Schwinger-Dyson equation

The Schwinger-Dyson equation, named after Julian Schwinger and Freeman Dyson, is an equation of quantum field theory (QFT). Given a polynomially bounded functional F over the field configurations, then, for any state vector (which is a solution of the QFT), |ψ>, we have

<math><\psi|\mathcal{T}\{\frac{\delta}{\delta\phi}F[\phi]\}|\psi>=-i<\psi|\mathcal{T}\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\}|\psi><math>

where S is the action functional and <math>\mathcal{T}<math> is the time ordering operation.

Equivalently, in the density state formulation, for any (valid) density state ρ, we have

<math>\rho(\mathcal{T}\{\frac{\delta}{\delta\phi}F[\phi]\})=-i\rho(\mathcal{T}\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\})<math>

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

To make the connection to diagramatical techniques (like feynman diagrams) clearer, it's often convenient to split the action S as S[φ]=1/2 D^-1_ij φⁱ φ^j+S_int[φ] where the first term is the quadratic part and D^-1 is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and S_int is the "interaction action". Then, we can rewrite the SD equations as

<math><\psi|\mathcal{T}\{F \phi^j\}|\psi>=<\psi|\mathcal{T}\{iF_{,i}D^{ij}-FS_{int,i}D^{ij}\}|\psi><math>

If F is a functional of φ, then for an operator K, F[K] is defined to be the operator which substitutes K for φ. For example, if

<math>F[\phi]=\frac{\partial^{k_1}}{\partial x_1^{k_1}}\phi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\phi(x_n)<math>

and G is a functional of J, then

<math>F[-i\frac{\delta}{\delta J}]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]<math>.

If we have an "analytic" (whatever that means for functionals) functional Z (called the generating functional) of J (called the source field) satisfying

<math>\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)><math>,

then, the Schwinger-Dyson equation for the generating functional is

<math>\frac{\delta S}{\delta \phi(x)}[-i \frac{\delta}{\delta J}]Z[J]+J(x)Z[J]=0<math>

If we expand this equation as a Taylor series about J = 0, we get the entire set of Schwinger-Dyson equations.

An example: φ⁴

To give an example, suppose

<math>S[\phi]=\int d^dx \left (\frac{1}{2} \partial^\mu \phi(x) \partial_\mu \phi(x) -\frac{1}{2}m^2\phi(x)^2 -\frac{\lambda}{4!}\phi(x)^4\right )<math>

for a real field φ.

Then,

<math>\frac{\delta S}{\delta \phi(x)}=-\partial_\mu \partial^\mu \phi(x) -m^2 \phi(x) - \frac{\lambda}{3!}\phi(x)^3<math>.

The Schwinger-Dyson equation for this particular example is:

<math>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<math>

Note that since

<math>\frac{\delta^3}{\delta J(x)^3}<math>

is not well-defined because

<math>\frac{\delta^3}{\delta J(x_1)\delta J(x_2) \delta J(x_3)}Z[J]<math>

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 <math>-\partial^\mu \partial_\mu-m^2<math> and so, the SD set of equation goes as

<math><\psi|\mathcal{T}\{\phi(x_0)\phi(x_1)\}|\psi>=iD(x_0,x_1)+\frac{\lambda}{3!}\int d^dx_2 D(x_0,x_2)<\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\phi(x_2)\phi(x_2)\}|\psi><math>

<math><\psi|\mathcal{T}\{\phi(x_0)\phi(x_1)\phi(x_2)\phi(x_3)\}|\psi>=iD(x_0,x_1)<\psi|\mathcal{T}\{\phi(x_2)\phi(x_3)\}|\psi>+iD(x_0,x_2)<\psi|\mathcal{T}\{\phi(x_1)\phi(x_3)\}|\psi>+iD(x_0,x_3)<\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\}|\psi>+\frac{\lambda}{3!}\int d^dx_4 D(x_0,x_4)<\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\phi(x_4)\phi(x_4)\}|\psi><math>

etc.

(unless there is spontaneous symmetry breaking, the odd correlation functions vanish)fr:Équation de Schwinger-Dyson