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TheSchwinger–Dyson equations (SDEs) orDyson–Schwinger equations, named afterJulian Schwinger andFreeman Dyson, are general relations betweencorrelation functions inquantum field theories (QFTs). They are also referred to as theEuler–Lagrange equations of quantum field theories, since they are theequations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.

In his paper "TheS-Matrix in Quantum electrodynamics",^[1] Dyson derived relations between differentS-matrix elements, or more specific "one-particle Green's functions", inquantum electrodynamics, by summing up infinitely manyFeynman diagrams, thus working in a perturbative approach. Starting from hisvariational principle, Schwinger derived a set of equations for Green's functions non-perturbatively,^[2] which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions ofquantum field theories. Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields oftheoretical physics, such assolid-state physics andelementary particle physics.

Schwinger also derived an equation for the two-particle irreducible Green functions,^[2] which is nowadays referred to as the inhomogeneousBethe–Salpeter equation.

Given a polynomially boundedfunctional${\displaystyle F}$ over the field configurations, then, for anystate vector (which is a solution of the QFT),${\displaystyle |\psi ⟩}$, we have

${\displaystyle ⟨\psi \left|\mathcal{T}\left\{\frac{\delta }{\delta \phi }F[\phi ]\right\}\right|\psi ⟩=-i⟨\psi \left|\mathcal{T}\left\{F[\phi ]\frac{\delta }{\delta \phi }S[\phi ]\right\}\right|\psi ⟩}$

where$\frac{\delta }{\delta \phi }$ is thefunctional derivative with respect to${\displaystyle \phi }$,${\displaystyle S}$ is theaction functional and${\displaystyle \mathcal{T}}$ is thetime ordering operation.

Equivalently, in thedensity state formulation, for any (valid) density state${\displaystyle \rho }$, we have

${\displaystyle \rho \left(\mathcal{T}\left\{\frac{\delta }{\delta \phi }F[\phi ]\right\}\right)=-i\rho \left(\mathcal{T}\left\{F[\phi ]\frac{\delta }{\delta \phi }S[\phi ]\right\}\right).}$

Thisinfinite set of equations can be used to solve for the correlation functionsnonperturbatively.

To make the connection to diagrammatic techniques (likeFeynman diagrams) clearer, it is often convenient to split the action${\displaystyle S}$ as

${\displaystyle S[\phi ]=\frac{1}{2}{\phi }^i{D}_{ij}^{-1}{\phi }^j+S_{\text{int}}[\phi ],}$

where the first term is the quadratic part and${\displaystyle D^{-1}}$ is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in thedeWitt notation whose inverse,${\displaystyle D}$ is called the bare propagator and${\displaystyle S_{\text{int}}[\phi ]}$ is the "interaction action". Then, we can rewrite the SD equations as

${\displaystyle ⟨\psi \left|\mathcal{T}\left\{F{\phi }^j\right\}\right|\psi ⟩=⟨\psi \left|\mathcal{T}\left\{i{F}_{,i}{D}^{ij}-F{S}_{\text{int},i}{D}^{ij}\right\}\right|\psi ⟩.}$

If${\displaystyle F}$ is a functional of${\displaystyle \phi }$, then for anoperator${\displaystyle K}$,${\displaystyle F[K]}$ is defined to be the operator which substitutes${\displaystyle K}$ for${\displaystyle \phi }$. For example, if

${\displaystyle F[\phi ]=\frac{{\mathrm{\partial }}^{k_1}}{\mathrm{\partial }{x}_1^{k_1}}\phi (x_1)\cdots \frac{{\mathrm{\partial }}^{k_n}}{\mathrm{\partial }{x}_n^{k_n}}\phi (x_n)}$

and${\displaystyle G}$ is a functional of${\displaystyle J}$, then

${\displaystyle F\left[-i\frac{\delta }{\delta J}\right]G[J]=(-i{)}^n\frac{{\mathrm{\partial }}^{k_1}}{\mathrm{\partial }{x}_1^{k_1}}\frac{\delta }{\delta J(x_1)}\cdots \frac{{\mathrm{\partial }}^{k_n}}{\mathrm{\partial }{x}_n^{k_n}}\frac{\delta }{\delta J(x_n)}G[J].}$

If we have an "analytic" (a function that is locally given by a convergent power series)functional${\displaystyle Z}$ (called thegenerating functional) of${\displaystyle J}$ (called thesource field) satisfying

${\displaystyle \frac{{\delta }^{n}Z}{\delta J(x_1)\cdots \delta J(x_n)}[0]=i^{n}Z[0]⟨\phi (x_1)\cdots \phi (x_n)⟩,}$

then, from the properties of the functional integrals

${\displaystyle {⟨\frac{\delta \mathcal{S}}{\delta \phi (x)}\left[\phi \right]+J(x)⟩}_J=0,}$

the Schwinger–Dyson equation for the generating functional is

${\displaystyle \frac{\delta S}{\delta \phi (x)}\left[-i\frac{\delta }{\delta J}\right]Z[J]+J(x)Z[J]=0.}$

If we expand this equation as aTaylor series about${\displaystyle J=0}$, we get the entire set of Schwinger–Dyson equations.

To give an example, suppose

${\displaystyle S[\phi ]=\int d^{d}x\left(\frac{1}{2}{\mathrm{\partial }}^{\mu }\phi (x){\mathrm{\partial }}_{\mu }\phi (x)-\frac{1}{2}{m}^2\phi (x{)}^2-\frac{\lambda }{4!}\phi (x{)}^4\right)}$

for a real field ${\displaystyle \phi }$.

Then,

${\displaystyle \frac{\delta S}{\delta \phi (x)}=-{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }\phi (x)-m^2\phi (x)-\frac{\lambda }{3!}{\phi }^3(x).}$

The Schwinger–Dyson equation for this particular example is:

${\displaystyle i{\mathrm{\partial }}_{\mu }{\mathrm{\partial }}^{\mu }\frac{\delta }{\delta J(x)}Z[J]+i{m}^2\frac{\delta }{\delta J(x)}Z[J]-\frac{i\lambda }{3!}\frac{{\delta }^3}{\delta J(x{)}^3}Z[J]+J(x)Z[J]=0}$

Note that since

${\displaystyle \frac{{\delta }^3}{\delta J(x{)}^3}}$

is not well-defined because

${\displaystyle \frac{{\delta }^3}{\delta J(x_1)\delta J(x_2)\delta J(x_3)}Z[J]}$

is adistribution in

${\displaystyle x_1}$,${\displaystyle x_2}$ and${\displaystyle x_3}$,

this equation needs to beregularized.

In this example, the bare propagator D is theGreen's function for${\displaystyle -{\mathrm{\partial }}^{\mu }{\mathrm{\partial }}_{\mu }-m^2}$ and so, the Schwinger–Dyson set of equations goes as

and

etc.

(Unless there isspontaneous symmetry breaking, the odd correlation functions vanish.)

• Functional renormalization group
• Dyson equation
• Path integral formulation
• Source field

There are not many books that treat the Schwinger–Dyson equations. Here are three standard references:

• Claude Itzykson, Jean-Bernard Zuber (1980).Quantum Field Theory.McGraw-Hill.ISBN 9780070320710.
• R.J. Rivers (1990).Path Integral Methods in Quantum Field Theories. Cambridge University Press.
• V.P. Nair (2005).Quantum Field Theory A Modern Perspective. Springer.

There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics. For applications toQuantum Chromodynamics there are

• R. Alkofer and L. v.Smekal (2001). "On the infrared behaviour of QCD Green's functions".Phys. Rep.353 (5–6): 281.arXiv:hep-ph/0007355.Bibcode:2001PhR...353..281A.doi:10.1016/S0370-1573(01)00010-2.S2CID 119411676.
• C.D. Roberts and A.G. Williams (1994). "Dyson-Schwinger equations and their applications to hadron physics".Prog. Part. Nucl. Phys.33:477–575.arXiv:hep-ph/9403224.Bibcode:1994PrPNP..33..477R.doi:10.1016/0146-6410(94)90049-3.S2CID 119360538.

• Quantum field theory
• Differential equations
• Freeman Dyson

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