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effective quantum field theory

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Perturbative QFT

Contents

Idea

While fundamental physics is at some level well described by quantum field theory, a typical Lagrangian used to define such a QFT can reasonably be expected to define only degrees of freedom and interactions that are relevant up to some given energy scale. In this perspective one speaks of the theory as being the effective quantum field theory of some – possibly known but possibly unspecified – more fundamental theory.

An example (historically the first to be successfully considered) is the Fermi theory of beta decay of hadrons: this contains interactions of four fermions at a time, for instance a process in which a neutron decays into a collection consisting of a proton, an electron and a neutrino. Later it was discovered that, more fundamentally, this is not a single reaction but is composed out of several other interactions that involve exchanges of W-bosons between these four particles. Nevertheless, Fermi’s original effective theory made very precise predictions at energy scales less than 10 MeV. The reason is that the WW-boson has mass several orders of magnitude higher than that (about 80 GeV) and was thus effectively invisible at these low energies.

The low energy expansion of any unitary, relativistic, crossing symmetric S-matrix can be described by an effective quantum field theory.

In the perspective of effective field theory notably non-renormalizable interaction Lagrangians can still make perfect sense as effective theories and give rise to well defined predictions: they can be effective approximations to renormalizable more fundamental theories. This is sometimes called a UV completion of the given effective theory.

For instance quantum gravity – which is notoriously non-renormalizable? – makes perfect sense as an effective field theory (see for instance the introduction in (Donoghue). It is in principle possible that there is some more fundamental theory with plenty of excitations at high energies that is however degreewise finite in perturbation theory, whose effective description at low energy is given by the non-renormalizable Einstein-Hilbert action. (For instance, string theory is meant to be such a theory.)

Details

The concept of effective perturbative QFT has a precise formulation in the rigoruous context of causal perturbation theory/perturbative AQFT:

Effective quantum field theory has traditioanlly been discussed informally, referring to path integral intuition:

In causal perturbation theory

We discuss the rigorous formulation of effective perturbative QFT in terms of causal perturbation theory/perturbative AQFT, due to (Brunetti-Dütsch-Fredenhagen 09, section 5.2, Dütsch 10), reviewed in Dütsch 18, section 3.8).

(“Re”-)Normalization via UV-Regularization

Definition

(UV cutoffs)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum over Minkowski spacetime Σ\Sigma (according to this def.), where Δ H=i2(Δ +Δ )+H\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H is the corresponding Wightman propagator inducing the Feynman propagator

Δ FΓ Σ×Σ(E BV-BRSTE BV-BRST) \Delta_F \in \Gamma'_{\Sigma \times \Sigma}(E_{\text{BV-BRST}} \boxtimes E_{\text{BV-BRST}})

by Δ F=i2(Δ ++Δ )+H\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 Δ F,Λ\Delta_{F,\Lambda} parameterized by positive real numbers

(0,) Γ Σ×Σ,cp(E BV-BRSTE BV-BRST) Λ Δ F,Λ \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 Δ F,Λ\Delta_{F,\Lambda} satisfies the following basic properties

    1. (translation invariance)

      Δ F,Λ(x,y)=Δ F,Λ(xy) \Delta_{F,\Lambda}(x,y) = \Delta_{F,\Lambda}(x-y)
    2. (symmetry)

      Δ F,Λ ba(y,x)=Δ F,Λ ab(x,y) \Delta^{b a}_{F,\Lambda}(y, x) \;=\; \Delta^{a b}_{F,\Lambda}(x, y)

      i.e.

      Δ F,Λ ba(x)=Δ F,Λ ab(x) \Delta_{F,\Lambda}^{b a}(-x) \;=\; \Delta_{F,\Lambda}^{a b}(x)
  2. the limit of the Δ F,Λ\Delta_{F,\Lambda} as Λ0\Lambda \to 0 exists and is zero

    limΛΔ F,Λ=0. \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; 0 \,.
  3. the limit of the Δ F,Λ\Delta_{F,\Lambda} as Λ\Lambda \to \infty exists and is the Feynman propagator:

    limΛΔ F,Λ=Δ F. \underset{\Lambda \to \infty}{\lim} \Delta_{F,\Lambda} \;=\; \Delta_F \,.

(Dütsch 10, section 4)

example: relativistic momentum cutoff with ϵ\epsilon-regularization (Keller-Kopper-Schophaus 97, section 6.1, Dütsch 18, example 3.126)

Definition

(effective S-matrix scheme)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum (according to this def.) and let {Δ F,Λ} Λ[0,)\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 𝒮 Λ\mathcal{S}_\Lambda at cutoff scale Λ[0,)\Lambda \in [0,\infty)

PolyObs(E BV-BRST) mc[[,g,j]] 𝒮 Λ PolyObs(E BV-BRST) mc[[,g,j]] O 𝒮 Λ(O) \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

(1)𝒮 Λ(O) exp F,Λ(1iO) =1+1iO+121(i) 2O F,ΛO+13!1(i) 3O F,ΛO F,Λ0+. \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 F,Λ\star_{F,\Lambda} induced by the Δ F,Λ\Delta_{F,\Lambda} (this 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 Δ F,Λ\Delta_{F,\Lambda} are non-singular distributions, by definition of UV cutoff.

(Dütsch 10, (4.2))

Proposition

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

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

Let moreover {Δ F,Λ} Λ[0,)\{\Delta_{F,\Lambda}\}_{\Lambda \in [0,\infty)} be a UV cutoff (def. ); with 𝒮 Λ\mathcal{S}_\Lambda the induced effective S-matrix schemes (1).

Then

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

    𝒮 limΛ(𝒮 Λ𝒵 Λ) \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 (this def.);

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

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

Hence UV-regularization via counterterms is a method of ("re"-)normalization of perturbative QFT (this 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 ρ k} k\{p_{\rho_{k}}\}_{k \in \mathbb{N}} be a sequence of projection maps as in (eq:ForExtensionOfDistributionsProjectionMaps) defining an Epstein-Glaser ("re"-)normalization (prop. ) of time-ordered products {T k} k\{T_k\}_{k \in \mathbb{N}} as extensions of distributions of the T kT_k, regarded as distributions via remark , by the choice q k α=0q_k^\alpha = 0 in (eq:ExtensionOfDitstributionsPointFixedAndChoice).

We will construct that 𝒵 Λ\mathcal{Z}_\Lambda in terms of these projections p ρp_\rho.

First consider some convenient shorthand:

For nn \in \mathbb{N}, write 𝒵 n1{1,,n}1n!Z n\mathcal{Z}_{\leq n} \coloneqq \underset{1 \in \{1, \cdots, n\}}{\sum} \frac{1}{n!} Z_n. Moreover, for kk \in \mathbb{N} write (T Λ𝒵 n) k(T_\Lambda \circ \mathcal{Z}_{\leq n})_k for the kk-ary coefficient in the expansion of the composite 𝒮 Λ𝒵 n\mathcal{S}_\Lambda \circ \mathcal{Z}_{\leq n}, as in equation (eq:MainTheoremPerturbativeRenormalizationInductionStep) in the proof of the main theorem of perturbative renormalization (theorem ).

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

(2)limΛ(T Λ𝒵 n,Λ) n=T n. \underset{\Lambda \to \infty}{\lim} \left( T_\Lambda \circ \mathcal{Z}_{\leq n, \Lambda} \right)_n \;=\; T_n \,.

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

Since by definition T 0=const 1T_0 = const_1, T 1=idT_1 = id and Z 0=const 0Z_0 = const_0, Z 1=idZ_1 = id the statement is trivially true for n=0n = 0 and n=1n = 1.

So assume now nn \in \mathbb{N} and {Z k} kn\{Z_{k}\}_{k \leq n} has been found such that (2) holds.

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

(3)T n+1(O,,O) =I{1,,n+1}I,I¯χ i(X)(T |I|(I)) F(T |I¯|(I¯)),p ρ k(OO) =limΛI{1,,n+1}I,I¯χ i(X)(T |I|(I)) F,Λ(T |I¯|(I¯)),p ρ k(OO). \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 (eq:TimeOrderedProductsAwayFromDiagonalByInduction) of T n+1T_{n+1} in terms of the {T k} kn\{T_k\}_{k \leq n} away from the diagonal, as in the proof of prop. , which is admissible because the image of p ρ n+1p_{\rho_{n+1}} vanishes on the diagonal. In the second step we replaced the star-product of the Feynman propagator Δ F\Delta_F with the limit over the star-products of the regularized propagators Δ F,Λ\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,ΛZ_{n+1,\Lambda} and K n+1,ΛK_{n+1,\Lambda} such that

(4)(T Λ𝒵 Λ) n+1,(,,) =I{1,,n+1}I,I¯χ i(X)(T |I|(I)) F,Λ(T |I¯|(I¯)),p ρ k(,,) =+K n+1,Λ(,,) \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. 𝒵 n+1,Λ\mathcal{Z}_{n+1,\Lambda} is local;

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

Now by expanding out the left hand side of (4) as

(T Λ𝒵 Λ) n+1=Z n+1,Λ+(T ΛZ n,Λ) n+1 (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=idT_1 = id) we find the unique solution of (4) for Z n+1,ΛZ_{n+1,\Lambda}, in terms of the {Z n,Λ}\{Z_{\leq n,\Lambda}\} and K n+1,ΛK_{n+1,\Lambda} (the latter still to be chosen) to be:

(5)Z n+1,Λ,(,,) =I{1,,n+1}I,I¯χ i(X)(T |I|(I)) F,Λ(T |I¯|(I¯)),p ρ n+1(,,) =(T Λ𝒵 n,Λ) n+1,(,,) =+K n+1,Λ,(,,). \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:

(6)K n+1,Λ(,,) (T Λ𝒵 n,Λ) n+1,p ρ n+1(,,) =I{1,,n+1}I,I¯χ i(X)(T |I|(I)) F,Λ(T |I¯|(I¯)),p ρ n+1(,,). \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,ΛZ_{n+1,\Lambda} is local and 2) the limit of K n+1,ΛK_{n+1,\Lambda} vanishes as Λ\Lambda \to \infty.

First regarding the locality of Z n+1,ΛZ_{n+1,\Lambda}: By inserting (6) into (5) we obtain

Z n+1,Λ,(,,) =(T Λ𝒵 n) n+1,p(,,)(T Λ𝒵 n) n+1,(,,) =(T Λ𝒵 n) n+1,(p ρ n+1id)(,,) \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 ρ n+1idp_{\rho_{n+1}} - id is the identity on test functions (adiabatic switchings) that vanish at the diagonal. This means that Z n+1,ΛZ_{n+1,\Lambda} is supported on the diagonal, and is hence local.

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

By applying the analogous causal decomposition (eq:TimeOrderedProductsAwayFromDiagonalByInduction) to the regularized products, we find

(7) (T Λ𝒵 n,Λ) n+1,p ρ n+1(,,) =I{1,,n+1}I,I¯χ i(X)((T Λ𝒵 n,Λ) |I|(I)) F,Λ((T Λ𝒵 n,Λ) |I¯|(I¯)),p ρ n+1(,,). \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:

(8) limΛ(T Λ𝒵 n,Λ) n+1,p ρ n+1(,,) =limΛI{1,,n+1}I,I¯χ i(X)((T Λ𝒵 n,Λ) |I|(I)) F,Λ((T Λ𝒵 n,Λ) |I¯|(I¯)),p ρ n+1(,,) =I{1,,n+1}I,I¯χ i(X)(limΛ(T Λ𝒵 n,Λ) |I|(I))T |I|(I)(limΛ F,Λ)(limΛ(T Λ𝒵 n,Λ) |I¯|(I¯))T |I¯|(I¯),p ρ n+1(,,) =limΛI{1,,n+1}I,I¯χ i(X)T |I|(I) F,ΛT |I¯|(I¯),p ρ n+1(,,). \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 (7); 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 (2) and the definition of UV cutoff (def. ).

Inserting this for the first summand in (6) shows that limΛK n+1,Λ=0\underset{\Lambda \to \infty}{\lim} K_{n+1, \Lambda} = 0.

In conclusion this shows that a consistent choice of counterterms 𝒵 Λ\mathcal{Z}_\Lambda exists to produce some S-matrix 𝒮=limΛ(𝒮 Λ𝒵 Λ)\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 𝒮˜\widetilde{\mathcal{S}} there exist counterterms 𝒵˜ λ\widetilde{\mathcal{Z}}_\lambda such that 𝒮˜=limΛ(𝒮 Λ𝒵˜ Λ)\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 𝒵\mathcal{Z} such that

𝒮˜ =𝒮𝒵 =limΛ(𝒮 Λ𝒵 Λ)𝒵 =limΛ(𝒮 Λ(𝒵 Λ𝒵𝒵˜ Λ)) \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 𝒵 Λ\mathcal{Z}_\Lambda for 𝒮\mathcal{S} given, then counterterms for any 𝒮˜\widetilde{\mathcal{S}} are given by the composite 𝒵˜ Λ𝒵 Λ𝒵\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}.

Remark

(counterterms)

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

Consider

gS int+jALocObs(E BV-BRST)[[,g,j]]g,j 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

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

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

𝒮 (gS int+jA)limΛ𝒮 Λ(𝒵 Λ(gS int+jA)) \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 gS int+jAg S_{int} + j A.

In this case the difference

S counter,Λ (gS int+jA)𝒵 Λ(gS int+jA)Loc(E BV-BRST)[[,g,j]]g 2,j 2,gj \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 this def. is at least of second order in the coupling constant/source field, as shown) is called a choice of counterterms at cutoff scale Λ\Lambda. These are new interactions which are added to the given interaction at cutoff scale Λ\Lambda

𝒵 Λ(gS int+jA)=gS int+jA+S counter,Λ. \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.

Effective quantum field theory

Proposition

(effective S-matrix schemes are invertible functions)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum (according to this def.) and let {Δ F,Λ} Λ[0,)\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 BV-BRST) mc[[,g,j]]g,jPolyObs(E BV-BRST) mc[[,g,j]] 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 ,g,j\hbar, g, j with coefficients polynomial observables on those which are at least of first order in g,jg,j, i.e. those that vanish for g,j=0g, j = 0 (as in this def.).

Write moreover

1+PolyObs(E BV-BRST) mc[[,g,j]]g,jPolyObs(E BV-BRST) mc[[,g,j]] 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,jg,j.

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

PolyObs(E BV-BRST) mc[[,g,j]]g,j𝒮 Λ1+PolyObs(E BV-BRST) mc[[,g,j]]g,j. PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \underoverset{\simeq}{\mathcal{S}_\Lambda}{\longrightarrow} 1 + PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]\langle g,j\rangle \,.

(Dütsch 10, (4.7))

Proof

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

A 1 F,ΛA 2=𝒯 Λ(𝒯 Λ 1(A 1)𝒯 Λ 1(A 2)) 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

𝒯 Λexp(12ΣΔ F,Λ ab(x,y)δ 2δΦ a(x)δΦ b(y)) \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 this equation).

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

𝒮 Λ=𝒯 Λexp (1i())𝒯 Λ 1 \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 this def.).

Now the exponential of the pointwise product on 1+PolyObs(E BV-BRST)[[,g,j]]g,j1 + PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle has as inverse function the natural logarithm? power series, and since 𝒯\mathcal{T} evidently preserves powers of g,jg,j this conjugates to an inverse at each UV cutoff scale Λ\Lambda:

(9)𝒮 Λ 1=𝒯 Λln(i())𝒯 Λ 1. \mathcal{S}_\Lambda^{-1} \;=\; \mathcal{T}_\Lambda \circ \ln\left( i \hbar (-) \right) \circ \mathcal{T}_\Lambda^{-1} \,.
Definition

(relative effective action)

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

Consider

gS int+jALocObs(E BV-BrST)[[,g,j]]g,j 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

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

two UV cutoff-scale parameters, we say the relative effective action S eff,Λ,Λ 0S_{eff, \Lambda, \Lambda_0} is the image of this interaction under the composite of the effective S-matrix scheme 𝒮 Λ 0\mathcal{S}_{\Lambda_0} at scale Λ 0\Lambda_0 (1) and the inverse function 𝒮 Λ 1\mathcal{S}_\Lambda^{-1} of the effective S-matrix scheme at scale Λ\Lambda (via prop. ):

(10)S eff,Λ,Λ 0𝒮 Λ 1𝒮 Λ 0(gS int+jA)AAAΛ,Λ 0[0,). 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 𝒮 \mathcal{S}_\infty (prop. ) this makes sense also for Λ 0=\Lambda_0 = \infty and we write:

(11)S eff,ΛS eff,Λ,𝒮 Λ 1𝒮 (gS int+jA)AAAΛ[0,) 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 BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum (according to this def.), let {Δ F,Λ} Λ[0,)\left\{ \Delta_{F,\Lambda}\right\}_{\Lambda \in [0,\infty)} be a choice of UV cutoffs for perturbative QFT around this vacuum (def. ), and let 𝒮 =limΛ𝒮 Λ𝒵 Λ\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

gS int+jALocObs(E BV-BrST)[[,g,j]]g,j 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 Λ(0,)\Lambda \in (0,\infty) the effective S-matrix (1) of the relative effective action (10) equals the genuine S-matrix 𝒮 \mathcal{S}_\infty of the genuine interaction gS int+jAg S_{int} + j A:

𝒮 Λ(S eff,Λ)=𝒮 (gS int+jA). \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,ΛS_{eff,\Lambda} encodes what the actual perturbative QFT defined by 𝒮 (gS int+jA)\mathcal{S}_\infty\left( g S_{int} + j A \right) effectively looks like at UV cutoff Λ\Lambda.

Therefore one says that S eff,ΛS_{eff,\Lambda} defines effective quantum field theory at UV cutoff Λ\Lambda.

Notice that in general S eff,ΛS_{eff,\Lambda} is not a local interaction anymore: By prop. the image of the inverse 𝒮 Λ 1\mathcal{S}^{-1}_\Lambda of the effective S-matrix is microcausal polynomial observables in 1+PolyObs(E BV-BRST) mc[[,g,j]]g,j1 + 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 Λ[0,)\Lambda \in [0,\infty) are in general not local field theories, even if their limit as Λ\Lambda \to \infty is, via prop. .

Proposition

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

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum (according to this def.) and let {Δ F,Λ} Λ[0,)\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 Λ=0\Lambda = 0 is the actual effective action (this def.) being ii \hbar times the Feynman perturbation series of Feynman amplitudes Γ(gS int+jA)\Gamma(g S_{int} + j A) for connected Feynman diagrams Γ\Gamma:

S eff,0 S eff,0, =S effΓΓ connΓ(gS int+jA). \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} \,.

(Dütsch 18, (3.473))

Proof

Observe that the effective S-matrix scheme at scale Λ=0\Lambda = 0 (1) is the exponential series with respect to the pointwise product (this def.)

𝒮 0(O)=exp (O). \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 (1iS eff)=𝒮 (gS int+jA). \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 this prop..

(“Re”-)Normalization via Wilsonian RG flow

The definition of the relative effective action 𝒮 eff,Λ𝒮 eff,Λ,\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty} in def. invokes a choice of UV regularization 𝒮 \mathcal{S}_\infty (prop. ). While (by that proposition and the main theorem of perturbative renormalization 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 𝒮 eff,Λ,Λ 0\mathcal{S}_{eff,\Lambda, \Lambda_0} for Λ 0<\Lambda_0 \lt \infty “flows” with the cutoff-parameters Λ\Lambda and in particular also with Λ 0\Lambda_0 (remark below) which suggests that examination of this flow yields information about full theory at 𝒮 \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 𝒮 \mathcal{S}_{\infty} and hence may be used to solve for the Wilsonian RG flow without knowing 𝒮 \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 𝒮 \mathcal{S}_\infty. In this sense “Wilsonian RG flow” is a method of ("re"-)normalization of perturbative QFT (this def.).

Remark

(Wilsonian groupoid of effective quantum field theories)

Let (E BV-BRST,L,Δ H)(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H ) be a gauge fixed relativistic free vacuum (according to this def.) and let {Δ F,Λ} Λ[0,)\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 𝒮 eff,Λ,Λ 0\mathcal{S}_{eff,\Lambda, \Lambda_0} (def. ) satisfy

S eff,Λ,Λ 0=(𝒮 Λ 1𝒮 Λ)(S eff,Λ,Λ 0)AAAforΛ,Λ[0,),Λ 0[0,){}. 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, following (Wilson 71).

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

Proposition

(Polchinski's flow equation)

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

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

ddΛS eff,Λ=121iddΛ(S eff,Λ F,ΛS eff,Λ)| Λ=Λ, \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 Δ F,Λ\Delta_{F,\Lambda'} (this 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 OPolyObs(E BV-BRST)[[,g,j]]O \in PolyObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ] we have

1(k+2)!ddΛ(O F,Λ F,ΛOk+2factors) =1(k+2)!ddΛ(prodexp(1i<jkΔ F,Λ,δδΦ iδδΦ j)(OOk+2factors)) =1(k+2)!(k+22)=121k!(ddΛO F,ΛO) F,ΛO F,Λ F,ΛOkfactors \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 δδΦ i\frac{\delta}{\delta \mathbf{\Phi}_i} denotes the functional derivative of the iith tensor factor of OO, 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+2k+2 tensor factors, where we use that the star product F,Λ\star_{F,\Lambda} is commutative (by symmetry of Δ F,Λ\Delta_{F,\Lambda}) and associative (by this prop.).

With this and the defining equality 𝒮 Λ(S eff,Λ)=𝒮(gS int+jA)\mathcal{S}_\Lambda(S_{eff,\Lambda}) = \mathcal{S}(g S_{int} + j A) (11) we compute as follows:

0 =ddΛ𝒮(gS int+jA) =ddΛ𝒮 Λ(S eff,Λ) =(1iddΛS eff,Λ) F,Λ𝒮 Λ(S eff,Λ)+(ddΛ𝒮 Λ)(S eff,Λ) =(1iddΛS eff,Λ) F,Λ𝒮 Λ(S eff,Λ)+12ddΛ(1iS eff,Λ F,Λ1iS eff,Λ)| Λ=Λ F,Λ𝒮 Λ(S eff,Λ) \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 () F,Λ𝒮 Λ(S eff,Λ)(-) \star_{F,\Lambda} \mathcal{S}_\Lambda( - S_{eff,\Lambda} ) (using that F,Λ\star_{F,\Lambda} is a commutative product, so that exponentials behave as usual) this yields the claimed equation.

\,

Traditional informal discussion

Traditional informal discussion of effective field theory proceeds from the following claim

For a given set of asymptotic states, perturbation theory with the most general Lagrangian containing all terms allowed by the assumed symmetries will yield the most general S-matrix elements consistent with analyticity, perturbative unitarity?, cluster decomposition and the assumed symmetries.

This is due to (Weinberg 1979) and (Leutwyler94); reviewed in Pich, p. 6.

Based on this, one argues to obtains an effective approximation to a given more fundamental theory (which may or may not be actually known) by

  1. choosing the (sub)set of fields to be considered;

  2. writing down a Lagrangian

    L eff= ic iO i L_{eff} = \sum_i c_i O_i

    that contains all the possible polynomial interaction terms O iO_i of these fields scaled by their expected/known energy scale [O i]=d i[O_i] = d_i, up to a maximal energy scale

    (this will in general contain lots of direct interaction that in the fundamental theory are really compound interactions)

    with c i1Λ d idimXc_i \propto \frac{1}{\Lambda^{d_i - dim X}};

  3. finally one fixes all the coupling constants of all these interactions by

    • either deriving them from a known fundamental theory by integrating out higher energy effects in that theory;

    • or, otherwise, measuring them in the laboratory. The point being that due to the energy cutoff, this is guaranteed to be a finite number of parameters. After these have been determined, all remaining quantities given by the Lagrangian are then predictions of the effective theory.

Examples

Light-by-light scattering

(Pich, section 2.1)

Rayleigh scattering

(Pich, section 2.2)

Fermi theory of weak interactions

(Pich, section 2.3)

Chiral perturbation theory

chiral perturbation theory? is an effective approximation of QCD in the light quark sector.

Heavy quark effective field theory

(…)

Neutrino masses (?)

On neutrino masses and the standard model of particle physics as an effective field theory:

I also noted at the same time that interactions between a pair of lepton doublets and a pair of scalar doublets can generate a neutrino mass, which is suppressed only by a factor M 1M^{-1}, and that therefore with a reasonable estimate of MM could produce observable neutrino oscillations. The subsequent confirmation of neutrino oscillations lends support to the view of the Standard Model as an effective field theory, with M somewhere in the neighborhood of 10 16GeV10^{16} GeV. (Weinberg 09, p. 15)

String theory and gravity coupled to gauge theory

The string scattering amplitudes for superstrings are finite (fully proven so for low loop order and with various plausibility arguments for higher loop order, see at string scattering amplitudes for more), hence define a UV-complete S-matrix. The corresponding low energy effective field theories are theories of supergravity coupled to gauge theory. (type II supergravity, heterotic supergravity).

See also at string theory FAQ – What is string theory?.

References

General

The modern picture of effective low-energy QFT goes back to

  • L. P. Kadanoff, Scaling laws for Ising models near T cT_c , Physica 2 (1966);

  • Kenneth Wilson, Renormalization group and critical phenomena 1. Renormalization group and the Kadanoff scaling picture , , Physical review B 4(9) (1971) (doi:10.1103/PhysRevB.4.3174)

  • Kenneth Wilson, Renormalization group and critical phenomena. 2. Phase space cell analysis of critical behavior, Phys. Rev. B4 , 3184 (1971). doi:10.1103/PhysRevB.4.3184

  • Joseph Polchinski, Renormalization and effective Lagrangians , Nuclear Phys. B B231, 1984 (pdf)

  • Steven Weinberg, Physica 96 A (1979) 327

  • H. Leutwyler, Ann. Phys., NY 235 (1994) 165.

Review includes

A standard textbook adopting this perspective is

  • Steven Weinberg, The Quantum Theory of Fields (Cambridge University

    Press,Cambridge,1995).

whose author describes his goal as:

This is intended to be a book on quantum field theory for the era of effective field theory.

Another book which takes the effective-field-theory approach to QFT is

  • Anthony Zee, Quantum Field Theory in a Nutshell (Princeton University Press, second edition, 2010).

Discussion with an eye towards condensed matter physics is in

The point that perturbatively non-renormalizable theories may be regarded as effective field theories at each energy scale was highligted in

Notably the theory of gravity based on the standard Einstein-Hilbert action may be regarded as just an effective QFT, which makes some of its notorious problems be non-problems:

and in the context of perturbation theory in AQFT:

Comments on this point are also in

See also

In causal perturbation theory

Discussion of perturbative effective QFT in the rigorous context of causal perturbation theory/perturbative AQFT and its relation to the Stückelberg-Petermann renormalization group is due to

reviewed in

See also

  • Georg Keller, Christoph Kopper, Clemens Schophaus, Perturbative Renormalization with Flow Equations in Minkowski Space, Helv.Phys.Acta 70 (1997) 247-274 (arXiv.hep-th/9605137)

For string theory

Discussion of the effective field theories induced by string theory includes the following:

Via string scattering amplitudes:

Via string field theory:

  • R. Brustein, S.P.De Alwis, Renormalization group equation and non-perturbative effects in string-field theory, Nuclear Physics B Volume 352, Issue 2, 25 March 1991, Pages 451-468 (doi:10.1016/0550-3213(91)90451-3)

  • Brustein and K. Roland, “Space-time versus world sheet renormalization group equation in string theory,” Nucl. Phys. B372, 201 (1992) (doi:10.1016/0550-3213(92)90317-5)

  • Ashoke Sen, Wilsonian Effective Action of Superstring Theory, J. High Energ. Phys. (2017) 2017: 108 (arXiv:1609.00459)

Last revised on April 14, 2018 at 07:42:11. See the history of this page for a list of all contributions to it.