Identification of the Polaron measure in strong coupling and the Pekar variational formula (1812.06927v3)

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.