Identification of the Polaron measure in strong coupling and the Pekar variational formula

Abstract

The path measure corresponding to the Fr\"ohlich Polaron appearing in quantum statistical mechanics is defined as the tilted measure $$ \widehat{\mathbb P}_{\varepsilon,T}= \frac{1}{Z(\varepsilon,T)}\exp\bigg(\frac{1}{2}\int_{-T}^T\int_{-T}^T \frac{\varepsilon\mathrm e^{-\varepsilon |t-s|}}{|\omega(t)-\omega(s)|} \mathrm d s \,\mathrm d t\bigg)\mathrm d\mathbb P. $$ Here $\varepsilon>0$ is the Kac parameter (or the inverse-coupling), and $\mathbb P$ is the law of $3d$ Brownian increments. In [13] it was shown that the (thermodynamic) limit $\lim_{T\to\infty}\widehat{\mathbb P}_{\varepsilon,T}=\widehat{\mathbb P}_\varepsilon$ exists as a process with stationary increments and this limit was identified explicitly as a mixture of Gaussian processes. In the present article, the strong coupling limit or the vanishing Kac parameter limit $\lim_{\varepsilon\to 0} \widehat{\mathbb P}_\varepsilon$ is investigated. It is shown that this limit exists and coincides with the increments of the Pekar process, which is a stationary diffusion process with generator $\frac 12 \Delta+ (\nabla\psi/\psi)\cdot \nabla$, where $\psi$ is the unique (modulo shifts) maximizer of the Pekar variational problem $$ g_0=\sup_{\|\psi\|_2=1} \Big\{\int_{\mathbb R^3}\int_{\mathbb R^3}\,\psi^2(x) \psi^2(y)|x-y|^{-1}\mathrm d x\mathrm d y -\frac 12\|\nabla \psi\|_2^2\Big\}. $$ As shown in [12,6,1], the Pekar process is itself approximated by the limiting "mean-field Polaron measures", and thus, the present identification of the strong coupling Polaron is a rigorous justification of the "mean-field approximation" (on the level of path measures) conjectured by Spohn in [15]. This approximation in the vanishing Kac limit ($\varepsilon\to 0$) is also shown to hold for a general class of Kac-Interaction of the form $H(t,x)=\varepsilon \mathrm e^{-\varepsilon|t|} V(x)$ where $V$ is any continuous function vanishing at infinity.