On the Ultraviolet Problem for the Ground State Energy of the Translation-Invariant Pauli--Fierz Model at Zero Total Momentum

TL;DR

This paper analyzes the ground state energy of the translation-invariant Pauli--Fierz model at zero total momentum, focusing on ultraviolet cutoff dependence and employing a convexified variational approach.

Contribution

It introduces a modified energy functional that is strictly convex and provides asymptotic estimates of the ground state energy growth with ultraviolet cutoff.

Findings

The energy functional is not convex in the original form.

Removing the non-convex term yields a strictly convex functional.

Ground state energy grows asymptotically as a3^{3/2} with the ultraviolet cutoff.

Abstract

We study the ground state energy of the Pauli--Fierz model in the absence of external potentials. We consider the fiber decomposition of the Pauli--Fierz operator with respect to the spectral values, $p$, of the total momentum operator and focus on the case $p = 0$. The corresponding variational problem is analyzed to estimate the dependence of the ground state energy on the ultraviolet cutoff $\Lambda$. We employ a Bogoliubov--Hartree--Fock approximation using pure, quasifree states generated by Bogolubov transformations (parametrized by a positive Hilbert--Schmidt operator $z$) and Weyl transformations (parametrized by a vector $\eta$) applied to the vacuum. We prove that the resulting energy functional is not a convex function of $\eta$ and $z$. We identify the non-convex term and remove it from the energy functional. The modified functional retains the full interaction term and is shown to be strictly convex. We study the ground state of the modified functional and prove the existence of a unique minimizer. Furthermore, we construct an explicit partial minimizer (with respect to $\eta$, for fixed $z$), which allows us to eliminate $z$ and reduce the minimization problem to a single variable, $\eta$. Finally, we estimate the minimum of the modified energy functional in terms of the ultraviolet cutoff $\Lambda$ and demonstrate that, up to a constant factor, it grows asymptotically as $\Lambda^{3/2}$, as $\Lambda \to \infty$.