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.

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

• (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:

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

• (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 $\phi _1,\phi _2\in \Gamma _s(\fancyscript{Y})$:

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

$\square$