Schwinger–Dyson equation

The Dyson -Schwinger equations ( DSGN ), also Schwinger -Dyson equations are relations found by Freeman Dyson and Julian S. Schwinger between different Green's functions of a quantum field theory. Since they represent the equations of motion for the Green's functions, they are often called the Euler -Lagrange equations of quantum field theory. There are an infinite number of functional differential equations, which are all, directly or indirectly coupled to each other. Therefore, a term often used by the infinite tower of Dyson -Schwinger equations.

The Dyson derived by summing an infinite number of Feynman diagrams Dyson - Schwinger equations were extended by in his quantum action principle to all the Green's functions of any quantum field theory. It can be Dyson -Schwinger equations for all n-point functions can be found. However, the most important are the equations for the 2 - and 3-position functions that represent solutions whose propagators and vertices. The presented by Schwinger 4-point function is a generalization of ( inhomogeneous form) of the Bethe -Salpeter equation.

The idea behind the DSGN is that reflected interactions of a theory in their green between functions or S- matrix elements. This dressed (of English: "dressed " ) or full green between functions, ie containing the interactions, should the corresponding bare ( = non-interacting ) Green's functions in the limiting case of free theories contain, and to interaction- dependent terms. The DSGN are a guide about how and what interaction- prone terms are to be considered.

The Dyson -Schwinger equations provide an access to phenomena that are not accessible with conventional perturbation theory. In the field of quantum chromodynamics, this is for example the low energy range, since the coupling constant is large.

Examples: Quantum electrodynamics

In quantum electrodynamics, these equations are found again and again:

The Dyson -Schwinger equation of the Elektronenpropagators

Of the photon propagator

And the electron-photon Vertex '

Here denote the variables with a subscript 0, respectively, the free terms, ie for vanishing interaction. denotes the four-electron interaction core, so the four- electron - T matrix.

These examples can already be some important properties of the DSGN show. It is added to each term a free interaction term. In addition, you can see that you need the dressed photon propagator to solve the Elektronenpropagators, which is the solution of his own DSG itself. Is required for both the clad electron-photon vertex which, in turn, coupled to the four-electron core, in turn, must be sufficient of its own DSG. So all DSG are directly or indirectly coupled to each other and it forms an infinite tower of coupled equations. If you want to solve these equations in practice, so you have to cut off this tower at a certain point ( truncate ) and the missing terms with similar approaches. modeling

If we identify the terms in the brackets with the electron or photon self-energy, so you can find the original Dyson equations in the above equations.

Derivation

There are several derivations for the Dyson -Schwinger equations. While Schwinger herleitete in the original publication, the DSGN using its quantum efficiency principle, the path integral formalism is now mostly used.

Analogously to the Euler -Lagrange equations, it is assumed that the path integral of the underlying quantum field is invariant under a transformation of the infinitesimal fields. Simplistic finished, we take here to a theory of a field. In the case of multiple fields, such as in the case of QED above, the fields and your sources must be identified (indexed) are. However, nothing changes with the general ideas of the derivation. so

Wherein, with an infinitesimal displacement and integral of the functional configurations of all of the field is (similar to the sum of the statistical state physics), the effect of the theory and the sources of the fields. This condition can now be translated into the requirement that an integration over a derivative with respect to the fields disappears:

Now, the clip can be pulled out of the integral. The fields in the action must then be replaced by derivatives and we obtain

And you get the master Dyson -Schwinger equation for the full Green's functions

From it all other Dyson -Schwinger equations for the full Green's functions by a functional derivative with respect to the fields now can be generated.

The Dyson -Schwinger equations for the connected Green's functions are obtained using the general relation and the definition of the generating functional of the Green's functions associated to equation ( 1) transform

And receives

The generating functional of the one -particle irreducible Green's functions is called effective action and is usually denoted by the symbols, the index cl indicates that it is not the original fields, but their vacuum expectation values ​​. Thus the effect is defined by a generalized Legendre transformation of the generating functionals. Then you can like above the master DSG for the 1 -particle irreducible Green's functions derived:

Using Equations (1 ), (2 ) and ( 3) we now have the master Dyson -Schwinger equations. The respective Dyson -Schwinger equations of the n-point Green's functions are calculated by functional derivatives of these equations.

Applications in current research

In the current research, the Dyson -Schwinger equations are used to Green's functions, such as Quark or Gluonenpropagatoren in QCD to calculate. Also one can find means of skilful combinatorics which fractions are dominant in the low energy range. So it is hoped that conclusions such as on the long-range behavior of the strong interaction, which is closely related to the confinement problem.

Together with the Bethe -Salpeter equation can calculate self-consistent properties of bound states. This is primarily used to to determine masses and electromagnetic form factors of mesons and baryons.