Schwinger–Dyson equation

The Schwinger–Dyson equations (SDEs) or Dyson–Schwinger equations, named after

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 "The S-Matrix in Quantum electrodynamics",

.

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

Derivation

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

operation.

Equivalently, in the

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

.

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

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

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

) 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.

An example: φ⁴

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.)

See also

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

References

Further reading

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.
  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.
  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.
  S2CID 119360538
  .