﻿ Feynman diagram

# Feynman diagram

Feynman diagrams present in particle physics and solid state physics, quantum field theoretical contributions to scattering processes depict and illustrate and so facilitate their calculation. They were developed in 1949 by Richard Feynman on the example of quantum electrodynamics, but were soon transferred to the solid state physics.

• 5.1 photons
• 5.2 Z- bosons
• 5.3 W bosons
• 5.4 gluons
• 5.5 Higgs bosons

## Importance

Feynman diagrams are a graphical representation of the interactions of particles. The interactions are described mathematically by Lagrangian densities. For example, the interaction between electrons and photons, is described by the following Lagrangian:

This is the electron ( or positron) corresponding spinor field as a column vector, the complex conjugate to the field as a row vector, the electric charge, the mass of the electron, the Dirac matrices and the photon corresponding electromagnetic four potential. The derivative is the spread of the particles in the space-time, while the term coupling the electric charge of the electron and the photon. The indices and represent the four dimensions of Minkowski space () dar. It is the Einstein summation convention. The product of the two fields is calculated spinor the purposes of the matrix multiplication, ie it corresponds to a scalar, because the left- spinor a row vector and the rightmost spinor is a column vector.

However, such expressions are very complicated. The Feynman diagrams are bidding on the one hand a simplified and descriptive presentation option. An exact solution, however, you can not just read from Feynman diagrams. Here it must be resorted to mathematical formulas.

On the other hand, can be calculated quantum mechanical amplitude for each Feynman diagram. The total amplitude of a spreading operation is the sum of the amplitudes of valid Feynman diagrams of the scattering process. Thus, the set of all Feynman diagrams associated with a fixed number of interaction points ( vertices ) for a term in the power series of the total amplitude in the coupling constant.

## Construction

Feynman diagrams are composed of basic symbols that represent each specific types of elementary particles of the Standard Model. The direction of the arrow indicates whether it is a normal particle or an antiparticle. In a normal particles, the arrow points in the direction of time and at a antiparticles, the arrow points to the direction of time. This does not mean that moves the antiparticle against time in the past. The direction of the arrow only serves to overview of particles and antiparticles. The number of arrows entering the Feynman diagram from left is always equal to the number of arrows that come out right again. Here, an arrow is counted against the time direction as negative and he lifts an arrow in time direction. The exchange particles, which allow interaction between particles are drawn by a different symbolism without arrows, as shown in the following table.

The time is read from left to right in the following.

The labeling of the particles defined by which particles it is concrete.

Feynman diagrams have external lines that one or leaving interaction points, and internal lines connecting pairs of points of interaction. The outer lines correspond to one - and outgoing particles. The interaction points where lines meet, are called vertices. At a vertex particles can be created, destroyed or scattered.

As the time passes from the left to the right, rotation of the vertex resulting in different interpretations.

• Confluence and splitting processes

Emission of a photon by the electron (energy lowering of the electron)

Absorption of a photon by the electron ( = emission of a photon by a positron energy increase of the electron)

Decomposing a photon into an electron and a positron

This is important to note that a Feynman diagram, there is only on the particles contiguous vertices. In addition, due to the charge conservation the number of incoming arrows in the diagram is equal to the number of outgoing arrows from the diagram.

In the solid-state physical interpretation corresponds to the second and third diagram from the last series of energy reduction or increase of the (in both cases from the left) incident electron by the emission or absorption of a phonon.

## Propagators

The inner lines are called propagators and interprets them as virtual particles. Virtual particles can not be observed. As a result, there is an ambiguity. All diagrams with the same inlet and outlet lines are equivalent and are summed.

### Examples

To calculate the scattering of two fermions - the Møller scattering - considering Feynman diagrams with two incoming and two outgoing electron.

The images show the scattering in lowest order ( " tree level "). The four outer lines represent the incoming and outgoing electron, the inner shaft line for the virtual photon, which causes the electromagnetic interaction.

An equivalent diagram for Møller scattering with another propagator.

Each of these diagrams corresponds to the contribution to scattering, the total spreading process is represented by the sum of all the plots.

As another example, here is the Compton effect is given in the lowest order. Again, the possible diagrams are summed.

Here, the neutrally charged fermion is in the right figure shows the sum of the other diagrams of the lowest order, on the one hand a negatively charged fermion (eg an electron) and on the other hand, positively charged fermion mediated (eg a proton ):

The calculation of scattering and more generally, the Feynman rules for the mathematical expressions that correspond to the lines and vertices can be found in many textbooks of particle physics (see links).

## Grind

The possible propagators form higher order loops ( loops ).

: A loop

: Two loops

: Two loops

The possible Feynman diagrams can be classified according to the number of inner loops. This number is called the loop order.

These diagrams are summed up in the wake of a series expansion.

There are any number of diagrams conceivable. However, the coupling of the coupling constant α with increasing order n is smaller. With sufficiently high order diagrams are numerically negligible according to the working hypothesis of perturbation theory, since they have an insignificant effect on the result.

Graphs without loops have the structure of a tree (trunk, branches of the branch ) and mean tree diagrams.

## Feynman rules

Describe the Feynman rules, which interactions are possible and which are not.

### Photons

Photons interact with all electrically charged elementary particles. Pictures for electrons and muons:

Interaction between muons and photon

### Z bosons

The Z boson interacts between all other elementary particles of the Standard Model with the exception of gluons; with photons they only interact simultaneously with W bosons. In particular, neutrinos ( and ) have no interaction on photon; Therefore one has to rely for their generation and detection of Z bosons and W - bosons.

Interaction between electron, positron and Z boson

Interaction between muon anti - muon and Z boson

### W bosons

The W boson mediates the one hand between neutrinos and the charged leptons l (electrons, muons and tauons ), and on the other hand. Between up -type quarks and down- type quarks The W boson is the carrier of a positive (W ) or negative (W- ) electric charge. Because of the electrical charge, the W - boson interaction is subject to the photon; it also interacts with the Z boson and other W - bosons.

Interaction between neutrino, positively charged lepton and W - boson

Interaction between two oppositely charged W bosons. Time axis runs from top to bottom

Interaction between two oppositely charged W bosons and two photons. Time axis runs from top to bottom

Interaction between two oppositely charged W bosons and two Z bosons. Time axis runs from top to bottom

Interaction between two oppositely charged W bosons, a Z boson and a photon. Time axis runs from top to bottom

W bosons are particularly interesting, since they allow a change of flavor. This means that the number of electrons, neutrinos, etc. may change. This characteristic plays an important role as the β -decay.

### Gluons

Gluons mediate the strong interactions of quarks.

Quarks have a color charge. In contrast to the electric charge from the charge "positive" ( ) and "negative" ( - ) which is the color charge from the charge "red", "green" and "blue", and the anti-color charges "anti- red " ( " cyan ")," Anti- Green " ( " magenta ") and" anti - blue " ( " yellow " ). Need to neutralize the color charge either quarks with color charges { Red, Green, Blue }, { cyan, magenta, yellow }, { Red, Cyan }, { green, magenta } or { Blue, Yellow } are connected via gluons.

Particles with three quarks are here as baryons ( including, for example, the protons and neutrons), respectively. Particles of two quarks are called mesons. Free quarks are connected immediately with gluons of the strong force to baryons or mesons.

Gluons are represented mathematically 3 × 3 matrices by adjoining the special unitary group SU (3) as a traceless Hermitian. Thus, there are eight (32-1 ) different gluons.

Gluons each carry two color charges. Thus they are themselves subject to strong interactions and are therefore able to reconnect with yourself. Theoretically, as well as gluon balls produce, which manage only of gluons and quarks without. However, these have not yet been demonstrated.

### Higgs bosons

The Higgs bosons interact with all massive elementary particles, ie also with other Higgs bosons ( self-interaction ). Only with photons and gluons there is no interaction. According to the standard model of particle physics, elementary particles get their mass only by this interaction (see Higgs mechanism ).

## Types of Feynman diagrams

• Related charts If each vertex is connected via internal lines and other vertices with every other vertex, so is referred to the chart as contiguous, otherwise than incoherent. At each related part of the chart, the sum of the energies of the incoming pulses, and charges the particles is equal to the sum of the energy pulses, and charges of the outgoing particles.
• One-particle irreducible diagrams Can not be divided by cutting an internal line into two disjointed plots a coherent diagram, it is said one-particle irreducible. In such diagrams occur integrations that can not be systematically simplify product as easier integrals.
• Amputees diagrams If we let in a chart corrections ( self- energy, see below ) toward the outer lines, as it is called amputated.
• Self-energy diagrams A diagram with a loop and outer lines of only two vertices is called (after amputation) self-energy diagram. Its value depends only on the energy and momentum of a vertex in and flows through to the external lines at the other addition.
• Skeleton diagrams A graph without a self - energy subdiagram is called skeleton graph.

## Applications

Applications of the Feynman diagrams, there are primarily in relativistic quantum field theories, such as quantum electrodynamics or quantum chromodynamics, but also in the non-relativistic solid state physics, especially in the many-body physics and statistical physics.

## Solids Physical analogy

The usual transfer to the solid state physics is obtained by by not taking in as positron from the cambered shown photon line, ie the quantum of the electromagnetic waves, goes to the so-called phonons, ie the quanta of sound waves, and the count-down electron sense of quantum electrodynamics, but interpreted as an electron hole in terms of the solid state theory. The essential diagrams are obtained in this way, inter alia, for the occurrence of superconductivity and in general for confluence and splitting processes by destroying or generating an elementary fermionic excitation (such as an electron or a hole or an electric polaron ) together with an incoming (or outgoing ) bosonic quasiparticle, such as the already mentioned phonon or a so-called magnons ( quantized spin waves ) or a plasmon ( a quantized plasma oscillation ).

For all interaction processes among Beiteiligung the suggestions mentioned the sum of the energies (frequencies times) or pulses ( wave number times) get so so correspond to the diagrams shown well-defined mathematical expressions for the amplitudes of the interaction

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