量子电动力学Hartree-Fock近似下的正负电子偶素
The positronium and the dipositronium in a Hartree-Fock approximation of quantum electrodynamics
摘要：
Bogoliubov-Dirac-Fock（BDF）模型是量子电动力学的无光子近似。它允许研究与狄拉克海相互作用的相对论电子。一个状态完全由它的单体密度矩阵，一个无限秩非负投影子来表征。我们证明了在无外场情况下用能量泛函的临界点表示的BDF模型中存在电子偶素（具有反平行自旋的电子和正电子的束缚态）。我们还证明了双电子偶素的存在，双电子偶素是由两个电子和两个正电子组成的分子，也是临界点。更一般地，对于任何半整数$j\in \tfrac{1}{2}+\mathbb{Z}_+$，我们证明了由$2j+1$电子和$2j+1$正电子组成的能量泛函的临界点的存在性。
英文摘要：
The Bogoliubov-Dirac-Fock (BDF) model is a no-photon approximation of quantum electrodynamics. It allows to study relativistic electrons in interaction with the Dirac sea. A state is fully characterized by its one-body density matrix, an infinite rank nonnegative projector. We prove the existence of the para-positronium, the bound state of an electron and a positron with antiparallel spins, in the BDF model represented by a critical point of the energy functional in the absence of external field. We also prove the existence of the dipositronium, a molecule made of two electrons and two positrons that also appears as a critical point. More generally, for any half integer $j\in \tfrac{1}{2}+\mathbb{Z}_+$, we prove the existence of a critical point of the energy functional made of $2j+1$ electrons and $2j+1$ positrons.
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