Schwinger–Dyson equation

Given a polynomially bounded functional F over the field configurations, then, for any state vector (which is a solution of the QFT), |\psi\rangle, we have :\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 \frac{\delta}{\delta \varphi} is the functional derivative with respect to \varphi, S is the action functional and \mathcal{T} is the time ordering operation. Equivalently, in the density state formulation, for any (valid) density state \rho, we have :\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 S as :S[\varphi] = \tfrac{1}{2}\varphi^{i}D^{-1}_{ij}\varphi^{j} + S_{\text{int}}[\varphi], 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_{\text{int}}[\varphi] is the "interaction action". Then, we can rewrite the SD equations as :\left\langle\psi\left\vert\mathcal{T}\left\{F \varphi^j\right\}\right\vert\psi\right\rangle=\left\langle\psi\left\vert\mathcal{T}\left\{iF_{,i}D^{ij}-FS_{\text{int},i}D^{ij}\right\}\right\vert\psi\right\rangle. If F is a functional of \varphi, then for an operator K, F[K] is defined to be the operator which substitutes K for \varphi. For example, if :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 G is a functional of J, then :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 Z (called the generating functional) of J (called the source field) satisfying :\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 :{\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 :\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 J = 0, we get the entire set of Schwinger–Dyson equations. ==An example: φ4==

To give an example, suppose :S[\varphi]=\int d^dx \left (\tfrac{1}{2} \partial^\mu \varphi(x) \partial_\mu \varphi(x) -\tfrac{1}{2}m^2\varphi(x)^2 -\frac{\lambda}{4!}\varphi(x)^4\right ) for a real field  \varphi . Then, :\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: :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 :\frac{\delta^3}{\delta J(x)^3} is not well-defined because :\frac{\delta^3}{\delta J(x_1)\delta J(x_2) \delta J(x_3)} Z[J] is a distribution in :x_1, x_2 and x_3, this equation needs to be regularized. In this example, the bare propagator D is the Green's function for -\partial^\mu \partial_\mu-m^2 and so, the Schwinger–Dyson set of equations goes as : \begin{align} & \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^dx_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{align} and : \begin{align} & \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^dx_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{align} etc. (Unless there is spontaneous symmetry breaking, the odd correlation functions vanish.) == See also==