 Selfenergy

In theoretical physics and quantum field theory a particle's selfenergy Σ 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 selfenergy 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 selfenergy 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 selfenergy.
Mathematically, this energy is equal to the socalled onthemassshell value of the proper selfenergy operator (or proper mass operator) in the momentumenergy representation (more precisely, to times this value). In this, or other representations (such as the spacetime representation), the selfenergy 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 particleparticle interaction; removing (or amputating) the leftmost and the rightmost straight lines in the diagram shown below (these socalled external lines correspond to prescribed values for, for instance, momentum and energy, or fourmomentum), one retains a contribution to the selfenergy operator (in, for instance, the momentumenergy representation). Using a small number of simple rules, each Feynman diagram can be readily expressed in its corresponding algebraic form.
In general, the onthemassshell value of the selfenergy operator in the momentumenergy representation is complex (see complex number). In such cases, it is the real part of this selfenergy that is identified with the physical selfenergy (referred to above as particle's selfenergy); 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 quasiparticle), in interacting systems are distinct from stable particles in vacuum; their state functions consist of complicated superpositions of the eigenstates of the underlying manyparticle system, which only, if at all, momentarily behave like those specific to isolated particles; the abovementioned lifetime is the time over which a dressed particle behaves as if it were a single particle with welldefined momentum and energy.
The selfenergy operator (often denoted by , and less frequently by ) is related to the bare and dressed propagators (often denoted by and respectively) via the Dyson equation (named after Freeman John Dyson):
 .
Multiplying on the left by the inverse of the operator G_{0} and on the right by G ^{− 1} yields
 .
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 Wboson and the Zboson 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.
In chemistry, the selfenergy or Born energy of an ion is the energy associated with the field of the ion itself.
In solid state and condensedmatter physics selfenergies and a myriad of related Quasiparticle properties are calculated by Green's function methods and Green's function (manybody theory) of interacting lowenergy excitations on the basis of electronic band structure calculations.
References
 A. L. Fetter, and J. D. Walecka, Quantum Theory of ManyParticle Systems (McGrawHill, New York, 1971); (Dover, New York, 2003)
 J. W. Negele, and H. Orland, Quantum ManyParticle 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: PrenticeHall.
 A. N. Vasil'ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004); ISBN10: 0415310024; ISBN13: 9780415310024
See also
 Quantum field theory
 Renormalization
 GW approximation
 WheelerFeynman absorber theory
Quantum electrodynamics anomalous magnetic dipole moment · Bhabha scattering · bremsstrahlung · Compton scattering · electron · GuptaBleuler formalism · Møller scattering · photon · positron · positronium · selfenergy · vacuum polarization · vertex function · Ward–Takahashi identity · ξ gauge
Categories: Quantum electrodynamics
 Quantum field theory
 Renormalization group
 Quantum physics stubs
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