nLab
Schwinger-Dyson equation

Context

Algebraic Quantum Field Theory

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

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 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, (15.52), 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.).

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

See also

Last revised on January 7, 2018 at 05:10:20. See the history of this page for a list of all contributions to it.