nLab retarded product



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 formalized as causal perturbation theory/perturbative AQFT the retarded products (Glaser-Lehmann-Zimmermann 57, Steinmann 71) is a system of operator-valued distributions which are the coefficients in the formal power series expression for the quantum observables on interacting fields.

If the quantum observables are obtained via Bogoliubov's formula from the S-matrix induced by time-ordered products, then the retarded products may be expressed in terms of the time-ordered products. But the retarded products may also be axiomatized directly (Dütsch-Fredenhagen 04), see (Collini 16, section 2.2).


Let Σ\Sigma be a spacetime of dimension p+1p + 1 and let EfbΣE \overset{fb}{\longrightarrow} \Sigma be a field bundle. Let L freeΩ Σ p+1,0(E)\mathbf{L}_{free}\in \Omega^{p+1,0}_\Sigma(E) be a local Lagrangian density for a free field theory with fields of type EE. Let 𝒲\mathcal{W} be the corresponding Wick algebra of quantum observables of the free field, with

loc:():𝒲 \mathcal{F}_{loc} \overset{:(-):}{\longrightarrow} \mathcal{W}

the corresponding quantization map from local observables (“normal ordering”).

Let then

S: locg,j𝒲[[g/]][[j/]] S \;\colon\; \mathcal{F}_{loc}\langle g,j\rangle \longrightarrow \mathcal{W}[ [ g/\hbar ] ][ [ j/\hbar ] ]

be a perturbative S-matrix. Moreover let

g swL int locg g_{sw} \mathbf{L}_{int} \in \mathcal{F}_{loc}\langle g\rangle

be an adiabatically switched interaction Lagrangian density, so that the full Lagrangian density is

L=L free+gL int. \mathbf{L} = \mathbf{L}_{free} + g \mathbf{L}_{int} \,.

For A locA \in \mathcal{F}_{loc} a local observable and jC cp (Σ)j \in C^\infty_{cp}(\Sigma), write

Z L int(jA)S 1(g swL int)S(g swL int+jA) Z_{\mathbf{L}_{int}}(j A) \; \coloneqq \; S^{-1}(g_{sw}\mathbf{L}_{int}) S( g_{sw}\mathbf{L}_{int} + j A )

for the generating function induced by the perturvbative S-matrix.


(retarded products induced from perturbative S-matrix)

It follows from the “perturbation” axiom on the S-matrix (see there) that there is a system of continuous linear functionals

R:( locg) k( loc) l𝒲[[g/]][[j/]] R \;\colon\; \left(\mathcal{F}_{loc}\langle g\rangle\right)^{\otimes^k} \otimes (\mathcal{F}_{loc})^{\otimes^l} \longrightarrow \mathcal{W}[ [ g/\hbar] ] [ [ j/\hbar] ]

for all k,lk,l \in \mathbb{N} such that the generating function induced by the S-matrix is expressed as

Z g swL(j swA)=k,l1k!l!R(g swLg swLkarguments,j swAj swAlarguments). Z_{g_{sw} L}(j_{sw} A) = \underset{k,l \in \mathbb{N}}{\sum} \frac{1}{k! l!} R( \underset{k \,\text{arguments}}{\underbrace{ g_{sw} L \cdots g_{sw} L } }, \underset{l \; \text{arguments}}{\underbrace{ j_{sw} A \cdots j_{sw} A }} ) \,.

Similarly there is

R:( locg) k( locj)𝒲[[g]][[]] R \;\colon\; \left(\mathcal{F}_{loc}\langle g \rangle\right)^{\otimes^k} \otimes \left(\mathcal{F}_{loc}\langle j \rangle\right) \longrightarrow \mathcal{W}[ [ g ] ] [ [ \hbar ] ]

such that

A^(h)=k1k!r(g swL intg swL intkarguments,hA). \widehat{A}(h) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} r( \underset{k \,\text{arguments}}{\underbrace{g_{sw}L_{int} \cdots g_{sw}L_{int}}}, h A ) \,.

These are called the (generating) retarded products (Glaser-Lehmann-Zimmermann 57, Epstein-Glaser 73, section 8.1).

The actual retarded products are, via Bogoliubov's formula, the derivatives of these generating retarded products with respect to the source field.

product in perturbative QFT\,\, induces
normal-ordered productWick algebra (free field quantum observables)
time-ordered productS-matrix (scattering amplitudes)
retarded productinteracting quantum observables


The concept goes back to

Discussion of retarded products as derived from time-ordered products in causal perturbation theory is due to

Direct axiomatization of retarded products is due to

reviewed for instance in

A textbook account is in

Last revised on May 8, 2020 at 10:46:13. See the history of this page for a list of all contributions to it.