Wick algebra

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A _Wick algebra_ is an [[algebra of quantum observables]] of [[free fields|free]] [[quantum field theory|quantum fields]].

In the [[quantum mechanics]] of [[harmonic oscillators]] or in [[quantum field theory]] of [[free fields]] on [[Minkowski spacetime]] one encounters [[linear operators]] $\{a_k, a^\ast_k\}_{k \in K}$ that satisfy the [[canonical commutation relation]] $[a_i, a^\ast_j] = diag((c_k))_{i j}$.
Then by a _normal ordered polynomial_ or _Wick polynomial_ ([Wick 50](#Wick50)) one means a [[polynomial]], denoted $:P:$, which is obtained from a polynomial $P((a_k, a^\ast_k))$ in these operators  by ordering all "creation operators" $a_k^\ast$ to the left of all "annihiliation operators". For example focusing on a single mode $k$ we have:

$$
  \array{
    :a^\ast: = a^\ast
    \\
    :a: = a
    \\
    :a^\ast a: = a^\ast a
    \\
    :a a^\ast: = a^\ast a
    \\
    :a a^\ast a: = a^\ast a a
    \\
    etc.
  }
  \,
$$

The intuitive idea is that these operators span a [[Hilbert space]] $\mathcal{H}$ of [[quantum states]] from a [[vacuum state]] $\vert vac \rangle \in \mathcal{H}$ characterized by the condition

$$
  a_k \vert vac \rangle = 0
  \phantom{AAA}
  \text{for all} \,\, k
$$

hence (if we think of $a_k$ as acting by "removing a quantum in mode $k$") by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the [[quantum observables]] with vanishing [[vacuum expectation value]]. In [[quantum field theory]] they model [[scattering]] processes where quanta enter a reaction process (the modes corresponding to the "annihilation" operators $a_k$) and other particles come out of the reaction (the modes corresppnding to the "creation" operators $a^\ast_k$).

The product of two Wick polynomials, computed in the ambient operator algebra and
then re-expressed as a Wick polynomial, is given by computing the relevant sequence of [[commutators]] by [[Wick's lemma]], for example

$$
  {:a^\ast a:} \, {:a^\ast a:}
  =
  :a^\ast a^\ast a a: + \hbar \, :a^\ast a:
  \,,
$$

where $\hbar = [a, a^\ast]$ is the value of the [[canonical commutation relations|canonical commutator]].

The [[associative algebra]] thus obtained is hence called the _algebra of normal ordered operators_ or _Wick polynomial algebra_ or just _Wick algebra_.

This plays a central role in [[perturbative quantum field theory]], where the [[quantization]] of [[quantum observables]] of [[free fields]] is traditionally _defined_ as the corresponding Wick algebra.

But the Wick algebra in [[quantum field theory]] may also be understood more systematically from first principles of [[quantization]]. It turns out that it is [[Moyal deformation quantization]] of the canonical [[Poisson bracket]] on the [[covariant phase space]] of the [[free field]], which is the [[Peierls bracket]] modified to an [[almost Kähler structure]] by the [[2-point function]] of a [[quasi-free Hadamard state]] ([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](#DutschFredenhagen01)).

In [[free field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations]] of motion, the analog of the star product tensor ([this equation](star+product#eq:InStarProductTensorInvertingHermitianForm))

$$
  \pi \;=\; \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}
$$

is the [[Wightman propagator]] according to [this equation](Hadamard+distribution#eq:DeompositionOfHadamardPropagatorOnMinkowkski)

$$
  \Delta_H
  \;=\;
  \tfrac{i}{2}\Delta_S + H
  \,.
$$

Understood in this form the construction directly generalizes to [[quantum field theory on curved spacetimes]] ([Brunetti-Fredenhagen 95](#BrunettiFredenhagen95), [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00), [Hollands-Wald 01](#HollandsWald01)).

Finally, the shift by the [[quasi-free Hadamard state]], which is the very source of the "normal ordering", was understood as an example of the almost-Kähler version of the quantization recipe of [[Fedosov deformation quantization]] ([Collini 16](#Collini16)). For more on this see at _[[locally covariant perturbative quantum field theory]]_.

$\,$

## Definition
 {#Definition}

Traditionally the Wick algebra is regarded as an [[operator algebra]] acting on a [[Fock space]].
However, it is useful to realize the Wick algebra directly as an [[associative algebra]] [[structure]]
on the space of [[microcausal polynomial observables]]. This "abstract" Wick algebra (meaning: not [[representation|represented]]
yet on a [[Hilbert space]]) we discuss in

* _[Abstract Wick algebra](#AbstractWickAlgebra)_.

That the abstract Wick algebra indeed has a [[faithful representation]] on [[Fock space]] is _[[Wick's lemma]]_.

Similarly there is the

* _[Abstract time-ordered product](#AbstractTimeOrderedProduct)_

The traditional notation for the [[operator products]] on the [[Fock space]] may be carried across the [[representation]] map to the abstract Wick algebra:

* _[Operator product notation](#OperatorProductAndNormalOrderedProduct)_

The abstract Wick algebra carries a canonical [[state on a star-algebra]], whose [[2-point function]] is just the
[[Wightman propagator]] that the abstract Wick algebra structure is constructed from. This we discuss in

* _[Hadamard vacuum states](#HadamardVacuumStatesOnWickAlgebras)_.

The only non-trivial part of the proof of the [[state on a star-algebra|state]]-property (prop. \ref{WickAlgebraCanonicalState}) below is
positivity. This however is immediate from the [[representation]] on the [[Fock space]], observing that
under this identification the state is represented by the [[inner product]] of the [[Hilbert space]] ([Dütsch 18, remark 2.20](#Duetsch18)).
From the point of view of [[algebraic quantum field theory]], the [[Hilbert space]]-structure mainly just serves
as a technical tool for establishing this positivity property.

### Abstract Wick algebra
 {#AbstractWickAlgebra}

The abstract [[Wick algebra]] of a [[free field theory]] with [[Green hyperbolic differential equation]] is directly analogous to the [[star product]]-algebra induced by a [[finite dimensional vector space|finite dimensional]] [[Kähler vector space]] ([this def.](star+product#WickAlgebraOfAlmostKaehlerVectorSpace))  under the following identification of the [[Wightman propagator]] with the [[Kähler space]]-[[structure]]:

+-- {: .num_remark #WightmanPropagatorAsKaehlerVectorSpaceStructure}
###### Remark
**([[Wightman propagator]] as [[Kähler vector space]]-[[structure]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[Green hyperbolic differential equation]]. Then the corresponding [[Wightman propagator]] is analogous to the rank-2 tensor on a [[Kähler vector space]] as follows:

| [[covariant phase space]]
 of [[free field theory|free]] [[Green hyperbolic differential equation|Green hyperbolic]]
 [[Lagrangian field theory]] | [[finite dimensional vector space|finite dimensional]]
 [[Kähler vector space]] |
|-------------------|-------------------------|
| [[space of field histories]]
  $\Gamma_\Sigma(E)$ |  $\mathbb{R}^{2n}$ |
| [[symplectic form]]
 $\tau_{\Sigma_p} \Omega_{BFV}$ | [[Kähler form]] $\omega$ |
| [[causal propagator]] $\Delta$ | $\omega^{-1}$ |
| [[Peierls-Poisson bracket]]
 $\{A_1,A_2\} = \int \Delta^{a_1 a_2}(x_1,x_2) \frac{\delta A_1}{\delta \mathbf{\Phi}^{a_1}(x_1)} \frac{\delta A_2}{\delta \mathbf{\Phi}^{a_2}(x_2)} dvol_\Sigma(x)$ | [[Poisson bracket]] |
| [[Wightman propagator]]
 $\Delta_H = \tfrac{i}{2} \Delta + H$  | [[Hermitian form]]
  $\pi = \tfrac{i}{2}\omega^{-1} + \tfrac{1}{2}g^{-1}$ |

=--

([Fredenhagen-Rejzner 15, section 3.6](pAQFT#FredenhagenRejzner15), [Collini 16, table 2.1](pAQFT#Collini16))

+-- {: .num_defn #MicrocausalObservable}
###### Definition
**([[microcausal observable|microcausal]] [[polynomial observables]])**
Let $E \overset{fb}{\to}$ be [[field bundle]] which is a [[vector bundle]]. An [[off-shell]] _[[polynomial observable]]_ is a [[smooth function]]

$$
  A
  \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \mathbb{C}
$$

on the [[on-shell]] [[space of sections]] of the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

$$
  \begin{aligned}
  A(\Phi)
  & =
  \phantom{+} \alpha^{(0)}
  \\
  & \phantom{=}
  +
  \int_\Sigma
    \alpha^{(1)}_a(x) \Phi^a(x)
   \,
  dvol_\Sigma(x)
  \\
  & \phantom{=}
  +
  \int_\Sigma \int_\Sigma
   \alpha^{(2)}_{a_1 a_2}(x_1, x_2)
   \Phi^{a_1}(x_1) \Phi^{a_2}(x_2)
  \,dvol_\Sigma(x_1)
  \, dvol_\Sigma(x_2)
  \\
  & \phantom{=}
  +
  \cdots
  \,,
  \end{aligned}
$$

where

$$
  \alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)
$$

is a [[compactly supported distribution]] [[distribution of two variables|of k variables]] on the $k$-fold graded-symmetric [[external tensor product of vector bundles]] of the [[field bundle]] with itself.

Write

$$
  PolyObs(E) \hookrightarrow Obs(E)
$$

for the [[subspace]] of off-shell polynomial observables onside all off-shell [[observables]].

Let moreover $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] whose [[equations of motion]] are [[Green hyperbolic differential equations]]. Then an _[[on-shell]] polynomial observable_ is the [[restriction]] of an off-shell polynomial observable along the inclusion of the [[on-shell]] [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

$$
  PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})
$$

for the subspace of all on-shell polynomial observables inside all on-shell [[observables]].

By [this prop.](Green+hyperbolic+partial+differential+equation#DistributionsOnSolutionSpaceAreTheGeneralizedPDESolutions) restriction yields an [[isomorphism]] between polynomial on-shell observables and polynomial off-shell observables modulo the image of the [[differential operator]] $P$:

$$
  PolyObs(E,\mathbf{L})
    \underoverset{\simeq}{\text{restriction}}{\longleftarrow}
  PolyObs(E)/im(P)
  \,.
$$

Finally a polynomial observable is a _[[microcausal observable]]_ if each [[coefficient]] $\alpha^{(k)}$ as above has [[wave front set]] away from those points where the $k$ [[wave vectors]] are all in the [[future cone]] or all in the [[past cone]]. We write

$$
  \array{
    PolyObs(E)_{mc}
     &\hookrightarrow&
    PolyObs(E)
    \\
    PolyObs(E,\mathbf{L})_{mc}
    \simeq
    PolyObs(E)_{mc}/im(P)
     &\hookrightarrow&
    PolyObs(E,\mathbf{L})
  }
$$

for the [[subspace]] of [[off-shell]]/[[on-shell]] [[microcausal observables]] inside all [[off-shell]]/[[on-shell]] [[polynomial observables]].

=--

+-- {: .num_prop #MoyalStarProductOnMicrocausal}
###### Proposition
**([[Hadamard distribution|Hadamard]]-[[Moyal star product]] on [[microcausal observables]] -- [[abstract Wick algebra]])**

Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$. Write $\Delta$ for the [[causal propagator]] and let

$$
  \Delta_H
  \;=\;
  \tfrac{i}{2}\Delta + H
$$

be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).

Then the [[star product]] induced by $\Delta_H$

$$
  A \star_H A
  \;\coloneqq\;
  prod
   \circ
  \exp\left(
    \int_{X^2} \hbar \Delta_H^{a b}(x_1, x_2) \frac{\delta}{\delta \mathbf{\Phi}^a(x_1)}
    \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(x_2)} dvol_g
  \right)
  (P_1 \otimes P_2)
$$

on [[off-shell]] [[microcausal observables]] $A_1, A_2 \in \mathcal{F}_{mc}$ (def. \ref{MicrocausalObservable}) is well defined in that the [[wave front sets]] involved in the [[products of distributions]] that appear in expanding out the [[exponential]] satisfy [[Hörmander's criterion]].

Hence by the general properties of [[star products]] ([this prop.](star+product#AssociativeAndUnitalStarProduct)) this yields a [[unital algebra|unital]] [[associative algebra]] [[structure]] on the space of [[formal power series]] in $\hbar$ of [[off-shell]] [[microcausal observables]]

$$
  \left(
    PolyObs(E)_{mc}[ [\hbar] ] \,,\, \star_H
  \right)
  \,.
$$

This is the _[[off-shell]] [[Wick algebra]]_ corresponding to the choice of [[Wightman propagator]] $H$.

Moreover the image of $P$ is an ideal with respect to this algebra structure, so that it descends to the [[on-shell]] [[microcausal observables]] to yield the _[[on-shell]] [[Wick algebra]]_

$$
  \left(
    PolyObs(E,\mathbf{L})_{mc}[ [ \hbar ] ] \,,\, \star_H
  \right)
  \,.
$$

Finally, under [[complex conjugation]] $(-)^\ast$ these are [[star algebras]] in that

$$
  \left(
    A_1 \star_H A_2
  \right)^\ast
  =
  A_2^\ast \star_H A_1^\ast
  \,.
$$

=--

([Dito 90](#Dito90), [Dütsch-Fredenhagen 00](#DuetschFredenhagen00) [Dütsch-Fredenhagen 01](#DuetschFredenhagen01), [Hirshfeld-Henselder 02](#HirschfeldHenselder02), see [Collini 16, p. 25-26](#Collini16))

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

By definition of the [[Wightman propagator]] (or else by [this prop.](Hadamard+distribution#WaveFronSetsForKGPropagatorsOnMinkowski)), the [[wave front set]] of powers of $\Delta_H$ has all cotangent [[wave vectors]] on the first variables in the [[closed future cone]] at the given base point (which itself is on the [[light cone]])

and hence all those on the second variables in the [[closed past cone]].

The first variables are integrated against those of $A_1$ and the second against $A_2$. By definition of [[microcausal observables]] (def. \ref{MicrocausalObservable}), the wave front sets of $A_1$ and $A_2$ are disjoint from the subsets where all components are in the [[closed future cone]] or all components are in the [[closed past cone]]. Therefore the relevant sum of of the wave front covectors never vanishes and hence [[Hörmander's criterion]] for partial [[products of distributions|products of]] [[distributions of several variables]] ([this prop.](product+of+distributions#PartialProductOfDistributionsOfSeveralVariables)) is met and the star product is well defined.

It remains to see that the star product $A_1 \star_H A_2$ is itself again a [[microcausal observable]]. It is clear that it is again a [[polynomial observable]] and that it respects the ideal generated by the equations of motion. That it still satisfies the condition on the [[wave front set]] follows directly from the fact that the wave front set of a [[product of distributions]] is inside the fiberwise sum of elements of the factor wave front sets ([this prop.](product+of+distributions#WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets), []()).

Finally the [[star algebra]]-structure follows via remark \ref{WightmanPropagatorAsKaehlerVectorSpaceStructure} as in [this prop.](star+product#StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra).

=--

+-- {: .num_remark #WickAlgebraIsFormalDeformationQuantization}
###### Remark
**([[Wick algebra]] is [[formal deformation quantization]] of [[Poisson-Peierls bracket|Poisson-Peierls algebra of observables]])**

Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] $P \Phi = 0$ with [[causal propagator]] $\Delta$ and let $\Delta_H \;=\; \tfrac{i}{2}\Delta + H$  be a corresponding [[Wightman propagator]] ([[Hadamard 2-point function]]).

Then the [[Wick algebra]]  $\left( PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ] \,,\, \star_H \right)$ from prop. \ref{MoyalStarProductOnMicrocausal} is a [[formal deformation quantization]] of the [[Poisson algebra]] on the [[covariant phase space]] given by the [[on-shell]] [[polynomial observables]] equipped with the [[Poisson-Peierls bracket]] $\{-,-\} \;\colon\; PolyObs(E,\mathbf{L})_{mc} \otimes PolyObs(E,\mathbf{L})_{mc}  \to PolyObs(E,\mathbf{L})_{mc}$ in that for all $A_1, A_2 \in PolyObs(E,\mathbf{L})_{mc}$ we have

$$
  A_1 \star_H A_2
  \;=\;
  A_1 \cdot A_2
  \;mod\;
  \hbar
$$

and

$$
  A_1 \star_H A_2
  -
  A_2 \star_H A_1
  \;=\;
  i \hbar \{A_1, A_2\}
  \;mod\;
  \hbar^2
  \,.
$$

=--

([Dito 90](#Dito90), [Dütsch-Fredenhagen 01](#DutschFredenhagen01))

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

By prop. \ref{MoyalStarProductOnMicrocausal} this is immediate from the general properties of the [[star product]] ([this example](A+first+idea+of+quantum+field+theory+--+Quantization#MoyalStarProductIsFormalDeformationQuantization)).

Explicitly, consider, without restriction of generality, $A_1 = \int (\alpha_1)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ and $A_2 = \int (\alpha_2)_a(x) \mathbf{\Phi}^a(x)\, dvol_\Sigma(x)$ be two linear observables. Then

$$
  \begin{aligned}
    A_1 \star_H A_2
    & =
    A_1 A_2
    +
    \hbar
    \int
    \left(
      \tfrac{i}{2} \Delta^{a_1 a_2}(x_1, x_2) + H^{a_1 a_2}(x_1,x_2)
     \right)
      \frac{\partial A_1}{\partial \mathbf{\Phi}^{a_1}(x_1)}
      \frac{\partial A_2}{\partial \mathbf{\Phi}^{a_2}(x_2)}
    \;mod\;
    \hbar^2
   \\
   & =
   A_1 A_2
   +
   \hbar
   \left(
    \int
      (\alpha_1)_{a_1}(x_1)
      \left(
       \tfrac{i}{2}\Delta^{a_1 a_2}(x_1, x_2)
       + H^{a_1 a_2}(x_1, x_2)
      \right)
      (\alpha_2)_{a_2}(x_2)
   \right)
   \;mod\;
   \hbar^2
  \end{aligned}
$$

Now since $\Delta$ is skew-symmetric while $H$ is symmetric is follows that

$$
  \begin{aligned}
    A_1 \star_H A_2
    -
    A_2 \star_H A_1
    & =
    i \hbar
    \left(
     \int
       (\alpha_1)_{a_1}(x_1)
       \Delta^{a_1 a_2}(x_1, x_2)
       (\alpha_2)_{a_2}(x_2)
    \right)
    \;mod\;
    \hbar^2
    \\
    & =
    i \hbar \, \left\{ A_1, A_2\right\}
  \end{aligned}
  \,.
$$

The right hand side is the [[integral kernel]]-expression for the [[Poisson-Peierls bracket]], as shown in the second line.

=--

### Abstract time-ordered product
 {#AbstractTimeOrderedProduct}

+-- {: .num_defn #OnRegularPolynomialObservablesTimeOrderedProduct}
###### Definition
**([[time-ordered product]] on [[regular polynomial observables]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the _[[time-ordered product]]_ on the space of [[off-shell]] [[regular polynomial observable]] $PolyObs(E)_{reg}$ is the [[star product]] induced by the [[Feynman propagator]] (via [this prop.](star+product#PropagatorStarProduct)):

$$
  \array{
    PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    (A_1, A_2)
    &\mapsto&
    \phantom{\coloneqq} A_1 \star_F A_2
  }
$$

hence

$$
  A_1 \star_F A_2
  \;
  \coloneqq
  \;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \underset{\Sigma \times \Sigma}{\int}
    \Delta_F^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
    \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
    \,
    dvol_\Sigma(x)
    \,
    dvol_\Sigma(y)
  \right)
$$

(Notice that this does not descend to the [[on-shell]] observables, since the [[Feynman propagator]] is not a solution to the _homogeneous_ [[equations of motion]].)

=--

+-- {: .num_prop #CausalOrderingTimeOrderedProductOnRegular}
###### Proposition
**([[time-ordered product]] is indeed causally ordered [[Wick algebra]] product)**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]]
over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] and with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[differential equations]]; write $\Delta_S = \Delta_+ - \Delta_-$ for the induced [[causal propagator]]. Let moreover $\Delta_H = \tfrac{i}{2}\Delta_S + H $ be a compatible [[Wightman propagator]] and write $\Delta_F = \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$ for the induced [[Feynman propagator]].

Then the [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is indeed a time-ordering of the [[Wick algebra]] product $\star_H$ in that for all [[pairs]] of [[regular polynomial observables]]

$$
  A_1, A_2 \in PolyObs(E)_{reg}[ [\hbar] ]
$$

with [[disjoint subset|disjoint]] [[spacetime]] [[support]] we have

$$
  A_1 \star_F A_2
  \;=\;
  \left\{
    \array{
      A_1 \star_H A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
      \\
      A_2 \star_H A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
    }
  \right.
  \,.
$$

Here $S_1 {\vee\!\!\!\wedge} S_2$ is the [[causal order]] relation ("$S_1$ does not intersect the [[past cone]] of $S_2$"). Beware that for general [[pairs]] $(S_1, S-2)$ of subsets neither $S_1 {\vee\!\!\!\wedge} S_2$ nor $S_2 {\vee\!\!\!\wedge} S_1$.

=--

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

Recall the following facts:

1. the [[advanced and retarded propagators]] $\Delta_{\pm}$ by definition are [[support|supported]] in the [[future cone]]/[[past cone]], respectively

   $$
     supp(\Delta_{\pm}) \subset \overline{V}^{\pm}
   $$

1. they turn into each other under exchange of their arguments ([this cor.](causal+propagator#CausalPropagatorIsSkewSymmetric)):

   $$
     \Delta_\pm(y,x) = \Delta_{\mp}(x,y)
     \,.
   $$

1. the real part $H$ of the [[Feynman propagator]], which by definition is the real part of the [[Wightman propagator]] is symmetric (by definition or else by [this prop.](Hadamard+distribution#SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)):

   $$
     H(x,y) = H(y,x)
   $$

Using this we compute as follows:

$$
  \label{CausallyOrderedWickProductViaFeynmanPropagator}
  \begin{aligned}
    A_1 \underset{\Delta_{F}}{\star} A_2
    \;
    & =
    A_1 \underset{\tfrac{i}{2}(\Delta_+ + \Delta_-) + H}{\star} A_2
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_1 \underset{\tfrac{i}{2}\Delta_- + H}{\star} A_2 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}\Delta_+ + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\tfrac{i}{2}(\Delta_+ - \Delta_-) + H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
    \\
    & =
    \left\{
      \array{
        A_1 \underset{\Delta_H}{\star} A_2 &\vert& supp(A_1) {\vee\!\!\!\wedge} supp(A_2)
       \\
       A_2 \underset{\Delta_H}{\star} A_1 &\vert& supp(A_2) {\vee\!\!\!\wedge} supp(A_2)
      }
    \right.
  \end{aligned}
$$

=--

+-- {: .num_prop #IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise}
###### Proposition
**([[time-ordered product]] on [[regular polynomial observables]] [[isomorphism|isomorphic]] to pointwise product)

The [[time-ordered product]] on [[regular polynomial observables]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is [[isomorphism|isomorphic]] to the pointwise product of [[observables]] ([this def.](A+first+idea+of+quantum+field+theory#Observable)) via the [[linear isomorphism]]

$$
  \mathcal{T}
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
$$

given by

$$
  \label{OnRegularPolynomialObservablesPointwiseTimeOrderedIsomorphism}
  \mathcal{T}A
  \;\coloneqq\;
  \exp\left(
    \tfrac{1}{2}
    \hbar
    \underset{\Sigma}{\int}
     \Delta_F(x,y)^{a b}
     \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
  A
$$

in that

$$
  \begin{aligned}
    T(A_1 A_2)
    & \coloneqq
    A_1 \star_{F} A_2
    \\
    & =
    \mathcal{T}( \mathcal{T}^{-1}(A_1) \cdot \mathcal{T}^{-1}(A_2) )
  \end{aligned}
$$

hence

$$
  \array{
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-)\cdot (-)}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T} \otimes \mathcal{T}}}_\simeq\Big\downarrow
      &&
    \downarrow^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E)_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E)_{reg}[ [\hbar] ]
      &\overset{(-) \star_F (-)}{\longrightarrow}&
    PolyObs(E)_{reg}[ [\hbar] ]
  }
$$

=--

([Brunetti-Dütsch-Fredenhagen 09, (12)-(13)](time-ordered+product#BrunettiDuetschFredenhagen09), [Fredenhagen-Rejzner 11b, (14)](time-ordered+product#FredenhagenRejzner11b))

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

Since the [[Feynman propagator]] is symmetric ([this prop.](A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator)), the statement is a special case of [this prop.](star+product#SymmetricContribution)).

=--

+-- {: .num_remark }
###### Remark
**([[renormalization]] of [[time-ordered product]])**

The [[time-ordered product]] on [[regular polynomial observables]] from prop. \ref{OnRegularPolynomialObservablesTimeOrderedProduct} extends to a product on [[polynomial observable|polynomial]] [[local observables]], then taking values in [[microcausal observables]]:

$$
  T
  \;\colon\;
  PolyLocObs(E)^{\otimes_n}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{mc}[ [\hbar] ]
  \,.
$$

This extension is not unique. A choice of such an extension, satisfying some evident compatibility conditions, is a choice of _[[renormalization scheme]]_ for the given [[perturbative quantum field theory]]. Every such choice corresponds to a choice of [[perturbative S-matrix]] for the theory. This construction is called _[[causal perturbation theory]]_.

=--

### Operator product notation
 {#OperatorProductAndNormalOrderedProduct}

+-- {: .num_defn #NormalOrderedProductNotation}
###### Definition
**(notation for [[operator product]] and [[normal-ordered product]])**

It is traditional to use the following alternative notation for the product structures on [[microcausal polynomial observables]]:

1. The [[Wick algebra]]-product, hence the [[star product]] $\star_H$ for the [[Wightman propagator]] (def. \ref{MoyalStarProductOnMicrocausal}), is rewritten as plain juxtaposition:

   $$
     \text{"operator product"}
     \phantom{AAA}
     A_1 A_2 \phantom{AA} \coloneqq \phantom{AA} A_1 \star_H A_1
     \phantom{AAAA}
     \array{
       \text{star product of}
       \\
       \text{Wightman propagator}
     }
     \,.
   $$

1. The pointwise product of observables ([this def.](A+first+idea+of+quantum+field+theory#Observable))  $A_1 \cdot A_2$
   is equivalently written as plain juxtaposition enclosed by colons:

   $$
     \array{
       \text{"normal-ordered}
       \\
       \text{product"}
     }
     \phantom{AAAA}
     :A_1 A_2:
      \phantom{AA}\coloneqq\phantom{AA}
     A_1 \cdot A_2
     \phantom{AAAA} \phantom{AAa}\text{pointwise product}\phantom{AAa}
   $$

1. The [[time-ordered product]], hence the [[star product]] for the [[Feynman propagator]] $\star_F$ (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) is equivalently written as plain juxtaposition prefixed by a "$T$"

   $$
     \array{
       \text{"time-ordered}
       \\
       \text{product"}
     }
     \phantom{AAAA}
     T(A_1 A_2)
      \phantom{AA}\coloneqq\phantom{AA}
     A_1 \star_F A_2
     \phantom{AAAA}
     \array{
       \text{star product of}
       \\
       \text{Feynman propagator}
     }
   $$

Under [[representation]] of the [[Wick algebra]] on a [[Fock space|Fock]] [[Hilbert space]] by [[linear operators]]
the first product become the _[[operator product]]_, while the second becomes the operator poduct applied after
suitable re-ordering, called "[[normal-ordered product|normal odering]]" of the factors.

Disregarding the [[Fock space]]-representation, which is [[faithful representation|faithful]], we may still
refer to these "abstract" products as the "operator product" and the "normal-ordered product", respectively.

=--

[[!include Wick algebra -- table]]

### Hadamard vacuum states
 {#HadamardVacuumStatesOnWickAlgebras}

+-- {: .num_prop #WickAlgebraCanonicalState}
###### Proposition
**(canonical [[Hadamard vacuum state|vacuum]] [[state on a star-algebra|states]] on abstract [[Wick algebra]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].

For

$$
  \Phi_0 \in \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
$$

any [[on-shell]] [[field history]] (i.e. solving the [[equations of motion]]), consider the function from the [[Wick algebra]] to [[formal power series]] in $\hbar$ with [[coefficients]] in the [[complex numbers]]
which evaluates any [[microcausal polynomial observable]] on $\Phi_0$

$$
  \array{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_{\Phi_0}}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    A &\mapsto& A(\Phi_0)
  }
$$

Specifically for $\Phi_0 = 0$ (which is a solution of the [[equations of motion]] by the assumption that $(E,\mathbf{L})$ defines a [[free field theory]]) this is the function

$$
  \array{
    PolyObs(E,\mathbf{L})_{mc}[ [[\hbar] ]
    &\overset{\langle -\rangle_0}{\longrightarrow}&
    \mathbb{C}[ [\hbar] ]
    \\
    \left.
    \begin{aligned}
      A & = \alpha^{(0)}
      \\ & \phantom{=} + \underset{\Sigma}{\int} \alpha^{(1)}_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)
      \\
      & \phantom{=} + \cdots
    \end{aligned}
    \right\}
    &\mapsto&
    A(0) = \alpha^{(0)}
  }
$$

which sends each [[microcausal polynomial observable]] to its value $A(\Phi = 0)$ on the zero [[field history]], hence to the constant contribution $\alpha^{(0)}$ in its [[polynomial]] expansion.

The function $\langle -\rangle_0$ is

1. [[linear function|linear]] over $\mathbb{C}[ [\hbar] ]$;

1. real, in that for all $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$

   $$
     \langle A^\ast \rangle = \langle A \rangle^\ast
   $$

1. positive, in that for every $A \in PolyObs(E,\mathbf{L})_{mc}[ [\hbar] ]$ there exist a $c_A \in \mathbb{C}[ [\hbar] ]$ such that

   $$
     \langle A^\ast \star_H A\rangle_{\Phi_0} = c_A^\ast \cdot c_A
     \,,
   $$

1. normalized, in that

   $$
     \langle 1\rangle_H = 1
   $$

where $(-)^\ast$ denotes componet-wise [[complex conjugation]].

This means that $\langle -\rangle_{0}$ is a [[state on a star-algebra|states]] on the [[Wick algebra|Wick]] [[star-algebra]] $\left( (PolyObs(E,\mathbf{L}))_{mc}[ [\hbar] ], \star_H\right)$ (prop. \ref{MoyalStarProductOnMicrocausal}). One says that

* $\langle - \rangle_0$ is a _[[Hadamard vacuum state]]_;

and generally

* $\langle - \rangle_{\Phi_0}$ is called a _[[coherent state]]_.

=--

([Dütsch 18, def. 2.12, remark 2.20, def. 5.28, exercise 5.30 and equations (5.178)](#Duetsch18))

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

The properties of linearity, reality and normalization are obvious, what requires proof is positivity. This is proven by exhibiting a [[representation]] of the Wick algebra on a [[Fock space|Fock]] [[Hilbert space]] (this algebra [[homomorphism]] is _[[Wick's lemma]]_), with formal powers in $\hbar$ suitably taken care of, and showing that under this representation the function $\langle -\rangle_0$ is represented, degreewise in $\hbar$, by the [[inner product]] of the [[Hilbert space]].

=--

+-- {: .num_example #HadamardMoyalStarProductOfTwoLinearObservables}
###### Example
**([[operator product]] of two [[linear observables]])**

Let

$$
  A_i \in LinObs(E,\mathbf{L})_{mc} \hookrightarrow PolyObs(E,\mathbf{L})_{mc}
$$

for $i \in \{1,2\}$ be two [[linear observable|linear]] [[microcausal observables]] represented by [[distributions]] which in [[generalized function]]-notation are given by

$$
  A_i
  \;=\;
  \int (\alpha_i)_{a_i}(x_i) \mathbf{\Phi}^{a_i}(x_i) \, dvol_\Sigma(x_i)
  \,.
$$

Then their Hadamard-Moyal [[star product]] (prop. \ref{MoyalStarProductOnMicrocausal}) is the [[sum]] of their pointwise product
with their value

$$
  \label{EvaluatingLinearObservablesInWightmanPropagator}
  \langle A_1 \star_H A_2 \rangle_0
  \;\coloneqq\;
  i \hbar \int \int
   (\alpha_1)_{a_1}(x_1) \Delta_H^{a_1 a_2}(x_1,x_2) (\alpha_2)_{a_2}(x_2)
   \,dvol_\Sigma(x_1)
   \,dvol_\Sigma(x_2)
$$

in the [[Wightman propagator]], which is the value of the [[Hadamard vacuum state]] from prop. \ref{WickAlgebraCanonicalState}

$$
  A_1 \star_H A_2
  \;=\;
  A_1 \cdot A_2
  \;+\;
  \langle A_1 \star_H A_2 \rangle_0
$$

In the [[operator product]]/[[normal-ordered product]]-notation of def. \ref{NormalOrderedProductNotation} this reads

$$
  A_1 A_2 \;=\; :A_1 A_2: \;+\; \langle A_1 A_2\rangle
  \,.
$$

=--

+-- {: .num_example #WeylRelations}
###### Example
**([[Weyl relations]])**

Let $(E,\mathbf{L})$ a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]] and with [[Wightman propagator]] $\Delta_H$.

Then for

$$
  A_1, A_2
  \;\in\;
  LinObs(E,\mathbf{L})_{mc}
  \hookrightarrow
  PolyObs(E,\mathbf{L})_{mc}
$$

two [[linear observables|linear]] [[microcausal observables]], the Hadamard-Moyal star product (def. \ref{MoyalStarProductOnMicrocausal}) of their [[exponentials]] exhibits the [[Weyl relations]]:

$$
  e^{A_1}
  \star_H
  e^{A_2}
  \;=\;
  e^{A_1 + A_2} \; e^{\langle A_1 \star_H A_2\rangle_0}
$$

where on the right we have the [[exponential]] of the value of the [[Hadamard vacuum state]] (prop. \ref{WickAlgebraCanonicalState}) as in example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

=--

(e.g. [Dütsch 18, exercise 2.3](#Duetsch18))

+-- {: .num_example }
###### Example
**([[Wightman propagator]] is [[2-point function]] in the [[Hadamard vacuum state]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]].

With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Wightman propagator]] $\Delta_H(x,y)$ itself is the _[[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the operator product of two [[field observables]]:

$$
  \left\langle
    \mathbf{\Phi}^a(x) \star_H \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_H(x,y) \delta(y-y')
    \right\rangle
  }}
$$

by example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:

$$
  \left\langle
    \mathbf{\Phi}^a(x)
    \mathbf{\Phi}^b(y)
  \right\rangle_0
  \;=\;
  \hbar \Delta_H(x,y)
  \,.
$$

=--

Similarly:

+-- {: .num_example }
###### Example
**([[Feynman propagator]] is time-ordered [[2-point function]] in the [[Hadamard vacuum state]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[Green hyperbolic differential equation|Green-hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]; and let $\Delta_H$ be a compatible [[Wightman propagator]] with induced [[Feynman propagator]] $\Delta_F$.

With respect to the induced [[Hadamard vacuum state]] $\langle - \rangle_0$ from prop. \ref{WickAlgebraCanonicalState}, the [[Feynman propagator]] $\Delta_F(x,y)$ itself is the _time-ordered [[2-point function]]_, namely the [[distribution|distributional]] [[vacuum expectation value]] of the [[time-ordered product]] (def. \ref{OnRegularPolynomialObservablesTimeOrderedProduct}) of two [[field observables]]:

$$
  \left\langle
    T\left(
      \mathbf{\Phi}^a(x) \star_F \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \underset{ = 0 }{
  \underbrace{
    \left\langle
       \mathbf{\Phi}(x) \cdot \mathbf{\Phi}(y)
    \right\rangle
  }}
  +
  \underset{ = \hbar \Delta^{a b}_H(x,y) }{
  \underbrace{
    \left \langle
      \hbar
      \underset{\Sigma \times \Sigma}{\int}
       \delta(x-x') \Delta^{a b}_F(x,y) \delta(y-y')
    \right\rangle
  }}
$$

analogous to example \ref{HadamardMoyalStarProductOfTwoLinearObservables}.

Equivalently in the [[operator product]]-notation of def. \ref{NormalOrderedProductNotation} this reads:

$$
  \left\langle
    T\left(
     \mathbf{\Phi}^a(x)
     \mathbf{\Phi}^b(y)
    \right)
  \right\rangle_0
  \;=\;
  \hbar \Delta_F(x,y)
  \,.
$$

=--

[[!include propagators - table]]

## Related concepts

[[!include products in pQFT -- table]]

[[!include states and observables -- content]]

## References

The construction goes back to

* {#Wick50} [[Gian-Carlo Wick]], _The evaluation of the collision matrix_, Phys. Rev. 80, 268-272 (1950)

Its realization as the [[Moyal deformation quantization]] of the [[Peierls bracket]] shifted by a [[quasi-free Hadamard state]] is due to

* {#Dito90} Joseph Dito, _Star-product approach to quantum field theory: The free scalar field_. Letters in Mathematical Physics, 20(2):125–134, 1990 ([spire](https://inspirehep.net/record/303898/))

further amplified in

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], section 5.1 of _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic field theory, and deformation quantization_, in [[Roberto Longo]] (ed.), _Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects_, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001 ([arXiv:hep-th/0101079](https://arxiv.org/abs/hep-th/0101079))

* {#HirschfeldHenselder02} A. C. Hirshfeld, P. Henselder, _Star Products and Perturbative Quantum Field Theory_, Annals Phys. 298 (2002) 382-393 ([arXiv:hep-th/0208194](https://arxiv.org/abs/hep-th/0208194))

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

and the generalization to [[quantum field theory on curved spacetime]] is discussed in

* {#BrunettiFredenhagen95} [[Romeo Brunetti]], [[Klaus Fredenhagen]], M. Köhler, _The microlocal spectrum condition and Wick polynomials on curved spacetimes_, Commun. Math. Phys. 180, 633-652, 1996 ([arXiv:gr-qc/9510056](https://arxiv.org/abs/gr-qc/9510056))

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

* {#HollandsWald01} [[Stefan Hollands]], [[Robert Wald]], _Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime_, Commun. Math. Phys. 223:289-326, 2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074))

Review is in

* {#Duetsch18} [[Michael Dütsch]], section 2.1 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects Wick algebras]]

[[!redirects Wick polynomial]]
[[!redirects Wick polynomials]]

[[!redirects normal ordering]]

[[!redirects normal-ordered product]]
[[!redirects normal-ordered products]]

[[!redirects normal ordered product]]
[[!redirects normal ordered products]]

[[!redirects abstract Wick algebra]]
[[!redirects abstract Wick algebras]]


Wick algebra -- table

| | [[free field theory|free field]] [[algebra of quantum observables]]  | [[physics]] terminology | [[mathematics|maths]] terminology |
|-|-|----------------------|--------------------------------|
| 1) | [[supercommutative algebra|supercommutative product]] |  $\phantom{AA} :A_1 A_2:$
 [[normal ordered product]] | $\phantom{AA} A_1 \cdot A_2$
 pointwise product of functionals |
| 2) | [[non-commutative algebra|non-commutative product]]
 ([[deformation quantization|deformation]] induced by [[Poisson bracket]]) |  $\phantom{AA} A_1 A_2$
 [[operator product]]  | $\phantom{AA} A_1 \star_H A_2$
 [[star product]] for [[Wightman propagator]]  |
| 3) |  | $\phantom{AA} T(A_1 A_2)$
 [[time-ordered product]] | $\phantom{AA} A_1 \star_F A_2$
 [[star product]] for [[Feynman propagator]] |
| | [[perturbative quantum field theory|perturbative expansion]]
 of 2) via 1) | [[Wick's lemma]]
  | [[Moyal star product|Moyal product]] for [[Wightman propagator]] $\Delta_H$
 $ \begin{aligned} & A_1 \star_H A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_H)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$ |
| | [[perturbative quantum field theory|perturbative expansion]]
 of 3) via 1) | [[Feynman diagrams]]
  | [[Moyal star product|Moyal product]] for [[Feynman propagator]] $\Delta_F$
 $ \begin{aligned} & A_1 \star_F A_2 = \\ & ((-)\cdot (-)) \circ \exp \left( \hbar \int (\Delta_F)^{ab}(x,y) \frac{\delta}{\delta \mathbf{\Phi}^a(x)} \otimes \frac{\delta}{\delta \mathbf{\Phi}^b(y)} \right)(A_1 \otimes A_2) \end{aligned}$ |


Wick rotation

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _Wick rotation_ (after [[Gian-Carlo Wick]]) is a method in [[physics]] for finding a construction in [[relativistic field theory]] on [[Minkowski spacetime]] or, more generally, on [[Lorentzian manifolds]], from a related construction in [[Euclidean field theory]] on [[Riemannian manifolds]] in a way that involves or generalizes the idea of [[analytic continuation]] for the _[[time]]_ [[coordinates]].

> The [[complex plane]] for a complexified [[time]] [[coordinate]] $t + i s$. After Wick rotation it is the [[imaginary part]] $s$ that replaces the "[[real part|real]]" time $t$.

>  graphics grabbed from [Fulling-Ruijsenaars 87](#FullingRuijsenaars87)

This is motivated by the observation that the [[Minkowski metric]] (with the $(-1,1,1,1)$ convention for [[spacetime signature]]) and the four-Euclidean metric are equivalent if the [[time]] components of either are allowed to have [[imaginary part|imaginary]] values. Hence Wick rotation, when it applies, involves [[analytic continuation]] of [[n-point functions]] to [[complex number|complex valued]] [[time]] [[coordinates]].

For [[relativistic field theory|relativistic]]/[[Euclidean field theory|Euclidean]] [[quantum field theory]] on [[Minkowski spacetime]] $\mathbb{R}^{d,1}$/[[Euclidean space]] $\mathbb{R}^{d+1}$ Wick rotation is rigorously understood and takes the form of a [[theorem]] relating the [[Wightman axioms]] for [[n-point functions]] in [[relativistic field theory]] to the _[[Osterwalder-Schrader axioms]]_ for [[correlators]] in [[Euclidean field theory]].

See for instance [Fulling-Ruijsenaars 87, section 2](#FullingRuijsenaars87) for a clear account.

More generally, in this context Wick rotation applies to relativistic field theory in [[thermal equilibrium]] [[states]] at [[positive number|positive]] [[temperature]] $T$ ("[[KMS states]]"), and then relates this to [[Euclidean field theory]] with compact/periodic Euclidean time of length $\beta = 1/T$ (see [Fulling-Ruijsenaars 87, section 3](#FullingRuijsenaars87)):

$$
  \array{
  \left.
  \array{
     \text{relativistic field theory}
     \\
     \text{on Minkowski spacetime}
     \\
     \mathbb{R}^{d,1}
     \\
     \text{in a thermal equilibrium state}
     \\
     \text{at temperature}\; T
  }
  \right\}
  &
  \;\;\;\;
  \overset{ \text{Wick rotation} }{\leftrightarrow}
  \;\;\;\;
  &
  \left\{
  \array{
    \text{Euclidean field theory}
    \\
    \text{on Euclidean space}
    \\
    \mathbb{R}^d \times S^1_{\beta}
    \\
    \text{with compact/periodic Euclidean time}
    \\
    \text{of length} \; \beta = 1/T
  }
  \right.
  \\
  \phantom{A}
  \\
  \underset{
    {
    \text{equal-time n-point function}
    \atop
    \text{of relativistic fields}
    }
    \atop
    \text{ in thermal equilibrium state } \; \vert T\rangle
  }{
  \underbrace{
    \left\langle  T\vert  :\mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t) : \vert T \right\rangle_{\mathbb{R}^{d,1}}
  }}
  &\;=\;&
  \underset{
    \text{correlator of Euclidean fields}
    \atop
    \text{ for "Euclidean time" of periodicity}\; \beta = 1/T
  }{
  \underbrace{
  \left\langle 0 \vert \mathbf{\Phi}(x_1,t) \mathbf{\Phi}(x_2,t) \cdots \mathbf{\Phi}(x_n,t)  \vert 0 \right\rangle_{\mathbb{R}^{d} \times S^1_{\beta}}
  }}
  }
$$

In this form, Wick rotation is also known as _[[thermal quantum field theory]]_. See there for more.

> graphics grabbed form [Frolov-Zelnikov 11](#FrolovZelnikov11)

Wick rotation also applies on suitable [[black hole]]-[[spacetimes]] spring
thermodynamics, such as the [[Bekenstein-Hawking entropy]], find elegant explanations, at least at the level of the manipulation of formulas (see e.g. [Fulling-Ruijsenaars 87, section 4](#FullingRuijsenaars87)).

### Example

Consider the Minkowski metric with the $-1,1,1,1$ convention for the tensor:

$d s^{2}= -(d t)^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$

and the four-dimensional Euclidean metric:

$d s^{2}= d \tau^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$.

Notice that if $d t = i\cdot d \tau$, the two are equivalent.

### Method

A typical method for employing Wick rotation would be to make the substitution $t=i\tau$ in a problem in Minkowski space.  The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem.

Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals.

## Related concepts

* [[reflection positivity]]

## References

Wick rotation between [[relativistic field theory]] in terms of [[Wightman axioms]] for [[n-point functions]] on [[Minkowski spacetime]] and [[Euclidean field theory]] in terms of [[Osterwalder-Schrader axioms]] for [[correlators]] on [[Euclidean space]] is due to

* {#OsterwalderSchrader73} [[Konrad Osterwalder]], [[Robert Schrader]], _Axioms for Euclidean Green's functions_, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 ([Euclid:1103858969](https://projecteuclid.org/euclid.cmp/1103858969))

The generalization of this for [[positive number|positive]] [[thermal equilibrium]] [[vacuum states]], relating to [[thermal quantum field theory]] with compact/periodic Euclidean time is discussed in

* {#KleinLandau81} Abel Klein, Lawrence Landau, _Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle_, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 ([
DOI: 10.2140/pjm.1981.94.341](https://msp.org/pjm/1981/94-2/p12.xhtml), [pdf](https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf))

The idea here apparently goes back to

* {#Bloch58} Claude Bloch, _Sur la détermination de l'état fondamental d'un système de particules_,  Nucl. Phys. 7 (1958) 451

This has maybe first been made precise, for the case of 1+1 dimensions, in

* {#HoeghKrohn74} [[Raphael Høegh-Krohn]], _Relativistic Quantum Statistical Mechanics in two-dimensional Space-Time_, Communications in Mathematical Physics 38.3 (1974): 195-224 ([pdf](https://www.duo.uio.no/bitstream/handle/10852/44072/1973-22.pdf))

Good review on the relation to [[thermal quantum field theory]] and [[black hole thermodynamics]] is in

* {#FullingRuijsenaars87} S.A. Fulling, S.N.M. Ruijsenaars, _Temperature, periodicity and horizons_, Physics Reports Volume 152, Issue 3, August 1987, Pages 135-176 ([pdf](https://www1.maths.leeds.ac.uk/~siru/papers/p26.pdf), doi:10.1016/0370-1573(87)90136-0)

* {#GibbonsPerry78} [[Gary Gibbons]], Malcolm J. Perry, _Black Holes and Thermal Green Functions_, Vol. 358, No. 1695 (1978) ([jstor:79482](https://www.jstor.org/stable/79482))

New formalization in terms of complex metrics:

* [[Maxim Kontsevich]], [[Graeme Segal]], *Wick rotation and the positivity of energy in quantum field theory*, The Quarterly Journal of Mathematics **72** 1-2  (2021) 673–699 [[arXiv:2105.10161](https://arxiv.org/abs/2105.10161), [doi:10.1093/qmath/haab027](https://doi.org/10.1093/qmath/haab027)]

review talks:

* [[Graeme Segal]], _Wick rotation and the positivity of energy in quantum field theory_, talk at Institut des Hautes Études Scientifiques (IHÉS), June 2014 ([video recording](https://www.youtube.com/watch?feature=player_embedded&v=vTvXHL6ZJik))

* [[Graeme Segal]], *Wick Rotation and the Positivity of Energy in Quantum Field Theory*, talk at IAS Physics Group Meeting, December 2021 ([video recording](https://www.ias.edu/video/wick-rotation-and-positivity-energy-quantum-field-theory))

Discussion of thermal Wick rotation on global [[anti-de Sitter spacetime]] (which is already periodic in _real_ time) is in

* {#AllenFolacciGibbons87} B. Allen, A. Folacci, [[Gary Gibbons]], _Anti-de Sitter space at finite temperature_, Physics Letters B Volume 189, Issue 3, 7 May 1987, Pages 304-310 (doi:10.1016/0370-2693(87)91437-7)

See also

* Dirk Schlingemann, _From euclidean field theory to quantum field theory_ ([arXiv:hep-th/9802035](http://arxiv.org/abs/hep-th/9802035))

* {#Witten13} [[Edward Witten]], _The Feynman $i \epsilon$ in String Theory_ ([arXiv:1307.5124](http://arxiv.org/abs/1307.5124))

On the (non-)existence of Wick rotation for [[quantum field theory on curved spacetimes]]:

MathOverflow comments:

* [[Igor Khavkine]] ([MO:a/165317](https://mathoverflow.net/a/165317/381))

* [[Robert Bryant]] ([MO:a/165345](https://mathoverflow.net/a/165345/381))

See also:

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], p. 7-8 in: _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001))

[[!redirects Wick rotations]]


Wick's lemma

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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Wick's lemma_ is a [[combinatorics|combinatorial]] formula for the product in [[associative algebras]] that appear in the context of [[free field theory]].

From the point of view of [[path integral quantization]], Wick's lemma is about the [[moments]] of [[Gaussian probability distributions]]. See at _[[Feynman diagram]]_ for more on this.

From the point of view of [[causal perturbation theory]] Wick's lemma expresses the [[Moyal deformation quantization]] of a [[free field theory]] ([[Wick algebras]]) in terms of [[operator products]] on.

[[!include Wick algebra -- table]]

From the point of view of [[BV-quantization]] Wick's lemma arises as a consequence of the [[homological perturbation lemma]]  ([Gwilliam, section 2.3](#Gwilliam)).

## Statement

Let $\mathcal{W}$ be the [[Wick algebra]] of the [[free field theory|free]] [[scalar field]], hence the space of [[microcausal observables]] with product the [[star product]] induced by the [[Wightman propagator]]:

$$
  \mathcal{W}
  \;\coloneqq\;
  \left(
    PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
  \right)
$$

Then the evident map from $\mathcal{W}$ to [[linear operators]] on the [[Fock space]] equipped with their [[operator product]] $\circ$ is an [[associative algebra]] [[isomorphism]] onto its image:

$$
  \left(
    PolyObs(E,\mathbf{L})_{mc}, \star_{\Delta_H}
  \right)
    \underoverset{\simeq}{\text{Wick's lemma}}{\hookrightarrow}
  \left(
    End(FockSpace), \circ
  \right)
$$

([Dütsch 18, theorem 2.17](#Duetsch18), following [Dütsch-Fredenhagen 00,, pages 10-11](#DuetschFredenhagen00), [Dütsch-Fredenhagen 01](#DuetschFredenhagen01))

## Related concepts

* [[perturbation theory]]

* [[Feynman diagram]]

## References

Original articles include

* {#Wick50} [[Gian-Carlo Wick]], _The evaluation of the collision matrix_, Phys. Rev. 80, 268-272 (1950)

* {#Hepp69} [[Klaus Hepp]], _Théorie de la Renormalisation_ Lect. Notes in Physics, Springer 1969

* {#BrunettiFredenhagenVerch03} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], theorem 2.4 in _The generally covariant locality principle -- A new paradigm for local quantum physics_, Commun.Math.Phys.237:31-68, 2003 ([arXiv:math-ph/0112041](https://arxiv.org/abs/math-ph/0112041))

The interpretation as an algebra isomorphism to the [[star product]] with respect to the [[Wightman propagator]] is made explicit in

* {#DuetschFredenhagen00} [[Michael Dütsch]], [[Klaus Fredenhagen]], pages 10-11 of _Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion_, Commun.Math.Phys. 219 (2001) 5-30 ([arXiv:hep-th/0001129](https://arxiv.org/abs/hep-th/0001129))

* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))

and [Dütsch 18, theorem 2.17](#Duetsch18)

Textbook accounts include

* {#Scharf95} [[Günter Scharf]], theorem 3.1 in _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Duetsch18} [[Michael Dütsch]], E.26 - E.30 in _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* Wikipedia, _[Wick's theorem](https://en.wikipedia.org/wiki/Wick%27s_theorem)_

Discussion of Wick's lemma as a consequence of the [[homological perturbation lemma]] for [[BV-complexes]] is in

* {#Gwilliam} [[Owen Gwilliam]], section 2.3 _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))

[[!redirects Wick lemma]]
[[!redirects Wick's theorem]]
[[!redirects Wick theorem]]


wide pullback

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### Context
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

\tableofcontents

## Definition

A **wide pullback** or **wide fiber product** or **wide fibre product** in a [[category]] $\mathcal{C}$ is a [[product]] (of arbitrary [[cardinality]]) in a [[over category|slice category]] $\mathcal{C} \downarrow C$.  In terms of $\mathcal{C}$, this can be expressed as a [[limit]] over a category obtained from a [[discrete category]] by adjoining a [[terminal object]].

Yet more explicitly, the wide pullback of a family of morphisms $f_i\colon A_i \to C$ (a **[[wide cospan]]**) is an object $P$ equipped with projection $p_i\colon P\to A_i$ such that $f_i p_i$ is independent of $i$, and which is universal with this property.

Binary wide pullbacks are the same as ordinary [[pullbacks]], a.k.a. fiber products.

Of course, a **wide pushout** is a wide pullback in the opposite category.

## Properties

* A category has wide pullbacks (of all [[small category|small]] cardinalities) if and only if it has (binary) [[pullbacks]] and [[cofiltered limits]].

* The [[saturated class of limits|saturation]] of the class of wide pullbacks is the class of limits over categories $C$ whose [[fundamental groupoid]] $\Pi_1(C)$ is trivial.

On the other hand, together with a terminal object, wide pullbacks generate all limits:

+-- {: .num_prop #WidePbToComplete}
###### Proposition
A [[category]] $C$ with all [[wide pullbacks]] and a [[terminal object]] $1$ is [[complete category|complete]]. If $C$ is complete and $F\colon C \to D$ preserves wide pullbacks and the terminal object, then it preserves all limits.
=--

+-- {: .proof}
######Proof
To build up arbitrary products $\prod_{i \in I} c_i$ in $C$, take the wide pullback of the family $c_i \to 1$. Then to build equalizers of diagrams $f, g\colon c \rightrightarrows d$, construct the pullback of the diagram
$$\array{
 & & d \\
 & & \downarrow \delta \\
c & \underset{\langle f, g \rangle}{\to} & d \times d
}$$
From products and equalizers, we can get arbitrary limits.

=--

## Related concepts

* [[connected limit]]

[[!include notions of pullback -- contents]]

**Analogues in [[dependent type theory]]**:

* [[wide pullback type]]

* [[wide pushout type]]

## References

The terminology *wide pullback* appears in:

* [[Paul Taylor]], _Quantitative domains, groupoids and linear logic_, Category Theory and Computer Science: Manchester, UK, September 5–8, 1989 Proceedings. Springer Berlin Heidelberg, 1989.

Wide pullbacks are considered under the term *fibred product* in:

* [[Robert Paré]], *Simply connected limits*.  Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746, [CMS](http://math.ca/10.4153/CJM-1990-038-6)

[[!redirects wide pullback]]
[[!redirects wide pullbacks]]

[[!redirects wide fiber product]]
[[!redirects wide fiber products]]

[[!redirects wide fibre product]]
[[!redirects wide fibre products]]

[[!redirects wide pullback type]]
[[!redirects wide pullback types]]

[[!redirects wide pushout]]
[[!redirects wide pushouts]]


wide subcategory

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###Context###
#### Category theory
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[[!include category theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### Of a general category

Traditionally, a **wide subcategory** of a [[category]] $C$ is a [[subcategory]] containing all the [[objects]] of $C$.

Equivalently, it is a subcategory through which the canonical [[functor]] $disc(Obj(C)) \to C$ (from the [[discrete category]] on the collection of [[objects]]) factors, or whose inclusion functor is [[bo functor|bijective on objects]].

Notice that the condition to contain all the objects is not invariant under [[equivalence of categories]] and so the definition of wide subcategory above violates the [[principle of equivalence]].  A variant of the definition which fixes this is:

an **essentially wide subcategory** contains at least one object from each [[isomorphism class]] of objects; that is, its inclusion functor is [[essentially surjective functor|essentially surjective on objects]].

A wide subcategory is also called a **lluf subcategory** ("lluf" being "[[full subcategory|full]]" spelled backwards).

### Of an abelian category

An unrelated definition of "wide subcategory" is commonly used in the study of [[derived categories]] and [[stability conditions]].

In this context, a [[full subcategory]] $\mathcal{W} \hookrightarrow \mathcal{A}$ of an [[abelian category]] $\mathcal{A}$ is called **wide** if it is closed under [[kernels]], [[cokernels]] and [[extensions]].

See, for example, [Hovey 01](#Hovey01), [Ingalls-Thomas 09](#IngallsThomas09), [Marks-Stovicek 15](#MarksStovicek15).

Given a wide subcategory $\mathcal{W}$ in this sense, one can consider the minimal [[torsion theory|torsion class]] $T(\mathcal{W})$ containing it. Conversely, if $\mathcal{T}$ is a torsion class, define $W(\mathcal{T})$ to be the [[full subcategory]] on those [[objects]] $X \in \mathrm{Ob}(\mathcal{T})$ such that, for any $Y \in \mathrm{Ob}(\mathcal{T})$ and any $g: Y \to X$, the [[kernel]] of $g$ is in $\mathcal{T}$. The [[composition]] $W \circ T$ is the [[identity functor|identity]], thus exhibiting the [[poset]] of wide subcategories as a [[retract]] of the poset of [[torsion theory|torsion classes]].

The [[surjection]] $W$ becomes an [[injection]] when restricted to [[functorially finite]] torsion classes, and is often a [[bijection]] between functorially finite torsion classes and functorially finite wide subcategories; see [Marks-Stovicek 15](#MarksStovicek15).

## Examples

### General

Given any category $C$, the maximal sub-groupoid of $C$ is the subcategory consisting of all objects of $C$ but with morphisms only the isomorphisms of $C$. This is the [[core]] of a category, and it is a wide subcategory.

## References

### General

...

### For abelian categories

* {#Hovey01} [[Mark Hovey]], _Classifying subcategories of modules_, Trans. Amer. Math. Soc. 353 (2001), 3181-3191  ([doi:10.1090/S0002-9947-01-02747-7](https://doi.org/10.1090/S0002-9947-01-02747-7))

* {#IngallsThomas09} Colin Ingalls, Hugh Thomas, _Noncrossing partitions and representations of quivers_,  Volume 145, Issue 6 November 2009 , pp. 1533-1562 ([doi:10.1112/S0010437X09004023](https://doi.org/10.1112/S0010437X09004023))

* {#MarksStovicek15} Frederik Marks, Jan Stovicek, _Torsion classes, wide subcategories and localisations_ [arXiv:1503.04639](https://arxiv.org/abs/1503.04639)

[[!redirects wide subcategory]]
[[!redirects wide subcategories]]
[[!redirects lluf subcategory]]
[[!redirects lluf subcategories]]
[[!redirects essentially wide subcategory]]
[[!redirects essentially wide subcategories]]
[[!redirects essentially lluf subcategory]]
[[!redirects essentially lluf subcategories]]


Wiebe van der Hoek

* [personal page](https://www.csc.liv.ac.uk/~wiebe/)

* [GoogleScholar page](https://scholar.google.com/citations?user=WNmaKaIAAAAJ&hl=en)

## Selected writings

On [[modal logic]] in [[game theory]] and [[information science]]:

* [[Wiebe van der Hoek]], [[Marc Pauly]], *Modal logic for games and information*, Ch. 20 in: *The Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 85-183 $[$doi:10.1016/S1570-2464(07)80023-1, [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/)$]$

On [[epistemic logic]] and application to [[game theory]]:

* [[Hans Ditmarsch]], [[Wiebe Hoek]], Barteld Kooi, *Dynamic Epistemic Logic*, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) **337**, Springer (2008) $[$[doi:10.1007/978-1-4020-5839-4_2](https://doi.org/10.1007/978-1-4020-5839-4_2), [pdf](http://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Epistemic-logic.pdf)$]$

## Related entries

* [[epistemic logic]]

category: people

[[!redirects Wiebe Hoek]]


Wiener measure

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### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Wiener measure_ is a [[measure]] on the [[topological space|space]] of [[continuous function|continuous paths]] in a given [[manifold]]. The [[Lebesgue integral]] with respect to Wiener's measure is called the __Wiener integral__.

The Wiener measure serves to make precise the [[path integral quantization]] for the (charged) non-relativistic [[particle]] (that of the [[relativistic particle]] may be amenable to Wiener measure methods via [[Wick rotation]], i.e. analytic continuation to [[imaginary number|imaginary]] time. ).

## Related concepts

* [[Brownian motion]]

## References

Named after [[Norbert Wiener]]'s discussion of [[Brownian motion]]:

* [[Norbert Wiener]], *The Average of an Analytic Functional and the Brownian Movement*, Proceedings of the National Academy of Sciences of the United States of America, **7** 10 (1921) 294-298 [[jstor:84434](https://www.jstor.org/stable/84434)]

Introduction:

* Tamas Szabados, *An elementary introduction to the Wiener process and stochastic integrals*, Studia Scientiarum Mathematicarum Hungarica **31** (1996) 249-297 [[arXiv:1008.1510](https://arxiv.org/abs/1008.1510)]

History:

* Arthur Genthon, *The concept of velocity in the history of Brownian motion -- From physics to mathematics and back*, Eur. Phys. J. H **45** (2020) 49-105 [[arXiv:2006.05399](https://arxiv.org/abs/2006.05399), [doi:10.1140/epjh/e2020-10009-8]( https://doi.org/10.1140/epjh/e2020-10009-8)]

See also:

* [[eom|Enc. Math
]], _[Wiener integral](https://encyclopediaofmath.org/wiki/Wiener_integral)_

* PlanetMath _[Wiener measure](http://planetmath.org/wienermeasure)_

A textbook account in the context of [[path integral quantization]]:

* [[Barry Simon]], around p. 49 in: _Functional integration and quantum physics_, AMS Chelsea Publ., Providence, 2005

[[!redirects Wiener measures]]

[[!redirects Wiener integral]]
[[!redirects Wiener integrals]]

[[!redirects Wiener process]]
[[!redirects Wiener processes]]


Wiener-Hopf decomposition

Decomposition of an analytic function into a product of a function analytic in upper half-plane and analytic function in lower half-plane. There is a version for stochastic processes. It can be related tp Riemann-Hilbert-[[Birkhoff decomposition]].

* wikipedia: [Wiener-Hopf method](http://en.wikipedia.org/wiki/Wiener%E2%80%93Hopf_method)
* [[eom]]: [Wiener-Hopf method](http://www.encyclopediaofmath.org/index.php/Wiener%E2%80%93Hopf_method)


Wieslaw Pawlowski

* [personal page](https://inf.ug.edu.pl/~wpawlowski/)

* [GoogleScholar page](https://scholar.google.pl/citations?user=mr7fct0AAAAJ&hl=en)

## Selected writings

On generalized [[congruences]] and [[coequalizers]] in the [[1-category]] [[Cat]] of [[small categories]]:

* {#BednarczykBorzyszkowskiPawlowski99} [[Marek A. Bednarczyk]], [[Andrzej M. Borzyszkowski]], [[Wieslaw Pawlowski]], Section 4 of: *Generalized congruences -- epimorphisms in $\mathcal{C}at$*, Theory and Applications of Categories **5** 11 (1999) 266-280 [[tac:5-11](http://www.tac.mta.ca/tac/volumes/1999/n11/5-11abs.html), [dml:120226](https://eudml.org/doc/120226?lang=fr&limit=5)]

category: people


Wightman axioms

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###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **Wightman axioms** are an attempt to [[axiom|axiomatize]] and thus formalize the notion of a [[quantum field theory]] on [[Minkowski spacetime]] ([[relativistic quantum field theory]]) in the sense of [[AQFT]], i.e. in terms of the assignment of field [[quantum observables]] to points or subsets of [[spacetime]] ([[operator-valued distributions]]).

They serve as the basis of what is known as **constructive quantum field theory** which seeks to provide a mathematically sound framework for quantum theory over the Minkowski space background of [[special relativity]].  [[Arthur Wightman]] first formulated them in the 1950s but they were not published until 1964 after advances in scattering theory confirmed their applicability.

The Wightman axioms served to establish rigorously several basic structural properties of quantum field theories on Minkowski spacetime, such as the [[spin-statistics theorem]]  or the [[Osterwalder–Schrader theorem]] relating Lorenzian and Euclidean quantum field theories ("Wick rotation").

They were later further abstracted to the [[Haag-Kastler axioms]] that characterize [[local net]]s of operator algebras and serve as the basis for [[AQFT|algebraic quantum field theory]]. See there for further details.

This page is a draft, see:

* Wikipedia on the [Wightman axioms] (http://en.wikipedia.org/wiki/Wightman_axioms)

## Wightman Axioms

Note: the numbering - and indeed the actual number - of axioms varies depending on the source.  In many sources, several of the axioms below are combined.

+-- {.num_defn}
###### Axiom
There is a physical Hilbert space $\mathcal{H}$ in which a unitary representation $U(a,\Lambda)$ of the [[Poincaré spinor group]], $P_{0}$ acts.
=--

+-- {.num_defn}
###### Axiom
The spectrum of the energy-momentum operator P is concentrated in the closed upper (forward) light cone $V^{+}$.
=--

+-- {.num_defn}
###### Axiom
There exists in $\mathcal{H}$ a unique unit vector $|0\rangle$ (the _[[vacuum state]]_), which is invariant with respect to the space-time translations $U(a,1)$.
=--

+-- {.num_defn}
###### Axiom
The components $\phi_{i}$ of the quantum field $\phi$ are [[operator-valued distributions]] $\phi_{i}(x)$ over the [[Schwartz space]] $S(M)$ ([[tempered distributions]]) with domain of definition $D$ which is common to all the operators and is dense in $\mathcal{H}$. $|0\rangle$ is contained in $D$ and $D$ is taken into itself under the action of $\phi (f)$ and $U(a,\Lambda)$.
=--

Note: As in distribution theory it is custom to abuse the notation and write $\phi (x)$ for a point $x$ of the Minkowski spacetime and talk about the function $\phi$, rather than the value of the distribution $\phi(f)$ of a test function $f$.

+-- {.num_defn}
###### Axiom
$U(a,\Lambda)\phi_{i}(x)U(a,\Lambda)^{-1}=\sum_{j}V_{ij}(\Lambda^{-1})\phi_{j}(\Lambda x + a)$ where $V_{ij}(\Lambda)$ is a complex or real finite-dimensional matrix representation of $SL(2,C)$.
=--

+-- {.num_defn}
###### Axiom
Any two field components $\phi_{i}(x)$ and $\phi_{j}(y)$ either commute or anticommute under a space-like separation of the arguments $x$ and $y$.
=--

+-- {.num_defn}
###### Axiom
The set $D_{0}$ of finite linear combinations of vectors of the form $\phi_{i_1}(f_{1})\ldots\phi_{i_n}(f_{n})|0\rangle$ is dense in $\mathcal{H}$.
=--

## Properties

### The Wightman Reconstruction Theorem

The **[[vacuum expectation values]]** ([[n-point functions]]) of the theory are all (tempered, by the axioms) distributions of the form
$$
\langle0| \phi_{i_1}(f_{1})\ldots\phi_{i_n}(f_{n})|0\rangle
$$
The term "distribution" is used interchangably with "function" and "value" in the physics literature. The Wightman reconstruction theorem states properties that a set of tempered distributions $\{\mathcal{W}^n | n \in \N \}$ need to have to be the set of vacuum expectation values of a Wightman theory, and all such Wightman theories are then unitarily equivalent.
...to be continued...

### Equivalence to Euclidean Field Theory

See [[Osterwalder–Schrader theorem]]

### Equivalence to the Haag--Kastler Axioms

See [[Haag–Kastler axioms]]. Since both the Wightman and the Haag-Kastler approach try to formulate an axiomatic approach to quantum field theory on Minkowski spacetime, the natural question to ask is what is their relationship? Three possible answers come to mind:

* They are equivalent.

* One contains the other, i.e. one set of axioms implies the other.

* At least one is wrong (from the physical viewpoint).

Unfortunatly the situation does not seem to be as clear as this list suggests. The current state of the affair seems to be that

* "the Haag-Kastler approach is more convenient, because it deals with algebras of bounded operators, while Wightman fields are allowed to generate unbounded operators"

and

* "a Wightman field theory is equivalent to a Haag-Kastler theory, if some mild additional assumptions are made".

Reference:

* [[Hans-Jürgen Borchers]], [[Jakob Yngvason]], *From quantum fields to local von Neumann algebras*, Rev. Math. Phys. **4** spec01 (1992) 15-47 [[doi:10.1142/S0129055X92000145](https://doi.org/10.1142/S0129055X92000145)]

+--{: .query}

[[Tim van Beek]]: The two statemants above are a condensate of diverse papers I read, any input about the true state of the art would be most welcome. My educated guess is that the "mildness" of the assumptions indicates the fact that all physically realistic models are supposed to satisfy these.

=--

Should it be bounded or unbounded operators/observables?

* Pro: Some observables do not have an upper bound, e.g. you can always increase the impulse of an electron, no matter how big that already is (in theory).

* Contra: A detector has an upper bound for every observable it can detect, and that is what is actually measured in an experiment. The situation of the pro-bullet won't arise because we measure observables in bounded regions of spacetime, and the energy contained in a bounded region is always finite (we are excluding artifacts like black holes from our consideration, of course).

* Contra to even asking the question: Theories of bounded and unbounded operators/observables are physically equivalent modulo mathematical subtleties, meaning they produce the same numbers that can be compared to experiments.

One simple situation where the construction of a Haag-Kastler net out of Wightman fields is straight forward is this: Suppose that all smeared field operators of the Wightman theory are essentially self adjoint for real test functions and commute strongly (their spectral projections commute) if the test functions have space-like separated support.
Then we can define local algebras by
$\mathcal{M}(\mathcal{O}) := \{F(\Psi(f)) | f$ is a real test function with support contained in $\mathcal{O}$ and F is a bounded Borel measurable function on $\R \}$.

(Our assumptions allow us to use the [Borel functional calculus] (http://en.wikipedia.org/wiki/Borel_functional_calculus)).

## Examples

### Neutral Real (uncharged) Scalar Field

A neutral real quantum scalar field on Minkowski spacetime $M$ with mass parameter $m \gt 0$ can be defined as follows:

+-- {: .un_def}
###### Definition
The **positive mass shell** is the subset of Minkowski spacetime defined by

$$
X^+_m := \{p | p^2 = m^2, \; p_0 \gt 0 \}
$$

The **normalized Lorentz-invariant measure** on the positive mass shell is defined with respect to the Lesbegue measure by
$$
d\lambda(p) = d\lambda(\omega_p, \vec p) = \frac{d^3 \vec p}{(2\pi)^3 \omega_p} \; \text{with} \; \omega_p = p^0 = \sqrt{|\vec p|^2 + m^2}
$$

=--

Let $H = L^2(X^+_m, \lambda)$ be the Hilbert space with $X^+_m$ the positive mass shell and $\lambda$ the normalized Lorentz-invariant measure on it as defined above. Construct the Boson [[Fock space]] $F_s(H)$.

Define the operator $R$ to be the Lorentz invariant Fourier transform restricted to $X^+_m$:
$$
R: \mathcal{S}(\mathbb{R}^4) \to H
$$

$$
R(f) := \hat f |_{X^+_m} \; \text{with} \; \hat f := \mathcal{F}(f) = \int_M e^{i p_{\mu} x^{\mu}} f(x) dx
$$

The quantum field $\Psi$ is now a real tempered distribution on $M$ with values in the space of operators of $F_s(H)$.

$$
\Psi:   \mathcal{S}(M) \to \mathcal{B}(F_s(H))
$$

$$
         f \mapsto \Psi(f) = \frac{1}{\sqrt(2)} (a(Rf) + a^*(Rf))
$$
$a$ and $a^*$ are the annihilation and creation operators on $F_s(H)$, that is $a(v)$ anihilates a single particle state $v$ and $a^*(v)$ creates a single particle state $v$.

$\Psi$ is a distribution solution of the [[Klein-Gordon equation]] by construction, that is for every test function $f$ we get

$$
\Psi((\Box + m^2) f) = 0
$$

The reason for this is that
$$
\mathcal{F}{[(\Box + m^2) f]} = (-p^2 + m^2) f = 0 \; \text{on} \; X^+_m
$$

### Further examples

The Wightman axioms have been established for the following theories.

* Massive 2d scalar theories with polynomial interactions, see [this article](http://www.jstor.org/stable/1970959) by Glimm, Jaffe and Spencer.

* Massive $\phi^4$ in 3d, see [this article](http://www.sciencedirect.com/science/article/pii/0003491676902232) by Feldman and Osterwalder as well as this one by Magnen and Seneor.

* Massive Gross-Neveu in 2d see [this article](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114300) by Gawedzki and Kupiainen and [this one](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104114625) by Feldman, Magnen, Rivasseau and Seneor.

* Massive Thirring model, see [this article](http://dx.doi.org/10.5169/seals-114796) by Frohlich and Seiler and this more recent one by Benfatto, Falco and Mastropietro.

## Related concepts

* [[AQFT]], [[local net]]

* [[conformal net]], [[conformal bootstrap]]

* analog for [[Euclidean field theory]]: [[Osterwalder-Schrader axioms]]

## References

The original texts are

* A. S. Wightman, L. Gårding, _Fields as Operator-valued Distributions in Relativistic Quantum Theory_, Arkiv f. Fysik, Kungl. Svenska Vetenskapsak. 28, 129–189 (1964).

* Raymond F. Streater, [[Arthur Wightman]], _PCT, spin and statistics, and all that_, Princeton University Press 1989  ([jstor](https://www.jstor.org/stable/j.ctt1cx3vcq), [ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1026.81027&format=complete) of the newest edition).

Raymond Streater relates some historical background about the book and the approach [on his webpage] (http://www.mth.kcl.ac.uk/~streater/wightman.html).

A review of [[QFT]] via Wightman axioms and [[AQFT]] is in

* [[Franco Strocchi]], _Relativistic Quantum Mechanics and Field Theory_, Found.Phys. 34 (2004) 501-527 ([arXiv:hep-th/0401143](http://arxiv.org/abs/hep-th/0401143))

The list of examples above draws from

* Abdesselam, _[TP.SE comment](http://theoreticalphysics.stackexchange.com/questions/308/which-qfts-were-rigorously-constructed/350#350)_

Discussion of the axioms for $\phi^4$-theory in 3d is in

* Joel Feldman, [[Konrad Osterwalder]], _The Wightman axioms and the mass
gap for weakly coupled $phi_3^4$ quantum field theories_, Ann. Physics 97 (1976), 80–135

See also

* Wikipedia, _[Wightman axioms](https://en.wikipedia.org/wiki/Wightman_axioms)_

[[!redirects Wightman theory]]
[[!redirects Wightman approach]]


Wigner 9j symbol

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=--
=--
=--

## Related concepts

* [[Wigner 3j symbols]] ([[Clebsch-Gordan coefficient]])

* [[Wigner 6j symbol]]

## References

* Wikipedia, _[9-j symbol](https://en.wikipedia.org/wiki/9-j_symbol)_

[[!redirects Wigner 9j symbols]]

[[!redirects Wigner 9-j symbol]]
[[!redirects Wigner 9-j symbols]]

[[!redirects 9-j symbol]]
[[!redirects 9-j symbols]]

[[!redirects 9j symbol]]
[[!redirects 9j symbols]]


Wigner classification

> not to be confused with [[Wigner's theorem]]

***

+-- {: .rightHandSide}
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#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The identification of [[particle]] species with [[irreducible representation|irreducible]] [[unitary representations]] of the [[symmetry group]] acting on [[matter]] [[field (physics)|fields]] over [[spacetime]]. Named after [[Eugene Wigner]].

For details see at

* _[[unitary representation of the Poincaré group]]_.

* _[[unitary representation of the super Poincaré group]]_.

It uses a method that goes back to Frobenius and came to be known as "Wigner's little group method" or the "Mackey machine" ([Mackey 68](#Mackey68)).

## Related concepts

* [[Doplicher-Roberts reconstruction theorem]]

* [[Deligne's theorem on tensor categories]] -- [Relevance for Supersymmetry](Deligne's+theorem+on+tensor+categories#RelevanceForSupersymmetry)

* [[supermultiplet]]

## References

The original article is

* {#Wigner39} [[Eugene Wigner]], _On unitary representations of the inhomogeneous Lorentz group_, Annals of Mathematics, 40 (1939) 149–204

The observation that Wigner's representations $U_{(-)}$ are those for which all the maps $g \mapsto \langle \Phi \vert U_g \Psi \rangle$ are [[measurable function|measurable]] and their correspondence of irreducible unitary representations to [[induced representations]] of irreps of the [[stabilizer group]] of a given momentum is (e.g. [Dragon 16](#Dragon16)) due to

* {#Mackey68} [[George Mackey]], _Induced Representations of Groups and Quantum Mechanics_, W. A. Benjamin, New York, 1968

Review includes

* {#Schroer96} [[Bert Schroer]], _Wigner Representation Theory of the Poincaré
Group, Localization, Statistics and the S-Matrix_, 1996  ([pdf](http://cds.cern.ch/record/308785/files/9608092.pdf))

* {#Dragon16} [[Norbert Dragon]], _Currents for Arbitrary Helicity_ ([arXiv:1601.07825](http://arxiv.org/abs/1601.07825))

* Wikipedia, _[Wigner's classification](http://en.wikipedia.org/wiki/Wigner%27s_classification)_

[[!redirects Wigner's classification]]

[[!redirects Wigner-Weyl theory]]


Wigner function

## Idea

The usual quantum mechanics based on Schroedinger's wave-function can be reformulated using a different distribution which behaves in some aspects like a probability, however it is negative for some points in space.

## Literature

* related $n$Lab entries include [[quantum mechanics]], [[Weyl calculus]]
* wikipedia/[Wigner quasiprobability distribution](https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution)

* E. P. Wigner, _On the quantum correction for thermodynamic equilibrium_,  Phys. Rev. 40 (5): 749–759 (1932)
* C. Zachos, D. Fairlie, T. Curtright, _Quantum mechanics in phase space: an overview with selected papers_, World Sci. 2005.
* Giuseppe Dito, Francisco J. Turrubiates, _The damped harmonic oscillator in deformation quantization_, Physics Letters A352:4–5 (2006) 309–316 [doi](http://dx.doi.org/10.1016/j.physleta.2005.12.013)
* B. Belchev, M.A. Walton, _On Wigner functions and a damped star product
in dissipative phase-space quantum mechanics_, Annals of Physics 324 (2009) 670–681
* B Molnár, M G Benedict, J Bertrand, _Coherent states and the role of the affine group in the quantum mechanics of the Morse potential_, J. Phys. A34, N.14 [doi](https://doi.org/10.1088/0305-4470/34/14/318)

category: physics

[[!redirects Wigner distribution]]


Wigner theorem

> not to be confused with _[[Wigner classification]]_

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#### AQFT
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=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

What is known as *Wigner's theorem* (in honor of the appendix of §20 in [Wigner 1931](#Wigner31)/[59](#Wigner1959)) is one of the basic results on the foundations of [[quantum mechanics]] (generally of [[quantum physics]]). The theorem characterizes the intrinsic notion of *[[symmetry|symmetries]]* in [[quantum physics]] ("quantum symmetries") assuming only observable properties of ([[pure state|pure]]) [[quantum states]] (namely their [[projective space|projectivity]] subject to the [[Born rule]]) and *deriving* from this that quantum symmetries are necessarily [[linear representation|represented]] by [[unitary operators]] *or* [[anti-unitary operators]] on the [[Hilbert space]] [[space of quantum states|of states]].

Notice that a prominent class of examples of [[anti-unitary operators]] appearing as [[quantum symmetries]] are [[time-reversal symmetry|time-reversal symmetries]] ([Wigner 1959, §26](#Wigner1959)) such as appearing in the [[CPT-theorem]].

A key applications of Wigner's theorem is within the [[K-theory classification of topological phases of matter]], where the (anti-)unitary quantum symmetries of [[effective field theory|effectively]] [[free field|free]] [[electrons]] in a [[crystal]] are identified with possible [[twisted equivariant K-theory|twistings]] of the [[equivariant K-theory]] of the crystal's [[Brillouin torus]] ([Freed & Moore 2013, §1](#FreedMoore13), see also [[schreiber:Anyonic topological order in TED K-theory|SS22, §2.2]]).

## Preliminaries
 {#Preliminaries}

Let $\mathcal{H}$ be a [[complex vector space|complex]] [[separable Hilbert space|separable]] [[Hilbert space]] and write

\[
  \label{ProjectiveSpaceOfHilbertSpace}
  P\mathcal{H}
  \;\coloneqq\;
  \big(
    \mathcal{H} \setminus \{0\}
  \big)/ \mathbb{C}^\times
\]

for its [[complex projective space]] (here $\mathbb{C}^\times = \mathbb{C} \setminus \{0\}$ denotes the [[group of units]] of the [[ring]] of [[complex numbers]] and $\mathrm{U}(1) \subset \mathbb{C}^\times$ will denote the [[unitary group]] [[U(1)]]).

In [[quantum physics]] one often says that a [[pure state|pure]] [[quantum state]] is an element (hence a [[vector]], then often called a "[[wavefunction]]") $\psi \,\in\, \mathcal{H}$, but by the [[Born rule]] it is really only the corresponding [[complex line]]

$$
  [\psi]
  \;\coloneqq\;
  \big\{
    c \cdot \psi
    \,\big\vert\,
    c \in \mathrm{U}(1)
  \big\}
  \;\;\;
  \in
  \;
  P\mathcal{H}
$$

which is physically observable. In fact, what is concretely observable (still according to the [[Born rule]]) are the transition [[probabilities]], given by the following [[function]] to the [[closed interval]] of possible [[probability]] values:

\[
  \label{ProbabilityFunctionOnRays}
  \array{
    P\mathcal{H}
    \times
    P\mathcal{H}
    &\xrightarrow{\;\;\;p\;\;\;}&
    [0,1]
    \\
    \big(
      [\psi_1]
      ,\,
      [\psi_2]
    \big)
    &
    \overset{\;\;\;\;}{\mapsto}
    &
    \frac{
       \langle \psi_1 \vert \psi_2 \rangle
       \langle \psi_2 \vert \psi_1 \rangle
    }{
       \langle \psi_1 \vert \psi_1 \rangle
       \langle \psi_2 \vert \psi_2 \rangle
    }
    \mathrlap{\,,}
  }
\]

where $\langle - \vert - \rangle$ denotes the given [[inner product]] on $\mathcal{H}$.

Hence a "quantum symmetry" in the sense of a re-shuffling of the [[pure states]] of a [[quantum system]] which leaves the observable physics [[invariant]] must be a [[bijection]] on the projective space $P\mathcal{H}$ (eq:ProjectiveSpaceOfHilbertSpace) which preserves the observable probabilities (eq:ProbabilityFunctionOnRays):

\[
  \label{QuantumSymmetry}
  S
  \;\colon\;
  P\mathcal{H}
  \xrightarrow{\;\sim\;}
  P\mathcal{H}
  \,,
  \;\;\;\;\;\;
  p
  \big(
    S(-)
    ,\,
    S(-)
  \big)
  \;=\;
  p(-,\,-)
  \,.
\]

An *example* of a quantum symmetry (eq:QuantumSymmetry) is provided by any [[complex numbers|complex]]-[[linear map|linear]] and [[unitary operator]] on the underlying [[Hilbert space]],

$$
  U \,\colon\,
  \mathcal{H}
  \xrightarrow{\;\;}
  \mathcal{U}
  \,,
  \;\;\;\;\;\;\;\;
  \big\langle
    U(-)
  \big\vert
    U(-)
  \big\rangle
  \;=\;
  \big\langle
    -
  \big\vert
    -
  \big\rangle
  \,,
$$

in that the induced projectivization

\[
  \label{ProjectivisedUnitaryOperator}
  \array{
    P\mathcal{H}
    &\xrightarrow{\;\;[U]\;\;}&
    P\mathcal{H}
    \\
    [\psi]
    &\mapsto&
    \big[
      U(\psi)
    \big]
  }
\]

evidently satisfies (eq:QuantumSymmetry).

On the one hand, there are more quantum symmetries than arise from unitary operators this way. Namely, if $U$ is an [[anti-linear map|anti-linear]] [[anti-unitary operator]], then (eq:ProjectivisedUnitaryOperator) still holds.

On the other hand, this does now already exhaust the most general situation: Wigner's theorem (Prop. \ref{WignerTheorem} below) says that for every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (eq:QuantumSymmetry) there exists a map $U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a [[unitary operator]] or an [[anti-unitary operator]], such that $S \,=\, [U]$ (eq:ProjectivisedUnitaryOperator).

## Statement
 {#Statement}

\begin{proposition}\label{WignerTheorem}
**(Wigner's theorem)**
\linebreak
  Every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (eq:QuantumSymmetry) is the projectivization (eq:ProjectivisedUnitaryOperator) of a map
$U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a [[unitary operator]] or an [[anti-unitary operator]].
\end{proposition}
The first full proof of this statement seems to be that due to [Bargmann 1964, §1.3](#Bargmann64), following indications in the appendix of §20 in [Wigner 1931](#Wigner31)/[59](#Wigner1959). A geometric proof via the [[Fubini-Study metric]] is given in [Freed 2012, Thm. 8](#Freed12). Statement and proof in the greater generality of possibly non-[[pure state|pure]] [[quantum states]] is given in [Moretti 2017, Thm. 12.11](#Moretti17).

## Related theorems

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Gleason's theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References
 {#References}

The original:

* {#Wigner31} [[Eugene P. Wigner]]: §20(app.) *Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren*, Springer (1931) [[doi:10.1007/978-3-663-02555-9](https://doi.org/10.1007/978-3-663-02555-9), [pdf](https://digbib.ubka.uni-karlsruhe.de/volltexte/wasbleibt/33355459/33355459.pdf)]

and its English translation:

* {#Wigner1959} [[Eugene P. Wigner]], §20(app.) and §26 in: *Group theory: And its application to the quantum mechanics of atomic spectra*, Academic Press (1959) [[doi:978-0-12-750550-3](https://www.elsevier.com/books/group-theory/wigner/978-0-12-750550-3)]

The first full proof:

* {#Bargmann64} [[Valentine Bargmann]], _Note on Wigner's theorem on symmetry transformations_, Journal of Mathematical Physics **5** 7 (1964) 862-868 [[doi:10.1063/1.1704188](https://doi.org/10.1063/1.1704188), [pdf](http://materias.df.uba.ar/t2a2019c2/files/2012/07/Bergman-1964.pdf), [[Bargmann-WignerTheorem.pdf:file]]]

New proof using the [[Fubini-Study metric]]:

* {#Freed12} [[Daniel Freed]], _On Wigner's theorem_, Geometry & Topology Monographs **18** (2012) 83–89 [[arXiv:1112.2133](http://arxiv.org/abs/1112.2133), [doi:10.2140/gtm.2012.18.83](http://dx.doi.org/10.2140/gtm.2012.18.83)]

A proof in the greater generality of possibly non-[[pure state|pure]] [[quantum states]]:

* {#Moretti17} [[Valter Moretti]], Thm. 12.11 of:  *Spectral Theory and Quantum Mechanics*, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]

See also:

* Wikipedia, _[Wigner's theorem](https://en.wikipedia.org/wiki/Wigner%27s_theorem#Statement)_

Discussion for [[quaternions|quaternionic]] Hilbert spaces:

* C. S. Sharma and D. F. Almeida, _Additive isometries on a quaternionic Hilbert space_, Journal of Mathematical Physics 31, 1035 (1990) ([doi:10.1063/1.528779](https://doi.org/10.1063/1.528779))

In the context of the [[K-theory classification of topological phases of matter]]:

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], §1 of: *Twisted equivariant matter*, Ann. Henri Poincaré **14** (2013)  1927-2023 [[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/10.1007/s00023-013-0236-x)]

[[!redirects Wigner theorem]]
[[!redirects Wigner's theorem]]
[[!redirects Wigner's theorem]]

[[!redirects quantum symmetry]]
[[!redirects quantum symmetries]]


wild category

## Definition

A **wild category** is a 1-dimensional approximation of an [[nlab:infinity-category]] that is definable in [[Book HoTT]].  It consists of

* A type $ob A$ of objects
* For objects $x,y$ a type $A(x,y)$ of morphisms
* For each object $x$ an identity $id_x : A(x,x)$
* For objects $x,y,z$ a composition function $\circ : A(y,z) \times A(x,y) \to A(x,z)$
* For objects $x,y$ and a morphism $f:A(x,y)$, equalities $f \circ id_x = f = id_y \circ f$
* For objects $x,y,z,w$ and morphisms $f:A(x,y)$ and $g:A(y,z)$ and $h:A(z,w)$, an equality $h\circ (g\circ f) = (h\circ g)\circ f$.

If each $A(x,y)$ is a [[set]], then a wild category reduces to a [[precategory]], but in general this condition is not imposed.  This means that, for instance, there is a nontrivial pentagon identity for the associativities that does not necessarily commute, and so on.  However, even lacking these coherence data, a wild category is sufficient for some purposes.

For example, we can define an *initial object* in a wild category to be an object $0$ such that $A(0,x)$ is contractible for all $x$.  In cases when the wild category "is" actually a coherent higher category, this still gives the right answer, and it is sufficient for applications such as producing induction principles for [[higher inductive types]].  See the references for more specific examples.

## References

* Paolo Capriotti, Nicolai Kraus, _Univalent Higher Categories via Complete Semi-Segal Types_, [arxiv](https://arxiv.org/abs/1707.03693), 2017

* Nicolai Kraus, Jakob von Raumer, _Path Spaces of Higher Inductive Types in Homotopy Type Theory_, [arxiv](https://arxiv.org/abs/1901.06022), 2019

* Mike Shulman, _Impredicative Encodings, Part 3_, [blog post](https://homotopytypetheory.org/2018/11/26/impredicative-encodings-part-3/), 2018


Wiles' proof of Fermat's last theorem

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [Frey82](#Frey82) [[Gerhard Frey]] showed that, if the equation $a^{p}+b^{p}=c^{p}$ had any nontrivial integer solutions $(a,b,c)$ for primes $p\gt 2$, the elliptic curve (now known as the _Frey-Hellegouarch curve_ after Frey and [[Yves Hellegouarch]])

$$y^{2}=x(x-a^{p})(x-b^{p})$$

would exhibit certain unusual properties. In particular in [Serer87](#Serre87) [[Jean-Pierre Serre]] showed, assuming the "epsilon conjecture", that if such a solution existed then the Frey-Hellegouarch curve would not be modular.

[[Andrew Wiles]] with [[Richard Taylor]] proved the [[modularity theorem]] stating that all elliptic curves over $\mathbb{Q}$ are modular. By [[Ken Ribet]]'s proof of the epsilon conjecture (now known as [[Ribet's theorem]]) in [Ribet90](#Ribet90) this implied a proof of [[Fermat's last theorem]].

## References

* {#Frey82} [[Gerhard Frey]], 1982, _Rationale Punkte auf Fermatkurven und getwisteten Modulkurven_, J. reine angew. Math., 331: 185–191

* {#Serre87} [[Jean-Pierre Serre]], 1987, _Sur les représentations modulaires de degré 2 de Gal(Q/Q)_, Duke Mathematical Journal, 54 (1): 179–230

* {#Ribet90} [[Ken Ribet]], 1990, _On modular representations of Gal(Q/Q) arising from modular forms_, Inventiones Mathematicae. 100 (2): 431–476.

See also the references at *[[modularity theorem]]* ([here](modularity+theorem#References))

See also:

* Wikipdia, _Wiles's proof of Fermat's last theorem_

[[!redirects Wiles's proof of Fremat's last theorem]]


Wilfrid Hodges

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

## related $n$Lab entries

* [[model theory]]

category: people


Wilhelm Ackermann

* [Wikipedia entry](http://en.wikipedia.org/wiki/Wilhelm_Ackermann)

## related $n$Lab entries

* [[Ackermann function]]

* [[mathematical logic]]

category: people


Wilhelm Killing

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

## Selected writings

On [[spherical space forms]]:

* {#Killing1891} [[Wilhelm Killing]], _Ueber die Clifford-Klein’schen Raumformen_, Math. Ann. 39 (1891), 257–278 ([doi:10.1007/BF01206655](https://doi.org/10.1007/BF01206655))

## Related $n$Lab entries

* [[Killing tensor]]

* [[Clifford-Klein space form]]

category: people


Wilhelm Magnus

[[!redirects W. Magnus]]
Wilhelm Magnus was a German mathematician who worked in [[combinatorial group theory]].

* [Wikipedia page](http://en.wikipedia.org/wiki/Wilhelm_Magnus)

* [St. Andrews history page](http://www-history.mcs.st-andrews.ac.uk/Biographies/Magnus.html)

## Selected writings

On understanding the [[braid group]] as a [[mapping class group]]:

* [[Wilhelm Magnus]], *Über Automorphismen von Fundamentalgruppen berandeter Flächen*, Mathematische Annalen **109** (1934) 617–646 [[doi:10.1007/BF01449158](https://doi.org/10.1007/BF01449158)]

On [[combinatorial group theory]] ([[presentations of groups]]):

* [[Wilhelm Magnus]], A. Karras, D. Solitar, *Combinatorial group theory: Presentation of groups in terms of generators and relations*, Courier Corporation (1966, 2004) [[ISBN:0-486-43830-9](https://www.maa.org/press/maa-reviews/combinatorial-group-theory-presentations-of-groups-in-terms-of-generators-and-relations), [GoogleBooks](https://books.google.com/books?hl=en&lr=&id=1LW4s1RDRHQC&oi=fnd&pg=PP1&ots=WszNa-0hEz&sig=Hn_MTbyLcEreBBNZ8jJtQUVKlZk)]

[[!redirects W. Magnus]]

category:people


Wilhemus A. J. Luxemburg

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

## Selected writings

On [[monads in nonstandard analysis]]:

* [[Wilhemus A. J. Luxemburg]], *A General Theory of Monads*, in: *Applications of Model Theory to Algebra, Analysis and Probability*, Holt, Rinehart and Minston (1966) 18–86 [[ark](https://archive.org/stream/in.ernet.dli.2015.141486/2015.141486.Applications-Of-Model-Theory-To-Algebra-analysis-And-Probability_djvu.txt)]

category: people

[[!redirects Wilhemus Luxemburg]]


Willard Miller

* [personal page](https://www-users.cse.umn.edu/~mille003/index.html)

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=7741)

## Selected writings

On [[symmetry groups]] in [[physics]] (such as [[crystallographic groups]]) and their [[group theory]]:

* [[Willard Miller]], *Symmetry Groups and Their Applications*, Pure and Applied Mathematics **50** (1972) 16-60 [[ISBN:9780080873657](https://www.elsevier.com/books/symmetry-groups-and-their-applications/miller/978-0-12-497460-9)]

category: people


Willard Quine

Willard Van Orman Quine (1908–2000) was an American [[philosophy|philosopher]] who worked in theoretical philosophy and in [[logic]].

## Related nLab entries

* [[logicality and invariance]]

## Selected writings

* 1951, _Two Dogmas of Empiricism_, Philosophical Review, 60: 20–43; reprinted in From a Logical Point of View, pp. 20–46.

* 1953, _From a Logical Point of View_, Cambridge, Mass.: Harvard University Press, revised edition 1980.


Willem van Est

* [German Wikipedia entry](http://de.wikipedia.org/wiki/Willem_Titus_van_Est)

## related $n$Lab entries

* [[van Est isomorphism]]

[[!redirects van Est]]

category: people


William Abram

* [webpage](https://sites.google.com/site/abrammath/home)

## related entries

* [[equivariant cohomology]]

* [[equivariant complex cobordism cohomology theory]]

category: people