Contents

# Contents

## Idea

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).

## Definition

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

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

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

Let then

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

be a perturbative S-matrix. Moreover let

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

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

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

For $A \in \mathcal{F}_{loc}$ a local observable and $j \in C^\infty_{cp}(\Sigm)$, write

$Z_L(\epsilon j A) \; \coloneqq \; S(g_{sw}\mathbf{L}_{int}) S( g_{sw}\mathbf{L}_{int} + j A )$

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

###### Definition

(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

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

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

$Z_{g_{sw} L}(j_{sw} A) = \underset{k,l \in \mathbb{N}}{\sum} \frac{1}{k! l!} R( \underset{k \,\text{arguments}}{\underbrace{ g_{sw} L \cdots g_{sw} L } }, \underset{l \; \text{arguments}}{\underbrace{ j_{sw} A \cdots j_{sw} A }} ) \,.$

Similarly there is

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

such that

$\widehat{A}(h) = \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} r( \underset{k \,\text{arguments}}{\underbrace{g_{sw}L_{int} \cdots g_{sw}L_{int}}}, h A ) \,.$

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

## References

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 April 29, 2018 at 00:59:06. See the history of this page for a list of all contributions to it.