nLab renormalization group

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.

(e.g. Brunetti-Dütsch-Fredenhagen 09, p. 10)

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

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

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

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. (…)

Related concepts

• renormalization

• renormalization group flow

• asymptotic safety

• cosmic Galois group

References

The Stückelberg-Petermann renormalization group is due to

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

The relation of the Stückelberg-Petermann renormalization group to renormalization group flow (Gell-Mann-Low renormalization cocycles)

• 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

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

• Michael Dütsch, Klaus Fredenhagen, Kai Keller, Katarzyna Rejzner, Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization, J. Math. Phy.

  55(12), 122303 (2014) (arXiv:1311.5424)

reviewed in

• Michael Dütsch, section 3.5.1 of From classical field theory to perturbative quantum field theory, 2018

See also

• Kiyoshi Higashijima, Kazuhiko Nishijima, Renormalization Groups of Gell-Mann and Low and of Callan and Symanzik, Progress in theoretical physics, vol. 64, no. 6, December 1980 (pdf)

• Assaf Shomer, A pedagogical explanation for the non-renormalizability of gravity (arXiv:0709.3555)

• Alessandro Giuliani, Vieri Mastropietro, Slava Rychkov, Gentle introduction to rigorous Renormalization Group: a worked fermionic example (arXiv:2008.04361)

• Wikipedia Renormalization group