Renormalization - Divergences in Quantum Electrodynamics

Divergences in Quantum Electrodynamics

When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative calculations many integrals were divergent.

One way of describing the divergences was discovered in the 1930s by Ernst Stueckelberg, in the 1940s by Julian Schwinger, Richard Feynman, and Shin'ichiro Tomonaga, and systematized by Freeman Dyson. The divergences appear in calculations involving Feynman diagrams with closed loops of virtual particles in them.

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy-momentum relation for the observed mass of that particle. (That is, is not necessarily the mass of the particle in that process (e.g. for a photon it could be nonzero).) Such a particle is called off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop must be balanced by an equal and opposite variation in the energy of another particle in the loop. So to find the amplitude for the loop process one must integrate over all possible combinations of energy and momentum that could travel around the loop.

These integrals are often divergent, that is, they give infinite answers. The divergences which are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one which comes from

• the region in the integral where all particles in the loop have large energies and momenta.
• very short wavelengths and high frequencies fluctuations of the fields, in the path integral for the field.
• Very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths.

So these divergences are short-distance, short-time phenomena.

There are exactly three one-loop divergent loop diagrams in quantum electrodynamics.

1. a photon creates a virtual electron-positron pair which then annihilate, this is a vacuum polarization diagram.
2. an electron which quickly emits and reabsorbs a virtual photon, called a self-energy.
3. An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in figure 2, and it is called a vertex renormalization. The Feynman diagram for this is also called a penguin diagram due to its shape remotely resembling a penguin (with the initial and final state electrons as the arms and legs, the second photon as the body and the first looping photon as the head).

The three divergences correspond to the three parameters in the theory:

1. the field normalization Z.
2. the mass of the electron.
3. the charge of the electron.

A second class of divergence, called an infrared divergence, is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the vertex function has both ultraviolet and infrared divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two on shell (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the bremsstrahlung process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a loop as in the vertex diagram and zero-energy photons emitted through bremsstrahlung. From a mathematical point of view the IR divergences can be regularized by assuming fractional differentiation with respect to a parameter, for example is well defined at p = a but is UV divergent, if we take the 3/2-th fractional derivative with respect to we obtain the IR divergence, so we can cure IR divergences by turning them into UV divergences.