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In [[theoretical physics]] and [[quantum field theory]] a particle's '''self-energy''' <math>\Sigma</math> represents the contribution to the particle's [[energy]], or [[Effective mass (solid-state physics)|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 [[Planck constant|<math>\hbar</math>]] 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 diagram]]s, such as the one shown below. In this particular diagram, the three arrowed straight lines represent particles, or particle ''[[propagator]]s'', 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 (see [[complex number]]). 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, if at all, momentarily 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 <math>\Sigma_{}^{}</math>, and less frequently by <math>M_{}^{}</math>) is related to the bare and dressed [[propagator]]s (often denoted by <math>G_0^{}</math> and <math>G_{}^{}</math> respectively) via the Dyson equation (named after [[Freeman John Dyson]]):

:<math>G = G_0^{} + G_0 \Sigma G.</math>

Multiplying on the left by the inverse <math>G_0^{-1}</math> of the operator <math>G_0</math>
and on the right by <math>G^{-1}</math> yields

:<math>\Sigma = G_0^{-1} - G^{-1}.</math>

:[[File:electron self energy.svg]]

:[[File:Dyson.svg]]

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 [[kaon]]s. Under appropriate simplifying assumptions this can be described [[Neutral particle oscillations|without quantum field theory]].

In [[chemistry]], the self-energy or ''Born energy'' of an ion is the energy associated with the field of the ion itself.

In [[Solid-state physics|solid state]] and [[Condensed-matter physics|condensed-matter]] physics self-energies and a myriad related [[quasiparticle]] properties are calculated by [[Green's function]] methods and [[Green's function (many-body theory)]] of '''interacting low-energy excitations''' on the basis of [[electronic band structure]] calculations.

== See also ==
* [[Quantum field theory]]
* [[QED vacuum]]
* [[Renormalization]]
* [[GW approximation]]
* [[Wheeler–Feynman absorber theory]]

== References ==
* A. L. Fetter, and J. D. Walecka, ''Quantum Theory of Many-Particle Systems'' (McGraw-Hill, New York, 1971); (Dover, New York, 2003)
* J. W. Negele, and H. Orland, ''Quantum Many-Particle Systems'' (Westview Press, Boulder, 1998)
* A. A. Abrikosov, L. P. Gorkov and I. E. Dzyaloshinski (1963): ''Methods of Quantum Field Theory in Statistical Physics'' Englewood Cliffs: Prentice-Hall.
* {{cite book
|last      = Alexei M. Tsvelik
|title     = Quantum Field Theory in Condensed Matter Physics
|edition   = 2nd
|publisher = Cambridge University Press
|year      = 2007
|isbn      = 0-521-52980-8
}}
* A. N. Vasil'ev ''The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics'' (Routledge Chapman & Hall 2004); ISBN 0-415-31002-4; ISBN 978-0-415-31002-4

{{QED}}

[[Category:Quantum electrodynamics]]
[[Category:Quantum field theory]]
[[Category:Renormalization group]]