Dyson-Schwinger variational approach to quantum field theory

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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.