Self-energy

In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy ${\displaystyle \Sigma }$, and represents the contribution to the particle's

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 ${\displaystyle \hbar }$ 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.

In general, the on-the-mass-shell value of the self-energy operator in the momentum-energy representation is

The self-energy operator (often denoted by ${\displaystyle \Sigma _{}^{}}$, 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):

${\displaystyle G=G_{0}^{}+G_{0}\Sigma G.}$

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

${\displaystyle \Sigma =G_{0}^{-1}-G^{-1}.}$

The

Neutral particles with internal quantum numbers can mix with each other through

Other uses

In chemistry, the self-energy or Born energy of an ion is the energy associated with the field of the ion itself.^{[citation needed]}

In

• Quantum field theory
• QED vacuum
• Renormalization
• Self-force
• 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.
• 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'ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004);
  ISBN 978-0-415-31002-4
• John E. Inglesfield (2015). The Embedding Method for Electronic Structure. IOP Publishing.
  ISBN 978-0-7503-1042-0
  .