effective quantum field theory

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## 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 [[fermion]]s 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-boson]]s 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 $W$-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 symmetry|crossing symmetric]] [[S-matrix]] can be described by an effective quantum field theory.

In the perspective of effective field theory notably  [[renormalizable interaction|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 interaction|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 interaction|non-renormalizable]] -- makes perfect sense as an effective field theory (see for instance the introduction in ([Donoghue](#DonoghueIntroduction)). 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 [[renormalizable interaction|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 pQFT in Causal perturbation theory](#InCausalPerturbationTheory)_

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

* _[Traditional informal arguments](#TraditionalInformalDiscussion)-

### In causal perturbation theory
 {#InCausalPerturbationTheory}

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](#BrunettiDuetschFredenhagen09), [Dütsch 10](#Duetsch10)), reviewed in [Dütsch 18, section 3.8](#Duetsch18)).

#### ("Re"-)Normalization via UV-Regularization
 {#UVRegularization}

+-- {: .num_defn #CutoffsUVForPerturbativeQFT}
###### Definition
**([[UV cutoffs]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$
be a [[gauge fixing|gauge fixed]] [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] over [[Minkowski spacetime]] $\Sigma$ (according to [this def.](S-matrix#VacuumFree)), 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)

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

   1. (symmetry)

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

      i.e.

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

1. the [[limit of a sequence|limit]] of the $\Delta_{F,\Lambda}$ as $\Lambda \to 0$ exists and is zero

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

1. the [[limit of a sequence|limit]] of the $\Delta_{F,\Lambda}$ as $\Lambda \to \infty$ exists and is the [[Feynman propagator]]:

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

=--

([Dütsch 10, section 4](#Duetsch10))

example:  relativistic momentum cutoff with $\epsilon$-regularization ([Keller-Kopper-Schophaus 97, section 6.1](#KellerKopperSchophaus97), [Dütsch 18, example 3.126](#Duetsch18))

+-- {: .num_defn #SMatrixEffective}
###### Definition
**([[effective S-matrix scheme]])**

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

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]]

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

with respect to the [[star product]] $\star_{F,\Lambda}$ induced by the $\Delta_{F,\Lambda}$ ([this def.](star+product#PropagatorStarProduct)).

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)](#Duetsch10))

+-- {: .num_prop #UVRegularization}
###### Proposition
**([[renormalization|("re"-)normalization]] via [[UV regularization]])**

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

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

Then

1. there exists a $[0,\infty)$-parameterized [[interaction vertex redefinition]] $\{\mathcal{Z}_\Lambda\}_{\Lambda \in \mathbb{R}_{\geq 0}}$ ([this def.](Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition))
such that the [[limit of a sequence|limit]] of [[effective S-matrix schemes]]  $\mathcal{S}_{\Lambda}$ (eq:EffectiveSMatrixScheme) applied to the $\mathcal{Z}_\Lambda$-[[vertex redefinition|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 ([this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering));

1. every [[S-matrix scheme]] around the given vacuum arises this way.

These $\mathcal{Z}_\Lambda$ are called _[[counterterms]]_ (remark \ref{TermCounter} 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 [[renormalization|("re"-)normalization]] of [[perturbative QFT]] ([this def.](S-matrix#ExtensionOfTimeOrderedProoductsRenormalization)).

=--

This was claimed in ([Brunetti-Dütsch-Fredenhagen 09, (75)](#BrunettiDuetschFredenhagen09)), a proof was indicated in ([Dütsch-Fredenhagen-Keller-Rejzner 14, theorem A.1](#DuetschFredenhagenKellerRejzner14)).

+-- {: .proof}
###### Proof

Let $\{p_{\rho_{k}}\}_{k \in \mathbb{N}}$ be a sequence of projection maps as in (eq:ForExtensionOfDistributionsProjectionMaps) defining an [[Epstein-Glaser renormalization|Epstein-Glaser ("re"-)normalization]] (prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal}) of [[time-ordered products]] $\{T_k\}_{k \in \mathbb{N}}$ as [[extensions of distributions]] of the $T_k$, regarded as distributions via remark \ref{TimeOrderedProductOfFixedInteraction}, by the choice $q_k^\alpha = 0$ in (eq:ExtensionOfDitstributionsPointFixedAndChoice).

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 (eq:MainTheoremPerturbativeRenormalizationInductionStep) in the proof of the [[main theorem of perturbative renormalization]] (theorem \ref{PerturbativeRenormalizationMainTheorem}).

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

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

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 (eq:CountertermsInductionAssumption) holds.

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

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

Here in the first step we inserted the causal decomposition (eq:TimeOrderedProductsAwayFromDiagonalByInduction)
of $T_{n+1}$ in terms of the $\{T_k\}_{k \leq n}$
away from the diagonal, as in the proof of prop. \ref{RenormalizationIsInductivelyExtensionToDiagonal},
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. \ref{CutoffsUVForPerturbativeQFT}).

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

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

subject to these two conditions:

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

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

Now by expanding out the left hand side of (eq:CountertermsAndCorrectionTerm) 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 (eq:CountertermsAndCorrectionTerm) 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:

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

We claim that the following choice works:

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

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 (eq:LocalityCorrection) into (eq:CountertermOrderByOrderInTermsOfCorrectionTerm)
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 [[support of a distribution|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 (eq:TimeOrderedProductsAwayFromDiagonalByInduction)
to the regularized products, we find

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

Using this we compute as follows:

$$
  \label{CorrectionTermForCountertermsVanishesAsCutoffIsRemoved}
  \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 (eq:InductionStepForCounterterms); 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 (eq:CountertermsInductionAssumption)
and the definition of [[UV cutoff]] (def. \ref{CutoffsUVForPerturbativeQFT}).

Inserting this for the first summand in (eq:LocalityCorrection) 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 $\mathcal{S} = \underset{\Lambda \to \infty }{\lim} (\mathcal{S}_\Lambda \circ \mathcal{Z}_\Lambda)$.

{#CountertermsForArbitrarySMatrixFromAnyGivenOnes} 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 \ref{PerturbativeRenormalizationMainTheorem})
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 $\widetilde{\mathcal{Z}}_\Lambda \coloneqq \mathcal{Z}_\Lambda \circ \mathcal{Z}$.

=--

+-- {: .num_remark #TermCounter}
###### Remark
**([[counterterms]])**

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

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 [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then prop. \ref{UVRegularization} 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 of a sequence|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 [this def.](Stückelberg-Petermann+renormalization+group#InteractionVertexRedefinition) 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. \ref{UVRegularization} 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 [[renormalization|("re"-)normalized]] [[perturbative QFT]], and every [[renormalization|(re"-)normalized]] [[perturbative QFT]] arises from some choice of counterterms.

=--

#### Effective quantum field theory
 {#RelativeEffectiveAction}

+-- {: .num_prop #EffectiveSmatrixSchemeInvertible}
###### Proposition
**([[effective S-matrix schemes]] are [[inverse|invertible functions]])**

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

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 [this def.](S-matrix#FormalParameters)).

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. \ref{SMatrixEffective}) [[restriction|restrict]] to
[[linear isomorphisms]] of the form

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

=--

([Dütsch 10, (4.7)](#Duetsch10))

+-- {: .proof}
###### Proof

Since each $\Delta_{F,\Lambda}$ is symmetric (def. \ref{CutoffsUVForPerturbativeQFT})
if follows by general properties of [[star products]] ([this prop.](star+product#SymmetricContribution))
just as for the genuine [[time-ordered product]] on [[regular polynomial observables]] ([this prop.](Wick+algebra#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise)) that
eeach the "effective time-ordered product" $\star_{F,\Lambda}$ is [[isomorphism|isomorphic]]
to the pointwise product $(-)\cdot (-)$ ([this def.](A+first+idea+of+quantum+field+theory#Observable))

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

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 [this equation](Wick+algebra#eq:OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism)).

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 [this def.](S-matrix#OnRegularObservablesPerturbativeSMatrix)).

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 [[conjugation|conjugates]]
to an inverse at each UV cutoff scale $\Lambda$:

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

=--

+-- {: .num_defn #EffectiveActionRelative}
###### Definition
**([[relative effective action]])**

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

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 [[adiabatic switching|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 [[composition|composite]]
of the [[effective S-matrix scheme]] $\mathcal{S}_{\Lambda_0}$ at scale $\Lambda_0$ (eq:EffectiveSMatrixScheme) and the [[inverse function]] $\mathcal{S}_\Lambda^{-1}$ of the [[effective S-matrix scheme]] at scale $\Lambda$ (via prop. \ref{EffectiveSmatrixSchemeInvertible}):

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

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

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

=--

([Dütsch 10, (5.4)](#Duetsch10))

+-- {: .num_remark #pQFTEffective}
###### Remark
**([[effective quantum field theory]])**

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

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 [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

Then def. \ref{CutoffsUVForPerturbativeQFT} and def. \ref{EffectiveActionRelative} say that
for any $\Lambda \in (0,\infty)$ the [[effective S-matrix]] (eq:EffectiveSMatrixScheme) of the
[[relative effective action]] (eq:RelativeEffectiveActionComposite) 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 observable|local]] [[interaction]]_ anymore:
By prop. \ref{EffectiveSmatrixSchemeInvertible} 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 of a sequence|limit]] as $\Lambda \to \infty$ is, via prop. \ref{UVRegularization}.

=--

+-- {: .num_prop #EffectiveActionAsRelativeEffectiveAction}
###### Proposition
**([[effective action]] is [[relative effective action]] at $\Lambda = 0$)**

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

Then the [[relative effective action]] (def. \ref{EffectiveActionRelative}) at $\Lambda = 0$ is the actual [[effective action]] ([this def.](S-matrix#InPerturbationTheoryActionEffective)) being $i \hbar$ times the [[Feynman perturbation series]] of
[[Feynman amplitudes]] $\Gamma(g S_{int} + j A)$ for [[connected graph|connected]] [[Feynman diagrams]] $\Gamma$:

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

=--

([Dütsch 18, (3.473)](#Duetsch18))

+-- {: .proof}
###### Proof

Observe that the [[effective S-matrix scheme]] at scale $\Lambda = 0$ (eq:EffectiveSMatrixScheme)
is the [[exponential series]] with respect to the pointwise product ([this def.](A+first+idea+of+quantum+field+theory#Observable))

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

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 [this prop.](S-matrix#LogarithmEffectiveAction).

=--

#### ("Re"-)Normalization via Wilsonian RG flow

The definition of the [[relative effective action]] $\mathcal{S}_{eff,\Lambda} \coloneqq \mathcal{S}_{eff,\Lambda, \infty}$ in def. \ref{EffectiveActionRelative}
invokes a choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}).
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]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ for $\Lambda_0 \lt \infty$
"flows" with the cutoff-parameters $\Lambda$ and in particular also with $\Lambda_0$ (remark \ref{GroupoidOfEFTs} 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. \ref{FlowEquationPolchinski} below),
which is the [[infinitesimal]] version of the "[[Wilsonian RG flow]]" (remark \ref{GroupoidOfEFTs}).
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 [[renormalization|("re"-)normalization freedom]] in choosing the [[UV regularization]] $\mathcal{S}_\infty$.
In this sense "[[Wilsonian RG flow]]" is a method of [[renormalization|("re"-)normalization]] of [[perturbative QFT]]
([this def.](S-matrix#ExtensionOfTimeOrderedProoductsRenormalization)).

+-- {: .num_remark #GroupoidOfEFTs}
###### Remark
**(Wilsonian [[groupoid]] of [[effective quantum field theories]])**

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

Then the [[relative effective actions]] $\mathcal{S}_{eff,\Lambda, \Lambda_0}$ (def. \ref{EffectiveActionRelative}) 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]] [[groupoid action|action]].

This is often called the _Wilsonian RG_, following ([Wilson 71](#Wilson71)).

=--

We now consider the [[infinitesimal]] version of this "flow":

+-- {: .num_prop #FlowEquationPolchinski}
###### Proposition
**([[Polchinski's flow equation]])**

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

Then for _every_ choice of [[UV regularization]] $\mathcal{S}_\infty$ (prop. \ref{UVRegularization}) the
corresponding [[relative effective actions]] $S_{eff,\Lambda}$ (def. \ref{EffectiveActionRelative}) 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 $\Delta_{F,\Lambda'}$ ([this def.](star+product#PropagatorStarProduct)).

=--

This goes back to ([Polchinski 84, (27)](#Polchinski84)). The rigorous formulation and proof is due to ([Brunetti-Dütsch-Fredenhagen 09, prop. 5.2](#BrunettiDuetschFredenhagen09), [Dütsch 10, theorem 2](#Duetsch10)).

+-- {: .proof}
###### 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 [this prop.](star+product#AssociativeAndUnitalStarProduct)).

With this and the defining equality $\mathcal{S}_\Lambda(S_{eff,\Lambda}) =  \mathcal{S}(g S_{int} + j A)$ (eq:RelativeEffectiveActionRelativeToInfinity) 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.

=--

$\,$

### Traditional informal discussion
 {#TraditionalInformalDiscussion}

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](#Weinberg79)) and ([Leutwyler94](#Leutwyler94)); reviewed in [Pich, p. 6](#Pich).

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;

1. writing down a [[Lagrangian]]

   $$
     L_{eff} = \sum_i c_i O_i
   $$

   that contains _all_ the possible polynomial interaction terms $O_i$ of these fields scaled by their expected/known energy scale $[O_i] = d_i$, 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 $c_i \propto \frac{1}{\Lambda^{d_i - dim X}}$;

1. 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](#Pich))

### Rayleigh scattering

([Pich, section 2.2](#Pich))

### Fermi theory of weak interactions
 {#FermiTheoryOfWeakInteractions}

* [[Fermi theory of beta decay]]

([Pich, section 2.3](#Pich))

### For nuclear physics

[[!include effective field theories of nuclear physics -- contents]]

### In lattice QCD

In [[lattice QCD]]: see _[[Symanzik 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 $M^{-1}$, and that therefore with a reasonable estimate of $M$ 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 $10^{16} GeV$. ([Weinberg 09, p. 15](#Weinberg09))

### String theory and gravity coupled to gauge theory
 {#StringTheoryAndGravityCoupledToGaugeTheory}

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?](string+theory+FAQ#WhatIsStringTheory)_.

## Related concepts

* [[effective action]]

* [[quasiparticle]]

* [[higher order curvature corrections]]

* [[Stückelberg-Petermann renormalization group]]

* [[threshold correction]]

* [[swampland]]

* [string theory FAQ -- What is the relationship between quantum field theory and string theory?](string+theory+FAQ#RelationshipBetweenQuantumFieldTheoryAndStringTheory)

## References

### General

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

* L. P. Kadanoff, _Scaling laws for Ising models near $T_c$_ , Physica 2 (1966);

* {#Wilson71} [[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](https://doi.org/10.1103/PhysRevB.4.3174))

* {#Wilson71b} [[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](https://doi.org/10.1103/PhysRevB.4.3184))

* {#Weinberg79} [[Steven Weinberg]], _Phenomenological Lagrangians_, Physica A: Statistical Mechanics and its Applications Volume 96, Issues 1–2, April 1979, Pages 327-340 (<a href="https://doi.org/10.1016/0378-4371(79)90223-1">doi:10.1016/0378-4371(79)90223-1</a>)

* {#Polchinski84} [[Joseph Polchinski]], _Renormalization and effective Lagrangians_ , Nuclear Phys. B B231, 1984 ([pdf](http://max2.physics.sunysb.edu/~rastelli/2016/Polchinski.pdf))

* {#Leutwyler94} H. Leutwyler, Ann. Phys., NY 235 (1994) 165.

Early history:

* [[Steven Weinberg]], _On the Development of Effective Field Theory_ ([arXiv:2101.04241](https://arxiv.org/abs/2101.04241))

Review:

* {#Weinberg09} [[Steven Weinberg]], _Effective Field Theory, Past and Future_ ([arXiv:0908.1964](http://arxiv.org/abs/0908.1964))

* {#Pich} A. Pich, _Effective Field Theory_ ([arXiv:hep-ph/9806303](http://arxiv.org/abs/hep-ph/9806303))

* {#Abdesselam13} [[Abdelmalek Abdesselam]], _QFT, RG, and all that, for mathematicians, in eleven pages_ ([arXiv:1311.4897](https://arxiv.org/abs/1311.4897))

* [[Daniel Freed]], _Lecture 5 of [[Five lectures on supersymmetry]]_

* Andrey Grozin, _Effective field theories_ ([arXiv:2001.00434](https://arxiv.org/abs/2001.00434))

* Riccardo Penco, _An Introduction to Effective Field Theories_ ([arXiv:2006.16285](https://arxiv.org/abs/2006.16285))

* C. P. Burgess, _Introduction to effective field theory_, Cambridge University Press 2020 ([arXiv:hep-th/0701053](https://arxiv.org/abs/hep-th/0701053), [ISBN:9781139048040](https://www.cambridge.org/core/books/introduction-to-effective-field-theory/A9CDB35F4AA7921E3A9CFD573EBA8B64))

A classical textbook adopting the EFT 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 for [[nuclear physics]]:

* [[Mannque Rho]], _Lectures on Effective Field Theories for Nuclei, Nuclear Matter and Dense Matter_, 2002 ([arXiv:nucl-th/0202078](https://arxiv.org/abs/nucl-th/0202078), [cds:539674](https://cds.cern.ch/record/539674))

Discussion with an eye towards [[condensed matter physics]] is in

* {#Shankar99} [[Ramamurti Shankar]], _Effective Field Theory in Condensed Matter Physics_ in _Conceptual Foundations of Quantum Field Theory_, 1999 ([arXiv:cond-mat/9703210](http://arxiv.org/abs/cond-mat/9703210))

and with an eye towards [[particle physics]] and the [[standard model of particle physics]]:

* {#PetrovBlechman16} [[Alexey Petrov]], Andrew E Blechman, _Effective Field Theories_, World Scientific 2016 ([doi:10.1142/8619](https://doi.org/10.1142/8619))

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

* {#GomisWeinberg95} J. Gomis and [[Steven Weinberg]], _Are nonrenormalizable gauge theories renormalizable?_, Nucl. Phys. B 469 (1996) 473 ([arXiv:hep-th/9510087](https://arxiv.org/abs/hep-th/9510087))

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:

* {#DonoghueIntroduction} [[John Donoghue]], _Introduction to the Effective Field Theory Description of Gravity_ ([arXiv:gr-qc/9512024](http://arxiv.org/abs/gr-qc/9512024))

* {#AtanceCortes96} Mario Atance, Jose Luis Cortes, _Effective Field Theory of pure Gravity and the Renormalization Group_ ([arXiv:hep-th/9604076](http://arxiv.org/abs/hep-th/9604076))

* [[John F. Donoghue]], *Quantum General Relativity and Effective Field Theory*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.09902](https://arxiv.org/abs/2211.09902)]

and in the context of [[perturbation theory]] in [[AQFT]]:

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Quantum gravity from the point of view of locally covariant quantum field theory_ ([arXiv:1306.1058](http://arxiv.org/abs/1306.1058))

Comments on this point are also in

* [[Jacques Distler]], blog posts

  * _[Motivation](http://golem.ph.utexas.edu/~distler/blog/archives/000639.html)_

  * _[Effective field theory and gravity](http://golem.ph.utexas.edu/~distler/blog/archives/001255.html)_

* [[Kevin Costello]], _[[Renormalization and Effective Field Theory]]_

See also

* [[Alastair Hamilton]], _On two constructions of an effective field theory_ ([arXiv:1502.05790](http://arxiv.org/abs/1502.05790))

### In causal perturbation theory
 {#ReferencesInCausalPerturbationTheory}

Discussion of [[perturbative QFT|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

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.2 of _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* {#Duetsch10} [[Michael Dütsch]], _Connection between the renormalization groups of Stückelberg-Petermann and Wilson_, Confluentes Mathematici, Vol. 4, No. 1 (2012) 12400014 ([arXiv:1012.5604](https://arxiv.org/abs/1012.5604))

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], appendix A of _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy. 55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.8 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* {#KellerKopperSchophaus97} 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](https://arxiv.org/abs/hep-th/9605137))

### For string theory

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

Via [[string scattering amplitudes]]:

* [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _String Scattering Amplitudes and Low Energy Effective Field Theory_, chapter 16 in _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics pp 585-639 Springer 2013 ([TOC pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-29497-6%2F1.pdf), [publisher page](http://www.springer.com/gp/book/9783642294969))

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 (<a href="https://doi.org/10.1016/0550-3213(91)90451-3">doi:10.1016/0550-3213(91)90451-3</a>)

* Brustein and K. Roland, “Space-time versus world sheet renormalization group equation in string theory,” Nucl. Phys. B372, 201 (1992) (<a href="https://doi.org/10.1016/0550-3213(92)90317-5">doi:10.1016/0550-3213(92)90317-5</a>)

* {#Sen16} [[Ashoke Sen]], _Wilsonian Effective Action of Superstring Theory_, J. High Energ. Phys. (2017) 2017: 108 ([arXiv:1609.00459](https://arxiv.org/abs/1609.00459))

Discussion of possible criteria for which effective field theory do _not_ arise as effective field theories of a string theory:

* [[Eran Palti]], _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))

For more see at _[[landscape of string theory vacua]]_.

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effective topological space

+-- {: .rightHandSide}
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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A kind of [[topological space]] about which one can reason "effectively", hence [[constructive mathematics|constructively]]. Used in [[computable analysis]], see at _[[computable function (analysis)]]_.

## Related concepts

* [[formal topology]] Every effective topological space $X$ defines a formal space. If $X$ is 'constructively complete', then the formal points of $X$ coincide with its effective points; see [Spreen10](#Spreen10).
* [[equilogical space]]

## References

* Dieter Spreen, _On effective topological spaces_, The Journal of Symbolic Logic, Vol. 63, No. 1, Mar., 1998  ([JSTOR](http://www.jstor.org/stable/2586596))

* {#Spreen10} Dieter Spreen, Effectively Given Spaces, Domains, and Formal Topology, 2010 [PDF](http://www.uni-siegen.de/fb6/tcs/team/spreen/public/frauenwoerth2010.pdf)

See the references at _[[computable analysis]]_.

Discussion in relation to [[equilogical spaces]] is in

* {#Bauer00} [[Andrej Bauer]], section 4.1.2 _The Realizability Approach to
Computable Analysis and Topology_, PhD thesis CMU (2000) ([pdf](http://andrej.com/thesis/thesis.pdf))

With an eye twoards application in [[computable physics]] the definition also appears as def. 2.2 in

* {#WeihrauchZhong02} [[Klaus Weihrauch]],  Ning Zhong, _Is wave propagation computable or can wave computers beat the Turing machine?_, Proc. of the London Math. Soc. (3) 85 (2002) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.95.5994))

[[!redirects effective topological spaces]]


effective topos

+-- {: .rightHandSide}
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### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The effective topos $Eff$ is an example of an [[elementary topos]] with a [[natural numbers object]] which is not a [[Grothendieck topos]] (and doesn't even have a [[geometric morphism]] to [[Set]]).

It is an environment for [[higher-order logic|higher order]] [[recursion]] theory, where, in the [[internal logic]], it is provable that every [[total function]] from [[natural numbers]] to natural numbers is [[computable function|recursive]] (furthermore, the functor $Hom(1, -)$ from the effective topos into [[Set]], the inverse image part of a geometric morphism $Set \to Eff$ preserves the natural numbers object, giving a suitable version of this result in the external logic as well).

It can be specified as the [[realizability topos]] for [[Kleene's first algebra]].

The effective topos construction alluded in the above paragraph can be performed more generally, in every topos $E$ with a natural numbers object, replacing [[Set]]. To every such topos one constructs the corresponding "external" effective topos $e E$ and the correspondence $E \mapsto e E$ extends to a functor admitting a [[full and faithful functor|fully faithful]] [[right adjoint]].  Kleene's first algebra can also be replaced by any [[partial combinatory algebra]], or even some more general types of gadgets; toposes obtained in this way are called [[realizability topos]]es.

The effective topos is the category obtained from the category of sets by first freely adjoining recursively-indexed coproducts (but being careful to preserve the empty set), and then adding quotients of (pseudo-)equivalence relations. ([RobinsonRosolini](#RobinsonRosolini)).

## Related concepts

* [[realizability]]

* [[topos of recursive sets|recursive topos]]

## References

Reviews include

* {#Vermeeren09} [[Stijn Vermeeren]], section 3 of _Realizability Toposes_, 2009 ([pdf](http://stijnvermeeren.be/download/mathematics/essay.pdf))

* J.M.E. [[Martin Hyland|Hyland]], _The effective topos_ in  A. S. Troelstra (ed.)  D. van Dalen (ed.) , _The L.E.J. Brouwer Centenary Symposium_, North-Holland  (1982)  pp. 165--216.

* [[Bart Jacobs]], Chapter 6 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

In the context of [[triposes]]:

* Andy Pitts, _The theory of triposes_, thesis, [pdf](http://www.cl.cam.ac.uk/~amp12/papers/thet/thet.pdf)

Then

* Sori Lee, [[Jaap van Oosten]], _Basic subtoposes of the effective topos_, [arxiv/1201.2571](http://arxiv.org/abs/1201.2571)

* Edmund Robinson, Giuseppe Rosolini, _Colimit completions and the effective topos_, The Journal of symbolic logic, 55, no 2 (1990) ([JSTOR](http://www.jstor.org/discover/10.2307/2274658?uid=3738736&uid=2129&uid=2134&uid=2&uid=70&uid=4&sid=47698789906927))
 {#RobinsonRosolini}

* [[Aurelio Carboni]], _Some free constructions in realizability and proof theory_, J. Pure App. Alg., Vol. 103 Issue 2 (1995), 117-233. ([web](http://www.sciencedirect.com/science/journal/00224049/103/2))

[[!redirects effective toposes]]


effects of foundations on core mathematics

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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
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#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[foundations]] of [[mathematics]] may seem to be a topic curiously disconnected from the core of mathematical practice. But in fact, the foundations of mathematics has a significant impact upon core mathematics; different [[foundational axiom|foundational rules and axioms]] included in the foundations of mathematics would result in certain [[theorems]] being [[true]] in one foundation of mathematics, [[false]] in another foundation of mathematics, and [[independent]] in a third foundation of mathematics. This page lists numerous examples where questions of foundations do affect questions and problems in core mathematics.

The term “core mathematics” is discussed in [Quinn 2012](#Quinn12).

## Examples

### Large cardinal axioms
 {#LargeCardinalAxiom}

[[large cardinal axiom|Large cardinal axioms]], traditionally not assumed in various foundations, have effects in real mathematics:

* That every [[extremally disconnected]] [[profinite set]] has a [[hypercover]] whose terms are extremally disconnected requires that beth cardinals exist. See Peter Scholze's comment [here](https://golem.ph.utexas.edu/category/2021/06/large_sets_2.html#c060032). In particular, it is provable in Zermelo-Frankel set theory, but not provable in Zermelo set theory or [[ETCS]], because replacement implies that beth cardinals exist, and while Zermelo-Frankel set theory has replacement, neither Zermelo set theory nor ETCS have replacement or beth cardinals.

### Excluded middle

* That every [[net]] in a [[compact space]] has a [[subnet]] that [[converges]] is equivalent to [[excluded middle]].

### Axiom of choice

See the nLab's [[axiom of choice#equivalents|list of statements equivalent to the axiom of choice]].

See also Wikipedia's [list of statements equivalent to the axiom of choice](https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents).

There are a number of theorems that are strictly weaker than the axiom of choice but traditionally proved using the axiom of choice; these could be assumed as [[foundational axioms]] of intermediate strength between nothing and the [[axiom of choice]]. Wikipedia has a list of them [here](https://en.wikipedia.org/wiki/Axiom_of_choice#Weaker_forms).

The [Consequences of the Axiom of Choice Project](http://consequences.emich.edu/conseq.htm) provides an interactive data base that can be used to search for implications between various (weakened) forms of the Axiom of Choice. [Choiceless grapher](http://cgraph.inters.co/) builds on this data and provides a graphical presentation.

See also Wikipedia\'s [list of statements undecidable in ZFC](http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC).

### Whitehead's theorem

In [[set theory]], [[Whitehead's theorem]] is a theorem valid for all [[infinity-groupoids]], modeled as [[Kan complexes]] or [[CW-complexes]]. However, in [[homotopy type theory]], Whitehead's theorem is not provable when regarded as a statement about types in homotopy type theory, since it admits models in non-[[hypercomplete (∞,1)-toposes]]. As a result, the truth of Whitehead's theorem is a [[foundational axiom]] that may be regarded as a "classicality" property, akin to [[excluded middle]] or the [[axiom of choice]], and is usually called *Whitehead's principle* by homotopy type theorists.

### Univalence

* Given the [[univalence axiom]] in [[homotopy type theory]], one could prove that every [[proset]] is a [[poset]].

### Whitehead problem

The following question is called the **Whitehead problem**

Every [[free abelian group]] $A$ satisfies $Ext^1(A,\mathbb{Z}) = 0$. Is, conversely, every abelian group $A$ that satisfies $Ext^1(A,\mathbb{Z}) = 0$ a free abelian group?

When formalizing [[group theory]] within [[ETCS]] as a [[set theory]], then this question is undecidable.  (It is still undecidable in [[ZFC]], equivalently in ETCS + Collection.)  In some models it is true, while other models have counterexamples. As a result, it is also undecidable in [[homotopy type theory]].

### Fermat's last theorem

It was [conjectured](http://en.wikipedia.org/wiki/Elementary_function_arithmetic#Friedman.27s_grand_conjecture) by Harvey Friedman that all theorems involving finitary objects published in _Annals of Mathematics_ (for argument's sake) can be proved in [[Elementary Function Arithmetic]] (EFA), a weak fragment of [[Peano arithmetic]]. One implication is that [[Fermat's last theorem]] is provable in PA. There is a current program of research by Angus MacIntyre to show this last fact directly. From a category theoretic point of view, [[Colin McLarty]] has shown that all of the material in [[SGA]] relies only on quite weak foundations, namely MacLane set theory ([McLarty 2020](#McLarty20)) with one [[universe]] (weaker than the theory $V_{\omega\cdot 3}$ considered in [[ZFC]]).

[McLarty 2010](#McLarty10) comments extensively on the possibility of proving Fermat's last theorem, and more generally the Modularity theorem, in PA.

## See also

* [[foundations of mathematics]]

## References

* {#McLarty10} [[Colin McLarty]], _What does it take to prove Fermat's last theorem? Grothendieck and the logic of number theory_, Bulletin of Symbolic Logic, **16** 3 (2010) 359 - 377 $[$[doi:10.2178/bsl/1286284558](https://projecteuclid.org/journals/bulletin-of-symbolic-logic/volume-16/issue-3/What-does-it-take-to-prove-Fermats-Last-Theorem-Grothendieck/10.2178/bsl/1286284558.short)$]$, [pdf](http://www.cwru.edu/artsci/phil/Proving_FLT.pdf)$]$

* {#McLarty20} [[Colin McLarty]], _The large structures of Grothendieck founded on finite-order arithmetic_, The Review of Symbolic Logic **13**  2 (2020) 296 - 325  $[$[arXiv:1102.1773v4](https://arxiv.org/abs/1102.1773v4), [doi:10.1017/S1755020319000340](https://doi.org/10.1017/S1755020319000340)$]$

* {#Quinn12} [[Frank Quinn]], *A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today*, Notices of the AMS (2012) $[$[pdf](http://www.ams.org/notices/201201/rtx120100031p.pdf)$]$

[[!redirects effects of foundations on "real" mathematics]]


Efim Fradkin

* [Wikipedia entry](https://en.wikipedia.org/wiki/Efim_Fradkin)

* [German Wikipedia entry](https://de.wikipedia.org/wiki/Jefim_Samoilowitsch_Fradkin)

* [Cern courirer obituary](https://www.tamm.lpi.ru/confs/esfconf/fradkin.html)

## Selected writings

On [[BFV-formalism]]:

* [[Igor Batalin]], [[Efim Fradkin]], _A generalized canonical formalism and quantization of reducible gauge theories_ , Phys. Lett. B122 (1983) 157-164.

On the [[DBI-action]] for [[D-branes]]:

* [[Efim Fradkin]], [[Arkady Tseytlin]], _Non-linear electrodynamics from quantized strings_, Physics Letters B Volume 163, Issues 1–4, 21 November 1985 (doi:10.1016/0370-2693(85)90205-9)

## Related $n$Lab entries

* [[BV-formalism]]

category: people


Efimov K-theory

## Idea

A generalization of [[Waldhausen K-theory]] to dualizable [[dg-categories]] and dualizable [[stable ∞-categories]].

For compactly generated inputs, recovers the [[Waldhausen K-theory]] of the full subcategory of compact objects.

The formalism is applicable to $\lambda$-presentable stable ∞-categories, where $\lambda$ can be uncountable (for example, various categories of sheaves, or categories occurring in functional analysis).

## Related concepts

* [[K-theory]], [[Waldhausen K-theory]]

## References

* [[Alexander Efimov]], _On the K-theory of large triangulated categories_, ICM 2022, <https://www.youtube.com/watch?v=RUDeLo9JTro>

* [[Marc Hoyois]], _K-theory of dualizable categories (after A. Efimov)_, <https://hoyois.app.uni-regensburg.de/papers/efimov.pdf>.


EGA

This entry is about the text

* [[Alexander Grothendieck]], _Éléments de géométrie algébrique_ ([numdam](http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short))

#Contents#
* table of contents
{:toc}

## EGA

EGA stands for _Éléments de géométrie algébrique_ (Elements of algebraic geometry), which was written by [[Alexandre Grothendieck]] and co-edited with [[Jean Dieudonne|Jean Dieudonné]]. These volumes (and a list is given below) were among his many works attempting to build the foundations for [[algebraic geometry]] in the language of [[schemes]]. They were preceded by [[FGA]] and were followed up by the SGA, more details of these can be found below.

The published part of _EGA_ is in Publ. IHÉS, now free online at [numdam](http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short) (detailed links to chapters, and their contents will be added here later, see also below for a list of the volumes in the series). We plan to list here the grand plan and some remarks and links.

### List of volumes of EGA

Note that there is a chapter 0 that continues at the beginning of each chapter, establishing preliminaries from [[topology]], [[category theory]], [[commutative algebra]], [[homological algebra]], etc.

* I. Le langage des schémas,
Inst. Hautes Études Sci. Publ. Math. 4 (1960), 5–228. Ch. 0.§1–7 ; Ch I.§1–10.

* II. Étude globale élémentaire de quelques classes de morphismes,  Inst. Hautes Études Sci. Publ. Math. 8 (1961), 5–222. Ch. II.§1–8.

* III_1. Étude cohomologique des faisceaux cohérents (Première partie)
Inst. Hautes Études Sci. Publ. Math. 11 (1961), 5–167. Ch. 0.§8–13 ; Ch.III.§1–5.

* III_2. Étude cohomologique des faisceaux cohérents (Seconde partie)  Inst. Hautes Études Sci. Publ. Math. 17 (1963), 5–91. Ch. III .§6–7.

* IV_1. Étude locale des schémas et des morphismes de schémas (Première partie)  Inst. Hautes Études Sci. Publ. Math. 20 (1964), 5–259. Ch.0.§14–23 ; Ch. IV.§1.

* IV_2. Étude locale des schémas et des morphismes de schémas (Seconde partie) Inst. Hautes Études Sci. Publ. Math. 24 (1965), 5–231. Ch. IV2.§2–7 .

* IV_3. Étude locale des schémas et des morphismes de schémas (Troisième partie) Inst. Hautes Études Sci. Publ. Math. 28 (1966), 5–255. Ch. IV.§8–15 ([numdam:PMIHES_1966__28__5_0/](http://www.numdam.org/item/PMIHES_1966__28__5_0/))

* IV_4. Étude locale des schémas et des morphismes de schémas (Quatrième partie)
Inst. Hautes Études Sci. Publ. Math. 32 (1967), 5–361. Ch.IV.§16–21 ([numdam:PMIHES_1967__32__5_0](http://www.numdam.org/item/PMIHES_1967__32__5_0))

There was a second edition of EGA I:

* I. Le langage des schémas. (2nd ed.)
Grundlehren der Mathematischen Wissenschaften (1971). 166. Springer-Verlag.

In this edition, the term "prescheme" (now outdated but used in the first edition for what we now call schemes) was replaced by "scheme", and the term "scheme" became [[separated scheme]].  Also, there was a transition to the [[functor of points]] approach to scheme theory.

### EGA V and beyond

EGA was never completed. The listed volumes I-IV are just a part of the original plan. Grothendieck outlined what was meant to be in chapters V-VII, at least, and some handwritten prenotes existed for a small part of those. See e.g. these prenotes for some parts of [EGA V](http://www.jmilne.org/math/Documents/EGA-V.pdf).

[Wikipedia](http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique) lists the titles of planned chapters I-XIII.

## The other series:

### FGA (Fondements de la Géometrie Algébrique)

 The earliest of the series is _[[FGA]]_ written alone at the end of the 1950s; it is sort of quick outline of the theory and some deep results with proofs at the beginning of the period of the development of [[scheme theory]]. A modern version of the material in [[FGA]] appeared as a book [[FGA explained]] (ICTP, Trieste 2003--2005). The original is available [here](http://www.math.jussieu.fr/~leila/grothendieckcircle/FGA.pdf).

### SGA (Séminaire de Géométrie Algébrique du Bois Marie)

(Bois Marie is the name of an area of woodland at Bures-sur-Yvette, where the [[math institutions|IHÉS]] is situated.)

 _SGA_ consists of the writeups of seminars at IHÉS lead by Grothendieck; the work of the members of the seminar under the guidance of Grothendieck are included. In $n$lab we have pages on  [[SGA1]], and [[SGA4]].

The [Wikipedia entry](http://en.wikipedia.org/wiki/S%C3%A9minaire_de_G%C3%A9om%C3%A9trie_Alg%C3%A9brique_du_Bois_Marie) lists all of the seminars.

[[Pierre Deligne]] edited a supplementary volume  _SGA $4\frac{1}{2}$_ comprisings parts of [[SGA5]], his own articles proving new results and some substantially rewritten and expanded material exposed in less satisfactory form in SGA5 (these results were used for his proof of the [[Weil conjecture | Weil conjectures]] which appeared before SGA5).

An ongoing project aims to retype the SGA volumes in TeX.
So far the following volumes have been retyped:

* [SGA 1](http://arxiv.org/abs/math/0206203) - Edited by B. Edixhoven, 2004

* [SGA 2](http://arxiv.org/abs/math/0511279) - Edited by Y. Laszlo, 2004

* [SGA 3](http://webusers.imj-prg.fr/~patrick.polo/SGA3/) - Edited by P. Gilles and P. Polo, 2011

* SGA 4 - Edited by Y. Laszlo.

  [Part 1](http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-1/sga41.pdf),

  [Part 2](http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-2/sga42.pdf),

  [Part 3](http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-3/sga43.pdf).

* [SGA $4{1\over2}$](https://github.com/dkmiller/sga4.5) - Edited by Daniel Miller.

See also:

- [SGA 4](http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/index.html) Edited by F. Orgogozo

## References

There are Wikipedia entries

* Wikipedia, [Éléments de géométrie algébrique](http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique)

* Wikipedia, [Séminaire de Géométrie Algébrique du Bois Marie](http://en.wikipedia.org/wiki/S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique)

* Wikipedia, [Fondements de la Géometrie Algébrique](http://en.wikipedia.org/wiki/Fondements_de_la_G%C3%A9ometrie_Alg%C3%A9brique)

EGA and the other published texts of Grothendieck are available from

* [[Grothendieck Circle]], [Scans of published texts](http://www.math.jussieu.fr/~leila/grothendieckcircle/pubtexts.php).

and some of the EGA and SGA links are at this [html](http://www.cmls.polytechnique.fr/perso/chambert/DJVU/index.html).

David Madore has prepared a detailed table of contents of EGA:

* [Detailed table of contents of EGA](https://math.berkeley.edu/~mhaiman/math256-fall13-spring14/EGA-contents.pdf)

A translated table of contents has been prepared by Mark Haiman, available at

* [[Mark Haiman]], [English translation of table of contents of EGA](http://math.berkeley.edu/~mhaiman/math256/EGA-contents.pdf)

along with synopsises, in English, of many sections at

* [[Mark Haiman]], [English synopsises of most of EGA I and II](https://math.berkeley.edu/~mhaiman/math256-fall18-spring19/#ega)

There is an ongoing open-source project to translate EGA into English:

* [English translation of EGA I, and partial translations of the rest](https://github.com/ryankeleti/ega)

Daniel Murfet has written notes for parts of EGA, see

* [[Daniel Murfet]], [The Rising Sea](http://therisingsea.org/?page_id=3)

Project of classification and transcription

* [SGA, EGA, FGA](https://agrothendieck.github.io/SGAEGAFGA.html) By Mateo Carmona

For an essay on content of EGA (about 50 years after the publication) see

* U. Görtz, _Classics revisited: Éléments de Géométrie Algébrique_, Jahresbericht der Deutschen Mathematiker-Vereinigung __120__, 235–290 (2018) [doi](https://doi.org/10.1365/s13291-018-0181-1)

[[!redirects SGA]]
[[!redirects Éléments de géométrie algébrique]]
[[!redirects Elements de geometrie algebrique]]
[[!redirects Elements of algebraic geometry]]

category: reference


EGA II

This is a part of [[EGA]], now freely available at numdam.org.

* [[Alexandre Grothendieck]], _Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : II. Étude globale élémentaire de quelques classes de morphismes_, Publications Mathématiques de l'IHÉS __8__ (1961), p. 5-222, [numdam](http://www.numdam.org/item?id=PMIHES_1961__8__5_0)

 1. Morphismes affines.

 2. Spectres premiers homogènes.

 3. Spectre homogène d'un faisceau d'algèbres graduées.

 4. Fibres projectifs. Faisceaux amples.

 5. Morphismes quasi-affines; morphismes quasi-projectifs; morphismes propres;
morphismes projectifs.

 6. Morphismes entiers et morphismes finis.

 7. Critères valuatifs.

 8. Schémas éclatés; cônes projetants; fermeture projective.


EGA IV

Chapter IV of [[EGA]]

*  A. Grothendieck, _Éléments de géométrie algébrique IV (rédigés avec la collaboration de Jean Dieudonné). Étude locale des schémas et des morphismes de schémas_

now can be found online:

*  Première partie. Publications Mathématiques de l'IHÉS, 20 (1964), p. 5-259 [numdam](http://www.numdam.org/item?id=PMIHES_1964__20__5_0)
*  Seconde partie. Publications Mathématiques de l'IHÉS, 24 (1965), p. 5-231 [numdam](http://www.numdam.org/item?id=PMIHES_1965__24__5_0)
*  Troisième partie. Publications Mathématiques de l'IHÉS, 28 (1966), p. 5-255 [numdam](http://www.numdam.org/item?id=PMIHES_1966__28__5_0)
*  Quatrième partie. Publications Mathématiques de l'IHÉS, 32 (1967), p. 5-361 [numdam](http://www.numdam.org/item?id=PMIHES_1967__32__5_0)

category: reference

[[!redirects EGAIV]]
[[!redirects EGA4]]


Egbert Brieskorn

* [Wikipedia entry](https://en.wikipedia.org/wiki/Egbert_Brieskorn)

* Gert-Martin Greuel, Walter Purkert: *Life and work of Egbert Brieskorn (1936 - 2013)* $[$[arXiv:1711.09600](https://arxiv.org/abs/1711.09600)$]$

## Selected writings

On [[singularities]] and introducing what came to be called the [[Brieskorn sphere]]:

* {#Brieskorn66} [[Egbert Brieskorn]], *Beispiele zur Differentialtopologie von Singularitäten*, Inventiones mathematicae **2** (1966) 1–14 $[$[doi:10.1007/BF01403388](https://doi.org/10.1007/BF01403388)$]$

On [[Gauss-Manin connections]]:

* [[Egbert Brieskorn]], *Die Monodromie der isolierten Singularitäten von Hyperflächen*, Masucripta Math. **2** (1970) 103-161  $[$[doi:10.1007/BF01155695](https://doi.org/10.1007/BF01155695), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/brieskorn7.pdf)$]$

## Related entries

* [[Brieskorn sphere]]

category: people


Egbert Rijke

* [page on HoTT site](http://homotopytypetheory.org/author/egbertrijke/)

* [github page](https://github.com/EgbertRijke)

## Selected writings

Discussion of [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]]:

* {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], *[[Higher Groups in Homotopy Type Theory]]*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 [[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)]

On [[homotopy type theory]] ([[synthetic homotopy theory]]):

* [[Egbert Rijke]], _Homotopy type theory_ (2012) ([pdf](http://hottheory.files.wordpress.com/2012/08/hott2.pdf))

* {#Rijke22} [[Egbert Rijke]], _[[Introduction to Homotopy Type Theory]]_ (2022) ([pdf] (https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf), [GitHub](https://github.com/martinescardo/HoTTEST-Summer-School/blob/main/HoTT/hott-intro.pdf))

* {#Rijke19b} [[Egbert Rijke]], _Classifying Types_ ([arXiv:1906.09435](https://arxiv.org/abs/1906.09435))

On [[modal homotopy type theory]]:

* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], _Modalities in homotopy type theory_,  Logical Methods in Computer Science, January 8, 2020, Volume 16, Issue 1 ([arXiv:1706.07526](https://arxiv.org/abs/1706.07526), [episciences:6015](https://lmcs.episciences.org/6015))

* [[Felix Cherubini]], [[Egbert Rijke]], *Modal Descent*, Mathematical Structures in Computer Science **31** 4 (2021) 363-391  [[doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201), [arXiv:2003.09713](https://arxiv.org/abs/2003.09713)]

  > (such as concerning [[homotopy n-types]]/[[n-truncation modality]] and [[covering spaces]])

* {#ChristensenRijke20} [[J. Daniel Christensen]], [[Egbert Rijke]], _Characterizations of modalities and lex modalities_ ([arXiv:2008.03538](https://arxiv.org/abs/2008.03538))

On [[localization of a space|localization]] (in [[algebraic topology]]) formulated in [[modal homotopy type theory|modal]] [[homotopy type theory]]:

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

On [[delooping]], [[H-spaces]] and [[Eilenberg-MacLane spaces]] in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

On [[epimorphisms]] and [[acyclic types]] in [[univalent type theory]]:

* [[Ulrik Buchholtz]], [[Tom de Jong]], [[Egbert Rijke]], *Epimorphisms and Acyclic Types in Univalent Mathematics*, ([arXiv:2401.14106](https://arxiv.org/abs/2401.14106))

## Related $n$Lab entries

* [[homotopy type theory]]

* [[modal type theory]]

category: people


Ehlers transformation

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## Idea

An **Ehlers transformation** is a transformation that [[action|acts]] on the parameters of [[static spacetimes]] solving the [[Einstein field equations]] and generates other solutions of the field equations that are in general [[stationary spacetimes]].

## Related entries

* [[classical double copy]]

## References

See also:

* Wikipedia, _[Ehlers group](https://en.wikipedia.org/wiki/Ehlers_group)_

In the context of [[gravity]] regarded as a _[[double copy]]_ ([[classical double copy]]) of [[Yang-Mills theory]]:

* {#ABSP19} Rashid Alawadhi, [[David Berman]], Bill Spence, David Peinador Veiga, _S-duality and the Double Copy_ ([arXiv:1911.06797](https://arxiv.org/abs/1911.06797))


EHP spectral sequence

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## Idea

The _EHP spectral sequence_ (we follow [Mahowald 85](#Mahowald85)) is the [[spectral sequence]] for computation of [[homotopy groups of spheres]] induced from the [[filtration]] of the underlying homotopy type $\Omega^\infty \Sigma^\infty S^0 = \Omega^\infty \mathbb{S}$  of the [[sphere spectrum]]  by [[suspensions]] (German: _E_inhängung):

$$
  \Omega^n S^n \stackrel{E}{\longrightarrow} \Omega^{n+1} S^{n+1}
  \,.
$$

More concretely, ([James 57](#James57)) constructed maps

$$
  \Omega S^n \stackrel{H}{\longrightarrow} \Omega S^{2n-1}
$$

(for _H_opf as in [[Hopf invariant]]) and showed that  [[p-localization|2-locally]] these fit with $E$ into [[homotopy fiber sequences]]

$$
  \Omega^{n+2} S^{2n+1}
    \stackrel{P}{\longrightarrow}
  \Omega^n S^n
    \stackrel{E}{\longrightarrow}
  \Omega^{n+1} S^{n+1}
    \stackrel{H}{\longrightarrow}
  \Omega^{n+1}S^{2n+1}
  \,.
$$

(Here $P$ is by definition the [[homotopy fiber]] of $E$, the notation refers to _[[Whitehead product]]_.)

This "EHP-[[long homotopy fiber sequence]]" gives rise to the corresponding [[long exact sequence of homotopy groups]] and so to an [[exact couple]] of the form

$$
  \array{
     \underset{s,t}{\oplus} \pi_{s+t}(\Omega^{s+1}S^{s+1})
     && \stackrel{i}{\longrightarrow} &&
     \pi_{s+t}(\Omega^{s+1}S^{s+1})
     \\
     & \nwarrow && \swarrow
     \\
     &&
     \underset{s,t}{\oplus}
     \pi_{t+s}(\Omega^{s+1}S^{2s+1})
  }
  \,.
$$

The corresponding [[spectral sequence]] is the EHP spectral sequence proper. It converges, 2-locally, to the [[stable homotopy groups of spheres]], with $E_1$-page given by

$$
  E^{k,n}_1 = \pi_{k+n}(S^{2n-1}) \;\Rightarrow\; \pi_\bullet^{\mathbb{S}}
  \,.
$$

For more general [[prime numbers]] than just 2, ([Toda 62](#Toda62)) found analogous fibrations, which hence give EHP spectral sequences for general $p$.

The EHP spectral sequence is often used used in the context of the [[Adams-Novikov spectral sequence]] for [[p-localization]] at some prime $p$.

## Constructions of $H$

For James' fiber sequence, the essential property required of $H$ is to realize the isomorphism
$$ (the\:map\:H)^{cohomology\:pullback} : H^{2n}(\Omega\mathbb{S}^{2n+1})\overset{\sim}{\to} H^{2n}(\Omega \mathbb{S}^{n+1}) .$$
The remaining corollaries then follow using the fact cohomology pullback is a ring homomorphism, and the $mod 2$ [[Leray-Serre spectral sequence]].

### Via the James model

Using the James model of $\Omega\Sigma X$ as a quotient space of $colim_n X^n $, a candidate $H$ is constructed by recursion:
$$ H([x_0 , \dots, x_{n+1}]) = H([x_0,\dots,x_n]) \# ([x_0 \wedge x_{n+1} , \dots , x_n\wedge x_{n+1}]) $$
where $\#$ denotes concatenation and $\wedge$ [[smash product]].  One checks that the ordering of product terms $ x_i\wedge x_j $ w.r.t. $x_k\wedge x_l$ depends only on the relative orders of $i,j,k,l$, so that $H$ is well-defined on the quotient space $\Omega\Sigma X \to \Omega \Sigma(X\wedge X)$.

In particular, the restriction to $J_2 X$ factors through $ X\wedge X \to \Omega\Sigma X\wedge X$ as the cofiber of the inclusion $X \to J_2 X$.  In the case $X\simeq \mathbb{S}^n$, the desired cohomology isomorphism is immediate.

### Via pushout/pullback comparisons

Starting with the three-legged cospan $ X \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} * $, construct the cube of all pushouts
$$
 \array{
   X    &    &              & \to &       &     &    * \\
    & \searrow &             &   &     & \swarrow \\
            &       &   *   & \to &   \Sigma X \\
  \downarrow & & \downarrow &     & \downarrow & & \downarrow \\
            &    & \Sigma X & \to & \Sigma X \vee \Sigma X \\
         & \nearrow  & &     &   &       \nwarrow \\
   *       &      &         & \to &       &      & \Sigma X
 }
$$
Construct pullbacks in some pair of parallel squares, and compare them by naturality
$$
 \array{
   \Omega\Sigma X &   &           & \to &           &           & * \\
        & \searrow &           &     &           & \swarrow \\
        &   & \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) & \to & \Sigma X \\
   \downarrow   &    & \downarrow &     & \downarrow &        & \downarrow \\
             &       & \Sigma X   & \to & \Sigma X \vee \Sigma X \\
             & \nearrow &         &     &               & \nwarrow \\
       *     &          &         & \to &            &          & \Sigma X
 }
$$
where $\star$ is [[join|reduced join]]. On the other hand, the natural transformations $ \Sigma\Omega \to 1$ give natural maps, e.g.
$$\Omega( ev \circ \tau \circ ev ) : \Omega((\Omega\Sigma X) \star (\Omega\Sigma X)) \to \Omega( X \star X) .$$
The composite
$$ \Omega \Sigma X \to \Omega((\Omega\Sigma X)\star(\Omega\Sigma X)) \to \Omega(X\star X) $$
is a candidate $H$.

## Related concepts

* the [[Freudenthal suspension theorem]] may be obtained from the EHP spectral sequence;

* the [[Goodwillie spectral sequence]] of the identity functor at the point also computes homotopy groups of spheres, the interplay of the two is discussed in ([Behrens 10](#Behrens10))

## References

Original articles include

* [[Ioan Mackenzie James]], _Reduced product spaces_, Ann. of Math. (2) 62 (1955), 170-197.

* {#James57} [[Ioan Mackenzie James]], _On the Suspension Sequence_, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 ([jstor](http://www.jstor.org/stable/1969666))

* {#Toda62} [[Hiroshi Toda]], _Composition methods in homotopy groups of spheres_, Princeton University Press (1962)

* {#Mahowald85} [[Mark Mahowald]], _Lin's theorem and the EHP sequence_. Conference on algebraic topology in honor of [[Peter Hilton]], Contemp. Math. 37 (1985), 115–119. Amer. Math. Soc., Providence, RI.

* [[Marcel Bökstedt]], [[Anne Marie Svane]], _A generalization of the stable EHP spectral sequence_ ([arXiv:1208.3938](http://arxiv.org/abs/1208.3938))

Relation to the [[Goodwillie spectral sequence]] is discussed in

* {#Behrens10} [[Mark Behrens]], _The Goodwillie tower and the EHP sequence_ ([arXiv:1009.1125](http://arxiv.org/abs/1009.1125))

An algebraic version of the EHP spectral sequence for the [[Lambda-algebra]] and used for computation of the second page of the [[classical Adams spectral sequence]] (the _[[Curtis algorithm]]_), is discussed in

* {#Kochmann96} [[Stanley Kochman]], around prop. 5.2.6  of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Review:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lectures 6,7 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

* [[Mark Mahowald]], [[Doug Ravenel]], section 7 of _Towards  a Global Understanding of the Homotopy Groups of Spheres_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf))

* [[Doug Ravenel]], chapter 1, section 5 of _[[Complex cobordism and stable homotopy groups of spheres]]_

* [[eom]], _[EHP spectral sequence](https://www.encyclopediaofmath.org/index.php/EHP_spectral_sequence)_

See also:

* Wikipedia, _[EHP spectral sequence](http://en.wikipedia.org/wiki/EHP_spectral_sequence)_


Ehrenfeucht-Fraïssé comonad

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## Idea

The **Ehrenfeucht-Fraïssé comonad** is a [[comonad]] introduced by [[Samson Abramsky]] and [[Nihil Shah]] in [Abramsky-Shah 2018](#AbramskyShah), to study [[Ehrenfeucht-Fraïssé games]].

If $\mathcal{R}(\sigma)$ denotes the set of $\sigma$-structures, for $\sigma$ a relational signature, there is a comonad $\mathbb{E}_k$ on $\mathcal{R}(\sigma)$, where

- The elements of $\mathbb{E}_k\mathcal{A}$ are finite nonempty sequences in $\mathcal{A}$
- CoKleisli morphisms $\mathbb{E}_k \mathcal{A} \to \mathcal{B}$ correspond to winning strateges for duplicator in the **existential** Ehrenfeucht-Fraïssé game of length $k$, played from $\mathcal{A}$ to $\mathcal{B}$, i.e the one where spoiler only plays in $\mathcal{A}$, duplicator only plays in $\mathcal{B}$, and the two substructures are only required to satisfy the same formulas from the positive existential fragment. The correspondence between strategies and coKleisli morphisms is that, given a sequence of choices by Spoiler $a_1, \dots a_n \in \mathcal{A}$, $f(a_1, \dots a_n) \in \mathcal{B}$ is the choice by Duplicator (supposed to correspond to $a_n$.

There are also a description of the "proper" Ehrenfeucht-Fraïssé game in terms of this comonad, see [Abramsky-Shah 2018](#AbramskyShah) for more details.

## Definition

Let $\mathcal{R}(\sigma)$ be as above.

- The underlying set of $\mathbb{E}_k(\mathcal{A})$, given a $\sigma$-structure $\mathcal{A}$, is, as noted above, the set of nonempty sequences $a_1, \dots a_n$ with $n \leq k$.
- The action on functions is defined in the obvious way, $\mathbb{E}_k(f)(a_1, \dots a_n) = (f(a_1), \dots f(a_n))$.
- The counit is defined by $\epsilon(a_1, \dots a_n) = a_n$
- Given a relation symbol $R \in \sigma$, $R(s_1, \dots s_m)$ if and only if
  - For each $i,j$, either $s_i$ is a prefix of $s_j$ and vice versa
  - And $R(\epsilon(s_1), \dots, \epsilon(s_m))$.
- The comultiplication $\delta: \mathbb{E}_k(\mathcal{A}) \to \mathbb{E}_k\mathbb{E}_k\mathcal{A}$ is given by $\delta(a_1, \dots a_n) = ((a_1), (a_1,a_2), \dots (a_1, \dots a_n))$, i.e by taking the sequence of prefixes of the sequence.

## References

* {#AbramskyShah} [[Samson Abramsky]] and [[Nihil Shah]],
_Relating Structure and Power: Comonadic Semantics for Computational Resources_ ([arXiv:1806.09031](https://arxiv.org/abs/1806.09031))