Contents

Idea

In perturbative quantum field theory the Schwinger-Dyson equation (Dyson 49, Schwinger 51) equates, on-shell, the time-ordered product of the functional derivative of the action functional $S$ for a free field theory and another observable $A$ with the time ordering of the corresponding functional derivative of just $A$ itself, times $i \hbar$ (imaginary unit times Planck's constant):

(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

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

which makes the distributional Schwinger-Dyson equation read

\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}

(e.q. Dermisek 09)

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

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

Since (by the principle of extremal action) the equation

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

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.).

Details

For details and proof see at BV-operator this prop., following Rejzner 16, remark 7.7, following Henneaux-Teitelboim 92, section 15.5.3

References

The equation is originally due to

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

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