Self-energy

In classical physics , the self-energy of a charge distribution is understood as the energy that is required to compose the charge distribution from originally infinitely distant components.

In quantum field theory , the perturbative corrections to the propagator are called self-energy .

Classical physics

→ Main article : Classic electron radius

If the charge is evenly distributed over a spherical surface with a radius , the electric field inside the sphere disappears and the total field energy outside is: ${\displaystyle q}$ ${\displaystyle R}$

${\displaystyle E={\frac {1}{8\pi \varepsilon _{0}}}\cdot {\frac {q^{2}}{R}}}$

If you equate this energy with the rest energy of the electron , you get half the classical electron radius . If one were to concentrate the elementary charge on a spherical shell with less than half the classical electron radius, according to classical physics the self-energy of this charge distribution would already be greater than the rest energy of the electron. ${\displaystyle E=m_{e}c^{2}}$ ${\displaystyle R}$

Quantum field theory

In quantum field theory , the self-energy (also mass term) denotes the contributions of all diagrams with an incoming and an outgoing line.

As irreducible self-energy insertion ( ) is a self-energy insertion is referred to, which can not be disassembled by separating a line into two separate portions. In this case, for example, the diagrams in which several loops follow one another separately are not recorded. The self-energy is then defined as the sum of the contributions of all irreducible self-energy injections. The combination of the self-energy operator ( ) as well as the free propagator ( ) and attracted propagator ( ) describes the Dyson equation: ${\displaystyle \Sigma }$ ${\displaystyle \Sigma }$ ${\displaystyle G_{0}^{}}$ ${\displaystyle G}$

${\displaystyle G=G_{0}\sum _{j=0}^{\infty }(\Sigma G_{0})^{j}=G_{0}+G_{0}\Sigma G_{0}+G_{0}\Sigma G_{0}\Sigma G_{0}+\dots }$

This corresponds to the following diagrammatic representation:

Adding up the geometric series gives

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

A self-energy diagram is called a skeleton if it is built exclusively from propagators that do not contain any self-energy inserts, i.e. loops. An attracted skeleton is a skeleton from the development of self-energy, in which every free propagator has been replaced by a propagator that has been corrected for self-energy. So the self-energy is the sum of the contributions of all attracted skeletons.

The representation of self-energy as the sum of the contributions of all attracted skeletons and the Dyson equation form a system of equations that must be solved simultaneously (self-consistently). This can be done iteratively until self-consistency can be terminated. This leads to the self-consistent renormalization .

The simplest case of the Dyson equation, the two-point function, just looks at self-energy.

literature

• AL Fetter, and JD Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971); (Dover, New York, 2003)
• JW Negele, and H. Orland, Quantum Many-Particle Systems (Westview Press, Boulder, 1998)
• AA Abrikosov, LP Gorkov and IE Dzyaloshinski (1963): Methods of Quantum Field Theory in Statistical Physics Englewood Cliffs: Prentice-Hall.

See also

• Self-energy functional theory