nLab Schwinger-Dyson equation



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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 SS for a free field theory and another observable AA with the time ordering of the corresponding functional derivative of just AA itself, times ii \hbar (imaginary unit times Planck's constant):

(1)𝒯(ΣδSδΦ a(x)A a(x)dvol Σ(x))=i𝒯(ΣδA a(x)δΦ a(x)dvol Σ(x))AAAon-shell. \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} \,.

(e.q. Henneaux-Teitelboim 92, (15.25), Rejzner 16, remark 7.7).

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)δ(xx 0)δ a 0 aΦ a 1(x 1)Φ a n(x n) 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

T(δSδΦ a 0(x 0)Φ a 1(x 1)Φ a n(x n)) =on-shellikT(Φ a 1(x 1)Φ a k1(x k1)δ(x 0x k)δ a k a 0Φ a k+1(x k+1)Φ a n(x m)) \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)(x k,a k)(x_0,a_0) \neq (x_k, a_k) for all k{1,,n}k \in \{1,\cdots ,n\} then

T(δSδΦ a 0(x 0)Φ a 1(x 1)Φ a n(x n))=0AAAon-shell 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

δSδΦ a 0(x 0)=0 \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 0x_0”, this may be interpreted as saying that the classical equations of motion for fields at x 0x_0 still hold for time-ordered quantum expectation values, as long as all other observables are evaluated away from x 0x_0; while if observables do coincide at x 0x_0 then there is a correction (governed by the BV-operator of the theory, see this prop.).


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


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

See also

Discussion of round chord diagrams organizing Dyson-Schwinger equations:

  • Nicolas Marie, Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, Communications in Number Theory and Physics, 7(2):251291, 2013 (arXiv:1210.5457)

  • Markus Hihn, Karen Yeats, Generalized chord diagram expansions of Dyson-Schwinger equations, Ann. Inst. Henri Poincar Comb. Phys. Interact. 6 no 4:573-605 (arXiv:1602.02550)

Review in:

  • Ali Assem Mahmoud, Section 3 of: On the Enumerative Structures in Quantum Field Theory (arXiv:2008.11661)

Last revised on April 23, 2021 at 09:29:56. See the history of this page for a list of all contributions to it.