Schwinger–Dyson equation

Given a polynomially bounded functional ${\displaystyle F}$ over the field configurations, then, for any state vector (which is a solution of the QFT), ${\displaystyle |\psi \rangle }$, we have

${\displaystyle \left\langle \psi \left|{\mathcal {T}}\left\{{\frac {\delta }{\delta \varphi }}F[\varphi ]\right\}\right|\psi \right\rangle =-i\left\langle \psi \left|{\mathcal {T}}\left\{F[\varphi ]{\frac {\delta }{\delta \varphi }}S[\varphi ]\right\}\right|\psi \right\rangle }$

where ${\displaystyle S}$ is the action functional and ${\displaystyle {\mathcal {T}}}$ is the time ordering operation.

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

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

This infinite set of equations can be used to solve for the correlation functions nonperturbatively.

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

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

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 the deWitt notation whose inverse, ${\displaystyle D}$ is called the bare propagator and ${\displaystyle S_{\text{int}}[\varphi ]}$ is the "interaction action". Then, we can rewrite the SD equations as

${\displaystyle \langle \psi |{\mathcal {T}}\{F\varphi ^{j}\}|\psi \rangle =\langle \psi |{\mathcal {T}}\{iF_{,i}D^{ij}-FS_{int,i}D^{ij}\}|\psi \rangle .}$

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

${\displaystyle F[\varphi ]={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n})}$

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

${\displaystyle F\left[-i{\frac {\delta }{\delta J}}\right]G[J]=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G[J].}$

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

${\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}[0]=i^{n}Z[0]\langle \varphi (x_{1})\cdots \varphi (x_{n})\rangle ,}$

then, from the properties of the functional integrals

${\displaystyle {\left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\left[\varphi \right]+J(x)\right\rangle }_{J}=0,}$

the Schwinger–Dyson equation for the generating functional is

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

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

To give an example, suppose

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

for a real field φ.

Then,

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

The Schwinger–Dyson equation for this particular example is:

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

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 a distribution in

x₁, x₂ and x₃,

this equation needs to be regularized.

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

${\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\}\mid \psi \rangle \\[4pt]={}&iD(x_{0},x_{1})+{\frac {\lambda }{3!}}\int d^{d}x_{2}\,D(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{2})\varphi (x_{2})\}\mid \psi \rangle \end{aligned}}}$

and

${\displaystyle {\begin{aligned}&\langle \psi \mid {\mathcal {T}}\{\varphi (x_{0})\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle \\[6pt]={}&iD(x_{0},x_{1})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{2})\varphi (x_{3})\}\mid \psi \rangle +iD(x_{0},x_{2})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{3})\}\mid \psi \rangle \\[4pt]&{}+iD(x_{0},x_{3})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\}\mid \psi \rangle \\[4pt]&{}+{\frac {\lambda }{3!}}\int d^{d}x_{4}\,D(x_{0},x_{4})\langle \psi \mid {\mathcal {T}}\{\varphi (x_{1})\varphi (x_{2})\varphi (x_{3})\varphi (x_{4})\varphi (x_{4})\varphi (x_{4})\}\mid \psi \rangle \end{aligned}}}$

etc.

(Unless there is spontaneous 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 to Quantum 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.