algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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 $\Sigma$ 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
the corresponding quantization map from local observables (“normal ordering”).
Let then
be a perturbative S-matrix. Moreover let
be an adiabatically switched interaction Lagrangian density, so that the full Lagrangian density is
For $A \in \mathcal{F}_{loc}$ a local observable and $j \in C^\infty_{cp}(\Sigma)$, write
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
for all $k,l \in \mathbb{N}$ such that the generating function induced by the S-matrix is expressed as
Similarly there is
such that
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.
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
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.