nLab Schwinger-Dyson equation

Contents

Idea

(1)

\mathcal{T} \left( \underset{\Sigma}{\int} \frac{\delta S}{\delta \mathbf{\Phi}^a(x)} \cdot A^a(x) \, dvol_\Sigma(x) \right) \;=\; i \hbar \mathcal{T} \left( \underset{\Sigma}{\int} \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)} \, dvol_\Sigma(x) \right) \phantom{AAA} \text{on-shell} \,.

This may be understood as a special case of the quantum-correction of the BV-differential by the BV-operator in pQFT, which is hence also called the Schwinger-Dyson operator; see there.

Often (1) is displayed before spacetime-smearing of observables in terms of operator products of operator-valued distributions

A^a(x) \;\coloneqq\; \delta(x-x_0) \delta^a_{a_0} \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)

\begin{aligned} & T \left( \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \cdot \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n) \right) \\ & \underset{\text{on-shell}}{=} i \hbar \underset{k}{\sum} T \left( \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_{k-1}}(x_{k-1}) \cdot \delta(x_0 - x_k) \delta^{a_0}_{a_k} \cdot \mathbf{\Phi}^{a_{k+1}}(x_{k+1}) \cdots \mathbf{\Phi}^{a_n}(x_m) \right) \end{aligned}

In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then

T \left( \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \cdot \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n) \right) \;=\; 0 \phantom{AAA} \text{on-shell}

\frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)} \;=\; 0

is the Euler-Lagrange equation of motion (for the classical field theory) “at $x_0$ ”, this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for time-ordered quantum expectation values, as long as all other observables are evaluated away from $x_0$ ; while if observables do coincide at $x_0$ then there is a correction (governed by the BV-operator of the theory, see this prop.).

Precursor discussion for quantum mechanics (QFT in $1+0$ -dimensions) goes back to

Richard P. Feynman, equation (46) in: Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20 (1948) 367 [doi:10.1103/RevModPhys.20.367, pdf]

The Dyson-Schwinger equation is named after:

Freeman Dyson, The S Matrix in Quantum Electrodynamics, Phys. Rev. 75: 1736 (1949) (doi:10.1103/PhysRev.75.1736)
Julian Schwinger, On Green’s functions of quantized fields I + II, PNAS. 37: 452–459 (1951) (doi:10.1073/pnas.37.7.452)

The traditional informal account in terms of path integral-heuristics is reviewed for instance in

Marc Henneaux, Claudio Teitelboim, section 15.1.4, 15.5.1, 15.5.3 of Quantization of Gauge Systems, Princeton University Press 1992.
Radovan Dermisek, Schwinger-Dyson equations, 2009 (pdf)

A rigorous derivation in terms of BV-formalism in causal perturbation theory/pAQFT is provided in

Katarzyna Rejzner, remark 7.7 Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

For discussion in the context of the master Ward identity, see

Michael Dütsch, Klaus Fredenhagen, The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory, Commun.Math.Phys. 243 (2003) 275-314 (arXiv:hep-th/0211242)