On the Ultraviolet Problem for the Ground State Energy of the Translation-Invariant Pauli--Fierz Model at Zero Total Momentum
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2026
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| author | Bach, Volker Ballesteros, Miguel Mlinarzik, Merten |
| author_facet | Bach, Volker Ballesteros, Miguel Mlinarzik, Merten |
| contents | 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 $Λ$. 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 $η$) applied to the vacuum. We prove that the resulting energy functional is not a convex function of $η$ 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 $η$, for fixed $z$), which allows us to eliminate $z$ and reduce the minimization problem to a single variable, $η$. Finally, we estimate the minimum of the modified energy functional in terms of the ultraviolet cutoff $Λ$ and demonstrate that, up to a constant factor, it grows asymptotically as $Λ^{3/2}$, as $Λ\to \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Ultraviolet Problem for the Ground State Energy of the Translation-Invariant Pauli--Fierz Model at Zero Total Momentum Bach, Volker Ballesteros, Miguel Mlinarzik, Merten Mathematical Physics 81T10 (primary), 49R05 (secondary) 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 $Λ$. 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 $η$) applied to the vacuum. We prove that the resulting energy functional is not a convex function of $η$ 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 $η$, for fixed $z$), which allows us to eliminate $z$ and reduce the minimization problem to a single variable, $η$. Finally, we estimate the minimum of the modified energy functional in terms of the ultraviolet cutoff $Λ$ and demonstrate that, up to a constant factor, it grows asymptotically as $Λ^{3/2}$, as $Λ\to \infty$. |
| title | On the Ultraviolet Problem for the Ground State Energy of the Translation-Invariant Pauli--Fierz Model at Zero Total Momentum |
| topic | Mathematical Physics 81T10 (primary), 49R05 (secondary) |
| url | https://arxiv.org/abs/2605.02349 |