Wigner measures approach to the classical limit of the Nelson model: Convergence of dynamics and ground state energy

We consider the classical limit of the Nelson model, a system of stable nucleons interacting with a meson field. We prove convergence of the quantum dynamics towards the evolution of the coupled Klein-Gordon-Schroedinger equation. Also, we show that the ground state energy level of \(N\) nucleons, when \(N\) is large and the meson field approaches its classical value, is given by the infimum of the classical energy functional at a fixed density of particles. Our study relies on a recently elaborated approach for mean field theory and uses Wigner measures.

[The introduction is copied here for rapid consultation]

The Nelson model refers to a quantum dynamical system describing a nucleon field interacting with a meson scalar Bose field. When an ultraviolet cut off is put in the interaction, the Hamiltonian becomes a self-adjoint operator and so the quantum dynamics is well defined. In the early sixties, E. Nelson showed that the quantum dynamics of this system exists even when the ultraviolet cut off is removed [see Nelson 1964]. It is indeed one of the simplest examples in non-relativistic quantum field theory (QFT) where renormalization is needed and successfully performed using only basic tools of functional analysis.

Over the past two decades, there has been considerable effort devoted to the study of the Nelson model that have led to a thorough investigation of its spectral and scattering properties [see AbdHas 2012, Arai 2001, BacFroSig 1999, BetHirLorMinSpo 2002, DerGer 1999, FroGriSch 2004, GeoGerMol 2004, GerHirPanSuz 2011, Pizzo 2003, Spohn 1998 to mention but a few]. However, the fact that the Nelson Hamiltonian is a Wick quantization of a classical Hamiltonian system is quite often neglected except in few references [Falconi 2013, GinNirVel 2006]. We believe that the study of the classical limit of such quantum dynamical systems is a significant question leading to an unexplored phase-space point of view in QFT. This for sure will enrich the subject and may also provide some insight on some of the remaining open problems.

In this article, we neglect the spin and isospin for nucleons, so we are considering a scalar Yukawa field theory. We also suppose that an ultraviolet cut off is imposed on the interaction. We prove two main results stated in Theorem 1.1 and Theorem 1.2, namely:

1. Convergence of the quanutm dynamics towards the classical evolution of the coupled Klein-Gordon-Schroedinger equation.
2. Convergence of the ground state energy level of \(N\) nucleons to the infimum of the classical energy functional with fixed density of particles, when \(N\) tends to infinity and the Bose field approaches its classical limit.

There are basically two schemes for proving 1): either one studies the propagation of states or those of observables. The latter strategy being very difficult for systems with unconserved number of particles, we rely on the first scheme. To establish 1), we follow indeed a Wigner measures approach, recently elaborated in [AmmNie 2008-2011] for the purpose of mean field limit in many-body theory. This method turns out to be quite general and flexible. It can be adapted to quantum electrodynamics (QED) and relativistic quantum field theory (QFT) and it gives a fair description of the propagation of general states in the classical limit (see Theorem 1.1). Actually, the convergence 1) is known in the particular case of coherent states [see Falconi 2013, GinNirVel 2006] by Hepp's method [Hepp 1974] which relies on the special structure of those states. The result in Theorem 1.1 says that the convergence of the Neslon quantum dynamics towards the classical one has nothing to do with any particular structure or choice of states but it is rather a general (Bohr) quantum-classical correspondence principle for a system with an infinite number of degrees of freedom. In this sense, Theorem 1.1 is more general and provides a better understanding of the classical limit of Nelson Hamiltonians.

In addition to the fact that the Wigner measures approach gives a stronger convergence result compared to the coherent state method, it also proves to be a powerful tool for tackling variational questions of type 2). Indeed, asymptotic properties of a given minimizing sequence can be derived by looking to its Wigner measures and it turns out that some a priori information on these Wigner measures are crucial in the proof of Theorem 1.2. However, both our results give only the limit of quantum quantities in terms of their classical approximations and provide no error bound on the difference. This is of course an interesting question, among several others, and it is beyond the scope of this article. Actually, our work is also meant to stimulate further investigations and to underline some open problems. For instance, removal of the momentum cutoff and drop of the confining potential as well as time asymptotics and scattering theory within the classical limit are quite interesting open questions. We believe indeed that our work provides a basis for further developments on the above-mentioned problems.

The phase space of the theory is \(\mathscr{Z}:=L^2(\mathbb{R}^d)\oplus L^2(\mathbb{R}^d)\), and we consider the symmetric Fock space \(\mathscr{H}:=\Gamma_s(\mathscr{Z})\sim \Gamma_s(L^2(\mathbb{R}^d))\otimes\Gamma_s(L^2(\mathbb{R}^d))\). We denote by \(\psi^{\#}\) the annihilation and creation of the non-relativistic particles (nucleons), by \(a^{\#}\) the annihilation and creation of the relativistic meson field. We recall that for each \(\varepsilon\in(0,\bar \varepsilon)\), with \(\bar\varepsilon>0\) fixed once and for all, we choose the algebra:

\begin{equation*} [\psi(x_1),\psi^*(x_2)]=\varepsilon \delta(x_1-x_2)\;,\quad [a(k_1),a^*(k_2)]=\varepsilon \delta(k_1-k_2)\;. \end{equation*}

This fixes the scaling so that each \(\psi^{\#}\) and \(a^{\#}\) behaves like \(\sqrt{\varepsilon}\). For instance, the second quantization operators \(d\Gamma(\cdot)=\int_{\mathbb{R}^{d}} a^{*}(k) (\,\cdot\,) a(k)dk\) or \(\int_{\mathbb{R}^{d}} \psi^{*}(x) (\,\cdot\,) \psi(x) dx\) scale like \(\varepsilon\). This is also the case for the number operators \(N_{1}=d\Gamma(1)\otimes 1, N_{2}=1\otimes d\Gamma(1)\) and \(N=N_{1}+N_{2}\). The Weyl operators are \(W(\xi)=W(\xi_{1})\otimes W(\xi_{2})\), for \(\xi=\xi_{1}\oplus\xi_{2}\in\mathscr{Z}\), with \(W(\xi_{1})=e^{i\frac{\psi^{*}(\xi_{1})+\psi(\xi_{1})}{\sqrt{2}}}\) and \(W(\xi_{2})=e^{i\frac{a^{*}(\xi_{2})+a(\xi_{2})}{\sqrt{2}}}\) being the Weyl operators on \(\Gamma_s(L^2(\mathbb{R}^d))\).

In the Fock representation of these canonical commutation relations, the Nelson Hamiltonian takes the form:

\begin{equation*} \begin{split} H=d\Gamma(-\frac{\Delta}{2M}+V)\otimes 1+1\otimes d\Gamma(\omega)+\int_{\mathbb{R}^{2d}}^{}\frac{\chi(k)}{\sqrt{\omega(k)}}\psi^{*}(x)\bigl(a^{*}(k)e^{-ik\cdot x}+a(k)e^{ik\cdot x}\bigr)\psi(x) dkdx\; ; \end{split} \end{equation*}

where \(\omega(k)=\sqrt{k^2+m_0^2}\) and \(m_0\geq 0\). Here \(m_0\) and \(M\) are respectively the meson and nucleon mass at rest. It is useful to split \(H\) in a free part \(H_0\), and an interaction part \(H_I\), with:

\begin{align*} H_0&=d\Gamma(-\frac{\Delta}{2M}+V)\otimes 1+1\otimes d\Gamma(\omega)\; ,\\ H_I&=\int_{\mathbb{R}^{2d}}^{}\frac{\chi(k)}{\sqrt{\omega(k)}}\psi^{*}(x)\bigl(a^{*}(k)e^{-ik\cdot x}+a(k)e^{ik\cdot x}\bigr)\psi(x) dkdx\; . \end{align*}

We assume the potential \(V(x)\) to be in \(L^2_{loc}(\mathbb{R}^{d},\mathbb{R}_+)\), so that \(-\Delta +V\) is a positive self-adjoint operator on \(L^2(\mathbb{R}^d)\), by Kato inequality, and essentially self-adjoint on \(C_0^\infty(\mathbb{R}^d)\). The main assumption we require on the cut off function \(\chi\) is that \(\omega^{-1/2}\chi\in L^2(\mathbb{R}^{d})\). This is enough to define \(H\) as self-adjoint operator (see Proposition 2.5). To recapitulate, we assume throughout the article the assumption

\begin{equation} \label{ass:a}\tag{A} V\in L^2_{loc}(\mathbb{R}^{d},\mathbb{R}_+)\text{ and } \omega^{-1/2}\chi\in L^2(\mathbb{R}^{d})\,. \end{equation}

Actually, the Nelson Hamiltonian is a Wick quantization of the classical energy functional

\begin{equation*} h(z_1\oplus z_2)=\langle z_1,-\frac{\Delta}{2M}+V \,z_1\rangle+\langle z_2,\omega(k) z_2\rangle+ \int_{\mathbb{R}^{2d}}^{}\frac{\chi(k)}{\sqrt{\omega(k)}} |z_1|^2(x) \bigl(\bar{z}_2(k) e^{-ik\cdot x}+z_2(k)e^{ik\cdot x}\bigr) dkdx\;. \end{equation*}

The Hamiltonian \(h\) describes the coupled Klein-Gordon-Schroedinger system with an Yukawa type interaction subject to a momentum cut off. With the assumption above, the related Cauchy problem is well posed in \(\mathscr{Z}\) (see Propositions 2.7 and 2.8).

The main point in the proof of 1) is to understand the propagation of normal states on the Fock space \(\mathscr{H}\) with the appropriate scaling. The idea is to encode the oscillations of any family of states with respect to the semiclassical parameter \(\varepsilon\) by classical quantities, namely probability measures on the phase space (Wigner measures). Then 1) can be restated as the propagation of these measures along the classical flow of the Klein-Gordon-Schroedinger equation.

We say that a Borel probability measure \(\mu\) on \(\mathscr{Z}\) is a Wigner measure of a family of normal states \((\varrho_{\varepsilon})_{\varepsilon\in (0,\bar \varepsilon)}\) on \(\mathscr{H}\) if there exists a sequence \((\varepsilon_k)_{k\in\mathbb{N}}\) in \((0,\bar\varepsilon)\), such that \(\varepsilon_k\to 0\) and for any \(\xi\in\mathscr{Z}\),

\begin{equation} \label{quan1} \lim_{k\to\infty} \mathrm{Tr}[\varrho_{\varepsilon_k} W(\xi)]= \int_{\mathscr{Z}} e^{i\sqrt{2} {\Re}\langle\xi, z\rangle} \, d\mu(z)\,, \end{equation}

where \(W(\xi)\) refers to the Weyl operator on the Fock space \(\mathscr{H}\) which depends on \(\varepsilon_{k}\) (here \(\Re \langle \cdot , \cdot\rangle_{}\) is the real part of the scalar product on \(\mathscr{Z}\)). We denote the set of all Wigner measures of a given family of states \((\varrho_{\varepsilon})_{\varepsilon\in (0,\bar \varepsilon)}\) by \(\mathscr{M}(\varrho_{\varepsilon}, \varepsilon\in (0,\bar \varepsilon))\). It was proved in [AmmNie 2008] that the assumption \[ \exists \delta>0, \exists C>0, \forall \varepsilon\in (0,\bar\varepsilon)\quad\mathrm{Tr}[\varrho_{\varepsilon}N^{\delta}]< C \] ensures that the set of Wigner measures \(\mathscr{M}(\varrho_{\varepsilon}, \varepsilon\in (0,\bar \varepsilon))\) is non empty. Notice that the \(\mathrm{Tr}[\,\cdot\,]\) is understood as \(\sum_{i=0}^{\infty}\lambda_i\langle \varphi_i , N^{\delta}\varphi_i \rangle_{}\), where \(\{\varphi_i\}_{i\in\mathbb{N}}\) is an O.N.B. of eigenvectors of \(\varrho_{\varepsilon}\) associated to the eigenvalues \(\{\lambda_i\}_{i\in\mathbb{N}}\).

The Nelson Hamiltonian \(H\) has a fibred structure with respect to the number of nucleons. So, it can be written as \(H=\oplus_{n=0}^{\infty} H_{|L^{2}_s(\mathbb{R}^{nd})\otimes\Gamma_s(L^{2}(\mathbb{R}^{d}))}\) where \(L_s^2(\mathbb{R}^{nd})\) denotes the space of symmetric square integrable functions (see Section 2). It also turns out that \(H\) is unbounded from below while \(H_{|L^{2}_s(\mathbb{R}^{nd})\otimes\Gamma_s(L^{2}(\mathbb{R}^{d}))}\) is bounded from below.

Under the aforementioned assumptions on the potential \(V\) and the cut off function \(\chi\), we are in position to precisely state our two main results.

• Theorem 1.1. Assume \eqref{ass:a} holds. Let \((\varrho_{\varepsilon})_{\varepsilon\in (0,\bar \varepsilon)}\) be a family of normal states on the Hilbert space \(\mathscr{H}\) satisfying the assumption: \[ \exists\delta>0,\exists C>0,\forall\varepsilon\in(0,\bar\varepsilon)\quad\mathrm{Tr} [\varrho_{\varepsilon}(H_0+N+1)^{\delta}]< C . \] Then for any \(t\in\mathbb{R}\) the set of Wigner measures associated with the family \((e^{-i \frac{t}{\varepsilon} H}\varrho_{\varepsilon}e^{i \frac{t}{\varepsilon} H})_{\varepsilon\in (0,\bar \varepsilon)}\) is \[ \mathscr{M}(e^{-i \frac{t}{\varepsilon} H}\varrho_{\varepsilon}e^{i \frac{t}{\varepsilon} H}, \varepsilon\in (0,\bar \varepsilon))=\{\Phi(t,0)_{\#}\mu_{0}, \mu_{0}\in \mathscr{M}(\varrho_{\varepsilon}, \varepsilon\in (0,\bar \varepsilon))\}\,, \] where \(\Phi(t,0)_{\#}\mu_{0}\) is the push forward of \(\mu_{0}\) by the classical flow \(\Phi(t,s)\) of the coupled Klein-Gordon-Schroedinger equation well defined and continuous on \(\mathscr{Z}\) by Propositions 2.7 and 2.8 .

• Theorem 1.2. Assume that \eqref{ass:a} holds and additionally \(m_0>0\) and \(V\) is a confining potential, i.e.: \(\lim_{|x|\to\infty} V(x)=+\infty \). Then the ground state energy of the restricted Nelson Hamiltonian has the following limit, for any \(\lambda>0\),

  \begin{equation} \displaystyle\lim_{\varepsilon\to 0, n\varepsilon=\lambda^2} \inf \sigma(H_{|L^{2}_s(\mathbb{R}^{dn})\otimes \Gamma_{s}(L^{2}(\mathbb{R}^{d}))})= \displaystyle\inf_{||z_{1}||_{L^2(\mathbb{R}^d)}=\lambda} h(z_{1}\oplus z_{2})\,, \end{equation}

  where the infimum on the right hand side is taken over all \(z_1\in D(\sqrt{\frac{-\Delta}{2M}+V})\) and \(z_2\in D(\omega^{1/2})\) with the constraint \(||z_{1}||_{L^2(\mathbb{R}^d)}=\lambda\).

The proof of Theorem 1.1 is given in Sections 3 and 4 and uses the properties of the quantum and classical dynamics proved in Section 2. It is rather lengthy, so for reader's convenience we outline its key arguments below. The proof of Theorem 1.2, given in Section 5, relies on an upper bound derived by using coherent states localized around the infimum of the classical energy and a lower bound resulting from the a priori information on the Wigner measures of a given minimizing sequence. So, we conclude that these measures are a fortiori concentrated around the infimum of the classical energy.

• Proof of Theorem 1.1 (Outline) Our goal is to identify the Wigner measures of the evolved state \({\varrho}_{\varepsilon}(t)=e^{-i \frac{t}{\varepsilon} H}\varrho_{\varepsilon}e^{i \frac{t}{\varepsilon} H}\) given in Theorem 1.1 . However, instead of considering \({\varrho}_{\varepsilon}(t)\), we work in the interaction representation with \[ \tilde{\varrho}_{\varepsilon}(t)=e^{i \frac{t}{\varepsilon} H_0} {\varrho}_{\varepsilon}(t) e^{-i \frac{t}{\varepsilon} H_0}\,. \] By doing so, we require less regularity on the state \(\varrho_{\varepsilon}\) and it is still easy to recover Wigner measures of \({\varrho}_{\varepsilon}(t)\) from those of \(\tilde{\varrho}_{\varepsilon}(t)\). The main point now is that Wigner measures of the latter states are determined through all possible "limit points", when \(\varepsilon\to 0\), of the map

  \begin{equation} \label{quan2} \xi\mapsto \mathrm{Tr} \Bigl[ \tilde{\varrho}_{\varepsilon}(t) W(\xi)\Bigr] \,. \end{equation}

  Despite its apparent simplicity, there is no straightforward way to compute such limit explicitly. Moreover, uniqueness of Wigner measures at each time \(t\) is not guarantied even if it is assumed at the initial time \(t=0\) (i.e.: the map above may have several limit points though it has one single limit at \(t=0\)). To overcome the last difficulty, we use a diagonal extraction (or Ascoli type) argument which implies that for any sequence \((\tilde{\varrho}_{\varepsilon_n})_{n\in\mathbb{N}}\), \(\varepsilon_n\to 0\), we can extract a subsequence \((\tilde{\varrho}_{\varepsilon_{n_k}})_{k\in\mathbb{N}}\) such that for each time, \(t\in\mathbb{R}\), \((\tilde{\varrho}_{\varepsilon_{n_k}}(t))_{k\in\mathbb{N}}\) admits a unique Wigner measure denoted by \(\tilde\mu_t\). The next step is to observe that the map satisfies a dynamical equation which when \(\varepsilon\to 0\) leads to a well behaved classical dynamical equation on the inverse-Fourier transform of the Wigner measures \(\tilde\mu_t\). By integrating with respect to appropriate trial functions, we obtain a natural transport (Liouville) equation satisfied by \(\tilde\mu_t\). Therefore, it is possible to identify the measures \(\tilde\mu_t\) if we can prove that such transport equation has a unique solution for each data \(\tilde\mu_0\) given by the push-forward of \(\tilde\mu_0\) by the corresponding classical dynamics. To sum up, the outline of the proof goes as follows:

  1. We justify the integral (or Duhamel) formula

     \begin{equation*} \mathrm{Tr} \Bigl[ \tilde{\varrho}_{\varepsilon}(t) W(\xi)\Bigr] = \mathrm{Tr} \Bigl[\varrho_{\varepsilon} W(\xi)\Bigr] + \frac{i}{\varepsilon}\int_0^t\mathrm{Tr} \Bigl[ \varrho_{\varepsilon}(s) [H_I,W(\tilde{\xi}(s))] \Bigr] ds\; , \end{equation*}

     in Proposition 3.5 for states \({\varrho}_{\varepsilon}\) satisfying a strong regularity condition, namely that it belongs to the space \(\mathcal{T}_{\varepsilon}^{1}\) given in Definition 3.1.

  2. By explicit computation and taking care of domain problems, we show in Proposition 3.9 that

     \begin{equation} \label{eq.29} \mathrm{Tr} \Bigl[ \tilde{\varrho}_{\varepsilon}(t) W(\xi)\Bigr] = \mathrm{Tr} \Bigl[\varrho_{\varepsilon} W(\xi)\Bigr] + \sum_{j=0}^2\varepsilon^j\int_0^t\mathrm{Tr} \Bigl[ \varrho_{\varepsilon}(s)W( \tilde{\xi}(s)) B_j( \tilde{\xi}(s)) \Bigr] ds\;; \end{equation}

     where \(B_j( \tilde{\xi}(s))\) are operators given in (22)-(24).

  3. There is no loss of generality if we assume that \(({\varrho}_{\varepsilon})_{\varepsilon\in(0,\bar{\varepsilon})}\) has a single Wigner measure \(\mu_0\). Moreover, we prove as explained before that from any sequence \(\varepsilon_n\to 0\) we can extract a subsequence \((\varepsilon_{n_k})_{k\in\mathbb{N}}\) such that \((\tilde{\varrho}_{\varepsilon_{n_k}}(t))_{k\in\mathbb{N}}\) has a single Wigner measure \(\tilde\mu_t\) for each time \(t\in\mathbb{R}\) (see Subsection 4b).

  4. Letting \(\varepsilon_{n_k}\to 0\) in the equation above and using some elementary $ε$-uniform estimates proved in Section 2 with some Wigner measures properties; we show in Proposition 4.10 that

     \begin{equation*} \begin{split} \tilde\mu_t(e^{i \sqrt{2}\Re\langle {\xi} ,\,\cdot\, \rangle_{}})= \mu_0(e^{i \sqrt{2}\Re\langle {\xi}, \,\cdot\,\rangle_{}})+i\sqrt{2}\int_0^t \tilde\mu_{s}\left(e^{i \sqrt{2}\Re\langle {\xi} , z\rangle_{}} {\Re}\langle \xi, \mathscr{V}_{s}(z)\rangle\right)\; ds; \end{split} \end{equation*}

     with a velocity vector field \(\mathscr{V}_{s}(z)\) defined by (16).

  5. In Proposition 4.11, we show that \(t\in\mathbb{R}\mapsto\tilde{\mu}_{t}\) is a weakly narrowly continuous map valued on probability measures satisfying the transport equation

     \begin{equation*} \partial_{t}\tilde{\mu}_t+\nabla^T\left(\mathscr{V}_{t} \tilde{\mu}_{t}\right)=0\,, \end{equation*}

     understood in the weak sense,

     \begin{equation*} \int_{\mathbb{R}}\int_{\mathscr{Z}} \left(\partial_{t}f+\Re\langle \nabla f, \mathscr{V}_{t}\rangle\right)~d\tilde{\mu}_{t}dt=0\,. \end{equation*}
  6. To identify the measures \(\tilde\mu_t\) we rely on an argument worked out in finite dimension by Ambrosio, Gigli and Savare' [2005] for the purpose of optimal transport theory and extended in [AmmNie 2011] to an infinite dimensional Hilbert space setting. This yields, in Proposition 4.12, the result of Theorem 1.1 but under a strong assumption on \((\varrho_{\varepsilon})_{\varepsilon\in(0,\bar{\varepsilon})}\in\cap_{\delta>0} \mathcal{T}^{\delta}\cap \mathcal{S}^1\), given in Definition 4.1.
  7. To complete the proof, we use an approximation argument allowing to extend the previous result to states satisfying the weak assumption (2) in Theorem 1.1 (see Section 4e).