Contents
Context
Algebraic Quantum Field Theory
Contents
Idea
In perturbative quantum field theory an (adiabatically switch) interaction action functional is a local observable of the form
In the Feynman perturbation series this is interpreted as the label of vertices in multigraphs underlying Feynman diagrams which encode field interactions at that point.
In the process of (re-)normalization of perturbative QFT one considers redefinitions of such “interaction vertices” by terms of higher order in the coupling constant ad source field :
In the context of effective QFT via UV cutoffs, the correction term is called a counterterm.
Under mild assumptions, such interaction vertex redefinitions form a group, called the Stückelberg-Petermann renormalization group. See there for more.
Definition
Definition
(perturbative interaction vertex redefinition)
Let be a gauge fixed free field vacuum (this def.).
A perturbative interaction vertex redefinition (or just vertex redefinition, for short) is an endofunction
on local observables with formal parameters adjoined (this def.) such that there exists a sequence of continuous linear functionals, symmetric in their arguments, of the form
such that for all the following conditions hold:
-
(perturbation)
-
-
-
and
-
(field independence) The local observable depends on the field histories only through its argument , hence by the chain rule:
(1)
Properties
The following proposition should be compared to the axiom of causal additivity of the S-matrix scheme (this equation):
Proposition
(local additivity of vertex redefinitions)
Let be a gauge fixed free field vacuum (this def.) and let be a vertex redefinition (def. ).
Then for all local observables with spacetime support denoted (this def.) we have
-
(local additivity)
-
(preservation of spacetime support)
hence in particular
(Dütsch 18, exercise 3.98)
Proof
Under the inclusion
of local observables into polynomial observables we may think of each as a generalized function, as for time-ordered products in this remark.
Hence if
is the transgression of a Lagrangian density we get
Now by definition is in the subspace of local observables, i.e. those polynomial observables whose coefficient distributions are supported on the diagonal, which means that
Together with the axiom “field independence” (1) this means that the support of these generalized functions in the integrand here must be on the diagonal, where .
By the assumption that the spacetime supports of and are disjoint, this means that only the summands with and those with contribute to the above sum. Removing the overcounting of those summands where all we get
This directly implies the claim.
References
The original discussion is due to
in the context of the Stückelberg-Petermann renormalization group.
For more see the references there and those at main theorem of perturbative renormalization.