外磁场中类石墨烯介质的单圈真空极化;库仑势的修正
The 1-loop vacuum polarization for a graphene-like medium in an external magnetic field; corrections to the Coulomb potential
摘要：
我计算的1圈真空极化$Π_{μν}(k,B,a)$的动量$k=(\hat k,k_3)$的光子与电子的薄介质的厚度$2a$模拟石墨烯，在存在一个恒定的和均匀的外部磁场$B$正交于它（平行于$k_3$）。计算是用施温格的技术完成的，适应于仔细研究的几何和哈密顿。这种情况比电子自能的情况更复杂，因为光子现在也可以在介质外传播。这使得$Π_{μν}$分解为量子、“简化”的$T_{μν}(\hat k,B)$和透射函数$V(k,a)$，其中样品的几何形状和由此产生的$γ\,e^+\,e^-$顶点的限制起主要作用。这使得结果偏离了2-膜上的约化QED $_{3+1}$。$V$在$k^2=0$上的有限性是满足$Π_{μν}$重整化条件和确定相应反项的必要条件。研究了它们与$Π_{μν}$横截性的关系。对库仑势的修正及其对$B$的依赖性与QED $_{3+1}$有很大的不同。
英文摘要：
I calculate the 1-loop vacuum polarization $Π_{μν}(k,B,a)$ for a photon of momentum $k=(\hat k,k_3)$ interacting with the electrons of a thin medium of thickness $2a$ simulating graphene, in the presence of a constant and uniform external magnetic field $B$ orthogonal to it (parallel to $k_3$). Calculations are done with the techniques of Schwinger, adapted to the geometry and Hamiltonian under scrutiny. The situation gets more involved than for the electron self-energy because the photon is now allowed to also propagate outside the medium. This makes $Π_{μν}$ factorize into a quantum, "reduced" $T_{μν}(\hat k,B)$ and a transmittance function $V(k,a)$, in which the geometry of the sample and the resulting confinement of the $γ\,e^+\,e^-$ vertices play major roles. This drags the results away from reduced QED$_{3+1}$ on a 2-brane. The finiteness of $V$ at $k^2=0$ is an essential ingredient to fulfill suitable renormalization condition for $Π_{μν}$ and to fix the corresponding counterterms. Their connection with the transversality of $Π_{μν}$ is investigated. The corrections to the Coulomb potential and their dependence on $B$ strongly differ from QED$_{3+1}$.
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