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Wigner Measures Approach to the Classical Limit of the Nelson Model: Convergence of Dynamics and Ground State Energy

Abstract

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–Schrödinger 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.

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Acknowledgments

Marco Falconi has been supported by the Centre Henri Lebesgue (programme ANR-11-LABX-0020-01).

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Correspondence to Marco Falconi.

Appendix: Estimates on Fock Space

Appendix: Estimates on Fock Space

We provide some technical results used throughout the paper and proved here for general Hilbert spaces.

Lemma 6.1

Let \(\fancyscript{Y}\) be an Hilbert space, \(\Gamma _s(\fancyscript{Y})\) the corresponding symmetric Fock space (with \(a^{\#}\), \(N\), \(W(\xi )\) the annihilation/creation, number and Weyl operators respectively).

Let \(y\) be a positive self-adjoint operator on \(\fancyscript{Y}\) with domain \(D(y)\); and let \(d\Gamma (y)\) be the second quantization of \(y\), with form domain \(D(Y^{1/2})\). Then for all \(\xi \in D(y^{1/2})\), and \(\phi _1,\phi _2\in D(Y^{1/2})\):

$$\begin{aligned} \langle \phi _1 , W^{*}(\xi ) d\Gamma (y) W(\xi )\phi _2 \rangle _{}=\langle \phi _1 , \bigl (d\Gamma (y)+\frac{i\varepsilon }{\sqrt{2}}(a^{*}(y\xi )-a(y\xi ))+\frac{\varepsilon ^2}{2}\langle \xi ,y\xi \rangle _{\fancyscript{Y}} \bigr )\phi _2\rangle _{}. \end{aligned}$$

Proof

Let \(\xi \in D(y^{1/2})\) be fixed, let \(\phi _1,\phi _2\in D(N)\). Furthermore, let \((y_m)_{m\in \mathbb {N}}\in \mathcal {L}(\fancyscript{Y})\) be a sequence of bounded operators that converges strongly to \(y\) on \(D(y)\), with \(y_m\le y\) for all \(m\). Then we define, for all \(\lambda \in \mathbb {R}\),

$$\begin{aligned} M(\lambda ):=\langle \phi _1 , W(\lambda \xi ) \bigl (d\Gamma (y_m)+\frac{i\lambda \varepsilon }{\sqrt{2}}(a^{*}(y_m\xi )-a(y_m\xi ))+\frac{\lambda ^2\varepsilon ^2}{2}\langle \xi ,y_m\xi \rangle _{\fancyscript{Y}} \bigr )W^{*}(\lambda \xi )\phi _2 \rangle _{}. \end{aligned}$$

We remark that for every \(\delta \ge 0\) the Weyl operator maps \(D(N^{\delta })\) into itself. Taking the derivative in \(\lambda \), we obtain

$$\begin{aligned} \frac{d}{d\lambda }M(\lambda )&= \langle W^{*}(\lambda \xi )\phi _1 , i\bigl [\varphi (\lambda \xi )\; ,\; d\Gamma (y_m)+\frac{i\lambda \varepsilon }{\sqrt{2}}(a^{*}(y_m\xi )-a(y_m\xi ))\bigr ]W^{*}(\lambda \xi )\phi _2 \rangle _{}\\&+\langle W^{*}(\lambda \xi )\phi _1 , \bigl (\frac{i\varepsilon }{\sqrt{2}}(a^{*}(y_m\xi )-a(y_m\xi )) +\lambda \varepsilon ^2\langle \xi ,y_m\xi \rangle _{\fancyscript{Y}}\bigr )W^{*}(\lambda \xi )\phi _2 \rangle _{}=0. \end{aligned}$$

Hence for all \(\phi _1,\phi _2\in D(N)\) we obtain, by \(M(0)=M(1)\), for all \(m\in \mathbb {N}\):

$$\begin{aligned} \langle \phi _1 , W^{*}(\xi )d\Gamma (y_m)W(\xi )\phi _2 \rangle _{}&= \langle \phi _1 , \bigl (d\Gamma (y_m)+\frac{i\varepsilon }{\sqrt{2}}(a^{*}(y_m\xi )-a(y_m\xi ))\nonumber \\&+\frac{\varepsilon ^2}{2}\langle \xi ,y_m\xi \rangle _{\fancyscript{Y}} \bigr )\phi _2\rangle _{}. \end{aligned}$$
(6.1)

Choose now \(\phi _1=\phi _2=\phi \in D(Y^{1/2})\cap D(N)\). Then

$$\begin{aligned} \langle \phi , W^{*}(\xi )d\Gamma (y_m)W(\xi )\phi \rangle&\le ||d\Gamma (y)^{1/2}\phi ||_{}^2+\sqrt{2}\varepsilon ||y^{1/2}\xi ||_{\fancyscript{Y}}^{}||d\Gamma (y)^{1/2}\phi ||_{}^{}||\phi ||_{}^{}\\&+\frac{\varepsilon ^2}{2}||y^{1/2}\xi ||_{\fancyscript{Y}}^2||\phi ||_{}^{}. \end{aligned}$$

By monotone convergence theorem, the left hand side converges to \(\langle \phi , W^{*}(\xi ) d\Gamma (y) W(\xi ) \phi \rangle \) when \(m\rightarrow \infty \), since \(d\Gamma (y)\) is a closed operator. The result extends by density to all \(\phi \in D(Y^{1/2})\); so the Weyl operator \(W\) maps the form domain of \(d\Gamma (y)\) into itself. Then for all \(\phi _1,\phi _2\in D(Y^{1/2})\cap D(N)\), we can take the limit \(m\rightarrow \infty \) in (6.1). The result is then extended by density to all \(\phi _1,\phi _2\in D(Y^{1/2})\). \(\square \)

Corollary 6.2

  1. (i)

    Let \(\xi \in D(y)\). Then \((d\Gamma (y)+1)^{-1}W(\xi )(d\Gamma (y)+1)\in \mathcal {L}(\Gamma _s(\fancyscript{Y}))\). Furthermore, there exists \(C(||y \xi ||_{\fancyscript{Y}}^{},||\xi ||_{\fancyscript{Y}}^{})>0\) independent of \(\varepsilon \) such that:

    $$\begin{aligned} |(d\Gamma (y)+1)^{-1}W(\xi )(d\Gamma (y)+1)|_{\mathcal {L}(\Gamma _s(\fancyscript{Y}))}^{}\le C(||y \xi ||_{\fancyscript{Y}}^{},||\xi ||_{\fancyscript{Y}}^{})(1+O(\varepsilon )). \end{aligned}$$
  2. (ii)

    Let \(y\) be a positive bounded operator and let \(\xi \in \fancyscript{Y}\). Then for any \(\delta _1>0\) and \(\delta _2\in \mathbb {R}\), \((d\Gamma (y)^{\delta _1}+1)^{-\delta _2}W(\xi )(d\Gamma (y)^{\delta _1}+1)^{\delta _2}\in \mathcal {L}(\Gamma _s(\fancyscript{Y}))\). Furthermore, there exists a constant \(C(\delta _1,\delta _2,||\xi ||_{\fancyscript{Y}}^{},|y|_{\mathcal {L}(\fancyscript{Y})}^{})>0\) independent of \(\varepsilon \) such that:

    $$\begin{aligned} |(d\Gamma (y)^{\delta _1}+1)^{-\delta _2}W(\xi )(d\Gamma (y)^{\delta _1}+1)^{\delta _2}|_{\mathcal {L}(\Gamma _s(\fancyscript{Y}))}^{}\!\le \! C(\delta _1,\delta _2,||\xi ||_{\fancyscript{Y}}^{},|y|_{\mathcal {L}(\fancyscript{Y})})(1\!+\!O(\varepsilon )). \end{aligned}$$

The following proposition is a useful adaptation of [3, LemmasB.4 and B.6]:

Proposition 6.3

Let \(\fancyscript{Y}\) be an Hilbert space, \(\Gamma _s(\fancyscript{Y})\) the corresponding symmetric Fock space.

Let \(y_1,y_2\) be two operators on \(\fancyscript{Y}\) such that \((y_2+1)^{-1}y_1\in \mathcal {L}(\fancyscript{Y})\). Then \((d\Gamma (y_2^{*}y_2+1)+1)^{-1}d\Gamma (y_1) \in \mathcal {L}(\Gamma _s(\fancyscript{Y}))\), with:

$$\begin{aligned} |(d\Gamma (y_2^{*}y_2+1)+1)^{-1}d\Gamma (y_1)|_{\mathcal {L}(\Gamma _s(\fancyscript{Y}))}^{}\le (1+\sqrt{2})|(y_2+1)^{-1}y_1 |_{\mathcal {L}(\fancyscript{Y})}^{}. \end{aligned}$$

Proof

Let \(\phi _1,\phi _2\in D(d\Gamma (y_1))\). Then (\(y(j)\) is the operator acting on the \(j\)-th variable):

$$\begin{aligned} |\langle \phi _1 , d\Gamma (y_1)\phi _2 \rangle _{}|_{}^{}&\le \sum _n|\varepsilon \langle \phi _{1n} , \sum _{j=1}^ny_{1}(j)\phi _{2n}\rangle _{}|_{}^{}\\&\le \sum _n|\varepsilon n\langle \phi _{1n} , (y_{2}(1)+1)(y_{2}(1)+1)^{-1}y_{1}(1) \phi _{2n}\rangle _{}|_{}^{}\\&\le |(y_2+1)^{-1}y_1|_{\mathcal {L}(\fancyscript{Y})}^{}\sum _n^{}||\phi _{2n} ||_{}^{}\bigl (||\varepsilon n\phi _{1n} ||_{}^{}+||\varepsilon n y_2(1)\phi _{1n}||_{}^{}\bigr ). \end{aligned}$$

However, we have that:

$$\begin{aligned} ||\varepsilon n y_2(1)\phi _{1n}||^2&= \langle \phi _{1n} , \varepsilon ^2n^2y_2^{*}(1)y_2(1)\phi _{1n} \rangle _{}=\langle \phi _{1n} , d\Gamma (1) d\Gamma (y_2^{*}y_2)\phi _{1n} \rangle \\&\le \frac{1}{2}\langle \phi _{1n} , \Bigl ( \bigl (d\Gamma (1)\bigr )^2+ \bigl (d\Gamma (y_2^{*}y_2)\bigr )^2\Bigr )\phi _{1n} \rangle _{}\\&\le \frac{1}{2}\Bigl (||d\Gamma (1)\phi _{1n} ||_{}^2+||d\Gamma (y_2^{*}y_2)\phi _{1n} ||_{}^2\Bigr ). \end{aligned}$$

Hence, we obtain for any \(\phi _1,\phi _2\in \Gamma _s(\fancyscript{Y})\):

$$\begin{aligned} |\langle \phi _1 , (d\Gamma (y_2^{*}y_2+1)+1)^{-1}d\Gamma (y_1)\phi _2 \rangle _{}|_{}^{}&\le (1+\sqrt{2})|(y_2+1)^{-1}y_1|_{\mathcal {L}(\fancyscript{Y})}^{}\sum _n||\phi _{1n} ||_{}^{}||\phi _{2n} ||_{}^{}\\&\le (1+\sqrt{2})|(y_2+1)^{-1}y_1|_{\mathcal {L}(\fancyscript{Y})}||\phi _{1} ||_{}^{}||\phi _{2} ||_{}^{}. \end{aligned}$$

\(\square \)

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Ammari, Z., Falconi, M. Wigner Measures Approach to the Classical Limit of the Nelson Model: Convergence of Dynamics and Ground State Energy. J Stat Phys 157, 330–362 (2014). https://doi.org/10.1007/s10955-014-1079-7

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Keywords

  • Classical limit
  • Wigner measures
  • Nelson model
  • Klein–Gordon–Schrödinger equation
  • Ground state energy