nLab A first idea of quantum field theory -- Renormalization

Renormalization

In this chapter we discuss the following topics:

• Epstein-Glaser normalization

• Stückelberg-Petermann re-normalization

• UV-Regularization via Counterterms

• Wilson-Polchinski effective QFT flow

• Renormalization group flow

• Gell-Mann & Low RG Flow

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In the previous chapter we have seen that the construction of interacting perturbative quantum field theories is given by perturbative S-matrix schemes (def. ), equivalently by time-ordered products (def. ) or equivalently by Feynman amplitudes (prop. ). These are uniquely fixed away from coinciding interaction points (prop. ) by the given local interaction (prop. ), but involve further choices of interactions whenever interaction vertices coincide (prop. ). This choice is called the choice of ("re"-)normalization (def. ) in perturbative QFT.

In this rigorous discussion no “infinite divergent quantities” (as in the original informal discussion due to Schwinger-Tomonaga-Feynman-Dyson) that need to be “re-normalized” to finite well-defined quantities are ever considered, instead finite well-defined quantities are considered right away, and the available space of choices is determined. Therefore making such choices is rather a normalization of the time-ordered products/Feynman amplitudes (as prominently highlighted in Scharf 95, see title, introduction, and section 4.3). Actual re-normalization is the the change of such normalizations.

The construction of perturbative QFTs may be explicitly described by an inductive extension of distributions of time-ordered products/Feynman amplitudes to coinciding interaction points. This is called

• Epstein-Glaser renormalization.

This inductive construction has the advantage that it gives accurate control over the space of available choices of (“re”-)normalizations (theorem below) but it leaves the nature of the “new interactions” that are to be chosen at coinciding interaction points somwewhat implicit.

Alternatively, one may re-define the interactions explicitly (by adding “counterterms”, remark below), depending on a chosen UV cutoff-scale (def. below), and construct the limit as the “cutoff is removed” (prop. below). This is called (“re”-)normalization by

• UV-Regularization via Counterterms.

This still leaves open the question how to choose the counterterms. For that it serves to understand the relative effective action induced by the choice of UV cutoff at any given cutoff scale (def. below). This is the perspective of effective quantum field theory (remark below).

The infinitesimal change of these relative effective actions follows a universal differential equation, known as Polchinski's flow equation (prop. below). This makes the problem of (“re”-)normalization be that of solving this differential equation subject to chosen initial data. This is the perspective on (“re”-)normalization called

• Wilson-Polchinski effective QFT flow.

The main theorem of perturbative renormalization (theorem below) states that different S-matrix schemes are precisely related by vertex redefinitions. This yields the

• Stückelberg-Petermann renormalization group.

If a sub-collection of renormalization schemes is parameterized by some group $RG$, then the main theorem implies vertex redefinitions depending on pairs of elements of $RG$ (prop. below). This is known as

• Renormalization group flow

Specifically scaling transformations on Minkowski spacetime yield such a collection of renormalization schemes (prop. below); the corresponding renormalization group flow is known as

• Gell-Mann & Low RG flow.

The infinitesimal behaviour of this flow is known as the beta function, describing the running of the coupling constants with scale (def. below).

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Epstein-Glaser normalization

The construction of perturbative quantum field theories around a given gauge fixed relativistic free field vacuum is equivalently, by prop. , the construction of S-matrices $\mathcal{S}(g S_{int} + j A)$ in the sense of causal perturbation theory (def. ) for the given local interaction $g S_{int} + j A$. By prop. the construction of these S-matrices is inductively in $k \in \mathbb{N}$ a choice of extension of distributions (remark and def. below) of the corresponding $k$-ary time-ordered products of the interaction to the locus of coinciding interaction points. An inductive construction of the S-matrix this way is called Epstein-Glaser-("re"-)normalization (def. ).

By paying attention to the scaling degree (def. below) one may precisely characterize the space of choices in the extension of distributions (prop. below): For a given local interaction $g S_{int} + j A$ it is inductively in $k \in \mathbb{N}$ a finite-dimensional affine space. This conclusion is theorem below.

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Proposition

(("re"-)normalization is inductive extension of time-ordered products to diagonal)

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

Assume that for $n \in \mathbb{N}$, time-ordered products $\{T_{k}\}_{k \leq n}$ of arity $k \leq n$ have been constructed in the sense of def. . Then the time-ordered product $T_{n+1}$ of arity $n+1$ is uniquely fixed on the complement

$$\Sigma^{n+1} \setminus diag(n) \;=\; \left\{ (x_i \in \Sigma)_{i = 1}^n \;\vert\; \underset{i,j}{\exists} (x_i \neq x_j) \right\}$$

of the image of the diagonal inclusion $\Sigma \overset{diag}{\longrightarrow} \Sigma^{n}$ (where we regarded $T_{n+1}$ as a generalized function on $\Sigma^{n+1}$ according to remark ).

This statement appears in (Popineau-Stora 82), with (unpublished) details in (Stora 93), following personal communication by Henri Epstein (according to Dütsch 18, footnote 57). Following this, statement and detailed proof appeared in (Brunetti-Fredenhagen 99).

Proof

We will construct an open cover of $\Sigma^{n+1} \setminus \Sigma$ by subsets $\mathcal{C}_I \subset \Sigma^{n+1}$ which are disjoint unions of non-empty sets that are in causal order, so that by causal factorization the time-ordered products $T_{n+1}$ on these subsets are uniquely given by $T_{k}(-) \star_H T_{n-k}(-)$. Then we show that these unique products on these special subsets do coincide on intersections. This yields the claim by a partition of unity.

We now say this in detail:

For $I \subset \{1, \cdots, n+1\}$ write $\overline{I} \coloneqq \{1, \cdots, n+1\} \setminus I$. For $I, \overline{I} \neq \emptyset$, define the subset

$$\mathcal{C}_I \;\coloneqq\; \left\{ (x_i)_{i \in \{1, \cdots, n+1\}} \in \Sigma^{n+1} \;\vert\; \{x_i\}_{i \in I} {\vee\!\!\!\wedge} \{x_j\}_{j \in \{1, \cdots, n+1\} \setminus I} \right\} \;\subset\; \Sigma^{n+1} \,.$$

Since the causal order-relation involves the closed future cones/closed past cones, respectively, it is clear that these are open subsets. Moreover it is immediate that they form an open cover of the complement of the diagonal:

$$\underset{ { I \subset \{1, \cdots, n+1\} \atop { I, \overline{I} \neq \emptyset } } }{\cup} \mathcal{C}_I \;=\; \Sigma^{n+1} \setminus diag(\Sigma) \,.$$

(Because any two distinct points in the globally hyperbolic spacetime $\Sigma$ may be causally separated by a Cauchy surface, and any such may be deformed a little such as not to intersect any of a given finite set of points. )

Hence the condition of causal factorization on $T_{n+1}$ implies that restricted to any $\mathcal{C}_{I}$ these have to be given (in the condensed generalized function-notation from remark ) on any unordered tuple $\mathbf{X} = \{x_1, \cdots, x_{n+1}\} \in \mathcal{C}_I$ with corresponding induced tuples $\mathbf{I} \coloneqq \{x_i\}_{i \in I}$ and $\overline{\mathbf{I}} \coloneqq \{x_i\}_{i \in \overline{I}}$ by

(1)$$T_{n+1}( \mathbf{X} ) \;=\; T(\mathbf{I}) T(\overline{\mathbf{I}}) \phantom{AA} \text{for} \phantom{A} \mathcal{X} \in \mathcal{C}_I \,.$$

This shows that $T_{n+1}$ is unique on $\Sigma^{n+1} \setminus diag(\Sigma)$ if it exists at all, hence if these local identifications glue to a global definition of $T_{n+1}$. To see that this is the case, we have to consider any two such subsets

$$I_1, I_2 \subset \{1, \cdots, n+1\} \,, \phantom{AA} I_1, I_2, \overline{I_1}, \overline{I_2} \neq \emptyset \,.$$

By definition this implies that for

$$\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$$

a tuple of spacetime points which decomposes into causal order with respect to both these subsets, the corresponding mixed intersections of tuples are spacelike separated:

$$\mathbf{I}_1 \cap \overline{\mathbf{I}_2} \; {\gt\!\!\!\!\lt} \; \overline{\mathbf{I}_1} \cap \mathbf{I}_2 \,.$$

By the assumption that the $\{T_k\}_{k \neq n}$ satisfy causal factorization, this implies that the corresponding time-ordered products commute:

(2)$$T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2}) \, T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2) \;=\; T(\overline{\mathbf{I}_1} \cap \mathbf{I}_2) \, T(\mathbf{I}_1 \cap \overline{\mathbf{I}_2}) \,.$$

Using this we find that the identifications of $T_{n+1}$ on $\mathcal{C}_{I_1}$ and on $\mathcal{C}_{I_2}$, accrding to (1), agree on the intersection: in that for $\mathbf{X} \in \mathcal{C}_{I_1} \cap \mathcal{C}_{I_2}$ we have

$$\begin{aligned} T( \mathbf{I}_1 ) T( \overline{\mathbf{I}_1} ) & = T( \mathbf{I}_1 \cap \mathbf{I}_2 ) T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} ) \, T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 ) T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} ) \\ & = T( \mathbf{I}_1 \cap \mathbf{I}_2 ) \underbrace{ T( \overline{\mathbf{I}_1} \cap \mathbf{I}_2 ) T( \mathbf{I}_1 \cap \overline{\mathbf{I}_2} ) } T( \overline{\mathbf{I}_1} \cap \overline{\mathbf{I}_2} ) \\ & = T( \mathbf{I}_2 ) T( \overline{\mathbf{I}_2} ) \end{aligned}$$

Here in the first step we expanded out the two factors using (1) for $I_2$, then under the brace we used (2) and in the last step we used again (1), but now for $I_1$.

To conclude, let

(3)$$\left( \chi_I \in C^\infty_{cp}(\Sigma^{n+1}), \, supp(\chi_I) \subset \mathcal{C}_i \right)_{ { I \subset \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }$$

be a partition of unity subordinate to the open cover formed by the $\mathcal{C}_I$:

Then the above implies that setting for any $\mathbf{X} \in \Sigma^{n+1} \setminus diag(\Sigma)$

(4)$$T_{n+1}(\mathbf{X}) \;\coloneqq\; \underset{ { I \in \{1, \cdots, n+1\} } \atop { I, \overline{I} \neq \emptyset } }{\sum} \chi_i(\mathbf{X}) T( \mathbf{I} ) T( \overline{\mathbf{I}} )$$

is well defined and satisfies causal factorization.

Remark

(time-ordered products of fixed interaction as distributions)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge-fixed relativistic free vacuum according to def. , and assume that the field bundle is a trivial vector bundle (example )

and let

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

be a polynomial local observable as in def. , to be regarded as a adiabatically switched interaction action functional. This means that there is a finite set

$$\left\{ \mathbf{L}_{int,i}, \mathbf{\alpha}_{i'} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}}) \right\}_{i,i'}$$

of Lagrangian densities which are monomials in the field and jet coordinates, and a corresponding finite set

$$\left\{ g_{sw,i} \in C^\infty_{cp}(\Sigma)\langle g \rangle \,,\, j_{sw,i'} \in C^\infty_{cp}(\Sigma)\langle j \rangle \right\}$$

of adiabatic switchings, such that

$$g S_{int} + j A \;=\; \tau_{\Sigma} \left( \underset{i}{\sum} g_{sw,i} \mathbf{L}_{int,i} \;+\; \underset{i'}{\sum} j_{sw,i'} \mathbf{\alpha}_{i'} \right)$$

is the transgression of variational differential forms (def. ) of the sum of the products of these adiabatic switching with these Lagrangian densities.

In order to discuss the S-matrix $\mathcal{S}(g S_{int} + j A)$ and hence the time-ordered products of the special form $T_k\left( \underset{k \, \text{factors}}{\underbrace{g S_{int} + j A, \cdots, g S_{int} + j A }} \right)$ it is sufficient to restrict attention to the restriction of each $T_k$ to the subspace of local observables induced by the finite set of Lagrangian densities $\{\mathbf{L}_{int,i}, \mathbf{\alpha}_{i'}\}_{i,i'}$.

This restriction is a continuous linear functional on the corresponding space of bump functions $\{g_{sw,i}, j_{sw,i'}\}$, hence a dstributional section of a corresponding trivial vector bundle.

In terms of this, prop. says that the choice of time-ordered products $T_k$ is inductively in $k$ a choice of extension of distributions to the diagonal.

If $\Sigma = \mathbb{R}^{p,1}$ is Minkowski spacetime and we impose the renormalization condition “translation invariance” (def. ) then each $T_k$ is a distribution on $\Sigma^{k-1} = \mathbb{R}^{(p+1)(k-1)}$ and the extension of distributions is from the complement of the origina $0 \in \mathbb{R}^{(p+1)(k-1)}$.

Therefore we now discuss extension of distributions (def. below) on Cartesian spaces from the complement of the origin to the origin. Since the space of choices of such extensions turns out to depend on the scaling degree of distributions, we first discuss that (def. below).

Definition

(rescaled distribution)

Let $n \in \mathbb{N}$. For $\lambda \in (0,\infty) \subset \mathbb{R}$ a positive real number write

$$\array{ \mathbb{R}^n &\overset{s_\lambda}{\longrightarrow}& \mathbb{R}^n \\ x &\mapsto& \lambda x }$$

for the diffeomorphism given by multiplication with $\lambda$, using the canonical real vector space-structure of $\mathbb{R}^n$.

Then for $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution on the Cartesian space $\mathbb{R}^n$ the rescaled distribution is the pullback of $u$ along $m_\lambda$

$$u_\lambda \coloneqq s_\lambda^\ast u \;\in\; \mathcal{D}'(\mathbb{R}^n) \,.$$

Explicitly, this is given by

$$\array{ \mathcal{D}(\mathbb{R}^n) &\overset{ \langle u_\lambda, - \rangle}{\longrightarrow}& \mathbb{R} \\ b &\mapsto& \lambda^{-n} \langle u , b(\lambda^{-1}\cdot (-))\rangle } \,.$$

Similarly for $X \subset \mathbb{R}^n$ an open subset which is invariant under $s_\lambda$, the rescaling of a distribution $u \in \mathcal{D}'(X)$ is is $u_\lambda \coloneqq s_\lambda^\ast u$.

Definition

(scaling degree of a distribution)

Let $n \in \mathbb{N}$ and let $X \subset \mathbb{R}^n$ be an open subset of Cartesian space which is invariant under rescaling $s_\lambda$ (def. ) for all $\lambda \in (0,\infty)$, and let $u \in \mathcal{D}'(X)$ be a distribution on this subset. Then

1. The scaling degree of $u$ is the infimum

   $$sd(u) \;\coloneqq\; inf \left\{ \omega \in \mathbb{R} \;\vert\; \underset{\lambda \to 0}{\lim} \lambda^\omega u_\lambda = 0 \right\}$$

   of the set of real numbers $\omega$ such that the limit of the rescaled distribution $\lambda^\omega u_\lambda$ (def. ) vanishes. If there is no such $\omega$ one sets $sd(u) \coloneqq \infty$.

2. The degree of divergence of $u$ is the difference of the scaling degree by the dimension of the underlying space:

$$deg(u) \coloneqq sd(u) - n \,.$$

Example

(scaling degree of non-singular distributions)

If $u = u_f$ is a non-singular distribution given by bump function $f \in C^\infty(X) \subset \mathcal{D}'(X)$, then its scaling degree (def. ) is non-positive

$$sd(u_f) \leq 0 \,.$$

Specifically if the first non-vanishing partial derivative $\partial_\alpha f(0)$ of $f$ at 0 occurs at order ${\vert \alpha\vert} \in \mathbb{N}$, then the scaling degree of $u_f$ is $-{\vert \alpha\vert}$.

Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any bump function that

$$\begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega-n} \underset{\mathbb{R}^n}{\int} f(x) g(\lambda^{-1} x) d^n x \\ & = \lambda^{\omega} \underset{\mathbb{R}^n}{\int} f(\lambda x) g(x) d^n x \end{aligned} \,,$$

where in last line we applied change of integration variables.

The limit of this expression is clearly zero for all $\omega \gt 0$, which shows the first claim.

If moreover the first non-vanishing partial derivative of $f$ occurs at order ${\vert \alpha \vert} = k$, then Hadamard's lemma says that $f$ is of the form

$$f(x) \;=\; \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} + \underset{ {\beta \in \mathbb{N}^n} \atop { {\vert \beta\vert} = {\vert \alpha \vert} + 1 } }{\sum} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x)$$

where the $h_{\beta}$ are smooth functions. Hence in this case

$$\begin{aligned} \left\langle \lambda^{\omega} (u_f)_\lambda, n \right\rangle & = \lambda^{\omega + {\vert \alpha\vert }} \underset{\mathbb{R}^n}{\int} \left( \underset{i}{\prod} \alpha_i ! \right)^{-1} (\partial_\alpha f(0)) \underset{i}{\prod} (x^i)^{\alpha_i} b(x) d^n x \\ & \phantom{=} + \lambda^{\omega + {\vert \alpha\vert} + 1} \underset{\mathbb{R}^n}{\int} \underset{i}{\prod} (x^i)^{\beta_i} h_{\beta}(x) b(x) d^n x \end{aligned} \,.$$

This makes manifest that the expression goes to zero with $\lambda \to 0$ precisely for $\omega \gt - {\vert \alpha \vert}$, which means that

$$sd(u_f) = -{\vert \alpha \vert}$$

in this case.

Example

(scaling degree of derivatives of delta-distributions)

Let $\alpha \in \mathbb{N}^n$ be a multi-index and $\partial_\alpha \delta \in \mathcal{D}'(X)$ the corresponding partial derivatives of the delta distribution $\delta_0 \in \mathcal{D}'(\mathbb{R}^n)$ supported at $0$. Then the degree of divergence (def. ) of $\partial_\alpha \delta_0$ is the total order the derivatives

$$deg\left( {\, \atop \,} \partial_\alpha\delta_0{\, \atop \,} \right) \;=\; {\vert \alpha \vert}$$

where ${\vert \alpha\vert} \coloneqq \underset{i}{\sum} \alpha_i$.

Proof

By definition we have for $b \in C^\infty_{cp}(\mathbb{R}^n)$ any bump function that

$$\begin{aligned} \left\langle \lambda^\omega (\partial_\alpha \delta_0)_\lambda, b \right\rangle & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega-n} \left( \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(\lambda^{-1}x) \right)_{\vert x = 0} \\ & = (-1)^{{\vert \alpha \vert}} \lambda^{\omega - n - {\vert \alpha\vert}} \frac{ \partial^{{\vert \alpha \vert}} }{ \partial^{\alpha_1} x^1 \cdots \partial^{\alpha_n}x^n } b(0) \end{aligned} \,,$$

where in the last step we used the chain rule of differentiation. It is clear that this goes to zero with $\lambda$ as long as $\omega \gt n + {\vert \alpha\vert}$. Hence $sd(\partial_{\alpha} \delta_0) = n + {\vert \alpha \vert}$.

Example

(scaling degree of Feynman propagator on Minkowski spacetime)

Let

$$\Delta_F(x) \;=\; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k$$

be the Feynman propagator for the massive free real scalar field on $n = p+1$-dimensional Minkowski spacetime (prop. ). Its scaling degree is

$$\begin{aligned} sd(\Delta_{F}) & = n - 2 \\ & = p -1 \end{aligned} \,.$$

(Brunetti-Fredenhagen 00, example 3 on p. 22)

Proof

Regarding $\Delta_F$ as a generalized function via the given Fourier-transform expression, we find by change of integration variables in the Fourier integral that in the scaling limit the Feynman propagator becomes that for vannishing mass, which scales homogeneously:

$$\begin{aligned} \underset{\lambda \to 0}{\lim} \left( \lambda^\omega \; \Delta_F(\lambda x) \right) & = \underset{\lambda \to 0}{\lim} \left( \lamba^{\omega} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n} \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - (\lambda^{-2}) k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon } \, d k_0 \, d^p \vec k \right) \\ & = \underset{\lambda \to 0}{\lim} \left( \lambda^{\omega-n + 2 } \; \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \lambda x^\mu} }{ - k_\mu k^\mu + i \epsilon } \, d k_0 \, d^p \vec k \right) \,. \end{aligned}$$

Proposition

(basic properties of scaling degree of distributions)

Let $X \subset \mathbb{R}^n$ and $u \in \mathcal{D}'(X)$ be a distribution as in def. , such that its scaling degree is finite: $sd(u) \lt \infty$ (def. ). Then

1. For $\alpha \in \mathbb{N}^n$, the partial derivative of distributions $\partial_\alpha$ increases scaling degree at most by ${\vert \alpha\vert }$:

   $$deg(\partial_\alpha u) \;\leq\; deg(u) + {\vert \alpha\vert}$$

2. For $\alpha \in \mathbb{N}^n$, the product of distributions with the smooth coordinate functions $x^\alpha$ decreases scaling degree at least by ${\vert \alpha\vert }$:

   $$deg(x^\alpha u) \;\leq\; deg(u) - {\vert \alpha\vert}$$

3. Under tensor product of distributions their scaling degrees add:

   $$sd(u \otimes v) \leq sd(u) + sd(v)$$

   for $v \in \mathcal{D}'(Y)$ another distribution on $Y \subset \mathbb{R}^{n'}$;

4. $deg(f u) \leq deg(u) - k$ for $f \in C^\infty(X)$ and $f^{(\alpha)}(0) = 0$ for ${\vert \alpha\vert} \leq k-1$;

(Brunetti-Fredenhagen 00, lemma 5.1, Dütsch 18, exercise 3.34)

Proof

The first three statements follow with manipulations as in example and example .

For the fourth…

Proposition

(scaling degree of product distribution)

Let $u,v \in \mathcal{D}'(\mathbb{R}^n)$ be two distributions such that

1. both have finite degree of divergence (def. )

   $$deg(u), deg(v) \lt \infty$$

2. their product of distributions is well-defined

   $$u v \in \mathcal{D}'(\mathbb{R}^n)$$

   (in that their wave front sets satisfy Hörmander's criterion)

then the product distribution has degree of divergence bounded by the sum of the separate degrees:

$$deg(u v) \;\leq\; deg(u) + deg(v) \,.$$

With the concept of scaling degree of distributions in hand, we may now discuss extension of distributions:

Definition

(extension of distributions)

Let $X \overset{\iota}{\subset} \hat X$ be an inclusion of open subsets of some Cartesian space. This induces the operation of restriction of distributions

$$\mathcal{D}'(\hat X) \overset{\iota^\ast}{\longrightarrow} \mathcal{D}'(X) \,.$$

Given a distribution $u \in \mathcal{D}'(X)$, then an extension of $u$ to $\hat X$ is a distribution $\hat u \in \mathcal{D}'(\hat X)$ such that

$$\iota^\ast \hat u \;=\; u \,.$$

Proposition

(unique extension of distributions with negative degree of divergence)

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a distribution on the complement of the origin, with negative degree of divergence at the origin

$$deg(u) \lt 0 \,.$$

Then $u$ has a unique extension of distributions $\hat u \in \mathcal{D}'(\mathbb{R}^n)$ to the origin with the same degree of divergence

$$deg(\hat u) = deg(u) \,.$$

(Brunetti-Fredenhagen 00, theorem 5.2, Dütsch 18, theorem 3.35 a))

Proof

Regarding uniqueness:

Suppose $\hat u$ and ${\hat u}^\prime$ are two extensions of $u$ with $deg(\hat u) = deg({\hat u}^\prime)$. Both being extensions of a distribution defined on $\mathbb{R}^n \setminus \{0\}$, this difference has support at the origin $\{0\} \subset \mathbb{R}^n$. By prop. this implies that it is a linear combination of derivatives of the delta distribution supported at the origin:

$${\hat u}^\prime - \hat u \;=\; \underset{ {\alpha \in \mathbb{N}^n} }{\sum} c^\alpha \partial_\alpha \delta_0$$

for constants $c^\alpha \in \mathbb{C}$. But by example the degree of divergence of these point-supported distributions is non-negative

$$deg( \partial_\alpha \delta_0) = {\vert \alpha\vert} \geq 0 \,.$$

This implies that $c^\alpha = 0$ for all $\alpha$, hence that the two extensions coincide.

Regarding existence:

Let

$$b \in C^\infty_{cp}(\mathbb{R}^n)$$

be a bump function which is $\leq 1$ and constant on 1 over a neighbourhood of the origin. Write

$$\chi \coloneqq 1 - b \;\in\; C^\infty(\mathbb{R}^n)$$

graphics grabbed from Dütsch 18, p. 108

and for $\lambda \in (0,\infty)$ a positive real number, write

$$\chi_\lambda(x) \coloneqq \chi(\lambda x) \,.$$

Since the product $\chi_\lambda u$ has support of a distribution on a complement of a neighbourhood of the origin, we may extend it by zero to a distribution on all of $\mathbb{R}^n$, which we will denote by the same symbols:

$$\chi_\lambda u \in \mathcal{D}'(\mathbb{R}^n) \,.$$

By construction $\chi_\lambda u$ coincides with $u$ away from a neighbourhood of the origin, which moreover becomes arbitrarily small as $\lambda$ increases. This means that if the following limit exists

$$\hat u \;\coloneqq\; \underset{\lambda \to \infty}{\lim} \chi_\lambda u$$

then it is an extension of $u$.

To see that the limit exists, it is sufficient to observe that we have a Cauchy sequence, hence that for all $b\in C^\infty_{cp}(\mathbb{R}^n)$ the difference

$$(\chi_{n+1} u - \chi_n u)(b) \;=\; u(b)( \chi_{n+1} + \chi_n )$$

becomes arbitrarily small.

It remains to see that the unique extension $\hat u$ thus established has the same scaling degree as $u$. This is shown in (Brunetti-Fredenhagen 00, p. 24).

Proposition

(space of point-extensions of distributions)

For $n \in \mathbb{N}$, let $u \in \mathcal{D}'(\mathbb{R}^n \setminus \{0\})$ be a distribution of degree of divergence $deg(u) \lt \infty$.

Then $u$ does admit at least one extension (def. ) to a distribution $\hat u \in \mathcal{D}'(\mathbb{R}^n)$, and every choice of extension has the same degree of divergence as $u$

$$deg(\hat u) = deg(u) \,.$$

Moreover, any two such extensions $\hat u$ and ${\hat u}^\prime$ differ by a linear combination of partial derivatives of distributions of order $\leq deg(u)$ of the delta distribution $\delta_0$ supported at the origin:

$${\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq deg(u) } }{\sum} q^\alpha \partial_\alpha \delta_0 \,,$$

for a finite number of constants $q^\alpha \in \mathbb{C}$.

This is essentially (Hörmander 90, thm. 3.2.4). We follow (Brunetti-Fredenhagen 00, theorem 5.3), which was inspired by (Epstein-Glaser 73, section 5). Review of this approach is in (Dütsch 18, theorem 3.35 (b)), see also remark below.

Proof

For $f \in C^\infty(\mathbb{R}^n)$ a smooth function, and $\rho \in \mathbb{N}$, we say that $f$ vanishes to order $\rho$ at the origin if all partial derivatives with multi-index $\alpha \in \mathbb{N}^n$ of total order ${\vert \alpha\vert} \leq \rho$ vanish at the origin:

$$\partial_\alpha f (0) = 0 \phantom{AAA} {\vert \alpha\vert} \leq \rho \,.$$

By Hadamard's lemma, such a function may be written in the form

(5)$$f(x) \;=\; \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x)$$

for smooth functions $r_\alpha \in C^\infty_{cp}(\mathbb{R}^n)$.

Write

$$\mathcal{D}_\rho(\mathbb{R}^n) \hookrightarrow \mathcal{D}(\mathbb{R}^n) \coloneqq C^\infty_{cp}(\mathbb{R}^n)$$

for the subspace of that of all bump functions on those that vanish to order $\rho$ at the origin.

By definition this is equivalently the joint kernel of the partial derivatives of distributions of order ${\vert \alpha\vert}$ of the delta distribution $\delta_0$ supported at the origin:

$$b \in \mathcal{D}_\rho(\mathbb{R}^n) \phantom{AA} \Leftrightarrow \phantom{AA} \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } } {\forall} \left\langle \partial_\alpha \delta_0, b \right\rangle = 0 \,.$$

Therefore every continuous linear projection

(6)$$p_\rho \;\colon\; \mathcal{D}(\mathbb{R}^n) \longrightarrow \mathcal{D}_\rho(\mathbb{R}^n)$$

may be obtained from a choice of dual basis to the $\{\partial_\alpha \delta_0\}$, hence a choice of smooth functions

$$\left\{ w^\beta \in C^\infty_{cp}(\mathbb{R}^n) \right\}_{ { \beta \in \mathbb{N}^n } \atop { {\vert \beta\vert} \leq \rho } }$$

such that

$$\left\langle \partial_\alpha \delta_0 \,,\, w^\beta \right\rangle \;=\; \delta_\alpha^\beta \phantom{AAA} \Leftrightarrow \phantom{AAA} \partial_\alpha w^\beta(0) \;=\; \delta_\alpha^\beta \phantom{AAAA} \text{for}\, {\vert \alpha\vert} \leq \rho \,,$$

by setting

(7)$$p_\rho \;\coloneqq\; id \;-\; \left\langle \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} w^\alpha \partial_\alpha \delta_0 \,,\, (-) \right\rangle \,,$$

hence

$$p_\rho \;\colon\; b \mapsto b - \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha\vert}} w^\alpha \partial_\alpha b(0) \,.$$

Together with Hadamard's lemma in the form (5) this means that every $b \in \mathcal{D}(\mathbb{R}^n)$ is decomposed as

(8)$$\begin{aligned} b(x) & = p_\rho(b)(x) \;+\; (id - p_\rho)(b)(x) \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} x^\alpha r_\alpha(x) \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} \leq \rho } }{\sum} (-1)^{{\vert \alpha \vert}} w^\alpha \partial_\alpha b(0) \end{aligned}$$

Now let

$$\rho \;\coloneqq\; deg(u) \,.$$

Observe that (by prop. ) the degree of divergence of the product of distributions $x^\alpha u$ with ${\vert \alpha\vert} = \rho + 1$ is negative

$$\begin{aligned} deg\left( x^\alpha u \right) & = \rho - {\vert \alpha \vert} \leq -1 \end{aligned}$$

Therefore prop. says that each $x^\alpha u$ for ${\vert \alpha\vert} = \rho + 1$ has a unique extension $\widehat{ x^\alpha u}$ to the origin. Accordingly the composition $u \circ p_\rho$ has a unique extension, by (8):

(9)$$\begin{aligned} \left\langle \hat u \,,\, b \right\rangle & = \left\langle \hat u , p_\rho(b) \right\rangle + \left\langle \hat u , (id - p_\rho)(b) \right\rangle \\ & = \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha \vert} = \rho + 1 } }{\sum} \underset{ \text{unique} }{ \underbrace{ \left\langle \widehat{x^\alpha u} \,,\, r_\alpha \right\rangle } } \;+\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} \underset{ { q^\alpha } \atop { \text{choice} } }{ \underbrace{ \langle \hat u \,,\, w^\alpha \rangle } } \left\langle \partial_\alpha \delta_0 \,,\, b \right\rangle \end{aligned}$$

That says that $\hat u$ is of the form

$$\hat u \;=\; \underset{ \text{unique} }{ \underbrace{ \widehat{ u \circ p_\rho } } } + \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} c^\alpha \, \partial_\alpha \delta_0$$

for a finite number of constants $c^\alpha \in \mathbb{C}$.

Notice that for any extension $\hat u$ the exact value of the $c^\alpha$ here depends on the arbitrary choice of dual basis $\{w^\alpha\}$ used for this construction. But the uniqueness of the first summand means that for any two choices of extensions $\hat u$ and ${\hat u}^\prime$, their difference is of the form

$${\hat u}^\prime - \hat u \;=\; \underset{ { \alpha \in \mathbb{N}^n } \atop { {\vert \alpha\vert} \leq \rho } }{\sum} ( (c')^\alpha - c^\alpha ) \, \partial_\alpha \delta_0 \,,$$

where the constants $q^\alpha \coloneqq ( (c')^\alpha - c^\alpha ) \in \mathbb{C}$ are independent of any choices.

It remains to see that all these $\hat u$ in fact have the same degree of divergence as $u$.

By example the degree of divergence of the point-supported distributions on the right is $deg(\partial_\alpha \delta_0) = {\vert \alpha\vert} \leq \rho$.

Therefore to conclude it is now sufficient to show that

$$deg\left( \widehat{ u \circ p_\rho } \right) \;=\; \rho \,.$$

This is shown in (Brunetti-Fredenhagen 00, p. 25).

Remark

(“W-extensions”)

Since in Brunetti-Fredenhagen 00, (38) the projectors (7) are denoted “$W$”, the construction of extensions of distributions via the proof of prop. has come to be called “W-extensions” (e.g Dütsch 18).

In conclusion we obtain the central theorem of causal perturbation theory:

Theorem

(existence and choices of ("re"-)normalization of S-matrices/perturbative QFTs)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge-fixed relativistic free vacuum, according to def. , such that the underlying spacetime is Minkowski spacetime and the Wightman propagator $\Delta_H$ is translation-invariant.

Then:

1. an S-matrix scheme $\mathcal{S}$ (def. ) around this vacuum exists;

2. for $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a local observable as in def. , regarded as an adiabatically switched interaction action functional, the space of possible choices of S-matrices

   $$\mathcal{S}(g S_{int} + j A) \;\in\; PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$$

   hence of the corresponding perturbative QFTs, by prop. , is, inductively in $k \in \mathbb{N}$, a finite dimensional affine space, parameterizing the extension of the time-ordered product $T_k$ to the locus of coinciding interaction points.

Proof

By prop. the Feynman propagator is finite scaling degree of a distribution, so that by prop. the binary time-ordered product away from the diagonal $T_2(-,-)\vert_{\Sigma^2 \setminus diag(\Sigma)} = (-) \star_{F} (-)$ has finite scaling degree.

By prop. this implies that in the inductive description of the time-ordered products by prop. , each induction step is the extension of distributions of finite scaling degree of a distribution to the point. By prop. this always exists.

This proves the first statement.

Now if a polynomial local interaction is fixed, then via remark each induction step involved extending a finite number of distributions, each of finite scaling degree. By prop. the corresponding space of choices is in each step a finite-dimensional affine space.

$\,$

Stückelberg-Petermann renormalization group

A genuine re-normalization is the passage from one S-matrix ("re"-)normalization scheme $\mathcal{S}$ to another such scheme $\mathcal{S}'$. The inductive Epstein-Glaser ("re"-normalization) construction (prop. ) shows that the difference between any $\mathcal{S}$ and $\mathcal{S}'$ is inductively in $k \in \mathbb{N}$ a choice of extra term in the time-ordered product of $k$ factors, equivalently in the Feynman amplitudes for Feynman diagrams with $k$ vertices, that contributes when all $k$ of these vertices coincide in spacetime (prop. ).

A natural question is whether these additional interactions that appear when several interaction vertices coincide may be absorbed into a re-definition of the original interaction $g S_{int} + j A$. Such an interaction vertex redefinition (def. below)

$$\mathcal{Z} \;\colon\; g S_{int} + j A \;\mapsto\; g S_{int} + j A \;+\; \text{higher order corrections}$$

should perturbatively send local interactions to local interactions with higher order corrections.

The main theorem of perturbative renormalization (theorem below) says that indeed under mild conditions every re-normalization $\mathcal{S} \mapsto \mathcal{S}'$ is induced by such an interaction vertex redefinition in that there exists a unique such redefinition $\mathcal{Z}$ so that for every local interaction $g S_{int} + j A$ we have that scattering amplitudes for the interaction $g S_{int} + j A$ computed with the ("re"-)normalization scheme $\mathcal{S}'$ equal those computed with $\mathcal{S}$ but applied to the re-defined interaction $\mathcal{Z}(g S_{int} + j A)$:

$$\mathcal{S}' \left( {\, \atop \,} g S_{int} + j A {\, \atop \,} \right) \;=\; \mathcal{S}\left( {\, \atop \,} \mathcal{Z}(g S_{int} + j A) {\, \atop \,} \right) \,.$$

This means that the interaction vertex redefinitions $\mathcal{Z}$ form a group under composition which acts transitively and freely, hence regularly, on the set of S-matrix ("re"-)normalization schemes; this is called the Stückelberg-Petermann renormalization group (theorem below).

$\,$

Definition

(perturbative interaction vertex redefinition)

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

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

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

Proposition

(local additivity of vertex redefinitions)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ be a gauge fixed free field vacuum (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 $supp(O_i) \subset \Sigma$ (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 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” (10) 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 (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 (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” (prop. ), then so does $\mathcal{S} \circ \mathcal{Z}$.

(e.g Dütsch 18, theorem 3.99 (b))

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 (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 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 unique vertex redefinition)

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

Then for $\mathcal{S}, \mathcal{S}'$ any two S-matrix schemes (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

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

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 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 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

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

(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 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 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.

In conclusion this establishes the following pivotal statement of perturbative quantum field theory:

Theorem

(main theorem of perturbative renormalization – Stückelberg-Petermann renormalization group of vertex redefinitions)

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

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

2. the set of S-matrix ("re"-)normalization schemes (def. ), remark ) satisfying the renormalization condition “field independence” (prop. ) is a torsor over this group, hence equipped with a regular action in that

   1. the set of S-matrix schemes is non-empty;

   2. any two S-matrix ("re"-)normalization schemes $\mathcal{S}$, $\mathcal{S}'$ are related by a unique vertex redefinition $\mathcal{Z}$ via composition:

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

This group is called the Stückelberg-Petermann renormalization group.

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

Proof

The group-structure and regular action is given by prop. and prop. . The existence of S-matrices follows is the statement of Epstein-Glaser ("re"-)normalization in theorem .

$\,$

UV-Regularization via counterterms

While Epstein-Glaser renormalization (prop. ) gives a transparent picture on the space of choices in ("re"-)normalization (theorem ) the physical nature of the higher interactions that it introduces at coincident interaction points (via the extensions of distributions in prop. ) remains more implicit. But the main theorem of perturbative renormalization (theorem ), which re-expresses the difference between any two such choices as an interaction vertex redefinition, suggests that already the choice of ("re"-)normalization itself should have an incarnation in terms of interaction vertex redefinitions.

This may be realized via a construction of ("re"-)normalization in terms of UV-regularization (prop. below): For any choice of “UV-cutoff”, given by an approximation of the Feynman propagator $\Delta_F$ by non-singular distributions $\Delta_{F,\Lambda}$ (def. below) there is a unique “effective S-matrix” $\mathcal{S}_\Lambda$ induced at each cutoff scale (def. below). While the “UV-limit” $\underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda$ does not in general exist, it may be “regularized” by applying suitable interaction vertex redefinitions $\mathcal{Z}_\Lambda$; if the higher-order corrections that these introduce serve to “counter” (remark below) the coresponding UV-divergences.

This perspective of ("re"-)normalization via via counterterms is often regarded as the primary one. Its elegant proof in prop. below, however relies on the Epstein-Glaser renormalization via inductive extensions of distributions and uses the same kind of argument as in the proof of the main theorem of perturbative renormalization (theorem via prop. ) that establishes the Stückelberg-Petermann renormalization group.

$\,$

Definition

(UV cutoffs)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum over Minkowski spacetime $\Sigma$ (according to def. ), where $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ is the corresponding Wightman propagator inducing the Feynman propagator

$$\Delta_F \in \Gamma'_{\Sigma \times \Sigma}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})$$

by $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$.

Then a choice of UV cutoffs for perturbative QFT around this vacuum is a collection of non-singular distributions $\Delta_{F,\Lambda}$ parameterized by positive real numbers

$$\array{ (0, \infty) &\overset{}{\longrightarrow}& \Gamma_{\Sigma \times \Sigma,cp}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}}) \\ \Lambda &\mapsto& \Delta_{F,\Lambda} }$$

such that:

1. each $\Delta_{F,\Lambda}$ satisfies the following basic properties

   1. (translation invariance)

      $$\Delta_{F,\Lambda}(x,y) = \Delta_{F,\Lambda}(x-y)$$

   2. (symmetry)

      $$\Delta^{b a}_{F,\Lambda}(y, x) \;=\; \Delta^{a b}_{F,\Lambda}(x, y)$$

      i.e.

      $$\Delta_{F,\Lambda}^{b a}(-x) \;=\; \Delta_{F,\Lambda}^{a b}(x)$$

2. the $\Delta_{F,\Lambda}$ interpolate between zero and the Feynman propagator, in that, in the Hörmander topology:

   1. the limit as $\Lambda \to 0$ exists and is zero

      $$\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; 0 \,.$$

   2. the limit as $\Lambda \to \infty$ exists and is the Feynman propagator:

      $$\underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; \Delta_F \,.$$

(Dütsch 10, section 4)

Example

(relativistic momentum cutoff)

Recall from this prop. that the Fourier transform of distributions of the Feynman propagator for the real scalar field on Minkowski spacetime $\mathbb{R}^{p,1}$ is,

$$\begin{aligned} \widehat{\Delta}_F(k) & = \frac{+i}{(2\pi)^{p+1}} \frac{ 1 }{ - \eta(k,k) - \left( \tfrac{m c}{\hbar} \right)^2 + i 0 } \end{aligned}$$

To produce a UV cutoff in the sense of def. we would like to set this function to zero for wave numbers $\vert \vec k\vert$ (hence momenta $\hbar\vert \vec k\vert$) larger than a given $\Lambda$.

This needs to be done with due care: First, the Paley-Wiener-Schwartz theorem (prop. ) says that $\Delta_{F,\Lambda}$ to be a test function and hence compactly supported, its Fourier transform $\widehat{\Delta}_{F,\Lambda}$ needs to be smooth and of bounded growth. So instead of multiplying $\widehat{\Delta}_F$ by a step function in $k$, we may multiply it with an exponential damping.

(Keller-Kopper-Schophaus 97, section 6.1, Dütsch 18, example 3.126)

Definition

(effective S-matrix scheme)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

We say that the effective S-matrix scheme $\mathcal{S}_\Lambda$ at cutoff scale $\Lambda \in [0,\infty)$

$$\array{ PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] &\overset{\mathcal{S}_{\Lambda}}{\longrightarrow}& PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ] \\ O &\mapsto& \mathcal{S}_\Lambda(O) }$$

is the exponential series

(12)$$\begin{aligned} \mathcal{S}_\Lambda(O) & \coloneqq \exp_{F,\Lambda}\left( \frac{1}{i \hbar} O \right) \\ & = 1 + \frac{1}{i \hbar} O + \frac{1}{2} \frac{1}{(i \hbar)^2} O \star_{F,\Lambda} O + \frac{1}{3!} \frac{1}{(i \hbar)^3} O \star_{F,\Lambda} O \star_{F,\Lambda} 0 + \cdots \end{aligned} \,.$$

with respect to the star product $\star_{F,\Lambda}$ induced by the $\Delta_{F,\Lambda}$ (def. ).

This is evidently defined on all polynomial observables as shown, and restricts to an endomorphism on microcausal polynomial observables as shown, since the contraction coefficients $\Delta_{F,\Lambda}$ are non-singular distributions, by definition of UV cutoff.

(Dütsch 10, (4.2))

Proposition

(("re"-)normalization via UV regularization)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $g S_{int} + j A \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ a polynomial local observable as in def. , regarded as an adiabatically switched interaction action functional.

Let moreover $\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)}$ be a UV cutoff (def. ); with $\mathcal{S}_\Lambda$ the induced effective S-matrix schemes (12).

Then

1. there exists a $[0,\infty)$-parameterized interaction vertex redefinition $\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}$ (def. ) such that the limit of effective S-matrix schemes $\mathcal{S}_{\Lambda}$ (12) applied to the $\mathcal{Z}_\Lambda$-redefined interactions

   $$\mathcal{S}_\infty \;\coloneqq\; \underset{\Lambda \to \infty}{\lim} \left( \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda \right)$$

   exists and is a genuine S-matrix scheme around the given vacuum (def. );

2. every S-matrix scheme around the given vacuum arises this way.

These $\mathcal{Z}_\Lambda$ are called counterterms (remark below) and the composite $\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ is called a UV regularization of the effective S-matrices $\mathcal{S}_\Lambda$.

Hence UV-regularization via counterterms is a method of ("re"-)normalization of perturbative QFT (def. ).

This was claimed in (Brunetti-Dütsch-Fredenhagen 09, (75)), a proof was indicated in (Dütsch-Fredenhagen-Keller-Rejzner 14, theorem A.1).

Proof

Let $\{p_{\rho_{k}}\}_{k \in \mathbb{N}}$ be a sequence of projection maps as in (6) defining an Epstein-Glaser ("re"-)normalization (prop. ) of time-ordered products $\{T_k\}_{k \in \mathbb{N}}$ as extensions of distributions of the $T_k$, regarded as distributions via remark , by the choice $q_k^\alpha = 0$ in (9).

We will construct that $\mathcal{Z}_\Lambda$ in terms of these projections $p_\rho$.

First consider some convenient shorthand:

For $n \in \mathbb{N}$, write $\mathcal{Z}_{\leq n} \coloneqq \underset{1 \in \{1, \cdots, n\}}{\sum} \frac{1}{n!} Z_n$. Moreover, for $k \in \mathbb{N}$ write $(T_\Lambda \circ \mathcal{Z}_{\leq n})_k$ for the $k$-ary coefficient in the expansion of the composite $\mathcal{S}_\Lambda \circ \mathcal{Z}_{\leq n}$, as in equation (11) in the proof of the main theorem of perturbative renormalization (theorem , via prop. ).

In this notation we need to find $\mathcal{Z}_\Lambda$ such that for each $n \in \mathbb{N}$ we have

(13)$$\underset{\Lambda \to \infty}{\lim} \left( T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda} \right)_n \;=\; T_n \,.$$

We proceed by induction over $n \in \mathbb{N}$.

Since by definition $T_0 = const_1$, $T_1 = id$ and $Z_0 = const_0$, $Z_1 = id$ the statement is trivially true for $n = 0$ and $n = 1$.

So assume now $n \in \mathbb{N}$ and $\{Z_{k}\}_{k \leq n}$ has been found such that (13) holds.

Observe that with the chosen renormalizing projection $p_{\rho_{n+1}}$ the time-ordered product $T_{n+1}$ may be expressed as follows:

(14)$$\begin{aligned} T_{n+1}(O, \cdots, O) & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_k}(O \otimes \cdots \otimes O) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_k}(O \otimes \cdots \otimes O) \right\rangle \end{aligned} \,.$$

Here in the first step we inserted the causal decomposition (4) of $T_{n+1}$ in terms of the $\{T_k\}_{k \leq n}$ away from the diagonal, as in the proof of prop. , which is admissible because the image of $p_{\rho_{n+1}}$ vanishes on the diagonal. In the second step we replaced the star-product of the Feynman propagator $\Delta_F$ with the limit over the star-products of the regularized propagators $\Delta_{F,\Lambda}$, which converges by the nature of the Hörmander topology (which is assumed by def. ).

Hence it is sufficient to find $Z_{n+1,\Lambda}$ and $K_{n+1,\Lambda}$ such that

(15)$$\begin{aligned} \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\Lambda} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{k}}\left( -, \cdots, - \right) \right\rangle \\ & \phantom{=} + K_{n+1,\Lambda}(-, \cdots, -) \end{aligned}$$

subject to these two conditions:

1. $\mathcal{Z}_{n+1,\Lambda}$ is local;

2. $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$.

Now by expanding out the left hand side of (15) as

$$(T_\Lambda \circ \mathcal{Z}_\Lambda)_{n+1} \;=\; Z_{n+1,\Lambda} \;+\; (T_\Lambda \circ Z_{\leq n, \Lambda})_{n+1}$$

(which uses the condition $T_1 = id$) we find the unique solution of (15) for $Z_{n+1,\Lambda}$, in terms of the $\{Z_{\leq n,\Lambda}\}$ and $K_{n+1,\Lambda}$ (the latter still to be chosen) to be:

(16)$$\begin{aligned} \left\langle Z_{n+1,\Lambda} , (-,\cdots, -) \right\rangle & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \\ & \phantom{=} - \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n,\Lambda} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle \\ & \phantom{=} + \left\langle K_{n+1, \Lambda}, (-, \cdots, -) \right\rangle \end{aligned} \,.$$

We claim that the following choice works:

(17)$$\begin{aligned} K_{n+1, \Lambda}(-, \cdots, -) & \coloneqq \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda} \right)_{n+1} \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \\ & \phantom{=} - \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} T_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} T_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-, \cdots, -) \right\rangle \end{aligned} \,.$$

To prove this, we need to show that 1) the resulting $Z_{n+1,\Lambda}$ is local and 2) the limit of $K_{n+1,\Lambda}$ vanishes as $\Lambda \to \infty$.

First regarding the locality of $Z_{n+1,\Lambda}$: By inserting (17) into (16) we obtain

$$\begin{aligned} \left\langle Z_{n+1,\Lambda} \,,\, (-,\cdots,-) \right\rangle & = \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, p(-, \cdots, -) \right\rangle - \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, (-, \cdots, -) \right\rangle \\ & = \left\langle \left( T_{\Lambda} \circ \mathcal{Z}_{\leq n} \right)_{n+1} \,,\, ( p_{\rho_{n+1}} - id)(-, \cdots, -) \right\rangle \end{aligned}$$

By definition $p_{\rho_{n+1}} - id$ is the identity on test functions (adiabatic switchings) that vanish at the diagonal. This means that $Z_{n+1,\Lambda}$ is supported on the diagonal, and is hence local.

Second we need to show that $\underset{\Lambda \to \infty}{\lim} K_{n+1,\Lambda} = 0$:

By applying the analogous causal decomposition (4) to the regularized products, we find

(18)$$\begin{aligned} & \left\langle (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \,. \end{aligned}$$

Using this we compute as follows:

(19)$$\begin{aligned} & \left\langle \underset{\Lambda \to \infty}{\lim} (T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda})_{n+1} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { \mathbf{I}, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \!\!\chi_i(\mathbf{X})\, \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) {\, \atop \,} \right) \star_{F,\Lambda} \left( {\, \atop \,} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) {\, \atop \,} \right) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \chi_i(\mathbf{X})\, \underset{ T_{{\vert \mathbf{I}\vert}}(\mathbf{I}) }{ \underbrace{ \left( \underset{\Lambda \to \infty}{\lim} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{{\vert \mathbf{I}\vert}} (\mathbf{I}) \right) }} \left( \underset{\Lambda \to \infty}{\lim} \star_{F,\Lambda} \right) \underset{ T_{{\vert \overline{\mathbf{I}}\vert}}(\overline{\mathbf{I}}) }{ \underbrace{ \left( \underset{\Lambda \to \infty}{\lim} (T_{\Lambda} \circ \mathcal{Z}_{\leq n, \Lambda})_{ {\vert \overline{\mathbf{I}} \vert} } ( \overline{\mathbf{I}} ) \right) }} \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \\ & = \left\langle \underset{\Lambda \to \infty}{\lim} \underset{ {\mathbf{I} \in \{1, \cdots, n+1\} } \atop { I, \overline{\mathbf{I}} \neq \emptyset } }{\sum} \chi_i(\mathbf{X})\, T_{ { \vert \mathbf{I} \vert } }( \mathbf{I} ) \star_{F,\Lambda} T_{ {\vert \overline{\mathbf{I}} \vert} }( \overline{\mathbf{I}} ) \,,\, p_{\rho_{n+1}}(-,\cdots,-) \right\rangle \end{aligned} \,.$$

Here in the first step we inserted (18); in the second step we used that in the Hörmander topology the product of distributions preserves limits in each variable and in the third step we used the induction assumption (13) and the definition of UV cutoff (def. ).

Inserting this for the first summand in (17) shows that $\underset{\Lambda \to \infty}{\lim} K_{n+1, \Lambda} = 0$.

In conclusion this shows that a consistent choice of counterterms $\mathcal{Z}_\Lambda$ exists to produce some S-matrix $\mathcal{S} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)$. It just remains to see that for every other S-matrix $\widetilde{\mathcal{S}}$ there exist counterterms $\widetilde{\mathcal{Z}}_\lambda$ such that $\widetilde{\mathcal{S}} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \widetilde{\mathcal{Z}}_\Lambda)$.

But by the main theorem of perturbative renormalization (theorem ) we know that there exists a vertex redefinition $\mathcal{Z}$ such that

$$\begin{aligned} \widetilde{\mathcal{S}} & = \mathcal{S} \circ \mathcal{Z} \\ & = \underset{\Lambda \to \infty}{\lim} \left( \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda \right) \circ \mathcal{Z} \\ & = \underset{\Lambda \to \infty}{\lim} ( \mathcal{S}_\Lambda \circ ( \underset{ \widetilde{\mathcal{Z}}_\Lambda }{ \underbrace{ \mathcal{Z}_\Lambda \circ \mathcal{Z} } } ) ) \end{aligned}$$

and hence with counterterms $\mathcal{Z}_\Lambda$ for $\mathcal{S}$ given, then counterterms for any $\widetilde{\mathcal{S}}$ are given by the composite $\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}$.

Remark

(counterterms)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

Consider

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

a local observable, regarded as an adiabatically switched interaction action functional.

Then prop. says that there exist vertex redefinitions of this interaction

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

parameterized by $\Lambda \in [0,\infty)$, such that the limit

$$\mathcal{S}_\infty(g S_{int} + j A) \;\coloneqq\; \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda\left( \mathcal{Z}_\Lambda( g S_{int} + j A )\right)$$

exists and is an S-matrix for perturbative QFT with the given interaction $g S_{int} + j A$.

In this case the difference

$$\begin{aligned} S_{counter, \Lambda} & \coloneqq \left( g S_{int} + j A \right) \;-\; \mathcal{Z}_{\Lambda}(g S_{int} + j A) \;\;\;\;\;\in\; Loc(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g^2, j^2, g j\rangle \end{aligned}$$

(which by the axiom “perturbation” in def. is at least of second order in the coupling constant/source field, as shown) is called a choice of counterterms at cutoff scale $\Lambda$. These are new interactions which are added to the given interaction at cutoff scale $\Lambda$

$$\mathcal{Z}_{\Lambda}(g S_{int} + j A) \;=\; g S_{int} + j A \;+\; S_{counter,\Lambda} \,.$$

In this language prop. says that for every free field vacuum and every choice of local interaction, there is a choice of counterterms to the interaction that defines a corresponding ("re"-)normalized perturbative QFT, and every (re"-)normalized perturbative QFT arises from some choice of counterterms.

$\,$

Wilson-Polchinski effective QFT flow

We have seen above that a choice of UV cutoff induces effective S-matrix schemes $\mathcal{S}_\Lambda$ at cutoff scale $\Lambda$ (def. ). To these one may associated non-local relative effective actions $S_{eff,\Lambda}$ (def. below) which are such that their effective scattering amplitudes at scale $\Lambda$ coincide with the true scattering amplitudes of a genuine local interaction as the cutoff is removed. This is the Wilsonian picture of effective quantum field theory at a given cutoff scale (remark below). Crucially the “flow” of the relative effective actions with the cutoff scale satisfies a differential equation that in itself is independent of the full UV-theory; this is Polchinski's flow equation (prop. below). Solving this equation for given choice of initial value data is hence another way of choosing ("re"-)normalization constants.

$\,$

Proposition

(effective S-matrix schemes are invertible functions)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

Write

$$PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \hookrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$$

for the subspace of the space of formal power series in $\hbar, g, j$ with coefficients polynomial observables on those which are at least of first order in $g,j$, i.e. those that vanish for $g, j = 0$ (as in def. ).

Write moreover

$$1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \hookrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$$

for the subspace of polynomial observables which are the sum of 1 (the multiplicative unit) with an observable at least linear n $g,j$.

Then the effective S-matrix schemes $\mathcal{S}_\Lambda$ (def. ) restrict to linear isomorphisms of the form

$$PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \underoverset{\simeq}{\phantom{AA}\mathcal{S}_\Lambda \phantom{AA} }{\longrightarrow} 1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \,.$$

(Dütsch 10, (4.7))

Proof

Since each $\Delta_{F,\Lambda}$ is symmetric (def. ) if follows by general properties of star products (prop. ) just as for the genuine time-ordered product on regular polynomial observables (prop. ) that eeach the “effective time-ordered product” $\star_{F,\Lambda}$ is isomorphic to the pointwise product $(-)\cdot (-)$ (def. )

$$A_1 \star_{F,\Lambda} A_2 \;=\; \mathcal{T}_\Lambda \left( \mathcal{T}_\Lambda^{-1}(A_1) \cdot \mathcal{T}_\Lambda^{-1}(A_2) \right)$$

for

$$\mathcal{T}_\Lambda \;\coloneqq\; \exp \left( \tfrac{1}{2}\hbar \underset{\Sigma}{\int} \Delta_{F,\Lambda}^{a b}(x,y) \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)} \right)$$

as in (?).

In particular this means that the effective S-matrix $\mathcal{S}_\Lambda$ arises from the exponential series for the pointwise product by conjugation with $\mathcal{T}_\Lambda$:

$$\mathcal{S}_\Lambda \;=\; \mathcal{T}_\Lambda \circ \exp_\cdot\left( \frac{1}{i \hbar}(-) \right) \circ \mathcal{T}_\Lambda^{-1}$$

(just as for the genuine S-matrix on regular polynomial observables in def. ).

Now the exponential of the pointwise product on $1 + PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ has as inverse function the natural logarithm power series, and since $\mathcal{T}$ evidently preserves powers of $g,j$ this conjugates to an inverse at each UV cutoff scale $\Lambda$:

(20)$$\mathcal{S}_\Lambda^{-1} \;=\; \mathcal{T}_\Lambda \circ \ln\left( i \hbar (-) \right) \circ \mathcal{T}_\Lambda^{-1} \,.$$

Definition

(relative effective action)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

Consider

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

a local observable regarded as an adiabatically switched interaction action functional.

Then for

$$\Lambda,\, \Lambda_{vac} \;\in\; (0, \infty)$$

two UV cutoff-scale parameters, we say the relative effective action $S_{eff, \Lambda, \Lambda_0}$ is the image of this interaction under the composite of the effective S-matrix scheme $\mathcal{S}_{\Lambda_0}$ at scale $\Lambda_0$ (12) and the inverse function $\mathcal{S}_\Lambda^{-1}$ of the effective S-matrix scheme at scale $\Lambda$ (via prop. ):

(21)$$S_{eff,\Lambda, \Lambda_0} \;\coloneqq\; \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\Lambda_0}(g S_{int} + j A) \phantom{AAA} \Lambda, \Lambda_0 \in [0,\infty) \,.$$

For chosen counterterms (remark ) hence for chosen UV regularization $\mathcal{S}_\infty$ (prop. ) this makes sense also for $\Lambda_0 = \infty$ and we write:

(22)$$S_{eff,\Lambda} \;\coloneqq\; S_{eff,\Lambda, \infty} \;\coloneqq\; \mathcal{S}_{\Lambda}^{-1} \circ \mathcal{S}_{\infty}(g S_{int} + j A) \phantom{AAA} \Lambda \in [0,\infty)$$

(Dütsch 10, (5.4))

Remark

(effective quantum field theory)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ), and let $\mathcal{S}_\infty = \underset{\Lambda \to \infty}{\lim} \mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda$ be a corresponding UV regularization (prop. ).

Consider a local observable

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

regarded as an adiabatically switched interaction action functional.

Then def. and def. say that for any $\Lambda \in (0,\infty)$ the effective S-matrix (12) of the relative effective action (21) equals the genuine S-matrix $\mathcal{S}_\infty$ of the genuine interaction $g S_{int} + j A$:

$$\mathcal{S}_\Lambda( S_{eff,\Lambda} ) \;=\; \mathcal{S}_\infty\left( g S_{int} + j A \right) \,.$$

In other words the relative effective action $S_{eff,\Lambda}$ encodes what the actual perturbative QFT defined by $\mathcal{S}_\infty\left( g S_{int} + j A \right)$ effectively looks like at UV cutoff $\Lambda$.

Therefore one says that $S_{eff,\Lambda}$ defines effective quantum field theory at UV cutoff $\Lambda$.

Notice that in general $S_{eff,\Lambda}$ is not a local interaction anymore: By prop. the image of the inverse $\mathcal{S}^{-1}_\Lambda$ of the effective S-matrix is microcausal polynomial observables in $1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle$ and there is no guarantee that this lands in the subspace of local observables.

Therefore effective quantum field theories at finite UV cutoff-scale $\Lambda \in [0,\infty)$ are in general not local field theories, even if their limit as $\Lambda \to \infty$ is, via prop. .

Proposition

(effective action is relative effective action at $\Lambda = 0$)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

Then the relative effective action (def. ) at $\Lambda = 0$ is the actual effective action (def. ) in the sense of the the Feynman perturbation series of Feynman amplitudes $\Gamma(g S_{int} + j A)$ (def. ) for connected Feynman diagrams $\Gamma$:

$$\begin{aligned} S_{eff,0} & \coloneqq\; S_{eff,0,\infty} \\ & = S_{eff} \;\coloneqq\; \underset{\Gamma \in \Gamma_{conn}}{\sum} \Gamma(g S_{int} + j A) \,. \end{aligned}$$

More generally this holds true for any $\Lambda \in [0, \infty) \sqcup \{\infty\}$

$$\begin{aligned} S_{eff,0,\Lambda} & = \underset{\Gamma \in \Gamma_{conn}}{\sum} \Gamma_\Lambda(g S_{int} + j A) \,, \end{aligned}$$

where $\Gamma_\Lambda( g S_{int} + j A)$ denotes the evident version of the Feynman amplitude (def. ) with time-ordered products replaced by effective time ordered product at scale $\Lambda$ as in (def. ).

(Dütsch 18, (3.473))

Proof

Observe that the effective S-matrix scheme at scale $\Lambda = 0$ (12) is the exponential series with respect to the pointwise product (def. )

$$\mathcal{S}_0(O) = \exp_\cdot( O ) \,.$$

Therefore the statement to be proven says equivalently that the exponential series of the effective action with respect to the pointwise product is the S-matrix:

$$\exp_\cdot\left( \frac{1}{i \hbar} S_{eff} \right) \;=\; \mathcal{S}_\infty\left( g S_{int} + j A \right) \,.$$

That this is the case is the statement of prop. .

The definition of the relative effective action $\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty}$ in def. invokes a choice of UV regularization $\mathcal{S}_\infty$ (prop. ). While (by that proposition and the main theorem of perturbative renormalization, theorem )this is guaranteed to exist, in practice one is after methods for constructing this without specifying it a priori.

But the collection relative effective actions $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ for $\Lambda_0 \lt \infty$ “flows” with the cutoff-parameters $\Lambda$ and in particular also with $\Lambda_0$ (remark below) which suggests that examination of this flow yields information about full theory at $\mathcal{S}_\infty$.

This is made precise by Polchinski's flow equation (prop. below), which is the infinitesimal version of the “Wilsonian RG flow” (remark ). As a differential equation it is independent of the choice of $\mathcal{S}_{\infty}$ and hence may be used to solve for the Wilsonian RG flow without knowing $\mathcal{S}_\infty$ in advance.

The freedom in choosing the initial values of this differential equation corresponds to the ("re"-)normalization freedom in choosing the UV regularization $\mathcal{S}_\infty$. In this sense “Wilsonian RG flow” is a method of ("re"-)normalization of perturbative QFT (def. ).

Remark

(Wilsonian groupoid of effective quantum field theories)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ) and let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ).

Then the relative effective actions $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ (def. ) satisfy

$$S_{eff, \Lambda', \Lambda_0} \;=\; \left( \mathcal{S}_{\Lambda'}^{-1} \circ \mathcal{S}_\Lambda \right) \left( S_{eff, \Lambda, \Lambda_0} \right) \phantom{AAA} \text{for} \, \Lambda,\Lambda' \in [0,\infty) \,,\, \Lambda_0 \in [0,\infty) \sqcup \{\infty\} \,.$$

This is similar to a group of UV-cutoff scale-transformations. But since the composition operations are only sensible when the UV-cutoff labels match, as shown, it is really a groupoid action.

This is often called the Wilsonian RG.

We now consider the infinitesimal version of this “flow”:

Proposition

(Polchinski's flow equation)

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a gauge fixed relativistic free vacuum (according to def. ), let $\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)}$ be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ), such that $\Lambda \mapsto \Delta_{F,\Lambda}$ is differentiable.

Then for every choice of UV regularization $\mathcal{S}_\infty$ (prop. ) the corresponding relative effective actions $S_{eff,\Lambda}$ (def. ) satisfy the following differential equation:

$$\frac{d}{d \Lambda} S_{eff,\Lambda} \;=\; - \frac{1}{2} \frac{1}{i \hbar} \frac{d}{d \Lambda'} \left( S_{eff,\Lambda} \star_{F,\Lambda'} S_{eff,\Lambda} \right)\vert_{\Lambda' = \Lambda} \,,$$

where on the right we have the star product induced by $\Delta_{F,\Lambda'}$ (def. ).

This goes back to (Polchinski 84, (27)). The rigorous formulation and proof is due to (Brunetti-Dütsch-Fredenhagen 09, prop. 5.2, Dütsch 10, theorem 2).

Proof

First observe that for any polynomial observable $O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

$$\begin{aligned} & \frac{1}{(k+2)!} \frac{d}{d \Lambda} ( \underset{ k+2 \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } ) \\ & = \frac{1}{(k+2)!} \frac{d}{d \Lambda} \left( prod \circ \exp\left( \hbar \underset{1 \leq i \lt j \leq k}{\sum} \left\langle \Delta_{F,\Lambda} , \frac{\delta}{\delta \mathbf{\Phi}_i} \frac{\delta}{\delta \mathbf{\Phi}_j} \right\rangle \right) ( \underset{ k + 2 \, \text{factors} }{ \underbrace{ O \otimes \cdots \otimes O } } ) \right) \\ & = \underset{ = \frac{1}{2} \frac{1}{k!} }{ \underbrace{ \frac{1}{(k+2)!} \left( k + 2 \atop 2 \right) }} \left( \frac{d}{d \Lambda} O \star_{F,\Lambda} O \right) \star_{F,\Lambda} \underset{ k \, \text{factors} }{ \underbrace{ O \star_{F,\Lambda} \cdots \star_{F,\Lambda} O } } \end{aligned}$$

Here $\frac{\delta}{\delta \mathbf{\Phi}_i}$ denotes the functional derivative of the $i$th tensor factor of $O$, and the binomial coefficient counts the number of ways that an unordered pair of distinct labels of tensor factors may be chosen from a total of $k+2$ tensor factors, where we use that the star product $\star_{F,\Lambda}$ is commutative (by symmetry of $\Delta_{F,\Lambda}$) and associative (by prop. ).

With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) = \mathcal{S}(g S_{int} + j A)$ (22) we compute as follows:

$$\begin{aligned} 0 & = \frac{d}{d \Lambda} \mathcal{S}(g S_{int} + j A) \\ & = \frac{d}{d \Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) + \left( \frac{d}{d \Lambda} \mathcal{S}_{\Lambda} \right) \left( S_{eff, \Lambda} \right) \\ & = \left( \frac{1}{i \hbar} \frac{d}{d \Lambda} S_{eff,\Lambda} \right) \star_{F,\Lambda} \mathcal{S}_\Lambda(S_{eff,\Lambda}) \;+\; \frac{1}{2} \frac{d}{d \Lambda'} \left( \frac{1}{i \hbar} S_{eff,\Lambda} \star_{F,\Lambda'} \frac{1}{i \hbar} S_{eff, \Lambda} \right) \vert_{\Lambda' = \Lambda} \star_{F,\Lambda} \mathcal{S}_\Lambda \left( S_{eff, \Lambda} \right) \end{aligned}$$

Acting on this equation with the multiplicative inverse $(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} )$ (using that $\star_{F,\Lambda}$ is a commutative product, so that exponentials behave as usual) this yields the claimed equation.

$\,$

renormalization group flow

In perturbative quantum field theory the construction of the scattering matrix $\mathcal{S}$, hence of the interacting field algebra of observables for a given interaction $g S_{int}$ perturbing around a given free field vacuum, involves choices of normalization of time-ordered products/Feynman diagrams (traditionally called "re"-normalizations) encoding new interactions that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.

Whenever a group $RG$ acts on the space of observables of the theory such that conjugation by this action takes ("re"-)normalization schemes into each other, then these choices of ("re"-)normalization are parameterized by – or “flow with” – the elements of $RG$. This is called renormalization group flow (prop. below); often called RG flow, for short.

The archetypical example here is the group $RG$ of scaling transformations on Minkowski spacetime (def. below), which induces a renormalization group flow (prop. below) due to the particular nature of the Wightman propagator resp. Feynman propagator on Minkowski spacetime (example below). In this case the choice of ("re"-)normalization hence “flows with scale”.

Now the main theorem of perturbative renormalization (theorem ) states that (if only the basic renormalization condition called “field independence” is satisfied) any two choices of ("re"-)normalization schemes $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique interaction vertex redefinition $\mathcal{Z}$, as

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

Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an interaction vertex redefinition as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their (adiabatically switched) coupling constants as changing under such an interaction vertex redefinition, and hence then one has coupling constants $g_j$ that are parameterized by elements $\rho$ of $RG$:

$$\mathcal{Z}_{\rho_{vac}}^\rho \;\colon\; \{g_j\} \mapsto \{g_j(\rho)\}$$

This dependendence is called running of the coupling constants under the renormalization group flow (def. below).

One example of renormalization group flow is that induced by scaling transformations (prop. below). This is the original and main example of the concept (Gell-Mann & Low 54)

In this case the running of the coupling constants may be understood as expressing how “more” interactions (at higher energy/shorter wavelength) become visible (say to experiment) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be differentiable; its logarithmic derivative (denoted “$\psi$” in Gell-Mann & Low 54) is known as the beta function (Callan 70, Symanzik 70):

$$\beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho} \,.$$

The running of the coupling constants is not quite a representation of the renormalization group flow, but it is a “twisted” representation, namely a group 1-cocycle (prop. below). For the case of scaling transformations this may be called the Gell-Mann-Low renormalization cocycle (Brunetti-Dütsch-Fredenhagen 09).

$\,$

Proposition

(renormalization group flow)

Let

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

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

Consider a group $RG$ equipped with an action on the Wick algebra of off-shell microcausal polynomial observables with formal parameters adjoined (as in def. )

$$rg_{(-)} \;\colon\; RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ] \,,$$

hence for each $\rho \in RG$ a continuous linear map $rg_\rho$ which has an inverse $rg_\rho^{-1} \in RG$ and is a homomorphism of the Wick algebra-product (the star product $\star_H$ induced by the Wightman propagator of the given vauum $vac$)

$$rg_\rho( A_1 \star_H A_2 ) \;=\; rg_\rho(A_1) \star_H rg_\rho(A_2)$$

such that the following conditions hold:

1. the action preserves the subspace of off-shell polynomial local observables, hence it restricts as

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

2. the action respects the causal order of the spacetime support (def. ) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

   $$\left( supp(O_1) \,{\vee\!\!\!\wedge}\, supp(O_2) \right) \phantom{A} \Rightarrow \phantom{A} \left( supp(rg_\rho(O_1)) \,{\vee\!\!\!\wedge}\, supp(rg_\rho(O_2)) \right)$$

   for all $\rho \in RG$.

Then:

The operation of conjugation by this action on observables induces an action on the set of S-matrix renormalization schemes (def. , remark ), in that for

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

a perturbative S-matrix scheme around the given free field vacuum $vac$, also the composite

$$\mathcal{S}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ] \longrightarrow PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]$$

is an S-matrix scheme, for all $\rho \in RG$.

More generally, let

$$vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )$$

be a collection of gauge fixed free field vacua parameterized by elements $\rho \in RG$, all with the same underlying field bundle; and consider $rg_\rho$ as above, except that it is not an automorphism of any Wick algebra, but an isomorphism between the Wick algebra-structures on various vacua, in that

(23)$$rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 ) \;=\; rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)$$

for all $\rho, \rho_{vac} \in RG$

Then if

$$\{ \mathcal{S}_{\rho} \}_{\rho \in RG}$$

is a collection of S-matrix schemes, one around each of the gauge fixed free field vacua $vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the composite

(24)$$\mathcal{S}_{\rho_{vac}}^\rho \;\coloneqq\; rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}$$

is an S-matrix scheme around the vacuum labeled by $\rho_{vac}$.

Since therefore each element $\rho \in RG$ in the group $RG$ picks a different choice of normalization of the S-matrix scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$ a re-normalization group flow.

(Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1, Dütsch 18, section 3.5.3)

Proof

It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom “perturbation” (in def. ).

In order to verify the axiom “causal additivity”, observe, for convenience, that by prop. it is sufficient to check causal factorization.

So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$ two local observables whose spacetime support is in causal order.

$$supp(O_1) \;{\vee\!\!\!\wedge}\; supp(O_2) \,.$$

We need to show that the

$$\mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2) = \mathcal{S}_{\rho_{vac}}^\rho(O_1) \star_{H,\rho_{vac}} \mathcal{S}_{vac_e}^\rho(O_2)$$

for all $\rho, \rho_{vac} \in RG$.

Using the defining properties of $rg_{(-)}$ and the causal factorization of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:

$$\begin{aligned} \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2) & = rg_\rho \circ \mathcal{S}_{\rho^{-1} \rho_{vac}} \circ rg_\rho^{-1}( O_1 + O_2 ) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1}\rho_{vac}} \left( rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \left( \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right) \right) \star_{H, \rho^{-1} \rho_{vac}} \left( \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right) \right) {\, \atop \,} \right) \\ & = rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right) {\, \atop \,} \right) \star_{H, \rho_{vac}} rg_\rho \left( {\, \atop \,} \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right) {\, \atop \,} \right) \\ & = \mathcal{S}^\rho_{\rho_{vac}}( O_1 ) \, \star_{H, \rho_{vac}} \, \mathcal{S}_{\rho_{vac}}^\rho(O_2) \,. \end{aligned}$$

Definition

(running coupling constants)

Let

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

be a relativistic free vacuum (according to def. ) around which we consider interacting perturbative QFT, let $\mathcal{S}$ be an S-matrix scheme around this vacuum and let $rg_{(-)}$ be a renormalization group flow according to prop. , such that each re-normalized S-matrix scheme $\mathcal{S}_{vac}^\rho$ satisfies the renormalization condition “field independence”.

Then by the main theorem of perturbative renormalization (theorem , via prop. ) there is for every pair $\rho_1, \rho_2 \in RG$ a unique interaction vertex redefinition

$$\mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] \longrightarrow LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$$

which relates the corresponding two S-matrix schemes via

(25)$$\mathcal{S}_{\rho_{vac}}^{\rho} \;=\; \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^\rho \,.$$

If one thinks of an interaction vertex, hence a local observable $g S_{int}+ j A$, as specified by the (adiabatically switched) coupling constants $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding interaction Lagrangian densities $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as

$$g S_{int} \;=\; \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)$$

(where $\tau_\Sigma$ denotes transgression of variational differential forms) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the (adiabatically switched) coupling constants $g_j$ of the renormalization group flow parameterized by $\rho$. The corresponding functions

$$\mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int}) \;\colon\; (g_j) \mapsto (g_j(\rho))$$

are then called running coupling constants.

(Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1, Dütsch 18, section 3.5.3)

Proposition

(running coupling constants are group cocycle over renormalization group flow)

Consider running coupling constants

$$\mathcal{Z}_{\rho_{vac}}^{\rho} \;\colon\; (g_j) \mapsto (g_j(\rho))$$

as in def. . Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied by the “running functions” (25):

$$\mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} \;=\; \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \left( \sigma_{\rho_1} \circ \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \right) \,.$$

(Brunetti-Dütsch-Fredenhagen 09 (69), Dütsch 18, (3.325))

Proof

Directly using the definitions, we compute as follows:

$$\begin{aligned} \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2} & = \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 } \\ & = \sigma_{\rho_1} \circ \underset{ = \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} = \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2} }{ \underbrace{ \sigma_{\rho_2} \circ \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}} \circ \sigma_{\rho_2}^{-1} }} \circ \sigma_{\rho_1}^{-1} \\ & = \underset{ = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \sigma_{\rho_1} }{ \underbrace{ \sigma_{\rho_1} \circ \mathcal{S}_{\rho_1^{-1} \rho_{vac}} \circ \overset{ = id }{ \overbrace{ \sigma_{\rho_1}^{-1} \circ \sigma_{\rho_1} } } }} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} \\ & = \mathcal{S}_{\rho_{vac}} \circ \mathcal{Z}_{\rho_{vac}}^{\rho_1} \circ \underbrace{ \sigma_{\rho_1} \circ \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2} \circ \sigma_{\rho_1}^{-1} } \end{aligned}$$

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 (theorem ) 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.

$\,$

Gell-Mann Low RG flow

We discuss (prop. below) that, if the field species involved have well-defined mass dimension (example below) then scaling transformations on Minkowski spacetime (example below) induce a renormalization group flow (def. ). This is the original and main example of renormalization group flows (Gell-Mann& Low 54).

Example

(scaling transformations and mass dimension)

Let

$$E \overset{fb}{\longrightarrow} \Sigma$$

be a field bundle which is a trivial vector bundle over Minkowski spacetime $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.

For $\rho \in (0,\infty) \subset \mathbb{R}$ a positive real number, write

$$\array{ \Sigma &\overset{\rho}{\longrightarrow}& \Sigma \\ x &\mapsto& \rho x }$$

for the operation of multiplication by $\rho$ using the real vector space-structure of the Cartesian space $\mathbb{R}^{p+1}$ underlying Minkowski spacetime.

By pullback this acts on field histories (sections of the field bundle) via

$$\array{ \Gamma_\Sigma(E) &\overset{\rho^\ast}{\longrightarrow}& \Gamma_\Sigma(E) \\ \Phi &\mapsto& \Phi(\rho(-)) } \,.$$

Let then

$$\rho \mapsto vac_\rho \;\coloneqq\; (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )$$

be a 1-parameter collection of relativistic free vacua on that field bundle, according to def. , and consider a decomposition into a set $Spec$ of field species (def. ) such that for each $sp \in Spec$ the collection of Feynman propagators $\Delta_{F,\rho,sp}$ for that species scales homogeneously in that there exists

$$dim(sp) \in \mathbb{R}$$

such that for all $\rho$ we have (using generalized functions-notation)

(26)$$\rho^{ 2 dim(sp) } \Delta_{F, 1/\rho, sp}( \rho x ) \;=\; \Delta_{F,sp, \rho = 1}(x) \,.$$

Typically $\rho$ rescales a mass parameter, in which case $dim(sp)$ is also called the mass dimension of the field species $sp$.

Let finally

$$\array{ PolyObs(E) & \overset{ \sigma_\rho }{\longrightarrow} & PolyObs(E) \\ \mathbf{\Phi}_{sp}^a(x) &\mapsto& \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x ) }$$

be the function on off-shell polynomial observables given on field observables $\mathbf{Phi}^a(x)$ by pullback along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the mass dimension, and extended from there to all polynomial observables as an algebra homomorphism.

This constitutes an action of the group

$$RG \coloneqq \left( \mathbb{R}_+, \cdot \right)$$

of positive real numbers (under multiplication) on polynomial observables, called the group of scaling transformations for the given choice of field species and mass parameters.

(Dütsch 18, def. 3.19)

Example

(mass dimension of scalar field)

Consider the Feynman propagator $\Delta_{F,m}$ of the free real scalar field on Minkowski spacetime $\Sigma = \mathbb{R}^{p,1}$ for mass parameter $m \in (0,\infty)$; a Green function for the Klein-Gordon equation.

Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of scaling transformations $\rho \in \mathbb{R}_+$ on Minkowski spacetime (def. ) act on the mass parameter by inverse multiplication

$$(\rho , \Delta_{F,m}) \mapsto \Delta_{F,\rho^{-1}m}(\rho (-)) \,.$$

Then we have

$$\Delta_{F,\rho^{-1}m}(\rho (-)) \;=\; \rho^{-(p+1) + 2} \Delta_{F,1}(x)$$

and hence the corresponding mass dimension (def. ) of the real scalar field on $\mathbb{R}^{p,1}$ is

$$dim(\text{scalar field}) = (p+1)/2 - 1 \,.$$

Proof

By prop. the Feynman propagator in question is given by the Cauchy principal value-formula (in generalized function-notation)

$$\begin{aligned} \Delta_{F,m}(x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \,. \end{aligned}$$

By applying change of integration variables $k \mapsto \rho^{-1} k$ in the Fourier transform this becomes

$$\begin{aligned} \Delta_{F,\rho^{-1}m}(\rho x) & = \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu \rho x^\mu} }{ - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1)+2} \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim} \frac{+i}{(2\pi)^{p+1}} \int \int_{-\infty}^\infty \frac{ e^{i k_\mu x^\mu} }{ - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon } \, d k_0 \, d^p \vec k \\ & = \rho^{-(p+1) + 2} \Delta_{F,m}(x) \end{aligned}$$

Proposition

(scaling transformations are renormalization group flow)

Let

$$vac \coloneqq vac_m \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})$$

be a relativistic free vacua on that field bundle, according to def. equipped with a decomposition into a set $Spec$ of field species (def. ) such that for each $sp \in Spec$ the collection of Feynman propagators the corresponding field species has a well-defined mass dimension $dim(sp)$ (def. )

Then the action of the group $RG \coloneqq (\mathbb{R}_+, \cdot)$ of scaling transformations (def. ) is a renormalization group flow in the sense of prop. .

(Dütsch 18, exercise 3.20)

Proof

It is clear that rescaling preserves causal order and the renormalization condition of “field indepencen”.

The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two microcausal polynomial observables we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that

$$\sigma_\rho \left( A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2 \right) \;=\; \sigma_\rho(A_1) \star_{H,\rho_{vac}} \sigma_\rho(A_2) \,.$$

By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$. Moreover, by the nature of the star product on polynomial observables, which is given by iterated contractions with the Wightman propagator, it is sufficient to check this for one such contraction.

Observe that the scaling behaviour of the Wightman propagator $\Delta_{H,m}$ is the same as the behaviour (26) of the correspponding Feynman propagator. With this we directly compute as follows:

$$\begin{aligned} \sigma_\rho (\mathbf{\Phi}(x)) \star_{F, \rho_{vac} m} \sigma_\rho (\mathbf{\Phi}(y) & = \rho^{-2 dim } \mathbf{\Phi}(\rho^{-1} x) \star_{F, \rho_{vac} m} \mathbf{\Phi}(\rho^{-1} y) \\ & = \rho^{-2 dim } \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y)) \\ & = \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \\ & = rg_{\rho}\left( \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1} \right) \\ & = rg_{\rho} \left( \mathbf{\Phi}(x) \star_{F, \rho^{-1} \rho_{vac} m} \mathbf{\Phi}(y) \right) \end{aligned} \,.$$

$\,$

This concludes our discussion of renormalization.