq, D m :" F ´1 R,`p A m q .By construction, D `" ∅, while , for all m P t1, . . ., M u, there exists η m P h such that @ η mˇzD h P A m .For any given x P R d , there is a single m P t1, . . ., 2 M u such that F R,`p xq P A m. Therefore, uniformly with respect to (2.27) Repeating the same procedure for the real negative and complex positive and negative parts, we obtain collections of sets and elements, respectively Let us now define the collection tB k u M 4 k"1 of disjoint Borel sets of R d for the simple approximation of Fpzq.We first identify k P t1, . . ., M 4 u with the image pm 1 , m 2 , m 3 , m 4 q with respect to some fixed set bijection  : t1, . . ., M u 4 Ñ t1, . . ., M 4 u, and then set (2.28) Therefore, we define ϕ k :" η m1 ´ηḿ 2 `ipξ m3 ´ξḿ 4 q and By construction, and therefore the convergence is proved.
Corollary 2.8.The approximating function F M pzq can be rewritten as xz |λ M,1 px j q y h ¨¨¨xz |λ M,ℓ px j q y h xλ M,ℓ`1 px j q| zy h ¨¨¨xλ M,ℓ`m px j q| zy h , (2.31) where λ M,j P L 8 pR d ; hq, j P t1, . . ., ℓ `mu, and lim M Ñ`8 }λ j ´λM,j } L 8 pR d ;hq " 0 . (2.32) Proof.Again, it is sufficient to prove the corollary for N " 1 and n " 1, m " 0, the other cases being direct consequences.The function λ M approximating λ is defined in (2.25) in the proof of Lemma 2.7, i.e., λ M pxq :" From the same proof it also follows that, for all z P h and for all x P R d , ˇˇxz |λpxq ´λM pxq y h ˇˇď 4K z h

M .
Therefore, it follows that ess sup }λpxq ´λM pxq} h " ess sup and thence the convergence is proved.
Proof of Proposition 2.6.Let us prove (2.22).Let us approximate Fpzq with F M pzq, as dictated by Lemma 2.7.The advantage of F M pzq is that its dependence on the z and x variables is separated, and thus its Wick quantization is the finite sum of tensor products of operators: a : ε pλ M,1 px j qq ¨¨¨a : ε pλ M,ℓ px j qqa ε pλ M,ℓ`1 px j qq ¨¨¨a ε pλ M,ℓ`m px j qq.The first term on the r.h.s.can be estimated using well-known estimates for creation and annihilation operators, the hypothesis on the expectation of the number operator, and Corollary 2.8: The proof is then concluded by taking the limit M Ñ 8 of the last expression, that by dominated convergence yields the sought result.

The Microscopic Model
Our aim is to study systems of nonrelativistic particles in interaction with radiation.As discussed previously, the techniques developed in this paper allow to study some well-known classes of explicit models (Nelson, polaron, Pauli-Fierz).We carry out here the detailed analysis only for the simplest example, the Nelson model, in order to convey the general strategy without too many technical details.The main adaptations needed for the polaron and Pauli-Fierz systems are outlined in § 7.
Let H b K ε " L 2 pR dN q b G ε `L2 pR d q ˘be the Hilbert space of the theory, then the Nelson Hamiltonian H ε is explicitly given by where K 0 " K 0 b 1 is the part of the Hamiltonian acting on the particles alone, being such that K 0 is self-adjoint on DpK 0 q Ă L 2 pR dN q, νpεq ą 0 is a quasi-classical scaling factor to be discussed in detail below, ω is the operator on L 2 pR d q acting as the multiplication by the positive dispersion relation of the field ωpkq, and λ P L 8 `Rd ; L 2 pR d q ˘": L 8 x L 2 k is the interaction's form factor.In addition, let us define the set of vectors with a finite number of field's excitations C 8 0 `dG ε p1q ˘: The question of self-adjointness of H ε has already been addressed in the literature and indeed we have the following: The operator H ε is essentially self-adjoint on DpK 0 q X DpdG ε pωqq X C 8 0 `dG ε p1q ˘.
Therefore, there exists a unitary evolution generated by H ε , U ε ptq " e ´itHε .
(3.3) Now for any normalized density matrix Γ ε P L 1 `,1 pH b K ε q, we denote by Γ ε ptq its unitary evolution by means of U ε ptq, i.e., The main aim of this paper is to characterize the asymptotic behavior as ε Ñ 0 of γ ε ptq :" tr Kε `Γε ptq ˘" tr GεpL 2 pR d qq `Γε ptq ˘. (3.5) As stated in Definition 1.1 and characterized in Proposition 2.6, the quasi-classical limit of a sufficiently regular state is determined by the weak convergence of its vector-valued noncommutative Fourier transform p Note that consequently γ ε ptq " " p Γ ε ptq ‰ p0q.The regularity of the state is given by (2.12), that should thus be satisfied at any time.It is therefore necessary to ensure a proper propagation in time of such a regularity.An estimate of that kind is however readily available for the Nelson model with cutoff: where c δ pεq :" max ) .
Since the exponential in the above inequality is bounded uniformly with respect to ε P p0, 1q, it follows that the bound (2.12) is satisfied by the state at any time with a suitable timedependent constant, provided it is satisfied by the state at t " 0: using that, for any δ P R `, which guarantees that the a priori bound (2.12) is preserved by the time evolution.
In analogy with the dynamical semiclassical limit for bosonic field theories (see, e.g., [AF14, AN15a, AF17]), the quasi-classical dynamics is characterized studying the limit ε Ñ 0 of the integral equation of evolution for the microscopic system.Let us sketch the main ideas: consider the family of states tΓ ε ptqu εPp0,1q, tPR at time t " 0, satisfying the bound (2.12).Then, we know that for each fixed t P R, there exists a subsequence ε n Ñ 0 such that Γ εn ptq ÝÝÝÝÑ nÑ`8 m t in the sense of Definition 1.1 by Propositions 2.3 and 3.2.In the next section we prove that it is actually possible to extract a common subsequence ε n k Ñ 0 such that for all t P R, Hence one only needs to characterize the map t Þ Ñ m t , and this is done by studying the associated transport equation, that is obtained passing to the limit in the microscopic integral equation of evolution.Let us provide some intuition on such strategy.For later convenience let us pass to the interaction representation and set r Γ ε ptq :" e ´iνpεqtdGεpωq Γ ε ptqe iνpεqtdGεpωq . (3.9) Then, the microscopic evolution can be rewritten as an integral equation, using Duhamel's formula: where accordingly r H ε ptq :" e ´iνpεqtdGεpωq pH ε ´νpεqdG ε pωqq e iνpεqtdGεpωq . (3.11) In addition, H ε ´νpεqdG ε pωq is the Wick quantization of an operator-valued symbol K 0 `Vpzq.Therefore, the quasi-classical analysis developed in § 2 suggests that the integral equation (3.10) shall converge, in the limit ε Ñ 0, to an equation for the measure r m t , obtained by replacing r Γ ε ptq ù r m t , H ε ´νpεqdG ε pωq ù K 0 `Vpzq, and substituting the quantum flow e ´iνpεqtdGεpωq in the phase space h " L 2 pR 3 q by its classical counterpart, i.e., z Þ Ñ e ´iνtω z, @z P L 2 pR 3 q. (3.12) In conclusion, we get the equation and the classical measure m t associated with the original state Γ ε ptq is simply given by the push-forward of r m t through the flow (3.12), i.e., Γ ε ptq ù m t " e ´iνtω 7 r m t .
(3.14)Such an equation is the integral form of a Liouville-type equation.Once the convergence of the microscopic to the quasi-classical integral equation has been established (see § 4), the crucial point is to prove that equation (3.13) has a unique solution that satisfies some properties, given by the a priori information that we have on the quasi-classical measure (see § 5).As a final step ( § 6), we show that the convergence is in fact at any time t ě 0 along the same subsequence tε n u nPN .Let us remark that, in order to make this heuristic strategy rigorous, some technical modifications are necessary, in particular it is necessary to pass to the full interaction representation.
We conclude the section with the rigorous derivation of the microscopic integral evolution equation for the Fourier transform of Γ ε ptq.By definition, for any η P h, the Fourier transform " p Γ ε ptq ı pηq is a reduced microscopic complex state for the particles, and therefore, if Γ ε is regular enough, its time evolution can be described by means of the microscopic generator H ε .It is technically convenient to use the evolved state in interaction picture, i.e., Υ ε ptq :" e itpK 0 `νpεqdGεpωqq Γ ε ptqe ´itpK 0 `νpεqdGεpωqq , (3.15) in place of Γ ε ptq, and therefore study the integral equation for p Υ ε ptq.
Remark 3.3 (Regularity propagation for Υ ε ).Thanks to the commutativity of e itpK 0 `νpεqdGεpωqq with dG ε p1q, one can easily realize that the results stated in Proposition 3.2 and, consequently the bound propagation in (3.8), hold true also for the density matrix Υ ε ptq in the interaction picture with the same constants.
weakly in L 1 pL 2 pR N d q, where ϕ ε p ¨q " a : ε p ¨q `aε p ¨q is the Segal field.
Proof.The proof is obtained adapting [AF14, Proposition 3.5].The differences here are only the presence of an arbitrary bounded particle observable, and that the Weyl operator acts only on the field's degrees of freedom.Therefore, we omit the details.

The Quasi-Classical Limit of Time Evolved States
In this section we focus on the quasi-classical limit ε Ñ 0 of the Fourier transform p Υ ε ptq of time evolved states in the interaction picture.The first and most relevant step is the proof that it is possible to extract a common subsequence for the convergence of p Υ ε ptq at any time (Proposition 4.3), which in turn follows from the uniform equicontinuity of p Υ ε (Proposition 4.2).Finally, we show that the limit measure satisfies the transport equation (Proposition 4.5) of Lemma 3.4.
Let us start with a preparatory lemma.
For any 0 ă δ ď 1{2, there exists a finite constant c δ such that for all η, ξ P h, We are now able to prove uniform equicontinuity of Then, " p Υ ε p ¨qı p ¨q : R ˆh Ñ L 1 pH q is uniformly equicontinuous w.r.t.ε P p0, 1q on bounded sets of R ˆh, if we endow L 1 pH q with the weak-˚topology.
Proof.Let us fix B P L 8 pH q, and pt, ηq, ps, ξq P R ˆh, with 0 ď s ď t.Then, ˇˇtr H "´" p Υ ε ptq Let us consider the two terms separately.Making use of Lemma 3.4, we obtain where r Bpτ q :" e ´iτ K 0 Be iτ K 0 .Therefore, where we have used the identity Next, we apply [AF14, Corollary 6.2 (ii)] and (3.8), which follows from Proposition 3.2, to deduce The second term pIIq is bounded using again Proposition 3.2 and Lemma 4.1, and the fact that e itpK 0 `νpεqdGεpωqq commutes with dG ε p1q: This concludes the proof.
By means of Proposition 4.2, we are now in a position to prove the existence of a common subsequence, convergent for all times.
Then, for any sequence tε n u nPN , with ε n Ñ 0, there exists a subsequence tε n k u kPN , with ε n k Ñ 0, and a family of state-valued probability measures tn t u tPR indexed by time, such that for all t P R, Υ εn k ptq ÝÝÝÝÑ kÑ`8 n t . (4.5) Furthermore, for any T ą 0, there exists CpT q ą 0, such that, for any t P r´T, T s and for any δ 1 ď δ, ż Proof.Let E :" tt j u jPN Ă R be a dense countable subset of R, and let ε n Ñ 0. Using a diagonal extraction argument, and Propositions 2.3 and 3.2 (see also Remark 3.3 and (3.8)), there exists a subsequence ε n k Ñ 0 such that for all t j P E: In addition, since " p Υ ε pt j q ‰ pηq L 1 pH q ď 1, for any η P h and t j P E, it follows that p n t j pηq L 1 pH q ď 1, by Banach-Alaoglu's theorem.Furthermore, by Proposition 4.2, for any t j , t ℓ P E and for any B P L 8 pH q, ˇˇtr H "´" p Υ εn k pt j q ´p Υ εn k pt ℓ q where the constants on the r.h.s. are independent of k.Therefore, we can take the limit k Ñ `8 of the above inequality obtaining that, for all B P L 8 pH q, ˇˇtr H "`p n t j pηq ´p Now, let t P R be arbitrary.By density of E Ă R, there exists a sequence tt j u jPN of times in E, such that t j Ñ t.It follows that, for any η P h, p n t j pηq ( jPN is a weak-˚Cauchy sequence in the ultraweakly compact unit ball of the uniform space L 1 pH q.Thus, it converges when t j Ñ t.Hence, we define p n t pηq :" w lim jÑ`8 p n t j pηq , (4.8) where the limit is meant in the weak-˚topology.For any t P R, η Þ Ñ p n t pηq is an ultraweakly continuous function such that: Therefore, by Bochner's theorem for cylindrical vector measures [Fal18b, Theorem A.17], p n t is the Fourier transform of a unique state-valued cylindrical probability measure n t .Furthermore, by approximating Υ εn k ptq with Υ εn k pt j q and using the uniform equicontinuity of the noncommutative Fourier transform, one can prove that Here, we have used Proposition 2.6 to lift the convergence from the weak-˚to the weak topology.This in particular implies that n t is a probability Radon measure on h, because it is a Wigner measure of Υ ε ptq, satisfying the hypotheses of Proposition 2.3, thanks to Proposition 3.2.
To summarize, we have defined the common subsequence, and the family of state-valued probability measures obtained in the limit at any arbitrary time.The last inequality (4.6) is finally proved again combining Propositions 2.3 and 3.2.
Once rewritten for the density matrix Γ ε ptq, the result of Proposition 4.3 reads as follows: Corollary 4.4.If lim εÑ0 ενpεq " ν P R, then, under the same hypotheses of Proposition 4.3, there exists a common subsequence tn k u kPN , such that, for any t P R, Γ εn k ptq ÝÝÝÝÑ kÑ`8 m t :" e ´itK 0 `e´itνω 7 n t ˘eitK 0 , (4.9) where e ´itνω 7 n t is the measure obtained pushing forward n t by means of the unitary map e ´itνω : h Ñ h.Furthermore, for any T ą 0, any t P r´T, T s and any δ 1 ď δ, ż where CpT q is the same as in (4.6).
Proof.The result trivially follows from Proposition 4.3 by identifying e itK 0 Be ´itK 0 , with B P BpH q, as the bounded operator for the weak convergence, and using a very general result for linear symplectic maps, and their quantization as maps on algebras of canonical commutation relations [Fal18b, Proposition 6.1].
Therefore, we have obtained a common convergent subsequence, and a map t Þ Ñ n t of quasiclassical Wigner measures.The next step is to characterize such dynamical map explicitly by means of a transport equation, and study the uniqueness properties of the latter.In order to do that, we study the convergence of the integral equation provided in Lemma 3.4.

Proposition 4.5 (Transport equation for nptq).
Under the same assumptions of Proposition 4.3, the family of state-valued probability measures tn t u tPR as in (4.5) satisfies in weak sense, i.e., when tested against any B P BpH q, the integral equation e iτ K 0 2Re xλpx j q| ¨yh e ´iτ K 0 (4.12) is meant as a map from h to BpH q.
Proof.The existence of a common subsequence tε n k u kPN , ε n k Ñ 0, such that (4.5) holds true is guaranteed by Proposition 4.3.Let us now fix s, t P R: given the convergence along the subsequence at any time, it is possible to take the limit k Ñ 8 separately in all terms of the microscopic integral equation of evolution given in Lemma 3.4, traced against an arbitrary operator B P BpH q.
For the integral term (second term on the r.h.s. of (3.16)), we make use of Propositions 2.6 and 3.2, where the latter is used to prove that Υ ε pτ q satisfies the hypotheses of the former for all τ P rs, ts, using e ´iτ K 0 Be iτ K 0 as test operators.In order to do that, it is necessary to take the limit within the time integral.That is possible thanks to a dominated convergence argument, that makes use of the regularity assumption on Γ ε : for any bounded operator B, consider the integrand function Its absolute value is bounded, using standard Fock space estimates, as |Ipτ q| ď 2N }λ} L 8 pR d ;,hq B Tr ´Υε pτ qpdG ε p1q `1q Using Proposition 3.2 (see (3.8) and Remark 3.3) and the regularity assumption on Γ ε , it follows that the r.h.s. of the above expression is uniformly bounded by a finite constant.Hence, Ipτ q is integrable on any finite interval rs, ts, uniformly in ε.

Uniqueness for the Quasi-Classical Equation of Transport
In this section we study the properties of the transport equation for state-valued measures obtained in Proposition 4.5 as the quasi-classical limit of the microscopic evolution of states.
The first technical point is discussed in Lemma 5.1 below, where it is proven that it is possible to exchange freely the two of the aforementioned equation, which reads which implies the result via (5.3).
From now on, we assume that we are considering a solution of t Þ Ñ n t that satisfies (4.11).Let us introduce some terminology: a family of measures t Þ Ñ n t solving (4.11) in Proposition 4.5 for all η P h is called weak or weak-˚Fourier solution, if (4.11) holds true when tested against bounded or compact operators, respectively.Note that every weak or weak-F ourier solution is also a weak or weak-˚solution of (5.2), respectively, where the latter denote solutions of the equation obtained testing with smooth cylindrical scalar functions instead of Fourier characters.Let us specify further these last features.We first have to properly define the set of test cylindrical functions.

Definition 5.2 (Cylindrical functions).
A function f : h Ñ C is a smooth and compactly supported cylindrical function over Ph, where P is an orthogonal projector and dim Ph ă 8, iff there exists g P C 8 0 pPhq such that for all z P h, f pzq " gpPzq .
We denote by C 8 0,cyl phq the set of all smooth cylindrical functions.(5.5)

Now
Now, let us fix s P R as the initial time, and the corresponding n s " n as the initial datum.Then, the following t Þ Ñ n t is easily checked to be both a weak and weak-˚solution of (5.2): where r U t,s pzq is the two-parameter unitary group on H generated by the time-dependent generator r V τ pe ´itνω zq P LpL 2 q.Note that such an evolution two-parameter group exists for all z P h and t P R, since the r V t pe ´itνω zq is bounded operators on H (see, e.g., [RS75]).Furthermore, the solution given by (5.6) satisfies (5.3) at all times, provided the inequality is satisfied by the initial datum.
It just remains to prove the solution in (5.6) is actually unique.This of course might depend on the notion of solution we adopt, but proving weak-˚uniqueness, we get also uniqueness for stronger solutions (weak, Fourier weak-˚, and Fourier weak).As a matter of fact, the proof of uniqueness is actually independent of the notion of solution considered.
Proposition 5.3 (Uniqueness for the transport equation for n t ).Let s P R be the fixed initial time, and let n s " n P M ph; H q be a Borel state-valued measure such that ż h dµ n pzq z h ă C .
Then, the integral transport equation (5.1) admits a unique weak-˚solution n t , which satisfies (5.3), defined by its norm Radon-Nikodým decomposition pµ nt , γ nt pzqq " ´µn , r U t,s pzq γ n pzq r U : t,s pzq ¯. (5.7) Such solution is continuous and differentiable on every Borel set in the strong topology of L 1 pH q and its the derivative B t n t is a self-adjoint but in general not positive state-valued measure.
Proof.Any weak solution n t of the transport equation (5.1) or (5.2) satisfying (5.3) is continuous and can be weakly differentiated w.r.t.time on Borel sets.However, given the structure of equation (5.2), it is easy to realize that such a derivative actually exists in the strong topology of L 1 pH q and reads d B t n t pzq " ´i " r V t pe ´itνω zq, γ nt pzq ı dµ nt pzq . (5.8) To prove uniqueness, suppose that n t is a solution satisfying (5.3).Since we already know that (5.6) solves the equation, it is sufficient to prove that n t admits the Radon-Nikodým decomposition (5.6) (recall Proposition 2.2).In order to do that, let us set It is now possible to combine the results obtained in § 2 to 5, and thus prove Theorem 1.4.We first state and prove the result for the evolution in the interaction picture and under a stronger assumption on the initial datum, and then complete the proof by relaxing it and going back to the evolution for Γ ε ptq.Proposition 6.1 (Quasi-classical evolution in the interaction picture).Let Γ ε P L 1 `,1 pH b K ε q be such that there exists δ ą 1 2 , so that Clearly, n 0 " m.Moreover, by Proposition 4.5, n t is also a weak solution of (5.2), satisfying Lemma 5.1 and (5.3).The weak solution of (5.2) satisfying (5.3) is however unique by Proposition 5.3, and therefore n t has the Radon-Nikodým decomposition `µm , r U t,0 pzqγ m pzq r U : t,0 pzq ˘.
We now show that the convergence holds at any time along the original subsequence tε n u nPN .Let us take a convergent subsequence of Υ εn ptq at an arbitrary time t, i.e., such that Υ εn j ptq Ý ÝÝÝ Ñ ˘P L 1 `,1 pH b K ε q , (6.5) where r ą 0 and χ r p¨q " χp¨{rq, χ P C 8 0 pRq, with 0 ď χ ď 1 and χ " 1 in a neighborhood of zero.By functional calculus and (6.4), for any Therefore, ν t " m t .Since any subsequence extraction yields the same result, it follows that, for all t P R, Γ εn ptq Ñ m t , thus concluding the proof of Theorem 1.4.

Technical Modifications for Pauli-Fierz and Polaron Models
Theorem 1.4 is stated not only for the regularized Nelson model, but also for the Pauli-Fierz and polaron models as well.The strategy of the proof for these cases is identical to the one followed above for the Nelson model.However, one shall overcome some technical difficulties related to the fact that such models are "more singular".In particular, the foremost difficulty is given by the presence of terms of type ∇ ¨a# ε `λpxq ˘and their adjoints in the microscopic Hamiltonian H ε .In relation to that, one needs to propagate in time some further regularity of quantum states, in addition to what is done in Proposition 3.2 for the Nelson model.Finally, some care has to be taken in defining the effective limit dynamics U t,s pzq.We comment below on the technical adaptations needed to take care of such difficulties.
7.1.Quasi-Classical Analysis of Gradient Terms.In order to deal with terms of the form ∇ ¨a# ε `λpxq ˘, with λ P L 8 pR d ; h d q, one needs to extend the convergence proven in Proposition 2.6 to such kind of observables.This is done in two steps: first, it is possible to restrict the set of test observables using the set K defined in Lemma 2.4, for it separates points, and then we prove that with such a restriction the expectation values indeed converge (Proposition 7.1).In particular, Lemma 2.4, is used below for the convergence of gradient terms, to solve possible domain ambiguities whenever the gradient acts on the test operator: we end up with a form of the integral transport equation for the measure that holds only when tested with particle observables in K (recall (2.16)), setting T " K 0 , where K 0 is the self-adjoint free particle Hamiltonian.With such testing it still makes sense to study uniqueness of the solution, since the aforementioned set separates points.
Let us now consider the convergence of the expectation value of the gradient term.Let us recall that a # ε pf q stands for either a ε pf q or a : ε pf q, and correspondingly xf |zy # h stands for either xf |zy h or xz|f y h .Let us recall that in all the concrete models considered, we have that K 0 ě p ą ´8, and |∇|pK 0 `1 ´pq ´1 2 P BpH q. (7.1) Proposition 7.1 (Convergence of expectation values of gradient terms).
Let Γ ε P L 1 `,1 pH b K ε q be such that there exists δ ą 1 so that First of all, we observe that pK 0 ´pq 1 2 Γ εn pK 0 ´pq 1 2 is a positive operator and we can then consider its quasi-classical convergence as n Ñ `8: by Proposition 2.5, we have that pK 0 ´pq (7.5) The term Γ ε a ε pλpxqq ¨∇, in which the gradient acts directly on B converges by Proposition 2.6, since B j B P L 8 pH q for all j " 1, . . ., d and B P L 8 pH q.
It remains to discuss the term Γ ε ∇ ¨aε pλpxqq.This term requires suitable approximations.First of all, let us approximate each operator-valued symbol ) .
The operator a ε pϕ µ,k qW ε pηq is the product of the Weyl quantizations of two cylindrical albeit not compactly supported symbols, over the complex Hilbert subspace spanned by ϕ j,k and η.Therefore, by finite dimensional pseudodifferential calculus, for all M there exists a smooth compactly supported scalar symbol F which also allows to take the limit M Ñ `8.
7.2.Pull-Through Formula.In this section we discuss the so-called pull-through formula, needed both to characterize the dynamics in the quasi-classical limit for the polaron and Pauli-Fierz models: as we are going to see, the pull-through formula is key to propagate the a priori bounds on the initial state at later times.The formula holds for the Nelson, Pauli-Fierz, and polaron models, therefore H ε in this section stands for any of such Hamiltonians as defined above, although it is not needed for the Nelson model with ultraviolet cut off, as we are considering in this paper.Indeed, in that case, one can simply use the commutativity of H ε with dG ε p1q (see Remark 3.3).Before discussing the formula, let us remark that the Pauli-Fierz and polaron Hamiltonians are self-adjoint and bounded from below.There is an extensive literature concerning the self-adjointness of the Pauli-Fierz Hamiltonian (see, e.g., [Hir00, Hir02, Spo04, HH08, Fal15, Mat17] and references therein), which, under our assumptions, is self-adjoint on DpK 0 q X DpdG ε pωqq.The polaron Hamiltonian is also self-adjoint [FS14,GW16], but its domain of self-adjointness is not explicitly characterized.On the other hand, its form domain is known, and it coincides with the form domain of K 0 `νpεqdG ε p1q.
We do not prove the pull-through formula, since it is discussed in detail for the renormalized Nelson model in [Amm00], and its independence of the semiclassical parameter has been shown in [AF17].The models we consider here are "contained" in the renormalized Nelson model, namely all the terms in the Hamiltonians contained here are part or are analogous to some parts of the renormalized Nelson Hamiltonian.Therefore, they have already been discussed in the aforementioned papers.

Proposition 7.2 (Pull-through formula).
There exist two finite constants a, b independent of ε, such that for any ε P p0, 1q and for any Ψ ε P DpH ε q, dG ε p1qΨ ε H bKε ď a νpεq pH ε `bqΨ ε H bKε . (7.10) To study the quasi-classical limit of the Pauli-Fierz model for νpεq " 1 ε , it is not sufficient to use the pull-through formula, but it has to be combined with the following result (see [AFH19] for a detailed proof).

Proposition 7.3 (Propagation estimate).
Let H ε be the Pauli-Fierz Hamiltonian, with either νpεq " 1 or νpεq " 1 ε .Then, there exist two finite constants C 1 , C 2 independent of ε, such that for any ε P p0, 1q, for any Ψ ε P DpK 0 q X DpdG ε pωqq X DpdG ε p1qq and for any t P R, ` pK 0 `dG ε pωq `1q Ψ ε H bKε ‰ e C 2 |t| .(7.11) In addition, there exist two finite constants c, C ą 0 independent of ε, such that for any ε P p0, 1q and for any Ψ ε P DpH ε q " DpK 0 q X DpdG ε pωqq, c }pH ε `1qΨ ε } H bKε ď pK 0 `νpεqdG ε pωq `1q Ψ ε H bKε ď C pH ε `1qΨ ε H bKε .(7.12) Let us now outline in more detail how one can use the pull-through formula in the adaptations of the arguments to cover the other microscopic models.The main technicality is the propagation of the a priori bound and regularity of the state.For the polaron, this can be achieved by a direct application of Proposition 7.2: one can simply restrict the proof of Theorem 1.4 to states satisfying Tr `Γε `pK 0 `dG ε pωq `1q 2 `dG ε p1q 2 ˘˘ď C ; (7.13) Tr `Γε H 2 ε ˘ď Cνpεq 2 , (7.14) for any ε P p0, 1q.Let us remark that the regularity assumptions above are not propagated in time as they are, but they are rather used to control the following expectations at any time t P R: ‚ Tr `Γε ptqK 0 ˘; ‚ Tr `Γε ptqpdG ε p1q `1q 2 ˘.
The first expectation is bounded uniformly w.r.t.ε as in [CCFO19, Lemma 3.4], using the assumption (7.13).The second expectation is bounded using Proposition 7.2 and assumption (7.14).Once the bounds for the two quantities above are established at any time, it is possible to use Proposition 7.1 for the quasi-classical convergence of the interaction terms appearing in the integral equation.The result is then extended to general states satisfying (A1) by means of the procedure outlined in § 6.
For the Pauli-Fierz model one proceeds similarly.Theorem 1.4 is first proved for initial states such that Tr `Γε pK 0 `dG ε pωq `pdG ε p1q `1q 2 q ˘ď C (7.15) for ε P p0, 1q.The needed regularity of the expectation of the number operator at any time is then obtained thanks to Proposition 7.3.To bound the free particle part, one proceeds analogously as it was done for the polaron model in the aforementioned result [CCFO19, Lemma 3.4], the only difference being that instead of using KLMN-smallness, which would be true only for small values of the particles' charge, one uses again the number estimate of Proposition 7.3 to close the argument (see [Oli19] for additional details).Therefore, it is possible to apply Proposition 7.1 for the quasi-classical convergence of the gradient terms

Acknowledgements:
The authors would like to thank Z. Ammari for many helpful discussions during the redaction of the paper.M.C. and M.O. are especially grateful to the Institut Mittag-Leffler, where part of this work was completed.M.F. has been partially funded by the Swiss National Science Foundation via the grant "Mathematical Aspects of Many-Body Quantum Systems".M.O. has been partially supported by GNFM group of INdAM through the grant Progetto Giovani 2019 "Derivation of effective theories for large quantum systems".

F
pλq j pzq :" xλ j pxq| zy h j " 1, . . ., d, by means of Lemma 2.7, and let us denote its approximation by F pλq j,M .It follows that, using estimates analogous to the ones used in the proof of Proposition 2.6, ˇˇˇˇd ÿj"1 Tr !Γ ε B j " a ε pλ µ pxqq ´Op Wick ε ´F pλq j,M ¯ı B b W ε pηq) the r.h.s.does not depend on ε, and converges to zero when M Ñ `8.In addition, let us recall that the symbol F pλq j,M has the form: j,k | zy h 1 B k pxq , where JpM q P N, ϕ j,k P h, and B k Ď R d is a Borel set.Let us consider the convergence as ε n Ñ 0 of each term of the above sums separately, for M fixed.In other words, let us consider the convergence of Tr ´Γεn B j a εn pϕ j,k q1 B k pxqB b W εn pηq ¯" tr H ! tr Kε " Γ εn a εn pϕ j,k qW εn pηq ‰ B j 1 B k pxqB In other words, a generic normalized quasi-classical state consists of a probability measure µ m describing the environment, and a function γ m pzq describing how (almost) each configuration of the field affects the quantum particles' state.We provide more technical details about state-valued measures in § A. The quasi-classical equivalent of taking the partial trace with respect to the field's degrees of freedom is integrating w.r.t. the quasi-classical state-valued measure, i.e., for any operator-valued function Fpzq P BpH q, ż h dmpzq Fpzq " ż h dµ m pzq γ m pzqFpzq.
hq ¨¨¨ λ p ´λM,p L 8 pR d ,hq ¨¨¨ λ M,ℓ`m L 8 pR d ,hq Let us now discuss the limit n Ñ `8 of the second term on the r.h.s. of (2.34): for any B P L 8 pL 2 pR N d qq, using the first identity of (2.33), we obtain Tr ´Γεn T Op Wick εn pF M qSB b W εn pηq tr H ´tr Kε ´Γεn a : εn pϕ k,1 q ¨¨¨a ε pϕ k,ℓ`m qW εn pηq ¯1B k px j qSBT ¯.Now, on one hand we know that Γ εn Ñ m by Proposition 2.3, and on the other handa : εn pϕ k,1 q ¨¨¨a ε pϕ k,ℓ`m q " Op Wick εn ´xz |ϕ k,1 y h ¨¨¨xϕ k,ℓ`m | zy h ¯,where the scalar symbol on the r.h.s. is polynomial and cylindrical.Therefore, since1 B k px j qSBT P L 8 pL 2 pR N d qq, by the quasi-classical analogue of [AN08, Theorem 6.13], we Tr ´Γεn T Op Wick εn pF M qSB b W εn pηq ď CN pℓ `mq max pPt1,...,ℓ`mu }λ p ´λM,p } L 8 pR d ;hq , where we have used that the λ M,p L 8 pR d ,hq are all uniformly bounded with respect to M by (2.32).The r.h.s. of the above expression then converges to zero as M Ñ `8 by Corollary 2.8, uniformly w.r.t.ε n .
n t pηq " p n s pηq ´i ż t Vp ¨qe ´iτ K 0 " Nikodým decomposition n t " pµ nt , γ nt pzqq, γ nt pzqdµ nt pzq " γ ns pzqdµ ns pzq ´i ż t Let tn t u tPR be the family of state-valued measures as in Proposition 4.3, then " r V t pe ´itνω zq, γ nt pzq ‰ is Bochner µ nt -integrable for any t P R, and the norm of the integral is uniformly bounded w.r.t.t on compact sets.P R and for some Cptq ă `8.Moreover, for µ nt -almost all z P h, }γ nt pzq} L 1 pH q " 1, Tr ´Γε pdG ε p1q `1q δ ¯ď C .Let us consider Υ εn ptq: by Proposition 4.3, there exists a common subsequence ε n k Ñ 0 such that for all t P R, Υ εn k ptq ÝÝÝÝÑ ( jPN such that we have convergence at any time.Furthermore, again by Proposition 4.5, the limit points n 1 t are weak solutions of the transport equation.Therefore, by uniqueness of the solution, n 1 t " n t .Hence, all the convergent subsequences of Υ εn ptq have the same limit point, which implies that Υ εn ptq ÝÝÝÝÑ Proof.If Proposition 6.1 holds, then Corollary 6.2 is a direct consequence of Corollary 4.4.The proof of Theorem 1.4 is almost complete, it remains only to extend the result to states Γ ε P L 1 `,1 pH b K ε q satisfying the weaker condition that there exists δ ą 0 and C ă `8, such thatTr ´Γε pdG ε p1q `1q δ ¯ď C .pdG ε p1q `1qΓ ε χ r pdG ε p1q `1q Tr `χr pdG ε p1q `1qΓ ε χ r pdG ε p1q `1q (6.4)This is done by standard approximation techniques, using an argument originally proposed in [AN11, §2] (see also [AF14, §4.5]).Let us briefly reproduce the key ideas here.Let Γ Suppose now that Γ εn Ñ m, and for all r ą 0, let ε n k prq Ñ 0 be a subsequence and m prq a state-valued measure such that Γ prq ε n k prq Ñ m prq .Then, by Corollary 6.2, Γ , for any t P R, where the latter is defined by Theorem 1.4, with m prq in place of m.Finally, let us extract a subsequence ε n k ℓ pr,tq Ñ 0 such that Γ ε n k l pr,tq ptq Ñ ν t .By adapting the sum of its positive and negative parts.Therefore, denoting by m t the measure appearing in Theorem 1.4, we have ż 1 pH bKεq " o r p1q , when r Ñ 8, uniformly w.r.t.ε P p0, 1q.In addition, Γ t [Fal18b, §A.3], i.e., h d |µ νt ´µmt | ď ż h d ˇˇµ νt ´µm prq t ˇˇ`ż h d ˇˇµ m prq t ´µmt ˇˇ" o r p1q .
We prove the result for Γ ε `∇ ¨aε pλpxqq `aε pλpxqq ¨∇˘, the other cases being perfectly analogous.