QED-IR as Topological Quantum Theory of Dressed States

We investigate quantum electrodynamics in the infrared regime (QED-IR) using the adiabatic approximation and the framework of the functional Berry phase. In this approach, the physical state space is exact, nonperturbatively dressed, and endowed with a topological structure. Electrons do not exist as bare particles, but as topologically protected electron-photon clouds, defining a new kind of infrared quantum. These clouds are weakly bound in energy -- with a binding scale estimated at $Λ_{\text{IR}} \sim 0.2$ eV -- and remain stable provided photon energies remain below this threshold. Crucially, the theory becomes exactly solvable in this regime due to the quantization of the functional Berry flux, which governs the infrared dynamics of the dressed states. When hard (high-energy) processes are involved, this topological protection is lifted, and the theory smoothly recovers conventional perturbative QED. In contrast, in the deep infrared, the electromagnetic interaction never fully vanishes, leading to observable effects. We argue that the energy required to dissolve the infrared electron-photon cloud in QED is approximately 0.2 eV, well above the thermal energy of the cosmic microwave background (CMB). However, the observed temperature anisotropies correspond to fluctuations near $10^{-9}$ eV, which may lie at the threshold for perturbing, but not destroying, the topological structure of the dressed state. This suggests that CMB deviations could reflect residual topological imprints of the functional infrared dynamics. Finally, we propose that analogous cloud-like structures may manifest in other quantum systems governed by low-energy photon dynamics, such as atomic and molecular environments.