nLab interaction vertex redefinition

Idea

In perturbative quantum field theory an (adiabatically switch) interaction action functional is a local observable of the form

$$g S_{int} + j A \;\in\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \,.$$

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 $g$ ad source field $j$:

$$\mathcal{Z} \;\colon\; g S_{int} + j A \;\mapsto\; g S_{int} + j A + \underset{ \mathcal{O}(g^2, j^2, g j) }{ \underbrace{ S_{counter} }} \,.$$

In the context of effective QFT via UV cutoffs, the correction term $S_{counter}$ 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 $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a gauge fixed free field vacuum (this def.).

A perturbative interaction vertex redefinition (or just vertex redefinition, for short) is an endofunction

$$\mathcal{Z} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$$

on local observables with formal parameters adjoined (this def.) such that there exists a sequence $\{Z_k\}_{k \in \mathbb{N}}$ of continuous linear functionals, symmetric in their arguments, of the form

$$\left( {\, \atop \,} LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle {\, \atop \,} \right)^{\otimes^k_{\mathbb{C}[ [ \hbar, g, j] ]}} \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$$

such that for all $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j\rangle$ the following conditions hold:

1. (perturbation)

   1. $Z_0(g S_{int + j A}) = 0$

   2. $Z_1(g S_{int} + j A) = g S_{int} + j A$

   3. and

      $$\begin{aligned} \mathcal{Z}(g S_{int} + j A) & = Z \exp_\otimes( g S_{int} + j A ) \\ & \coloneqq \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!} Z_k( \underset{ k \, \text{args} }{ \underbrace{ g S_{int} + j A , \cdots, g S_{int} + j A } } ) \end{aligned}$$

2. (field independence) The local observable $\mathcal{Z}(g S_{int} + j A)$ depends on the field histories only through its argument $g S_{int} + j A$, hence by the chain rule:

   (1)$$\frac{\delta}{\delta \mathbf{\Phi}^a(x)} \mathcal{Z}(g S_{int} + j A) \;=\; \mathcal{Z}'_{g S_{int} + j A} \left( \frac{\delta}{\delta \mathbf{\Phi}^a(x)} (g S_{int} + j A) \right)$$

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 $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a gauge fixed free field vacuum (this def.) and let $\mathcal{Z}$ be a vertex redefinition (def. ).

Then for all local observables $O_0, O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j\rangle$ with spacetime support denoted $supp(O_i) \subset \Sigma$ (this def.) we have

1. (local additivity)

   $$\begin{aligned} & \left( supp(O_1) \cap supp(O_2) = \emptyset \right) \\ & \Rightarrow \phantom{AA} \mathcal{Z}( O_0 + O_1 + O_2) = \mathcal{Z}( O_0 + O_1 ) - \mathcal{Z}(O_0) + \mathcal{Z}(O_0 + O_2) \end{aligned} \,.$$

2. (preservation of spacetime support)

   $$supp \left( {\, \atop \,} \mathcal{Z}(O_0 + O_1) - \mathcal{Z}(O_0) {\, \atop \,} \right) \;\subset\; supp(O_1)$$

   hence in particular

   $$supp \left( {\, \atop \,} \mathcal{Z}(O_1) {\, \atop \,} \right) = supp(O_1)$$

(Dütsch 18, exercise 3.98)

Proof

Under the inclusion

$$LocObs(E_{\text{BV-BRST}}) \hookrightarrow PolyObs(E_{\text{BV-BRST}})$$

of local observables into polynomial observables we may think of each $Z_k$ as a generalized function, as for time-ordered products in this remark.

Hence if

$$O_j = \underset{\Sigma}{\int} j^\infty_\Sigma( \mathbf{L}_j )$$

is the transgression of a Lagrangian density $\mathbf{L}$ we get

$$Z_k( (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) ) = \underset{ j_1, \cdots, j_k \in \{0,1,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \,.$$

Now by definition $Z_k(\cdots)$ is in the subspace of local observables, i.e. those polynomial observables whose coefficient distributions are supported on the diagonal, which means that

$$\frac{\delta}{\delta \mathbf{\Phi}^a(x)} \frac{\delta}{\delta \mathbf{\Phi}^b(y)} Z_{k}(\cdots) = 0 \phantom{AA} \text{for} \phantom{AA} x \neq y$$

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 $x_1 = \cdots = x_k$.

By the assumption that the spacetime supports of $O_1$ and $O_2$ are disjoint, this means that only the summands with $j_1, \cdots, j_k \in \{0,1\}$ and those with $j_1, \cdots, j_k \in \{0,2\}$ contribute to the above sum. Removing the overcounting of those summands where all $j_1, \cdots, j_k \in \{0\}$ we get

$$\begin{aligned} & Z_k\left( {\, \atop \,} (O_1 + O_2 + O_3) , \cdots , (O_1 + O_2 + O_3) {\, \atop \,} \right) \\ & = \underset{ j_1, \cdots, j_k \in \{0,1\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & \phantom{=} - \underset{ j_1, \cdots, j_k \in \{0,2\} }{\sum} \underset{\Sigma^{k}}{\int} Z( \mathbf{L}_{j_1}(x_1) , \cdots , \mathbf{L}_{j_k}(x_k) ) \\ & = Z_k\left( {\, \atop \,} (O_0 + O_1), \cdots, (O_0 + O_1) {\, \atop \,}\right) - Z_k\left( {\, \atop \,} O_0, \cdots, O_0 {\, \atop \,} \right) + Z_k\left( {\, \atop \,} (O_0 + O_2), \cdots, (O_0 + O_2) {\, \atop \,} \right) \end{aligned} \,.$$

This directly implies the claim.

References

The original discussion is due to

• Ernst Stückelberg, André Petermann, La normalisation des constantes dans la theorie des quanta, Helv. Phys. Acta 26 (1953), 499–520

in the context of the Stückelberg-Petermann renormalization group.

For more see the references there and those at main theorem of perturbative renormalization.