Self-energy

Characteristics

Mathematically, this energy is equal to the so-called on mass shell value of the proper self-energy operator (or proper mass operator) in the momentum-energy representation (more precisely, to ${\displaystyle \hslash }$ times this value). In this, or other representations (such as the space-time representation), the self-energy is pictorially (and economically) represented by means of Feynman diagrams, such as the one shown below. In this particular diagram, the three arrowed straight lines represent particles, or particle propagators, and the wavy line a particle-particle interaction; removing (or amputating) the left-most and the right-most straight lines in the diagram shown below (these so-called external lines correspond to prescribed values for, for instance, momentum and energy, or four-momentum), one retains a contribution to the self-energy operator (in, for instance, the momentum-energy representation). Using a small number of simple rules, each Feynman diagram can be readily expressed in its corresponding algebraic form.

In general, the on-the-mass-shell value of the self-energy operator in the momentum-energy representation is complex. In such cases, it is the real part of this self-energy that is identified with the physical self-energy (referred to above as particle's "self-energy"); the inverse of the imaginary part is a measure for the lifetime of the particle under investigation. For clarity, elementary excitations, or dressed particles (see quasi-particle), in interacting systems are distinct from stable particles in vacuum; their state functions consist of complicated superpositions of the eigenstates of the underlying many-particle system, which only momentarily, if at all, behave like those specific to isolated particles; the above-mentioned lifetime is the time over which a dressed particle behaves as if it were a single particle with well-defined momentum and energy.

The self-energy operator (often denoted by ${\displaystyle {\mathrm{\Sigma }}_{}^{}}$, and less frequently by ${\displaystyle M_{}^{}}$) is related to the bare and dressed propagators (often denoted by ${\displaystyle G_0^{}}$ and ${\displaystyle G_{}^{}}$ respectively) via the Dyson equation (named after Freeman Dyson):

${\displaystyle G=G_0^{}+G_0\mathrm{\Sigma }G.}$

Multiplying on the left by the inverse ${\displaystyle G_0^{-1}}$ of the operator ${\displaystyle G_0}$ and on the right by ${\displaystyle G^{-1}}$ yields

${\displaystyle \mathrm{\Sigma }=G_0^{-1}-G^{-1}.}$

The photon and gluon do not get a mass through renormalization because gauge symmetry protects them from getting a mass. This is a consequence of the Ward identity. The W-boson and the Z-boson get their masses through the Higgs mechanism; they do undergo mass renormalization through the renormalization of the electroweak theory.

Neutral particles with internal quantum numbers can mix with each other through virtual pair production. The primary example of this phenomenon is the mixing of neutral kaons. Under appropriate simplifying assumptions this can be described without quantum field theory.