Derivation of the Maxwell–Schrödinger equations: A note on the infrared sector of the radiation field

In this short paper, we derive the Maxwell–Schrödinger system of equations as an effective model describing a Bose–Einstein condensate of charged particles immersed in a coherent electromagnetic field. More precisely, we prove quantitatively that the Maxwell–Schrödinger system approximates well the many-body quantum evolution generated by the Pauli–Fierz Hamiltonian, provided that the total number of particles N is large, the particles are initially in a Bose–Einstein condensate, and the quantum nature of the field—quantified by the semiclassical parameter ℏ—is negligible. In particular, we focus on the combined regime $N∼1ℏ→+∞$⁠. Equivalently, the same effective dynamics approximates well N → ∞ bosons weakly interacting with a quantized electromagnetic field; see the discussion below.

This problem has already been studied by one of the authors, together with Pickl, in Ref. 1. The main focus here is to build on the results and techniques introduced there and to strengthen them by studying the convergence for the photons’ reduced density matrix. In the previous work, the quantum fluctuations around the coherent state of photons have been classified only by means of their energy. The extension to the reduced density matrix is physically relevant and mathematically nontrivial because the coherence of photons with small frequencies cannot be shown by the energy of the electromagnetic field due to its massless nature. We often refer to Ref. 1 throughout this paper, hopefully striking a good balance between being concise and being self-contained.

The Maxwell–Schrödinger system of equations describes the wave function φ of a quantum particle (with a nontrivial charge distribution κ) interacting with the classical electromagnetic field, described by the vector potential A and the electric field $E=−Ȧ$⁠. We choose the Coulomb gauge

and this makes, indeed, A and E the only dynamical degrees of freedom of the field. Let us also preliminarily define the current

The Maxwell–Schrödinger system thus takes the form

where $V[φ]$ is an interaction term for the quantum particle and $1−∇(∇⋅Δ−1)$ is the Helmholtz projection onto the subspace of divergence free vector fields. The choice of gauge, Coulomb’s in this case, can be seen as a constraint, for it is preserved by the Maxwell–Schrödinger flow. A typical example for the particle interaction $V[φ]$ could be

where W is an external potential and (⁠$v*$ |φ|²)φ is a nonlinear term, usually originating from a microscopic pair interaction. With this choice, (1.3) can used as a system of mean-field equations to model the dynamics of identical particles interacting via the potential v and the classical electromagnetic field which they generate. The Cauchy problem associated with (1.3) is obtained by fixing an initial datum (φ₀, A₀, E₀), subjected to the constraint ∇ · A₀ = 0. In order to do so, it is convenient to introduce the complex scalar fields $α0(⋅,λ)λ=1,2$ by defining

where $ϵλ(k)λ=1,2$ are the polarization vectors satisfying

which implement the Coulomb gauge. In fact, there is such a unique decomposition for any time, i.e.,

which respects both the Coulomb gauge and $Ȧ=−E$⁠. This makes it possible to rewrite Maxwell’s equations (same as in Ref. 2, Chap. 1.C) in terms of α_t and to consider the equivalent system

with initial datum (φ₀, α₀(·, 1), α₀(·, 2)). Throughout this article, we use $F[f]$ to denote the Fourier transform of f. As it will be clarified shortly, φ_t and α_t appear naturally as the effective counterparts of the microscopic dynamical variables. Note that the energy functional of the Maxwell–Schrödinger system is given by

with A being defined in analogy to (1.5). Global well-posedness for the Maxwell–Schrödinger system with $V[φ]=v*|φ|2φ$⁠, κ(x) = e δ(x) (where e is the electric charge of the Schrödinger particle), and $v(x)=e2|x|$ has been proven in Refs. 3 and 4. We will also consider only the case $V[φ]=v*|φ|2φ$⁠, but we may require the charge distribution κ to be extended, in order to well-define the microscopic system, as discussed below. Typical examples of charge distributions that we will consider are of the form

representing a charged particle with total charge $e∈R$⁠, distributed in a Gaussian fashion (“smoothed” spherical distribution of “diameter” σ), or

representing a sharp cutoff in momentum space with total charge $e∈R$⁠. Let us remark that globally neutral particles can be considered as long as they have a nontrivial charge distribution: for example,

yields null total charge but nontrivial dipole and quadrupole interactions with the electromagnetic field.

If the charge distribution is not concentrated in a single point and the potential v represents an electrostatic mean-field self-interaction, then the form of the latter changes as well: a physically sensible choice would be $v=κ*1|⋅|*κ$⁠. We will allow some liberty in the choices of κ and v; the specific requirements on the two will be made precise in Assumption 2.1. Concerning global well-posedness, let us remark that compared to the literature,^3,4 our choices for κ and v will be at most “better” (i.e., more regular) and therefore do not affect the proof in any way since both κ and v act by convolution in the equation.

For $m∈R$⁠, let $hm$ denote the weighed $L2(R3)⊗C2$-space with norm

Throughout this work, we will rely on the following statement, which results almost immediately from Proposition 1.1 (see the  Appendix).

The microscopic model corresponding to the Maxwell–Schrödinger system with mean-field self-interaction $v*$ |φ|²φ consists of many identical nonrelativistic particles—obeying Bose–Einstein condensation—interacting among themselves by means of a weak pair potential and with a quantized electromagnetic field in Coulomb’s gauge. Contrarily to the “classical” case, the microscopic model is known to be well-defined only for extended charges. Let us start by defining a Hilbert space $Hℏ(N)$ depending on two parameters $N∈N,ℏ∈R+$ as follows:

where $Ls2(R3N)$ is the natural Hilbert space of N identical bosons (the subscript s indicates symmetry under the interchange of variables) and $Γℏ$ is the second quantization functor associating with any (pre-)Hilbert space $h=L2(R3)⊗C2$ the corresponding Fock representation of the canonical commutation relations $[aℏ(f),aℏ*(g)]=ℏ⟨f,g⟩h,$ with ℏ being a semiclassical parameter measuring the degree of noncommutativity of the quantum field. The Fock representation is the natural one to describe noninteracting or regularized quantum field theories, the latter being the case here with $h≔L2(R3)⊗C2$⁠. With this interpretation, the limits N → ∞ and ℏ → 0 describe, respectively, the regimes in which the bosons are many and the quantum effects of the field are negligible. The time evolution is dictated by the Schrödinger equation,

where the Hamiltonian $HN,ℏ$⁠, called Pauli–Fierz Hamiltonian, is given by

where $μN,ℏ$ describes the coupling strength between the particles and the field, g_N is the coupling strength between the particles, κ and v are the charge distribution and the pair potential introduced previously,

is the field’s kinetic energy, with $aℏ♯(k,λ)$ being the polarized creation and annihilation operators satisfying the canonical commutation relations (CCR),

and³⁷ 

is the smeared quantized electromagnetic vector potential in Coulomb’s gauge. Let us remark that both $Âκ(x)$ and H_f depend on ℏ through the creation and annihilation operators, which have ℏ-dependent CCRs. The Hamiltonian $HN,ℏ$ is self-adjoint on $D(HN,ℏ)=D(HN,ℏ(0))$⁠, where $HN,ℏ(0)=HN,ℏ|μN,ℏ=gN=0$ whenever $⋅−1+⋅1/2F[κ]∈L2(R3)$ and v is Kato-infinitesimal with respect to −Δ.^5–7 

Our aim is to prove that the Maxwell–Schrödinger system emerges in some limit N → ∞ and/or ℏ → 0 as an effective model of the microscopic Pauli–Fierz dynamics. This is true only if we couple the parameters N, ℏ suitably and choose the coupling constants $μN,ℏ$⁠, g_N accordingly. A possible choice is given by N → ∞, $ℏ=1N$⁠, $μN,ℏ=1$⁠, and $gN=1N$⁠. In this regime, the electromagnetic field becomes classical inverse proportionally to the increasing number of bosons. At the same time, the coupling between the particles and field is of order one, while the coupling between pairs of particles becomes weak (of order $1N$⁠). Defining $N1/2aN−1(k,λ)$ and $N1/2aN−1*(k,λ)$ as new creation and annihilation operators leads to the mathematically equivalent but physically different choice N → ∞, ℏ = 1, $μN,ℏ=1N$⁠, and $gN=1N$⁠. Here, the physical interpretation is of many bosons that interact weakly both with the quantized electromagnetic field (coupling of order $1N$⁠) and among themselves (pair coupling of order $1N$⁠). Our result reads as follows.

Provided that we choose an initial microscopic state that is “close enough” to a non-interacting state representing a complete condensate and a coherent field of minimal uncertainty, then at any time t ≥ 0, the evolution keeps the state “close” (in the same sense as above) to an analogous configuration in which the one-particle wave function and the argument of the coherent field have been evolved by the coupled Maxwell–Schrödinger equations.

The described scaling regime has been considered in earlier works for the Nelson model with an ultraviolet cutoff,^8–11 the renormalized Nelson model,¹² and the Fröhlich model.^13,14 In Ref. 15, the Nelson model with an ultraviolet cutoff has been studied in a limit of many weakly interacting fermions. The classical behavior of quantum fields has also been proven in different scaling regimes.^16–31 We also would like to mention Ref. 32, which derives the Maxwell–Schrödinger equations in a nonrigorous manner by neglecting certain terms in the Pauli–Fierz Hamiltonian.

From now on, we will keep N as the single parameter and choose ℏ = 1, $μN,ℏ=1N$⁠, and $gN=1N$⁠. We will use the notations $H(N)=Hℏ(N)$⁠, $Fp=Γ1L2(R3)⊗C2$ with vacuum Ω, H_N = H_N,1, and Ψ_N = Ψ_N,1. Concerning the interaction potential and charge distribution, we will make the assumptions.

In order to state our result, we define for $ΨN∈H(N)$ the one-particle reduced density matrix of the charged particles $γΨN(1,0):L2(R3)→L2(R3)$ by

where Tr_2,…,N denotes the partial trace over the coordinates x₂, …, x_N and $TrFp$ is the trace over Fock space. In addition, we introduce the number of photon operators,

and the unitary Weyl operator $(f∈h)$⁠,

satisfying

Our result is as follows.

The rest of the article outlines the Proof of Theorem 2.2. We will proceed as follows:

1. We define a functional β[Ψ_N, φ, α], which measures if the charges of the many-body state Ψ_N form a Bose–Einstein condensate with condensate wave function φ and if the photons are in a coherent state with mean photon number $Nαh2$⁠.

2. Next, we show that the domain of β contains the solutions (φ_t, α_t) of (1.10) and Ψ_N,t of (1.17) from Theorem 2.2 for all t ≥ 0.

3. Afterward, we compute the change of β[Ψ_N,t, φ_t, α_t] in time.

4. Finally, we control the growth of β[Ψ_N,t, φ_t, α_t] with the help of Grönwall’s inequality. This concludes the proof.

In doing so, we will rely on the findings from Ref. 1 and rather explain how the original proof of Ref. 1 has to be adapted. Most of the modifications are necessary to show the invariance of the domain in step 2 and to compute the time derivative of β in step 3.

We define a functional that consists of three parts.

The functional β^a measures if the charges of the many-body state are in a Bose–Einstein condensate (we refer to Refs. 33 and 34 for a comprehensive introduction). Its relation to the trace norm distance of the one-particle reduced density matrix is given by (see, e.g., Ref. 1, Lemma 5.3)

The functional β^b quantifies the fluctuations of Ψ_N around the coherent state $W(Nα)Ω$⁠. Using property (2.6) of the Weyl operators, it can be written as

showing that it is the same quantity as usually considered in the coherent state approach.³⁵ While β^a and β^b measure the deviation of Ψ_N from the product state $φ⊗N⊗W(Nα)Ω$⁠, the functional β^c is introduced for technical reasons. It quantifies the fluctuations of the many-body energy per particle around the energy of the Maxwell–Schrödinger system.

In the original proof of Ref. 1, the functional β was considered with $βbΨN,α$ being replaced by

This definition has the advantage that it can be defined for many-body states Ψ_N in the domain $DHN=H2(R3N,C)⊗F∩DHf$⁠, which is invariant under the time evolution $e−iHNt$⁠. The additional difficulties with respect to Ref. 1 actually originate from the fact that $DN1/2$ [in contrast to $DHf$] is not contained in the domain of the Pauli–Fierz Hamiltonian. On the contrary, $β̃b$ does not allow us to investigate the coherence of photons with small frequencies because the factor $k$ in the integral on the right-hand side of (3.5) suppresses contributions from photons with small energies.

Throughout the rest of this article, (φ_t, α_t) and Ψ_N,t denote the solutions of (1.10) and (1.17) from Theorem 2.2. In this section, we show that $(ΨN,t,φt,αt)∈D(HN)∩DN1/2×H2(R3)×h32$ for all t ≥ 0. The condition on the Maxwell–Schrödinger solutions is satisfied because of Corollary 1.2. While $DHN$ is invariant under the evolution of the Pauli–Fierz Hamiltonian, due to Stone’s theorem, the invariance of $DN1/2$ is less clear because the photon number is not conserved during the time evolution. The next statement, however, displays that the number of photons can be controlled by the energy of the system.³⁸