Equation of Schwinger-Dyson

The equation of Schwinger-Dyson , according to Julian Schwinger and Freeman Dyson, is an equation of the Quantum theory of the fields. Being given a function limited F on the configurations of the field, then for all Vector of state |ψ> (which is solution of the QFT), we have:

with S the function of action and $\backslash \; mathcal\; \{T\}$ the operation of ordonnation of time.

In the same manner, in the formulation of the state of density, for any state (valid) ρ, we have:

$\backslash \; rho\; (\backslash \; mathcal\; \{T\}\; \backslash \; \{\backslash \; frac\; \{\backslash \; delta\}\; \{\backslash \; delta\; \backslash \; phi\}\; F\; \backslash \})\; =-i\; \backslash \; rho\; (\backslash \; mathcal\; \{T\}\; \backslash \; \{F\; \backslash \; frac\; \{\backslash \; delta\}\; \{\backslash \; delta\; \backslash \; phi\}\; S\; \backslash \})$

These infinite equations can be used to solve the functions of correlation, without disturbance.

One can also reduce the action S by separating it: S=1/2 D^-1_ij φⁱ φ ^j+S_int with for first term the quadratic share and covariant D^-1 a symmetrical and reversible tensor (antisymmetric for the fermions) of row 2 in the Notation of deWitt. Then one can rewrite the equations as follows:

If F is a function of φ, then for an operator K , F is defined as an operator who replaces K by φ. For example, if

$F=\; \backslash \; frac\; \{\backslash \; partial^\; \{k\_1\}\}\; \{\backslash \; partial\; x\_1^\; \{k\_1\}\}\; \backslash \; phi\; (x\_1)\; \backslash \; cdots\; \backslash \; frac\; \{\backslash \; partial^\; \{k\_n\}\}\; \{\backslash \; partial\; x\_n^\; \{k\_n\}\}\; \backslash \; phi\; (x\_n)$

and that G is a function of J , then:

$F\; J\}\; G=\; (-\; I)\; ^n\; \backslash \; frac\; \{\backslash \; partial^\; \{k\_1\}\}\; \{\backslash \; partial\; x\_1^\; \{k\_1\}\}\; \backslash \; frac\; \{\backslash \; delta\}\; \{\backslash \; delta\; J\; (x\_1)\}\; \backslash \; cdots\; \backslash \; frac\; \{\backslash \; partial^\; \{k\_n\}\}\; \{\backslash \; partial\; x\_n^\; \{k\_n\}\}\; \backslash \; frac\; \{\backslash \; delta\}\; \{\backslash \; delta\; J\; (x\_n)\}\; G$.

If we have an analytical function Z (called generating function) of J (called field source) satisfying the equation:

,

then, the equation of Schwinger-Dyson for the generator Z is:

$\backslash \; frac\; \{\backslash \; delta\; S\}\; \{\backslash \; delta\; \backslash \; phi\; (X)\}\; \backslash \; frac\; \{\backslash \; delta\}\; \{\backslash \; delta\; J\}\; Z+J\; (X)\; Z=0$

If we develop this equation in Taylor series for J near to 0, one obtains the whole play of the equations of Schwinger-Dyson.

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