Dyson-Schwinger variational approach to quantum field theory

本研究项目的目标是发展一种基于非高斯真空波泛函变分原理的非微扰哈密顿方法来研究连续介质中的量子场论。为此，波泛函被写成“作用力”的指数，它在场幂中展开。非局域核(变分核)作为展开系数将通过最小化真空能量密度来确定。该方法的核心是广义Dyson-Schwinger方程(DES)，它允许用变分核来表示场(特别是能量)的n点函数。这种方法得到了一个运动方程组，它除了包含广义的DSE外，还包括通过最小化能量得到的变分核的间隙方程。这个积分方程组必须自洽地求解。这种方法不仅限于相对论量子场理论，而且非常适合于相互作用的多体系统的非微扰处理，并允许系统地改进平均场近似。具体地说，对于库仑规中的QCD，应制定相应的方法。为此，使用了真空波泛函，它在指数中包含规范场中最高可达四次项的项，以及夸克与胶子的耦合。在第一步中，对于IR和UV中的n点函数，将通过动量空间中的功率展开来解析地求解所得到的运动方程。随后，在整个动量区域内对这些方程进行数值求解。这应首先在所谓的淬火近似中进行，随后也应在非淬火计算中进行。

The goal of the intended research project is the development of a general non-perturbative Hamiltonian approach to quantum field theory in the continuum, which is based on a variational principle with non-Gaussian vacuum wave functionals. To this end, the wave functional is written as the exponent of an "action", which is expanded in powers of the fields. The non-local kernels (variational kernels) appearing as expansion coefficients will be determined by minimizing the vacuum energy density. The centerpiece of the approach are generalized Dyson-Schwinger equations (DSEs), which allow to express the n-point functions of the fields (and, in particular, the energy) by the variational kernels. This approach leads to a system of equations of motion, which contains, besides the generalized DSEs, also the gap equations for the variational kernels obtained by minimizing the energy. This system of integral equations has to be solved self-consistently. The approach is not restricted to relativistic quantum field theories, but is also well suited for a non-perturbative treatment of interacting many-body systems and allows a systematic improvement of the mean field approximation. Specifically, the approach shall be worked out for QCD in Coulomb gauge. For this purpose, a vacuum wave functional is used which contains in the exponent up to quartic terms in the gauge field, as well as the coupling of the quarks to the gluons. In a first step, the resulting equations of motions shall be solved analytically by power expansion in momentum space for the n-point functions in both the IR and the UV. Subsequently, these equations shall be solved numerically in the entire momentum region. This shall be done first in the so-called quenched approximation, and subsequently also in an unquenched calculation.