How can infrared divergences in the fermion propagator be cured in QED?

In summary, in QED, there are both Ultraviolet and Infrared divergences in the fermion propagator. While the former can be solved by regularization, the latter can be treated by considering the appropriate contributions to the electron-to-electron transition amplitude plus two real photons. This solution was first established by Bloch and Nordsieck and later extended to non-Abelian gauge theories. The standard treatment within perturbation theory can be found in Weinberg's Quantum Theory of Fields. Additionally, a paper by Kulish and Faddeev explains that the physical reason for these IR properties is the long-ranged Coulomb force due to the masslessness of photons, which requires the use of modified asymptotic free states known as
  • #1
Jamister
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how to cure infrared divergences in fermion propagator in QED?
Summary: how to cure infrared divergences in fermion propagator in QED?

In calculating the fermion propagator in QED, we identify Ultraviolet and Infrared divergences. the Ultraviolet divergences solved by regularization, but I don't understand how to treat the Infrared divergences. Infrared divergences also appear in the QED vertex, but the solution there to Infrared divergences is by soft photons. How is it done in the fermion propagator?
This is the result of the 1PI diagram of the fermion propagator to leading order:
1563652745795.png

1563652892044.png


mu is the mass of the photon.
 
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The IR divergency is canceled by considering the appropriate contributions to the electron-to-electron transition amplitude plus two (un-noticed) real photons. This is a very general feature, first established by Bloch and Nordsieck and later extended to non-Abelian gauge theories by Kinoshita, Lee, and Nauenberg. The standard treatment within usual perturbation theory can be found in

Weinberg, Quantum Theory of Fields, vol. 1 (Chpt. 13).

A very illuminating paper,

P. Kulish, L. Faddeev, Asymptotic conditions and infrared divergences in quantum
electrodynamics, Theor. Math. Phys. 4 (1970) 745.
http://dx.doi.org/10.1007/BF01066485
showing that the physical reason for the IR properties is that due to the long-ranged Coulomb force due to the masslessness of the photons the naive plane-wave single-particle states used as asymptotic-free states are not the right ones. Rather you have to dress the "free" bare particles with a cloud of (real and virtual) photons around them. This cloud cancels the IR divergences. These modified asymptotic free states are sometimes called "infraparticles".
 
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1. What are infrared divergences in the fermion propagator?

Infrared divergences in the fermion propagator refer to the infinite values that arise when calculating certain physical quantities in quantum electrodynamics (QED) involving the propagation of fermions (particles with half-integer spin) over long distances. These divergences arise due to the masslessness of the photon, which leads to an infinite number of low-energy modes that contribute to the propagator.

2. Why do infrared divergences need to be cured in QED?

Infrared divergences need to be cured in QED because they lead to unphysical results and prevent the theory from making accurate predictions. For example, without a cure, calculations involving the scattering of particles would yield infinite cross-sections, which do not match with experimental observations.

3. How can infrared divergences be cured in QED?

There are several methods for curing infrared divergences in QED, such as using a finite photon mass, introducing a cutoff in momentum space, or using a non-perturbative approach like the Schwinger-Dyson equation. Another commonly used method is the Bloch-Nordsieck (BN) theorem, which states that infrared divergences cancel out in certain physical observables when all orders of perturbation theory are taken into account.

4. What is the Bloch-Nordsieck theorem?

The Bloch-Nordsieck (BN) theorem is a mathematical theorem that states that infrared divergences in QED cancel out in certain physical observables when all orders of perturbation theory are taken into account. This theorem allows for the consistent calculation of physical quantities in QED, as it ensures that the unphysical effects of infrared divergences do not affect the final results.

5. Are there any limitations to the methods used to cure infrared divergences in QED?

Yes, there are limitations to the methods used to cure infrared divergences in QED. For example, introducing a finite photon mass may lead to inconsistencies with other physical principles, and using a cutoff in momentum space can affect the renormalizability of the theory. Additionally, the BN theorem only applies to certain physical observables and may not be applicable in all situations. Therefore, it is important for scientists to carefully consider the limitations and implications of each method when attempting to cure infrared divergences in QED.

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