Static fields described by coherent states of photons/gravitons with longitudinal/timelike polarization

There are formulations of QED where electrostatic Coulomb fields around a charge are expressed as coherent states of photons with longitudinal/timelike polarization.  See for example chapter 3 of Scharf’s "Finite Quantum Electrodynamics" and  https://arxiv.org/abs/1001.1387

1. How general is this claim?  Are there formulations of QED where a charged object’s Coulomb field is expressed only with transverse photons?

2. In string theory, a curved Schwarzchild metric around a massive object is similarly said to be expressed as a coherent state of graviton mode strings.  In this sense, a Schwarzchild metric is a static field akin to the Coulomb field.  By analogy to the above, what is the polarization of the strings giving rise to a Schwarzchild metric?