量子电动力学中的平均场近似。无光子的情况
The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case
摘要：
我们研究了量子电动力学的平均场近似，通过热力学极限。QED哈密顿量是用库仑规范写成的，不包含任何正规序或裸电子/正电子子空间的选择。忽略光子，我们适当地定义这个哈密顿量在一个有限的盒子$[-L/2;L/2)^3$，周期性的边界条件和紫外截止$Λ$。然后，我们研究的极限的基态（即真空）的能量和极小值的$L$走向无穷大，在Hartree-Fock近似。在没有外场的情况下，我们证明了单位体积能量的收敛性，并在极限下得到了一个描述自由Hartree-Fock真空的不变量投影。我们还定义了每单位体积的能量守恒不变的状态，并证明了自由真空是这个能量的唯一极小。在外场存在的情况下，我们证明了当$L$趋于无穷大时，最小能量与自由真空能量之差收敛。在极限下得到了所谓的Bogoliubov-Dirac-Fock泛函。Hartree-Fock（极化）真空是自由真空的希尔伯特-施密特微扰，它使Bogoliubov-Dirac-Fock能量最小。
英文摘要：
We study the mean-field approximation of Quantum Electrodynamics, by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal-ordering or choice of bare electron/positron subspaces. Neglecting photons, we define properly this Hamiltonian in a finite box $[-L/2;L/2)^3$, with periodic boundary conditions and an ultraviolet cut-off $Λ$. We then study the limit of the ground state (i.e. the vacuum) energy and of the minimizers as $L$ goes to infinity, in the Hartree-Fock approximation. In case with no external field, we prove that the energy per volume converges and obtain in the limit a translation-invariant projector describing the free Hartree-Fock vacuum. We also define the energy per unit volume of translation-invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as $L$ goes to infinity. We obtain in the limit the so-called Bogoliubov-Dirac-Fock functional. The Hartree-Fock (polarized) vacuum is a Hilbert-Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov-Dirac-Fock energy.
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