Variational quantum electrodynamics

A variational method is discussed, based on the principle of minimal variance. The method seems to be suited for gauge interacting fermions, and the simple case of quantum electrodynamics is discussed in detail. The issue of renormalization is addressed, and the renormalized propagators are shown to be the solution of a set of finite integral equations. The method is proven to be viable, and, by a spectral representation, the multidimensional integral equations are recast in one-dimensional equations for the spectral weights. The UV divergences are subtracted exactly, yielding a set of coupled Volterra integral equations that can be solved iteratively and are known to have a unique solution.

Figure 1

The three vertices in the interaction $S_I$ of Eq. (4) are shown in the first line. First and second order graphs for the self-energy and polarization function are shown in the second and third lines, respectively. For each two-point function, we recognize a first order graph, a reducible second order graph, and a one-loop 1PI second order graph.