Self-energy

In most theoretical physics such as quantum field theory, a particle's self-energy ${\displaystyle \Sigma }$ represents the contribution to the particle's energy, or effective mass, due to interactions between the particle and the system it is part of. For example, in electrostatics the self-energy of a given charge distribution is the energy required to assemble the distribution by bringing in the constituent charges from infinity, where the electric force goes to zero. In a condensed matter context relevant to electrons moving in a material, the self-energy represents the potential felt by the electron due to the surrounding medium's interactions with it: for example, the fact that electrons repel each other means that a moving electron polarizes (causes to displace) the electrons in its vicinity and this in turn changes the potential the moving electron feels; these and other effects are included in the self-energy. In basic terms, the self-energy is the energy that a particle has as a result of changes that it itself causes in its environment.

Mathematically, this energy is equal to the so-called on-the-mass-shell value of the proper self-energy operator (or proper mass operator) in the momentum-energy representation (more precisely, to ${\displaystyle \hbar }$ 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.

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