Inquantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines itsself-energy${\displaystyle \mathrm{\Sigma }}$. The self-energy represents the contribution to the particle'senergy, oreffective mass, due to interactions between the particle and its environment. Inelectrostatics, the energy required to assemble the charge distribution takes the form of self-energy by bringing in the constituent charges from infinity, where the electric force goes to zero. In acondensed matter context, self-energy is used to describe interaction induced renormalization ofquasiparticle mass (dispersions) and lifetime. Self-energy is especially used to describe electron-electron interactions inFermi liquids. Another example of self-energy is found in the context ofphonon softening due to electron-phonon coupling.

Mathematically, this energy is equal to the so-calledon mass shell value of the proper self-energyoperator (or proper massoperator) 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 ofFeynman diagrams, such as the one shown below. In this particular diagram, the three arrowed straight lines represent particles, or particlepropagators, and the wavy line a particle-particle interaction; removing (oramputating) the left-most and the right-most straight lines in the diagram shown below (these so-calledexternal lines correspond to prescribed values for, for instance, momentum and energy, orfour-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 iscomplex. 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, ordressed particles (seequasi-particle), in interacting systems are distinct from stable particles in vacuum; their state functions consist of complicated superpositions of theeigenstates 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 theDyson equation (named afterFreeman 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

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

Thephoton andgluon do not get a mass throughrenormalization becausegauge symmetry protects them from getting a mass. This is a consequence of theWard identity. TheW-boson and theZ-boson get their masses through theHiggs mechanism; they do undergo mass renormalization through the renormalization of theelectroweak theory.

Neutral particles with internal quantum numbers can mix with each other throughvirtual pair production. The primary example of this phenomenon is the mixing of neutralkaons. Under appropriate simplifying assumptions this can be describedwithout quantum field theory.

Inchemistry, the self-energy orBorn energy of an ion is the energy associated with the field of the ion itself.^{[citation needed]}

Insolid state andcondensed-matter physics self-energies and a myriad of relatedquasiparticle properties are calculated byGreen's function methods andGreen's function (many-body theory) ofinteracting low-energy excitations on the basis ofelectronic band structure calculations. Self-energies also find extensive application in the calculation of particle transport through open quantum systems and the embedding of sub-regions into larger systems (for example the surface of a semi-infinite crystal).^{[citation needed]}

• Quantum field theory
• QED vacuum
• Renormalization
• Self-force
• GW approximation
• Wheeler–Feynman absorber theory

• 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.
• Alexei M. Tsvelik (2007).Quantum Field Theory in Condensed Matter Physics (2nd ed.). Cambridge University Press.ISBN 978-0-521-52980-8.
• A. N. Vasil'evThe 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
• John E. Inglesfield (2015).The Embedding Method for Electronic Structure. IOP Publishing.ISBN 978-0-7503-1042-0.

• Quantum electrodynamics
• Quantum field theory
• Renormalization group

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