We study infrared divergences in quantum electrodynamics using geometric phases and the adiabatic approximation in quantum field theory. In this framework, the asymptotic in and out states are modified by Berry phases, <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mtext>∆</mtext> <msub> <mi>α</mi> <mtext>in</mtext> </msub> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {e}^{i\Delta {\alpha}_{\textrm{in}}} $</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mtext>∆</mtext> <msub> <mi>α</mi> <mtext>out</mtext> </msub> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$ {e}^{i\Delta {\alpha}_{\textrm{out}}} $</EquationSource> </InlineEquation>, which encode the infrared structure non-perturbatively and regulate soft-photon divergences.Unlike the Faddeev-Kulish formalism, which employs perturbative dressing with coherent states, our approach reformulates the effective action in terms of Berry connections in field space. This yields finite, gauge-invariant scattering amplitudes without requiring a sum over soft-photon emissions.We show that infrared divergences cancel to all orders in the bremsstrahlung vertex function Γμ(p1, p2), due to destructive interference among inequivalent Berry phases. As an application, we study the formation of positronium in the infrared regime and argue that the dressed S-matrix exhibits a functional singularity at <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mi>s</mi> <mo>=</mo> <mn>4</mn> <msubsup> <mi>m</mi> <mi>e</mi> <mn>2</mn> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$ s=4{m}_e^2 $</EquationSource> </InlineEquation>, corresponding to a physical pole generated by topological flux.

Topology and the infrared structure of quantum electrodynamics

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2025年7月16日

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J. Gamboa

摘要

We study infrared divergences in quantum electrodynamics using geometric phases and the adiabatic approximation in quantum field theory. In this framework, the asymptotic in and out states are modified by Berry phases, $e^{i ∆ α_{in}}$ $ {e}^{i\Delta {\alpha}_{\textrm{in}}} $ and $e^{i ∆ α_{out}}$ $ {e}^{i\Delta {\alpha}_{\textrm{out}}} $ , which encode the infrared structure non-perturbatively and regulate soft-photon divergences.

Unlike the Faddeev-Kulish formalism, which employs perturbative dressing with coherent states, our approach reformulates the effective action in terms of Berry connections in field space. This yields finite, gauge-invariant scattering amplitudes without requiring a sum over soft-photon emissions.

We show that infrared divergences cancel to all orders in the bremsstrahlung vertex function Γ^μ(p₁, p₂), due to destructive interference among inequivalent Berry phases. As an application, we study the formation of positronium in the infrared regime and argue that the dressed S-matrix exhibits a functional singularity at $s = 4 m_{e}^{2}$ $ s=4{m}_e^2 $ , corresponding to a physical pole generated by topological flux.