Contents

# Contents

## Idea

In perturbative quantum field theories various concepts of “renormalization groups” describe the choices of ("re"-)normalization and their behaviour under scaling transformations or choices of cutoffs.

There are at least three different concepts referred to as “the renormalization group”, only the first is in general really a group:

1. the Stückelberg-Petermann renormalization group (Stückelberg-Petermann 53, historically the origin of the concept)

this is literally the group of re-normalizations, whose elements relate any two given normalization schemes $\mathcal{S}$ and $\mathcal{S}'$ by precomposition with a transformation $\mathcal{Z}$ of the space of local interaction action functionals;

2. renormalization group flow, say along scaling transformations yielding the Gell-Mann-Low renormalization cocycle

3. the Wilsonian RG of effective quantum field theories defined with a UV cutoff.

In more detail:

Let

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

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

Then a perturbative S-matrix scheme/("re"-)normalization scheme around this vacuum (this def.) is a function

$\array{ LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle & \overset{\mathcal{S}_{vac}}{\longrightarrow} & PolyObs(E_{\text{BV-BRST}})_{mc}( ( \hbar ) )[ [ g, j ] ] \\ g S_{int} + j A &\mapsto& \mathcal{S}_{vac}(g S_{int} + j A) }$

from local observables, regared as adiatically switched interaction action functionals to Wick algebra-elements $\mathcal{S}( g S_{int} + j A)$, encoding scattering amplitudes in the given vacuum $\mathbf{L}'$ for the given interaction $g S_{int} + j A$, with formal parameters adjoined as indicated.

The Stückelberg-Petermann renormalization group is a group of transformations

$\array{ LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j \rangle &\overset{Z}{\longrightarrow}& LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle \\ g S_{int} + J A &\mapsto& \mathcal{Z}(g S_{int} + j A) }$

such that for $\mathcal{S}$ and $\mathcal{S}'$ two normalization schemes/S-matrix schemes, there is a unique $\mathcal{Z}$ relating them by precomposition, in that

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

for all $g S_{int} + j A$. This is the main theorem of perturbative renormalization. Hence this says that any two ways of choosing interactions at coincident interaction points are related by a re-definition of the original interaction $g S_{int} + j A$.

Now it may happen that

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

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

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

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

In this case the above statement of the main theorem of perturbative renormalization implies with (1) that there exists a unique transformation $\mathcal{Z}^m_\rho$ of the space of local interaction action functionals such that

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

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

These $\mathcal{Z}^m_\rho$ are the Gell-Mann-Low cocycle elements. They do not actually form a group, unless $m = 0$, but satisfy the relation

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

From the definition we have

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

To conclude, it is now sufficient to see that the perturbative S-matrix $S_{vac(m)}$, as a function form interaction Lagrangian densities to Wick algebra-elements, is an injective function. (…)

## References

The Stückelberg-Petermann renormalization group is due to

• Murray Gell-Mann and F. E. Low, Quantum Electrodynamics at Small Distances, Phys. Rev. 95 (5) (1954), 1300–1312 (pdf)

as well as to Wilsonian RG of effective quantum field theories is due to

reviewed in