nLab retarded product

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 $j \in C^\infty_{cp}(\Sigma)$, write

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

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

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.

Related concepts

product in perturbative QFT	$\,\,$ induces
normal-ordered product	Wick algebra (free field quantum observables)
time-ordered product	S-matrix (scattering amplitudes)
retarded product	interacting quantum observables

References

The concept goes back to

• Vladimir Glaser, Harry Lehmann, Wolfhart Zimmermann, Field operators and retarded functions, Il Nuovo Cimento 6, 1122-1128 (1957) (doi:10.1007/bf02747395)

• O. Steinmann, Perturbation Expansions in Axiomatic Field Theory, Lecture Notes in Physics, vol. 11, Springer, Berlin and Heidelberg, 1971.

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

• Henri Epstein, Vladimir Glaser, section 8.1 of The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

Direct axiomatization of retarded products is due to

• Michael Dütsch, Klaus Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity, Rev. Math. Phys. 16, 1291 (2004) (arXiv:hep-th/0403213)

reviewed for instance in

• Giovanni Collini, section 2.2 of Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

A textbook account is in

• Michael Dütsch, From classical field theory to perturbative quantum field theory, 2018