renormalization group


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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 𝒮\mathcal{S} and 𝒮\mathcal{S}' by precomposition with a transformation 𝒵\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:


vac(E BV-BRST,L kin,Δ H) 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

LocObs(E BV-BRST)[[,g,j]]g,j 𝒮 vac PolyObs(E BV-BRST) mc(())[[g,j]] gS int+jA 𝒮 vac(gS int+jA) \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 𝒮(gS int+jA)\mathcal{S}( g S_{int} + j A), encoding scattering amplitudes in the given vacuum L\mathbf{L}' for the given interaction gS int+jAg S_{int} + j A, with formal parameters adjoined as indicated.

The Stückelberg-Petermann renormalization group is a group of transformations

LocObs(E BV-BRST)[[,g,j]]g,j Z LocObs(E BV-BRST)[[,g,j]]g,j gS int+JA 𝒵(gS int+jA) \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 𝒮\mathcal{S} and 𝒮\mathcal{S}' two normalization schemes/S-matrix schemes, there is a unique 𝒵\mathcal{Z} relating them by precomposition, in that

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

for all gS int+jAg 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 gS int+jAg S_{int} + j A.

Now it may happen that

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

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

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

    is a perturbative S-matrix around L kin(m)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 𝒵 ρ m\mathcal{Z}^m_\rho of the space of local interaction action functionals such that

σ ρ𝒮 vac(m/ρ)σ ρ 1(gS int+jA) =𝒮 vac(m)(𝒵 ρ m(gS int+jA)) \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 gS int+jAg S_{int} + j A.

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

𝒵 ρ 1ρ 2 m=𝒵 ρ 1 m(σ ρ 1𝒵 ρ 2 m/ρ 1σ ρ 2) \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))


From the definition we have

𝒮 vac(m)𝒵 ρ 1ρ 2 m =σ ρ 1σ ρ 2𝒮 vac(m/ρ 1ρ 2)σ ρ 2 1𝒮 vac(m/ρ 1)𝒵 ρ 2 m/ρ 1σ ρ 1 1 =σ ρ 1𝒮 vac(m/ρ 1)σ ρ 1 1σ ρ 1=id=𝒮 vac(m)𝒵 ρ 1 mσ ρ 1𝒵 ρ 2 m/ρ 1σ ρ 1 1 =𝒮 vac(m)𝒵 ρ 1 mσ ρ 1𝒵 ρ 2 m/ρ 1σ ρ 1 1 \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)S_{vac(m)}, as a function form interaction Lagrangian densities to Wick algebra-elements, is an injective function. (…)


The Stückelberg-Petermann renormalization group is due to

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

reviewed in

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)

  • Wikipedia Renormalization group

Last revised on February 1, 2018 at 13:27:26. See the history of this page for a list of all contributions to it.