Contents

# Contents

## Idea

In perturbative quantum field theory the Stückelberg-Petermann renormalization group (Stückelberg-Petermann 53) is the (original) incarnation of the renormalization group in the perspective of causal perturbation theory: It is the group (def. below) of perturbative interaction vertex redefinitions (def. ) which is such that any two choices of S-matrix renormalization schemes $\mathcal{S}, \mathcal{S}'$ are related by a unique vertex redefinition $\mathcal{Z}$ via precomposition

$\mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z} \,.$

This statement (prop. below) is also called the main theorem of perturbative renormalization.

If scaling transformations on spacetime happen to transform renormalization schemes into each other, then this main theorem of perturbative renormalization (prop. ) directly implies that every scaling transformation uniquely corresponds to a interaction vertex redefinition, or conversely that the vertex redefinitions are given as funcitons of scale. As such they are also referred to as running coupling constants (def. below). Beware that these do not in general form a group but a group cocycle, the Gell-Mann-Low renormalization cocycle (prop. below.

## Definition

The elements of the Stückelberg-Petermann renormalization group are perturbative interaction vertex redefinitions (def. below). These act on S-matrix ("re"-)normalization schemes by precomposition (def. ) and this action is free and transitive (prop. below). In this way these vertex redefinitions translate between different choices of ("re"-)normalization, and as such they form the Stückelerg-Petermann renormalization group (def. ) below.

$\,$

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

The following proposition should be compared to the axiom of causal additivity of the S-matrix scheme (this equation):

###### Proposition

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

\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

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

As a corollary we obtain:

###### Proposition

(composition of S-matrix scheme with vertex redefinition is again S-matrix scheme)

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

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

and S-matrix scheme (this def.), the composite

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

is again an S-matrix scheme.

Moreover, if $\mathcal{S}$ satisfies the renormalization condition “field independence” (this prop.), then so does $\mathcal{S} \circ \mathcal{Z}$.

###### Proof

It is clear that causal order of the spacetime supports implies that they are in particular disjoint

$\left( {\, \atop \,} supp(O_1) {\vee\!\!\!\wedge} supp(O_2) {\, \atop \,} \right) \phantom{AA} \Rightarrow \phantom{AA} \left( {\, \atop \,} supp(O_1) \cap supp(O_) \;=\; \emptyset {\, \atop \,} \right)$

Therefore the local additivity of $\mathcal{Z}$ (prop. ) and the causal factorization of the S-matrix (this remark) imply the causal factorization of the composite:

\begin{aligned} \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1 + O_2) {\, \atop \,} \right) & = \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1) + \mathcal{Z}(O_2) {\, \atop \,} \right) \\ & = \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_1) {\, \atop \,} \right) \, \mathcal{S} \left( {\, \atop \,} \mathcal{Z}(O_2) {\, \atop \,} \right) \,. \end{aligned}

But by this prop. this implies in turn causal additivity and hence that $\mathcal{S} \circ \mathcal{Z}$ is itself an S-matrix scheme.

Finally that $\mathcal{S} \circ \mathcal{Z}$ satisfies “field indepndence” if $\mathcal{S}$ does is immediate by the chain rule, given that $\mathcal{Z}$ satisfies this condition by definition.

###### Proposition

(any two S-matrix renormalization schemes differ by a unique vertex redefinition)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a gauge fixed free field vacuum (this def.).

Then for $\mathcal{S}, \mathcal{S}'$ any two S-matrix schemes (this def.) which both satisfy the renormalization condition “field independence”, the there exists a unique vertex redefinition $\mathcal{Z}$ (def. ) relating them by composition, i. e. such that

$\mathcal{S}' \;=\; \mathcal{S} \circ \mathcal{Z} \,.$

(See any of the references at main theorem of perturbative renormalization.)

###### Proof

By applying both sides of the equation to linear combinations of local observables of the form $\kappa_1 O_1 + \cdots + \kappa_k O_k$ and then taking derivatives with respect to $\kappa$ at $\kappa_j = 0$ (as in this example) we get that the equation in question implies

$(i \hbar)^k \frac{ \partial^k }{ \partial \kappa_1 \cdots \partial \kappa_k } \mathcal{S}'( \kappa_1 O_1 + \cdots + \kappa_k O_k ) \vert_{\kappa_1, \cdots, \kappa_k = 0} \;=\; (i \hbar)^k \frac{ \partial^k }{ \partial \kappa_1 \cdots \partial \kappa_k } \mathcal{S} \circ \mathcal{Z}( \kappa_1 O_1 + \cdots + \kappa_k O_k ) \vert_{\kappa_1, \cdots, \kappa_k = 0}$

which in components means that

\begin{aligned} T'_k( O_1, \cdots, O_k ) & = \underset{ 2 \leq n \leq k }{\sum} \frac{1}{n!} (i \hbar)^{k-n} \underset{ { { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } } \atop { I_1, \cdots, I_n \neq \emptyset } }{\sum} T_n \left( {\, \atop \,} Z_{{\vert I_1\vert}}\left( (O_{i_1})_{i_1 \in I_1} \right), \cdots, Z_{{\vert I_n\vert}}\left( (O_{i_n})_{i_n \in I_n} \right), {\, \atop \,} \right) \\ & \phantom{=} + Z_k( O_1,\cdots, O_k ) \end{aligned}

where $\{T'_k\}_{k \in \mathbb{N}}$ are the time-ordered products corresponding to $\mathcal{S}'$ (by this example) and $\{T_k\}_{k \in \mathcal{N}}$ those correspondong to $\mathcal{S}$.

Here the sum on the right runs over all ways that in the composite $\mathcal{S} \circ \mathcal{Z}$ a $k$-ary operation arises as the composite of an $n$-ary time-ordered product applied to the ${\vert I_i\vert}$-ary components of $\mathcal{Z}$, for $i$ running from 1 to $n$; except for the case $k = n$, which is displayed separately in the second line.

This shows that if $\mathcal{Z}$ exists, then it is unique, because its coefficients $Z_k$ are inductively in $k$ given by the expressions

\begin{aligned} & Z_k( O_1,\cdots, O_k ) \\ & = T'_k( O_1, \cdots, O_k ) \;-\; \underset{ (T \circ \mathcal{Z}_{\lt k})_k }{ \underbrace{ \underset{ 2 \leq n \leq k }{\sum} \frac{1}{n!} (i \hbar)^{k-n} \underset{ { { I_1 \sqcup \cdots \sqcup I_n } \atop { = \{1, \cdots, k\}, } } \atop { I_1, \cdots, I_n \neq \emptyset } }{\sum} T_n \left( Z_{{\vert I_1\vert}}( (O_{i_1})_{i_1 \in I_1} ), \cdots, Z_{{\vert I_n\vert}}( (O_{i_n})_{i_n \in I_n} ), \right) } } \end{aligned}

(The symbol under the brace is introduced as a convenient shorthand for the term above the brace.)

Hence it remains to see that the $Z_k$ defined this way satisfy the conditions in def. .

The condition “perturbation” is immediate from the corresponding condition on $\mathcal{S}$ and $\mathcal{S}'$.

Similarly the condition “field independence” follows immediately from the assumoption that $\mathcal{S}$ and $\mathcal{S}'$ satisfy this condition.

It only remains to see that $Z_k$ indeed takes values in local observables. Given that the time-ordered products a priori take values in the larrger space of microcausal polynomial observables this means to show that the spacetime support of $Z_k$ is on the diagonal.

But observe that, as indicated in the above formula, the term over the brace may be understood as the coefficient at order $k$ of the exponential series-expansion of the composite $\mathcal{S} \circ \mathcal{Z}_{\lt k}$, where

$\mathcal{Z}_{\lt k} \;\coloneqq\; \underset{ n \in \{1, \cdots, k-1\} }{\sum} \frac{1}{n!} Z_n$

is the truncation of the vertex redefinition to degree $\lt k$. This truncation is clearly itself still a vertex redefinition (according to def. ) so that the composite $\mathcal{S} \circ \mathcal{Z}_{\lt k}$ is still an S-matrix scheme (by prop. ) so that the $(T \circ \mathcal{Z}_{\lt k})_k$ are time-ordered products (by this example).

So as we solve $\mathcal{S}' = \mathcal{S} \circ \mathcal{Z}$ inductively in degree $k$, then for the induction step in degree $k$ the expressions $T'_{\lt k}$ and $(T \circ \mathcal{Z})_{\lt k}$ agree and are both time-ordered products. By this prop. this implies that $T'_{k}$ and $(T \circ \mathcal{Z}_{\lt k})_{k}$ agree away from the diagonal. This means that their difference $Z_k$ is supported on the diagonal, and hence is indeed local.

###### Definition

(Stückelberg-Petermann renormalization group of vertex redefinitions)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a gauge fixed free field vacuum (this def.).

Then prop. and prop together says that (“main theorem of perturbative renormalization”):

1. the vertex redefinitions $\mathcal{Z}$ (def. ) form a group under composition;

2. the set of S-matrix ("re"-)normalization schemes (this def., this remark) satisfying the renormalization condition “field independence” (this prop.) is a torsor over this group, meaning that the action is regular on this set, in that any two are S-matrix (“re”-)normalization schemes are related by a unique vertex redefinition.

This group is called the (large) Stückelberg-Petermann renormalization group.

Typically one imposes a set of renormalization conditions (this def.) and then the corresponding subgroup of vertex redefinitions preserving these.

## Properties

### Scaling transformations and “running coupling constants”

A priori the Stückelberg-Petermann renormalization group is not about scaling transformations. But if scaling transformations happen to produce new S-matrices/renormalization schemes from given ones, then the main theorem of perturbative renormalization induces for each such scaling transformation a re-definition of interaction Lagrangian densities, this is the Gell-Mann-Low renormalization cocycle (Gell-Mann & Low 54, Brunetti-Dütsch-Fredenhagen 09) for review see (Dütsch 18, section 3.5.3).

###### Definition

(running coupling constants under scale transformations)

Let

$vac \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}_{kin}, \Delta_H )$

be a relativistic free vacuum (according to this def.) around which we consider interacting perturbative QFT.

Assume that in fact

1. the free field vacuum $vac = vac(m)$ depends on a mass parameter, and with it the choice $\mathcal{S}_{vac(m)}$ of normalization scheme,

2. under scaling transformations on local observables $\sigma_\rho$ (Dütsch 18, def. 3.19) we have that with $\mathcal{S}_{vac(m)}$ a perturbative S-matrix scheme perturbing around $vac(m)$ also

$\sigma_\rho \circ \left(\mathcal{S}_{vac(m/\rho)}\right) \circ \sigma_\rho^{-1}$

is a perturbative S-matrix around $L_{kin}(m)$.

In this case the main theorem of perturbative renormalization (prop. ) says that there exists for each scale $\rho$ a unique interaction vertex redefinition $\mathcal{Z}^m_\rho$ (def. ) such that

\begin{aligned} & \sigma_\rho \circ \mathcal{S}_{vac(m/\rho)} \circ \sigma_\rho^{-1}( g S_{int} + j A ) \\ & = \mathcal{S}_{vac(m)}(\mathcal{Z}^m_\rho(g S_{int} + j A)) \end{aligned}

for all interaction action functionals $g S_{int} + j A$.

These $\mathcal{Z}^m_\rho$ are the Gell-Mann-Low renormalization cocycle elements.

The interaction vertex redefinitions $\mathcal{Z}^m_\rho$ as a function of the rescaling is known as the running coupling constants.

###### Proposition

(cocycle property of running coupling constants)

In the situation of def. , the Gell-Mann-Low renormalization cocycles (running coupling constants) $\mathcal{Z}^m_\rho$ satisfy the relation

$\mathcal{Z}^m_{\rho_1 \rho_2} \;=\; \mathcal{Z}^m_{\rho_1} \circ \left( \sigma_{\rho_1} \circ \mathcal{Z}^{m/\rho_1}_{\rho_2} \circ \sigma_{\rho_2} \right)$

Hence only for vanishing mass do these “renormalization cocycles” themselves form an actual renormalization group.

###### Proof

From the definition we have

\begin{aligned} \mathcal{S}_{vac(m)} \circ \mathcal{Z}^m_{\rho_1 \rho_2} & = \sigma_{\rho_1} \circ \underset{ \mathcal{S}_{vac(m/\rho_1)} \circ \mathcal{Z}^{m/\rho_1}_{\rho_2} }{ \underbrace{ \sigma_{\rho_2} \circ \mathcal{S}_{vac(m/\rho_1\rho_2)} \circ \sigma_{\rho_2}^{-1} }} \circ \sigma_{\rho_1}^{-1} \\ & = \underset{ = \mathcal{S}_{vac(m)} \circ \mathcal{Z}^m_{\rho_1} \circ \sigma_{\rho_1} }{ \underbrace{ \sigma_{\rho_1} \circ \mathcal{S}_{vac(m/\rho_1)} \circ \overset{ = id }{ \overbrace{ \sigma_{\rho_1}^{-1} \circ \sigma_{\rho_1} } } }} \circ \mathcal{Z}^{m/\rho_1}_{\rho_2} \circ \sigma_{\rho_1}^{-1} \\ & = \mathcal{S}_{vac(m)} \circ \mathcal{Z}^m_{\rho_1} \circ \sigma_{\rho_1} \circ \mathcal{Z}^{m/\rho_1}_{\rho_2} \circ \sigma_{\rho_1}^{-1} \end{aligned}

This demonstrates the equation between vertex redefinitions to be shown after composition with an S-matrix scheme. But by the uniqueness-clause in the main theorem of perturbative renormalization the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from vertex redefinitions to S-matrix schemes is injective. This implies the equation itself.