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

#Contents#
* table of contents
{:toc}

## Idea

What are called _advanced_ and _retarded causal Green functions_ $\Delta_{A/R}$ are [[Green functions]] for [[hyperbolic differential operators]] on manifolds with [[causal structure]] (e.g. [[spacetimes]]) whose [[support]] is in the [[future cone]] or [[past cone]], respectively, of the source excitation. The corresponding [[integral kernels]] hence say how a [[delta distribution|point excitation]] _propagates_ into the future or past, respectively, via the given [[differential equation]], and therefore these are also called the advanced/future _propagators_.

If both advanced and retarded Green functions exist for a differential operator as well as for its [[formally adjoint differential operator|formal adjoint]], then the differental operator is called a _[[Green hyperbolic differential operator]]_.  The archetypical examples are, on [[globally hyperbolic spacetimes]]:

1. [[normally hyperbolic differential operators]] such as the [[wave operator]] and the [[Klein-Gordon operator]];

1. [[Dirac operators]] on [[spinor bundles]] whose square is a normally hyperbolic differential operator as above.

The advanced/retarded [[integral kernel]]

$$
\Delta_{A/R}
\in \mathcal{D}'(X \times X)
$$

is such that

1. $(x,y) \in supp(\Delta_R)$ precisely if $x$ is in the [[causal future]] of $y$;

1. $(x,y) \in supp(\Delta_A)$ precisely if $x$ is in the [[causal past]] of $y$.

Written as [[generalized functions]] these satisfy

$$
\Delta_A(x,y) = \Delta_R(y,x)
\,.
$$

This implies in particular that

1. the _[[causal propagator]]_, which is the difference of the two

$$
\Delta_S \coloneqq \Delta_R - \Delta_A
$$

is skew-symmetric in its arguments (reflecting the fact that this is the [[integral kernel]] for the [[Peierls-Poisson bracket]] for the [[free field|free]] [[scalar field]] on the given spacetime);

1. the _[[Dirac propagator]]_, which is the sum of the two

$$
\Delta_D \coloneqq \Delta_R + \Delta_A
$$

is symmetric in its arguments, reflecting the fact that this is the integral kernel for [[time-ordered products]] away from the [[diagonal]].

## Definition

+-- {: .num_defn #CompactlySourceCausalSupport}
###### Definition
**(compactly sourced causal support)

Given a [[vector bundle]] $E \overset{}{\to} \Sigma$ over a manifold $\Sigma$ with [[conal causal structure|causal structure]]

Write $\Gamma_{\Sigma}(-)$ for [[space of sections|spaces of smooth sections]], and write

$$
\array{
\Gamma_{cp}(-) & \text{compact support}
\\
\Gamma_{\Sigma,\pm cp}(-) & \text{compactly sourced future/past support}
\\
\Gamma_{\Sigma,scp}(-) & \text{spacelike compact support}
\\
\Gamma_{\Sigma,(f/p)cp}(-) & \text{future/past compact support}
\\
\Gamma_{\Sigma,tcp}(-) & \text{timelike compact support}
}
$$

for the [[linear subspaces]] on those smooth sections whose [[support]] is

1. ($cp$) inside a [[compact subset]]

1. ($\pm cp$) inside the [[closed future cone]]/[[closed past cone]], respectively, of a [[compact subset]],

1. ($scp$) inside the [[closed causal cone]] of a [[compact subset]], which equivalently means that the [[intersection]] with every ([[spacelike]]) [[Cauchy surface]] is compact ([Sanders 13, theorem 2.2](#Sanders12)),

1. ($fcp$) inside the past of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),

1. ($pcp$) inside the future of a Cauchy surface ([Sanders 13, def. 3.2](#Sanders12)),

1. ($tcp$) inside the future of one Cauchy surface and the past of another ([Sanders 13, def. 3.2](#Sanders12))

=--

([Bär 14, section 1](#Baer14), [Khavkine 14, def. 2.1](#Khavkine14))

+-- {: .num_defn #AdvancedAndRetardedGreenFunctions}
###### Definition
**([[advanced and retarded Green functions]], [[causal Green function]] and [[propagators]])**

Let $\Sigma$ be a [[smooth manifold]] with [[causal structure]], let $E \to \Sigma$ be a [[smooth vector bundle]] and let
$P \;\colon\;\Gamma_\Sigma(E) \to \Gamma_\Sigma(\tilde E^\ast)$ be a [[differential operator]] on its [[space of smooth sections]].

Then a [[linear map]]

$$
\mathrm{G}_{P,\pm}
\;\colon\;
\Gamma_{\Sigma, cp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, \pm cp}(E)
$$

from spaces of sections of [[compact support]] to spaces of sections of causally sourced future/past support (def. \ref{CompactlySourceCausalSupport}) is called an _[[advanced or retarded Green function]]_ for $P$, respectively, if

1. for all $\Phi \in \Gamma_{\Sigma,cp}(E_1)$ we have

$$
\label{AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator}
G_{P,\pm} \circ P(\Phi) = \Phi
$$

and

$$
\label{AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator}
P \circ G_{P,\pm}(\Phi) = \Phi
$$

1. the [[support]] of $G_{P.\pm}(\Phi)$ is in the [[closed future cone]] or [[closed past cone]] of the support of $\Phi$, respectively.

If the advanced/retarded Green functions $G_{P\pm}$ exists, then the difference

$$
\mathrm{G}_P \coloneqq \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$

is called the _[[causal Green function]]_.

If there are [[integral kernel]], hence [[distributions in two variables]]

$$
\Delta_{P,\pm} \in \Gamma'\Sigma( \tilde E^\ast \boxtimes_\Sigma E )
$$

such that these Green functions are given by the corresponding [[integral transform]], in that
(in [[generalized function]]-notation)

$$
(G_{P,\pm} \Phi)(x)
\;=\;
\underset{y \in \Sigma}{\int}
\Delta_{P, \pm}(x,y) \cdot \Phi(y)
$$

then these integral kernels are called the advanced/retarded _[[propagators]]_; similarly then their difference

$$
\label{CausalPropagator}
\Delta_{P,S} \coloneqq \Delta_{P,+} - \Delta_{P,-}
$$

is the corresponding _[[causal propagator]]_.

=--

(e.g. [Bär 14, def. 3.2, cor. 3.10](#Baer14}))

## Properties

### Existence and uniqueness

+-- {: .num_prop #AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}
###### Proposition
**([[advanced and retarded Green functions]] of [[Green hyperbolic differential operator]] are unique)**

The [[advanced and retarded Green functions]] (def. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}) of a [[Green hyperbolic differential operator]] are unique.

=--

([Bär 14, cor. 3.12](#Baer14})

### Continuity
{#Continuity}

+-- {: .num_defn #TVSStructureOnSpacesOfSmoothSections}
###### Definition
**([[Fréchet space|Fréchet]] [[topological vector space]] on [[spaces of smooth sections]] of a [[smooth vector bundle]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]]. On its [[real vector space]] $\Gamma_\Sigma(E)$ [[space of sections|of smooth sections]] consider the [[seminorms]] indexed by a [[compact subset]] $K \subset \Sigma$ and a [[natural number]] $N \in \mathbb{N}$ and given by

$$
\array{
\Gamma_\Sigma(E) &\overset{p_K^N}{\longrightarrow}&
[0,\infty)
\\
\Phi &\mapsto&  \underset{n \leq N}{max} \left( \underset{x \in K}{sup} {\vert \nabla^n \Phi(x)\vert}\right) \,,
}
$$

where on the right we have the [[absolute values]] of the [[covariant derivatives]] of $\Phi$ for any fixed choice of [[connection on a bundle|connection]] on $E$ and [[norm]] on the [[tensor product of vector bundles]] $(T^\ast \Sigma)^{\otimes_\Sigma^n} \otimes_\Sigma E $.

This makes $\Gamma_\Sigma(E)$ a [[Fréchet space|Fréchet]] [[topological vector space]].

For $K \subset \Sigma$ any [[closed subset]] then the sub-space of sections

$$
\Gamma_{\Sigma,K}(E) \hookrightarrow \Gamma_\Sigma(E)
$$

of sections whose [[support]] is inside $K$ becomes a [[Fréchet space|Fréchet]] [[topological vector spaces]] with the induced [[subspace topology]], which makes these be [[closed subspaces]].

=--

([Bär 14, 2.1, 2.2](#Baer14))

+-- {: .num_defn #DistributionalSections}
###### Definition
**([[distribution|distributional]] [[sections]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] over a [[smooth manifold]] with [[causal structure]].

The [[vector space|vector]] [[spaces of smooth sections]] with restricted support from def. \ref{CompactlySourceCausalSupport} structures of [[topological vector spaces]] via def. \ref{TVSStructureOnSpacesOfSmoothSections}. We denote the topological [[dual spaces]] by

$$
\Gamma'_{\Sigma}(\tilde{E}^*) \coloneqq (\Gamma_{\Sigma,cp}(E))^*
$$

etc.

This is the space of _distributional sections_ of the bundle $\tilde{E}^*$.

With this notations, smooth compactly supported sections of the same bundle, regarded as the [[non-singular distributions]], constitute a [[dense subset]]

$$
\Gamma_{\Sigma,cp}(\tilde{E}^*)
\underset{\text{dense}}{\hookrightarrow}
\Gamma'_{\Sigma}(\tilde{E}^*)
\,.
$$

Imposing the same restrictions to the [[supports of distributions]] as in def. \ref{CompactlySourceCausalSupport}, we have the following subspaces of distributional sections:

$$
\Gamma'_{\Sigma,cp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,\pm cp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,scp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,fcp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,pcp}(\tilde E^\ast) ,
\Gamma'_{\Sigma,tcp}(\tilde E^\ast)
\subset
\Gamma'_{\Sigma}(\tilde E^\ast) .
$$

=--

([Sanders 13](#Sanders12), [Bär 14](#Baer14))

+-- {: .num_prop #GreenFunctionsAreContinuous}
###### Proposition
**([[causal Green functions]] of [[Green hyperbolic differential operators]] are [[continuous linear maps]])**

Given a [[Green hyperbolic differential operator]] $P$ (def. \ref{GreenHyperbolicDifferentialOperator}), the advanced, retarded and causal Green functions of $P$ (def. \ref{AdvancedAndRetardedGreenFunctions}) are [[continuous linear maps]] with respect to the [[topological vector space]] structure from def. \ref{TVSStructureOnSpacesOfSmoothSections} and also have a unique continuous extension to the spaces of sections with .larger support (def. \ref{CompactlySourceCausalSupport}) as follows:

$$
\begin{aligned}
\mathrm{G}_{P,+}
&\;\colon\;
\Gamma_{\Sigma, pcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, pcp}(E) ,
\\
\mathrm{G}_{P,-}
&\;\colon\;
\Gamma_{\Sigma, fcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma, fcp}(E) ,
\\
\mathrm{G}_{P}
&\;\colon\;
\Gamma_{\Sigma, tcp}(\tilde E^\ast)
\longrightarrow
\Gamma_{\Sigma}(E) ,
\end{aligned}
$$

such that we still have the relation

$$
\mathrm{G}_P = \mathrm{G}_{P,+} - \mathrm{G}_{P,-}
$$

and

$$
P \circ \mathrm{G}_{P,\pm} = \mathrm{G}_{P,\pm} \circ P = id
$$

and

$$
supp \mathrm{G}_{P,\pm}(\tilde{\alpha}^*) \subseteq J^\pm(supp \tilde{\alpha}^*)
\,.
$$

=--

([Bär 14, thm. 3.8, cor. 3.11](#Baer14))

## Examples

### For Klein-Gordon operator on Minkowski spacetime
{#ForKleinGordonOperatorOnMinkowskiSpacetime}

On [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ consider the [[Klein-Gordon operator]]

$$
\eta^{\mu \nu} \frac{\partial}{\partial x^\mu} \frac{\partial}{\partial x^\nu} \Phi -  \left( \tfrac{m c}{\hbar} \right)^2 \Phi \;=\; 0 \,.
$$

Its [[Fourier transform]] is

$$
- k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2
\;=\;
(k_0)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
\,.
$$

The [[dispersion relation]] of this equation we write

$$
\label{DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime}
\omega(\vec k)
\;\coloneqq\;
+ c \sqrt{ {\vert \vec k \vert}^2 + \left( \tfrac{m c}{\hbar}\right)^2 }
\,,
$$

where on the right we choose the [[non-negative real number|non-negative]] [[square root]].

$\,$

We now discuss

1. _[Advanced and regarded propagators](#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime)_

1. _[Causal propagator](#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)_

1. _[Wightman propagator](#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime)_

1. _[Feynman propagator](#FeynmanPropagator)_

1. _[Singular support and Wave front sets](#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime)_

$\,$

**[[advanced and retarded propagators]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]]**
{#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime}

+-- {: .num_prop #AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}
###### Proposition
**(mode expansion of [[advanced and retarded propagators]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**

The [[advanced and retarded Green functions]] $G_\pm$ of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] are given by [[integral kernels]] ("[[propagators]]")

$$
\Delta_\pm \in \mathcal{D}'(\mathbb{R}^{p,1}\times \mathbb{R}^{p,1})
$$

by (in [[generalized function]]-notation)

$$
G_\pm(\Phi)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int}
\Delta_{\pm}(x,y) \Phi(y) \, dvol(y)
$$

where the [[advanced and retarded propagators]] $\Delta_{\pm}(x,y)$ have the following equivalent expressions:

$$
\label{ModeExpansionForMinkowskiAdvancedRetardedPropagator}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2  -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c  + i \vec k \cdot (\vec x - \vec  y) }
\right)
d^p \vec k
& \vert & \text{if} \,  \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec  y) }
d^p \vec k
& \vert & \text{if} \,  \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$

Here $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]].

=--

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

The [[Klein-Gordon operator]] is a [[Green hyperbolic differential operator]] ([this example](Green+hyperbolic+partial+differential+equation#GreenHyperbolicKleinGordonOperator)) therefore its advanced and retarded Green functions exist uniquely (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}).
Moreover,  prop. \ref{GreenFunctionsAreContinuous} says that they are [[continuous linear functionals]] with respect to the [[topological vector space]] [[structures]] on [[spaces of smooth sections]] (def. \ref{TVSStructureOnSpacesOfSmoothSections}). In the case of the [[Klein-Gordon operator]] this just means that

$$
G_{\pm}
\;\colon\;
C^\infty_{cp}(\mathbb{R}^{p,1})
\longrightarrow
C^\infty_{\pm cp}(\mathbb{R}^{p,1})
$$

are [[continuous linear functionals]] in the standard sense of [[distributions]]. Therefore the
[[Schwartz kernel theorem]]  implies the existence of [[integral kernels]] being [[distributions in two variables]]

$$
\Delta_{\pm} \in \mathcal{D}(\mathbb{R}^{p,1} \times \mathbb{R}^{p,1})
$$

such that, in the notation of [[generalized functions]],

$$
(G_\pm \alpha)(x)
\;=\;
\underset{\mathbb{R}^{p,1}}{\int} \Delta_{\pm}(x,y) \alpha(y) \, dvol(y)
\,.
$$

These integral kernels are the advanced/retarded "[[propagators]]". We now compute
these [[integral kernels]] by making an Ansatz and showing that it has the defining properties,
which identifies them by the uniqueness statement of prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.

We make use of the fact that the [[Klein-Gordon equation]] is [[invariant]] under the defnining [[action]]
of the [[Poincaré group]] on [[Minkowski spacetime]], which is a [[semidirect product group]] of the [[translation group]]
and the [[Lorentz group]].

Since the [[Klein-Gordon operator]] is invariant, in particular, under [[translations]] in $\mathbb{R}^{p,1}$ it is clear that the propagators, as a [[distribution in two variables]], depend only on the difference of its two arguments

$$
  \label{TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime}
  \Delta_{\pm}(x,y) = \Delta_{\pm}(x-y)
\,.
$$

Since moreover the [[Klein-Gordon operator]] is [[formally adjoint differential operator|formally self-adjoint]] ([this prop.](Klein-Gordon+equation#FormallySelfAdjointKleinGordonOperator)) this implies that for $P$ the Klein the equation (eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator)

$$
P \circ G_\pm  = id
$$

is equivalent to the equation (eq:AdvancedRetardedGreenFunctionIsLeftInverseToDiffOperator)

$$
G_\pm \circ P = id
\,.
$$

Therefore it is sufficient to solve for the first of these two equation, subject to
the defining support conditions. In terms of the [[propagator]] [[integral kernels]] this means that we have to solve the [[distribution|distributional]] equation

$$
\label{KleinGordonEquationOnAdvacedRetardedPropagator}
\left(
\eta^{\mu \nu}
\frac{\partial}{\partial x^\mu}
\frac{\partial}{\partial x^\nu}
-
\left( \tfrac{m c}{\hbar} \right)^2
\right)
\Delta_\pm(x-y)
\;=\;
\delta(x-y)
$$

subject to the condition that the [[support of a distribution|distributional support]] is

$$
supp\left( \Delta_{\pm}(x-y) \right)
\subset
\left\{
{\vert x-y\vert^2_\eta}\lt 0
\;\,,\;
\pm(x^0 - y^ 0) \gt 0
\right\}
\,.
$$

We make the _Ansatz_ that we assume that $\Delta_{\pm}$, as a distribution in a single variable $x-y$, is a [[tempered distribution]]

$$
\Delta_\pm \in \mathcal{S}'(\mathbb{R}^{p,1})
\,,
$$

hence amenable to [[Fourier transform of distributions]]. If we do find a solution this way, it is guaranteed to be the unique solution by prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.

By [this prop.](Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces)
the [[Fourier transform of distributions|distributional Fourier transform]] of equation (eq:KleinGordonEquationOnAdvacedRetardedPropagator) is

$$
\begin{aligned}
\label{FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_{\pm}}(k)
& =
\widehat{\delta}(k)
\\
& =
1
\end{aligned}
\,,
$$

where in the second line we used the [[Fourier transform of distributions|Fourier transform]] of the [[delta distribution]]
from [this example](Dirac+distribution#FourierTransformOfDeltaDistribution).

Notice that this implies that the [[Fourier transform]] of the [[causal propagator]]

$$
  \Delta_S \coloneqq \Delta_+ - \Delta_-
$$

satisfies the homogeneous equation:

$$
\label{FourierVersionOfPDEForKleinGordonCausalPropagator}
\left(
- \eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2
\right)
\widehat{\Delta_S}(k)
\;=\;
0
\,,
$$

Hence we are now reduced to finding solutions $\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})$ to (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator) such that their [[Fourier inversion theorem|Fourier inverse]] $\Delta_\pm$ has the required [[support of a distribution|support]] properties.

We discuss this by a variant of the [[Cauchy principal value]]:

Suppose the following [[limit of a sequence|limit]] of [[non-singular distributions]] in the [[variable]] $k \in \mathbb{R}^{p,1}$
exists in the space of [[distributions]]

$$
\label{LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 } }{\lim}
\frac{1}{ (k_0 \mp i \epsilon)^2  - {\vert \vec k\vert^2} - \left( \tfrac{m c}{\hbar} \right)^2 }
\;\in\;
\mathcal{D}'(\mathbb{R}^{p,1})
$$

meaning that for each [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ the [[limit of a sequence|limit]] in $\mathbb{C}$

$$
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 }   }{\lim}
\underset{\mathbb{R}^{p,1}}{\int} \frac{b(k)}{ (k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2  }
d^{p+1}k
\;\in\;
\mathbb{C}
$$

exists. Then this limit is clearly a solution to the distributional equation (eq:FourierVersionOfPDEForKleinGordonAdvancedRetardedPropagator)
because on those bump functions $b(k)$ which happen to be products with $\left(-\eta^{\mu \nu}k_\mu k-\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)$
we clearly have

$$
\begin{aligned}
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 }   }{\lim}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) b(k)
}{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2
}
d^{p+1}k
& =
\underset{\mathbb{R}^{p,1}}{\int}
\underset{= 1}{
\underbrace{
\underset{ {\epsilon \in (0,\infty)} \atop { \epsilon \to 0 }   }{\lim}
\frac{
\left( -\eta^{\mu \nu} k_\mu k_\nu - \left( \tfrac{m c}{\hbar} \right)^2 \right) }{
(k_0\mp i \epsilon)^2 - {\vert \vec k\vert}^2 - \left( \tfrac{m c}{\hbar} \right)^2  }
}
}
b(k)\, d^{p+1}k
\\
& =
\langle 1, b\rangle
\,.
\end{aligned}
$$

Moreover, if the limiting distribution (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator)
exists, then it is clearly a [[tempered distribution]], hence we may apply [[Fourier inversion theorem|Fourier inversion]]
to  obtain [[Green functions]]

$$
\label{AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets}
\Delta_{\pm}(x,y)
\;\coloneqq\;
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\frac{1}{(2\pi)^{p+1}}
\underset{\mathbb{R}^{p,1}}{\int}
\frac{e^{i k_\mu (x-y)^\mu}}{
(k_0 \mp i \epsilon  )^2 - {\vert \vec k\vert}^2  - \left(\tfrac{m c}{\hbar}\right)^2
}
d k_0 d^p \vec k
\,.
$$

To see that this is the correct answer, we need to check the defining support property.

Finally, by the [[Fourier inversion theorem]], to show that the [[limit of a sequence|limit]] (eq:LimitOverImaginaryOffsetForFourierTransformedAdvancedRetardedPropagator) indeed exists it is sufficient to show that the
limit in (eq:AdvancedRetardedPropagatorViaFourierTransformOfLLimitOverImaginaryOffsets) exists.

We compute as follows

$$
\label{TheSupportOfTheCandidateAdvancedRetardedPropagatorIsinTheFutureOrPastRespectively}
\begin{aligned}
\Delta_\pm(x-y)
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i\epsilon)^2 - {\vert \vec k\vert}^2  -\left( \tfrac{m c}{\hbar}\right)^2
}
\, d k_0 \, d^p \vec k
\\
& =
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
(k_0 \mp i \epsilon)^2 - \left(\omega(\vec k)/c\right)^2
}
\, d k_0 \, d^p \vec k
\\
&=
\frac{1}{(2\pi)^{p+1}}
\underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
\int \int
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
\left(
(k_0 \mp i\epsilon) - \omega(\vec  k)/c
\right)
\left(
(k_0 \mp i \epsilon) + \omega(\vec k)/c
\right)
}
\, d k_0 \, d^p \vec k
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec  y)}
\right)
d^p \vec k
& \vert & \text{if} \,  \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp 1}{(2\pi)^{p}}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec  y) }
d^p \vec k
& \vert & \text{if} \,  \pm (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\end{aligned}
$$

where $\omega(\vec k)$ denotes the [[dispersion relation]] (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) of the [[Klein-Gordon equation]]. The last step is simply the application of [[Euler's formula]] $\sin(\alpha) = \tfrac{1}{2 i }\left( e^{i \alpha} - e^{- i \alpha}\right)$.

Here the key step is the application of [[Cauchy's integral formula]] in the fourth step. We spell this out now for $\Delta_+$, the discussion for $\Delta_-$ is the same, just with the appropriate signs reversed.

1. If  $(x^0 - y^0) \gt 0$ thn the expression $e^{ik_0 (x^0 - y^0)}$ decays with _[[positive number|positive]] [[imaginary part]]_ of $k_0$, so that we may expand the [[integration]] [[domain]] into the [[upper half plane]] as

$$
\begin{aligned}
\int_{-\infty}^\infty d k_0
& = \phantom{+}
\int_{-\infty}^0 d k_0 + \int_{0}^{+ i \infty} d k_0
\\
& = + \int_{+i \infty}^0 d k_0  + \int_0^\infty d k_0
\,;
\end{aligned}
$$

Conversely, if $(x^0 - y^0) \lt 0$ then we may analogously expand into the [[lower half plane]].

1. This integration domain may then further be completed to two [[contour integrations]]. For the expansion into the [[upper half plane]] these encircle counter-clockwise the [[poles]] at $\pm \omega(\vec k)+ i\epsilon \in \mathbb{C}$, while for expansion into the [[lower half plane]] no poles are being encircled.

<img src="https://ncatlab.org/nlab/files/ContourForAdvancedPropagator.png" height="280">

1. Apply [[Cauchy's integral formula]] to find in the case $(x^0 - y^0)\gt 0$ the sum of the [[residues]] at these two [[poles]] times $2\pi i$, zero in the other case. (For the retarded propagator we get $- 2 \pi i$ times the residues, because now the contours encircling non-trivial poles go clockwise).

1. The result is now non-singular at $\epsilon = 0$ and therefore the [[limit of a sequence|limit]] $\epsilon \to 0$ is now computed by evaluating at $\epsilon = 0$.

This computation shows a) that the limiting distribution indeed exists, and b) that the [[support of a distribution|support]]
of $\Delta_+$ is in the future, and that of $\Delta_-$ is in the past.

Hence it only remains to see now that the support of $\Delta_\pm$ is inside the [[causal cone]].
But this follows from the previous argument, by using that the [[Klein-Gordon equation]] is invariant under
[[Lorentz transformations]]: This implies that the support is in fact in the [[future]] of _every_ spacelike
slice through the origin in $\mathbb{R}^{p,1}$, hence in the [[closed future cone]] of the origin.

=--

+-- {: .num_cor #CausalPropagatorIsSkewSymmetric}
###### Corollary
**([[causal propagator]] is skew-symmetric)**

Under reversal of arguments the [[advanced and retarded causal propagators]] from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} are related by

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

It follows that the [[causal propagator]] $\Delta \coloneqq \Delta_+ - \Delta_-$ is skew-symmetric in its arguments:

$$
\Delta_S(x-y) = - \Delta_S(y-x)
\,.
$$

=--

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

By prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} we have with (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator)

$$
\begin{aligned}
\Delta_\pm(y-x)
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c -  i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c  - i \vec k \cdot (\vec x - \vec  y) }
\right)
d^p \vec k
& \vert & \text{if} \,  \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\pm i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{-i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{+i \omega(\vec k)(x^0 - y^0)/c  - i \vec k \cdot (\vec x - \vec  y) }
\right)
d^p \vec k
& \vert & \text{if} \,  \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\left\{
\array{
\frac{\mp i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{+i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c  - i \vec k \cdot (\vec x - \vec  y) }
\right)
d^p \vec k
& \vert & \text{if} \,  \mp (x^0 - y^0) \gt 0
\\
0 & \vert & \text{otherwise}
}
\right.
\\
& =
\Delta_\mp(x-y)
\end{aligned}
$$

Here in the second step we applied [[change of integration variables]] $\vec k \mapsto - \vec k$ (which introduces _no_ sign because in addition to $d \vec k \mapsto - d \vec k$ the integration domain reverses [[orientation]]).

=--

$\,$

**[[causal propagator]]**
{#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}

+-- {: .num_prop #ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}
###### Proposition
**(mode expansion of [[causal propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**

The [[causal propagator]] (eq:CausalPropagator) for the [[Klein-Gordon equation]] for [[mass]] $m$ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ is given, in [[generalized function]] notation, by

$$
\label{CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec  y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\,,
\end{aligned}
$$

where in the second line we used [[Euler's formula]] $sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)$.

In particular this shows that the [[causal propagator]] is [[real part|real]], in that it is equal to its [[complex conjugation|complex conjugate]]

$$
  \label{CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal}
  \left(\Delta_S(x,y)\right)^\ast = \Delta_S(x,y)
  \,.
$$

=--

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

By definition and using the expression from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} for the [[advanced and retarded causal propagators]] we have

$$
\begin{aligned}
\Delta_S(x,y)
& \coloneqq
\Delta_+(x,y) - \Delta_-(x,y)
\\
& =
\left\{
\array{
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec  y)}
\right)
d^p \vec k
& \vert & \text{if} \,  + (x^0 - y^0) \gt 0
\\
\frac{(-1) (-1) i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec  y)}
\right)
d^p \vec k
& \vert & \text{if} \,  - (x^0 - y^0) \gt 0
}
\right.
\\
& =
\frac{+ i}{(2\pi)^{p}}
\int
\frac{1}{2\omega(\vec k)/c}
\left(
e^{i \omega(\vec k)(x^0 - y^0)/c +  i \vec k \cdot (\vec x -\vec y)}
-
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec  y)}
\right)
d^p \vec k
\\
& =
\frac{-1}{(2\pi)^p}
\int
\frac{1}{\omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x -\vec y)}
d^p \vec k
\end{aligned}
$$

For the reality, notice from the last line that

$$
  \begin{aligned}
    \left(\Delta_S(x,y)\right)^\ast
    & =
    \frac{-1}{(2\pi)^p}
    \int
    \frac{1}{\omega(\vec k)/c}
    \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
     e^{-i \vec k \cdot (\vec x -\vec y)}
    d^p \vec k
    \\
    & =
    \frac{-1}{(2\pi)^p}
    \int
    \frac{1}{\omega(\vec k)/c}
    \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
     e^{+i \vec k \cdot (\vec x -\vec y)}
    d^p \vec k
    \\
    & =
    \Delta_S(x,y)
    \,,
   \end{aligned}
$$

where in the last step we used the [[change of integration variables]] $\vec k \mapsto - \vec k$
(whih introduces no sign, since on top of $d \vec k \mapsto - d \vec k$ the [[orientation]] of the integration [[domain]] changes).

=--

We consider a couple of equivalent expressions for the causal propagator which are useful for computations:

+-- {: .num_prop #CausalPropagatorForKleinGordonOnMinkowskiAsContourIntegral}
###### Proposition
**([[causal propagator]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]] as a [[contour integral]])**

The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]], given as a [[contour integral]] along a curve $C(\vec k)$ going counter-clockwise around the two [[poles]] at $k_0 = \pm \omega(\vec k)/c$:

$$
\Delta_S(x,y)
\;=\;
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2g  }
\,d k_0
\,d^{p} k
\,.
$$

=--

<img src="https://ncatlab.org/nlab/files/ContourForCausalPropagator.png" height="160">

> graphics grabbed from [Kocic 16](#Kocic16)

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

By [[Cauchy's integral formula]] we compute as follows:

$$
\begin{aligned}
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{e^{i k_\mu (x^\mu - y^\mu)}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
\,d k_0
\,d^{p} k
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 x^0} e^{ i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
\int
\underset{C(\vec k)}{\oint}
\frac{
e^{i k_0 (x^0 - y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 + \omega(\vec k)/c )
( k_0 - \omega(\vec k)/c )
}
\,d k_0
\,d^p \vec k
\\
& =
(2\pi)^{-(p+1)}
2\pi i
\int
\left(
\frac{
e^{i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}
{
2 \omega(\vec k)/c
}
-
\frac{
e^{ - i \omega(\vec k) (x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
}{
2 \omega(\vec k)/c
}
\right)
\,d^p \vec k
\\
& =
i
(2\pi)^{-p}
\int
\frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\,d^p \vec k
\,.
\end{aligned}
$$

The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}

=--

+-- {: .num_prop #CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator}
###### Proposition
**([[causal propagator]] as [[Fourier transform]] of [[delta distribution]]  on the [[Fourier transform|Fourier transformed]] [[Klein-Gordon operator]])**

The [[causal propagator]] for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] has the following equivalent expression, as a [[generalized function]]:

$$
\Delta_S(x,y)
\;=\;
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
\,,
$$

where the [[integrand]] is the product of the [[sign function]] of $k_0$ with the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]] and a [[plane wave]] factor.

=--

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

By decomposing the integral over $k_0$ into its negative and its positive half, and applying the [[change of integration variables]] $k_0 = \pm\sqrt{h}$ we get

$$
\begin{aligned}
i (2\pi)^{-p} \int \delta\left( k_\mu k^\mu + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 ) e^{ i k_\mu (x-y)^\mu } d^{p+1} k
& =
+ i (2\pi)^{-p} \int \int_0^\infty \delta\left( -k_0^2 + \vec k^2 + \left( \tfrac{m c}{\hbar}\right)^2 \right) e^{ i k_0 (x^0 - y^0) + i \vec k \cdot (\vec x - \vec y)}  d k_0 \, d^p \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_{-\infty}^0 \delta\left( -k_0^2 + \vec k^2 + \left(\tfrac{m c}{\hbar}\right)^2 \right)  e^{ i k_0 (x^0 - y^0)+ i \vec k \cdot (\vec x - \vec y) } d k_0 \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \int_0^\infty \frac{1}{2 \sqrt{h}} \delta\left( -h + \omega(\vec k)^2/c^2  \right) e^{ + i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x } d h \, d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \int_0^\infty  \frac{1}{2 \sqrt{h}} \delta\left( - h + \omega(\vec k)^2/c^2 \right) e^{ - i \sqrt{h} (x^0 - y^0) + i \vec k \cdot \vec x }  d h \, d^{p} \vec k
\\
& =
+i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x}  d^{p} \vec k
\\
& \phantom{=}
- i (2\pi)^{-p} \int \frac{1}{2 \omega(\vec k)/c} e^{ - i \omega(\vec k) (x-y)^0/c + i \vec k \cdot \vec x }  d^{p} \vec k
\\
& = -(2 \pi)^{-p} \int \frac{1}{\omega(\vec k)/c}
sin\left( \omega(\vec k)(x-y)^0/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\end{aligned}
$$

The last line is the expression for the causal propagator from prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.

=--

$\,$

**[[Wightman propagator]]**
{#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime}

Prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} exhibits the [[causal propagator]] of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] as the difference of a contribution for [[positive real number|positive]] temporal [[angular frequency]] $k_0 \propto \omega(\vec k)$ (hence positive [[energy]] $\hbar \omega(\vec k)$ and a contribution of negative temporal [[angular frequency]].

The [[positive real number|positive]] [[frequency]] contribution to the [[causal propagator]] is called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below), also known as the the _[[vacuum state]] [[2-point function]] of the [[free field|free]] [[real scalar field]] on [[Minkowski spacetime]]_. Notice that the temporal component
of the [[wave vector]] is proportional to the _negative_ [[angular frequency]]

$$
k_0 = -\omega/c
$$

(see at _[[plane wave]]_), therefore the appearance of the [[step function]] $\Theta(-k_0)$ in (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) below:

+-- {: .num_defn #StandardHadamardDistributionOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] or [[vacuum state]] [[2-point function]] for [[Klein-Gordon operator]] on [[Minkowski spacetime]])**

The _[[Wightman propagator]]_ for the [[Klein-Gordon operator]] at [[mass]] $m$ on [[Minkowski spacetime]] is the [[tempered distribution|tempered]] [[distribution in two variables]] $\Delta_H \in \mathcal{S}'(\mathbb{R}^{p,1})$ which as a [[generalized function]] is given by the expression

$$
\label{HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime}
\begin{aligned}
\Delta_H(x,y)
& \coloneqq
\frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
\\
& =
\frac{1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
\, d^p \vec k
\,,
\end{aligned}
$$

Here in the first line we have in the [[integrand]] the [[delta distribution]] of the [[Fourier transform]] of the [[Klein-Gordon operator]]  times a [[plane wave]] and times the [[step function]] $\Theta$ of the temporal component of the [[wave vector]]. In the second line we used the [[change of integration variables]] $k_0 = \sqrt{h}$, then the definition of the [[delta distribution]] and the fact that $\omega(\vec k)$ is by definition the [[non-negative real number|non-negative]] solution to the Klein-Gordon [[dispersion relation]].

=--

(e.g. [Khavkine-Moretti 14, equation (38) and section 3.4](Hadamard+distribution#KhavineMoretti14))

+-- {: .num_prop #ContourIntegralForStandardHadamardPropagatorOnMinkowskiSpacetime}
###### Proposition
**([[contour integral]] representation of the [[Wightman propagator]] for the [[Klein-Gordon operator]] on [[Minkowski spacetime]])

The [[Wightman propagator]] from
def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} is
equivalently given by the [[contour integral]]

$$
\label{StandardHadamardPropagatorOnMinkowskiSpacetimeInTermsOfContourIntegral}
\Delta_H(x,y)
\;=\;
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{-i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
\,,
$$

where the [[Jordan curve]] $C_+(\vec k) \subset \mathbb{C}$ runs counter-clockwise, enclosing the point $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$, but not enclosing the point $- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$.

<img src="https://ncatlab.org/nlab/files/ContourForHadamardPropagator.png" height="200">

> graphics grabbed from [Kocic 16](#Kocic16)

=--

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

We compute as follows:

$$
\begin{aligned}
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{e^{ - i k_\mu (x-y)^\mu}}{ -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 }
d k_0
d^{p} k
& =
-i(2\pi)^{-(p+1)}
\int
\oint_{C_+(\vec k)}
\frac{
e^{ -i k_0 x^0} e^{i \vec k \cdot (\vec x - \vec y)}
}{
k_0^2 - \omega(\vec k)^2/c^2
}
d k_0
d^p \vec k
\\
& =
-i(2\pi)^{-(p+1)}
\int
\underset{C_+(\vec k)}{\oint}
\frac{
e^{ - i k_0 (x^0-y^0)} e^{i \vec k \cdot (\vec x - \vec y)}
}{
( k_0 - \omega_\epsilon(\vec k)  )
( k_0 + \omega_\epsilon(\vec k) )
}
d k_0
d^p \vec k
\\
& =
(2\pi)^{-p}
\int
\frac{1}{2 \omega(\vec k)}
e^{-i \omega(\vec k) (x^0-y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
d^p \vec k
\,.
\end{aligned}
$$

The last step is application of [[Cauchy's integral formula]], which says that the [[contour integral]] picks up the [[residue]] of the [[pole]] of the [[integrand]] at $+ \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$. The last line is $\Delta_H(x,y)$, by definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime}.

=--

+-- {: .num_prop #SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(skew-symmetric part of [[Wightman propagator]] is the [[causal propagator]])**

The [[Wightman propagator]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is of the form

$$
  \label{DeompositionOfHadamardPropagatorOnMinkowkski}
  \begin{aligned}
    \Delta_H
    & =
    \tfrac{i}{2} \Delta_S
    +
    H
    \\
    & =
    \tfrac{i}{2}
    \left(
      \Delta_+ - \Delta_-
    \right)
    +
    H
  \end{aligned}
  \,,
$$

where

1. $\Delta_S$ is the [[causal propagator]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}),
   which is real (eq:CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsReal) and skew-symmetric (prop. \ref{CausalPropagatorIsSkewSymmetric})

   $$
     (\Delta_S(x,y))^\ast = \Delta_S(x,y)
     \phantom{AA}
     \,,
     \phantom{AA}
     \Delta_S(y,x) = - \Delta_S(x,y)
   $$

1. $H$ is real and symmetric

   $$
     (H(x,y))^\ast = H(x,y)
     \phantom{AA}
     \,,
     \phantom{AA}
     H(y,x) = H(x,y)
   $$

=--

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

By applying [[Euler's formula]] to (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) we obtain

$$
  \label{SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime}
  \begin{aligned}
  \Delta_H(x,y)
  & =
  \frac{1}{(2\pi)^p}
   \int \frac{1}{2 \omega(\vec k)/c}
   e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
   \, d^p \vec k
  \\
  & =
   \tfrac{i}{2}
   \underset{= \Delta_S(x,y)}{
   \underbrace{
     \frac{-1}{(2\pi)^p}
     \int
     \frac{1}{\omega(\vec k)/c}
     \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
     e^{i \vec k \cdot (\vec x - \vec y) }
     \, d^p \vec k
   }}
   \;+\;
   \underset{
     \coloneqq H(x,y)
   }{
   \underbrace{
    \frac{1}{(2\pi)^p}
     \int \frac{1}{2 \omega(\vec k)/c}
     \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
     e^{i \vec k \cdot (\vec x - \vec y) }
     \, d^p \vec k
   }}
  \end{aligned}
$$

On the left this identifies the [[causal propagator]] by (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime), prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}.

The second summand changes, both under complex conjugation as well as under $(x-y) \mapsto (y-x)$,
via [[change of integration variables]] $\vec k \mapsto - \vec k$ (because the [[cosine]] is an even function).
This does not change the integral, and hence $H$ is symmetric.

=--

$\,$

**[[Feynman propagator]]**
 {#FeynmanPropagator}

We have seen that the [[positive real number|positive]] [[frequency]] component of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) is the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) given, according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}, by (eq:DeompositionOfHadamardPropagatorOnMinkowkski)

$$
  \begin{aligned}
    \Delta_H
    & =
    \tfrac{i}{2} \Delta_S
    +
    H
    \\
    & =
    \tfrac{i}{2}
    \left(
      \Delta_+ - \Delta_-
    \right)
    +
    H
  \end{aligned}
  \,.
$$

There is an evident variant of this combination, which will be of interest:

+-- {: .num_defn #FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**

The _[[Feynman propagator]]_ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[linear combination]]

$$
  \Delta_F
  \coloneqq
    \tfrac{i}{2}
    \left(
      \Delta_+ + \Delta_-
    \right)
    +
    H
$$

where the first term is proportional to the sum of the [[advanced and retarded propagators]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) and the second is the symmetric part of the [[Wightman propagator]] according to prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime}.

Similarly the _[[anti-Feynman propagator]]_ is

$$
  \Delta_{\overline{F}}
  \coloneqq
    \tfrac{i}{2}
    \left(
      \Delta_+ + \Delta_-
    \right)
    -
    H
  \,.
$$

=--

+-- {: .num_prop #ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**(mode expansion for [[Feynman propagator]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**

The [[Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is given by the following equivalent expressions

$$
  \begin{aligned}
  \Delta_F(x,y)
  & =
  \left\{
    \array{
    \frac{1}{(2\pi)^p}
    \int
    \frac{1}{\omega(\vec k)c}
    e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \gt 0
    \\
    \frac{1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
      \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
      \\
      \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
    }
  \right.
  \end{aligned}
$$

Similarly the [[anti-Feynman propagator]] is equivalently given by

$$
  \begin{aligned}
  \Delta_{\overline{F}}(x,y)
  & =
  \left\{
    \array{
    \frac{-}{(2\pi)^p}
    \int
    \frac{1}{\omega(\vec k)c}
    e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \gt 0
    \\
    \frac{-}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
      -\Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
      \\
      -\Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
    }
  \right.
  \end{aligned}
$$

=--

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

By the mode expansion of $\Delta_{\pm}$ from (eq:ModeExpansionForMinkowskiAdvancedRetardedPropagator) and the mode expansion of $H$ from (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have

$$
  \begin{aligned}
   \Delta_F(x,y)
   & =
  \left\{
    \array{
      \underset{
        = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
      }{
      \underbrace{
      \frac{- i}{(2\pi)^{p}}
      \int
      \frac{1}{2 \omega(\vec k)/c}
      \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
      e^{i \vec k \cdot (\vec x - \vec  y) }
      \, d^p \vec k
      }
      }
      +
      \underset{
        = H(x,y)
      }{
      \underbrace{
      \frac{1}{(2\pi)^p}
      \int \frac{1}{2 \omega(\vec k)/c}
      \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
      e^{i \vec k \cdot (\vec x - \vec y) }
      \, d^p \vec k
      }
      }
      &\vert&
      (x^0 - y^0) \gt 0
      \\
      \underset{
         = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
      }{
      \underbrace{
      \frac{+ i}{(2\pi)^{p}}
      \int
      \frac{1}{2 \omega(\vec k)/c}
      \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
      e^{i \vec k \cdot (\vec x - \vec  y) }
      \, d^p \vec k
      }
      }
      +
      \underset{ = H(x,y) }{
      \underbrace{
      \frac{1}{(2\pi)^p}
      \int \frac{1}{2 \omega(\vec k)/c}
      \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
      e^{i \vec k \cdot (\vec x - \vec y) }
      \, d^p \vec k
      }
      }
      &\vert&
      (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
    \frac{1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{-i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \gt 0
    \\
    \frac{1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
      \Delta_H(x,y) &\vert& (x^0 - y^0) \gt 0
      \\
      \Delta_H(y,x) &\vert& (x^0 - y^0) \lt 0
    }
  \right.
  \end{aligned}
$$

where in the second line we used [[Euler's formula]]. The last line follows by comparison with (eq:HadamardPropagatorForKleinGordonOperatorOnMinkowskiSpacetime) and using that the integral over $\vec k$
is invariant under $\vec k \mapsto - \vec k$.

The computation for $\Delta_{\overline{F}}$ is the same, only now with a minus sign in front of the [[cosine]]:

$$
  \begin{aligned}
   \Delta_{\overline{F}}(x,y)
   & =
  \left\{
    \array{
      \underset{
        = \tfrac{i}{2} \Delta_+(x,y) + 0 \;\text{for}\; (x^0 - y^0) \gt 0
      }{
      \underbrace{
      \frac{- i}{(2\pi)^{p}}
      \int
      \frac{1}{2 \omega(\vec k)/c}
      \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
      e^{i \vec k \cdot (\vec x - \vec  y) }
      \, d^p \vec k
      }
      }
      -
      \underset{
        = H(x,y)
      }{
      \underbrace{
      \frac{1}{(2\pi)^p}
      \int \frac{1}{2 \omega(\vec k)/c}
      \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
      e^{i \vec k \cdot (\vec x - \vec y) }
      \, d^p \vec k
      }
      }
      &\vert&
      (x^0 - y^0) \gt 0
      \\
      \underset{
         = 0 + \tfrac{i}{2}\Delta_-(x,y) \;\text{for}\; (x^0 - y^0) \lt 0
      }{
      \underbrace{
      \frac{+ i}{(2\pi)^{p}}
      \int
      \frac{1}{2 \omega(\vec k)/c}
      \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
      e^{i \vec k \cdot (\vec x - \vec  y) }
      \, d^p \vec k
      }
      }
      -
      \underset{ = H(x,y) }{
      \underbrace{
      \frac{1}{(2\pi)^p}
      \int \frac{1}{2 \omega(\vec k)/c}
      \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
      e^{i \vec k \cdot (\vec x - \vec y) }
      \, d^p \vec k
      }
      }
      &\vert&
      (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
    \frac{-1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{+i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \gt 0
    \\
    \frac{-1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{-1i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
      - \Delta_H(y,x) &\vert& (x^0 - y^0) \gt 0
      \\
      - \Delta_H(x,y) &\vert& (x^0 - y^0) \lt 0
    }
  \right.
  \end{aligned}
$$

=--

As before for the [[causal propagator]], there are equivalent reformulations of the [[Feynman propagator]], which are useful for computations:

+-- {: .num_prop #FeynmanPropagatorAsACauchyPrincipalvalue}
###### Proposition
**([[Feynman propagator]] as a [[Cauchy principal value]])**

The [[Feynman propagator]] and [[anti-Feynman propagator]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is equivalently given by the following expressions, respectively:

$$
  \begin{aligned}
  \left.
  \array{
    \Delta_F(x,y)
    \\
    \Delta_{\overline{F}}(x,y)
  }
  \right\}
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu (x^\mu - y^\mu)}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \end{aligned}
$$

where we have a [[limit of a sequence|limit]] of [[distributions]] as for the [[Cauchy principal value]] ([this prop](Cauchy+principal+vlue#CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta)).

=--

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

We compute as follows:

$$
  \begin{aligned}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu (x^\mu - y^\mu)}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu (x^\mu - y^\mu)}
  }{
    (k_0)^2
      -
    \underset{
      \coloneqq \omega_{\pm\epsilon}(\vec k)^2/c^2
    }{\underbrace{ \left( \omega(\vec k)^2/c^2 \pm i \epsilon \right) }}
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu (x^\mu - y^\mu)}
  }{
     \left(
       k_0 - \omega_{\pm \epsilon}(\vec k)/c
     \right)
     \left(
       k_0 + \omega_{\pm \epsilon}(\vec k)/c
     \right)
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \left\{
    \array{
    \frac{\mp 1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{\pm i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \gt 0
    \\
    \frac{\mp 1}{(2\pi)^p}
    \int
    \frac{1}{2\omega(\vec k)c}
    e^{\mp i\omega(\vec k)(x^0 - y^0)/c} e^{i \vec k \cdot (\vec x - \vec y)}
    \, d^p \vec k
    &\vert&
    (x^0 - y^0) \lt 0
    }
  \right.
  \\
  & =
  \left\{
    \array{
      \Delta_F(x,y)
      \\
      \Delta_{\overline{F}}(x,y)
    }
  \right.
  \end{aligned}
$$

Here

1. In the first step we introduced the [[complex number|complex]] [[square root]] $\omega_{\pm \epsilon}(\vec k)$. For this to be compatible with the choice of _non-negative_ square root for $\epsilon = 0$ in (eq:DispersionRelationForKleinGordonooeratorObMinkowskiSpacetime) we need to choose that complex square root whose [[complex phase]] is one half that of $\omega(\vec k)^2 - i \epsilon$ (instead of that plus [[π]]). This means that $\omega_{+ \epsilon}(\vec k)$ is in the _[[upper half plane]]_ and $\omega_-(\vec k)$ is in the [[lower half plane]].

1. In the third step we observe that

   1. for $(x^0 - y^0) \gt 0$ the [[integrand]] decays for [[positive real number|positive]] [[imaginary part]] and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[upper half plane]];

   1. for $(x^0 - y^0) \lt 0$ the integrand decays for [[negative real number|negative]] [[imaginary part]]  and hence the integration over $k_0$ may be deformed to a [[Jordan curve|contour]] which encircles the [[pole]] in the [[lower half plane]]

   and then apply [[Cauchy's integral formula]] which picks out $2\pi i$ times the [[residue]] a these poles.

   <img src="https://ncatlab.org/nlab/files/ContourForFeynmanPropagator.png" height="300">

   Notice that when completing to a contour in the [[lower half plane]] we pick up a minus signs from the fact that now the contour runs clockwise.

1. In the fourth step we used prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime}.

=--

$\,$

$\,$

**[[singular support]] and [[wave front sets]]**
{#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime}

We now discuss the [[singular support]] and the [[wave front sets]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]].

+-- {: .num_prop #SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[causal propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**

The [[singular support]] of the [[causal propagator]] $\Delta_S$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]],
regarded via [[translation]] [[invariant|invariance]] as a [[generalized function]] in a single variable (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
is the [[light cone]] of the origin:

$$
  supp_{sing}(\Delta_S)
  \;=\;
  \left\{
    x \in \mathbb{R}^{p,1}
    \,\vert\,
    {\vert x\vert}^2_\eta = 0
  \right\}
  \,.
$$

=--

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

By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate
Fourier transforms:

$$
  \begin{aligned}
    \Delta_S(x)
    & \propto
    \widehat{
      \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
    }
    \\
    &\propto
    \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
    \star
    \widehat{sgn( k_0 )}
  \end{aligned}
$$

By ([Gel'fand-Shilov 66, III 2.11 (7), p 294](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell), the [[singular support]]
of the first convolution factor is the [[light cone]].

The second factor is

$$
  \begin{aligned}
    \widehat{sgn(k_0)}
    & \propto
    \left(2\widehat{\Theta(k_0)} - \widehat{1}\right) \delta(\vec k)
    \\
    & \propto
    \left(2\tfrac{1}{i x^0 + 0^+}  - \delta(x^0)\right) \delta(\vec k)
  \end{aligned}
$$

(by [this example](Dirac+distribution#FourierTransformOfDeltaDistribution) and [this example](Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution)) and hence the [[wave front set]] of the second
factor is

$$
  WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$

(by [this example](wavefront+set#WaveFrontOfDeltaDistribution) and [this example](Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation)).

With this the statement follows, via a [[partition of unity]], from [this prop.](convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions).

For illustration we now make this general argument more explicit in the special case of [[spacetime]] [[dimension]]

$$
  p + 1 = 3 + 1
$$

by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].

We follow ([Scharf 95 (2.3.18)](causal+perturbation+theory#Scharf95)).

Consider the formula for the [[causal propagator]] in terms of the mode expansion (eq:CausalPropagatorModeExpansionForKleinGordonOnMinkowskiSpacetime). Since the [[integrand]] here depends on the [[wave vector]] $\vec k$ only via its [[norm]] ${\vert \vec k\vert}$ and the [[angle]] $\theta$ it makes with the given [[spacetime]] [[vector]] via

$$
\vec k \cdot (\vec x - \vec y)
\;=\;
{\vert \vec k\vert} \, {\vert \vec x\vert} \, \cos(\theta)
$$

we may express the [[integration]] in terms of [[polar coordinates]] as follws:

$$
\begin{aligned}
\Delta_S(x - y)
& =
\frac{-1}{(2\pi)^p}
\int \frac{1}{2 \omega(\vec k)/c}
\sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
e^{i \vec k \cdot (\vec x - \vec y)}
\, d^p \vec k
\\
& =
\frac{- vol_{S^{p-2}}}{(2\pi)^p}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\underset{ \theta \in [0,\pi] }{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
{\vert \vec k\vert} ({\vert \vec k\vert} \sin(\theta))^{p-2}
\,
d \theta
\wedge
d {\vert \vec k\vert}
\end{aligned}
$$

In the special case of [[spacetime]] [[dimension]] $p + 1 = 3 + 1$ this becomes

$$
\label{StepsInComputingCausalPropagatorIn3plus1Dimension}
\begin{aligned}
\Delta_S(x - y)
& =
\frac{- 2\pi}{(2\pi)^{3}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert}^2 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\underset{
=
\tfrac{1}{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} }
\left(
e^{i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
-
e^{-i {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert}}
\right)
}{
\underbrace{
\underset{ \cos(\theta) \in [-1,1] }{\int}
e^{ i {\vert \vec k\vert} {\vert \vec x - \vec y\vert} \cos(\theta) }
d \cos(\theta)
}
}
\wedge
d {\vert \vec k \vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ {\vert \vec k \vert} }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\sin\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 2}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{{\vert \vec k\vert} \in \mathbb{R}_{\geq 0}}{\int}
\frac{ 1 }{ \omega(\vec k)/c }
\sin\left( \omega(\vec k) (x^0 - y^0) /c \right)
\cos\left( {\vert \vec k\vert}\, {\vert \vec x - \vec y\vert} \right)
\, d {\vert \vec k\vert}
\\
& =
\frac{- 1}{(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y \vert } }
\underset{  \kappa \in \mathbb{R}  }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left( \omega(\kappa) (x^0 - y^0) /c \right)
\cos\left( \kappa\, {\vert \vec x - \vec y\vert} \right)
\, d \kappa
\\
& =
\frac{- 1}{2(2\pi)^{2} {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y} \vert }
\left(
\underset{\coloneqq I_+}{
\underbrace{
\underset{ \kappa \in \mathbb{R}  }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
+
\kappa\, {\vert \vec x - \vec y\vert}
\right)
d\kappa
}
}
+
\underset{ \coloneqq I_- }{
\underbrace{
\underset{ \kappa \in \mathbb{R}  }{\int}
\frac{ 1 }{ \omega(\kappa)/c }
\sin\left(
\omega(\kappa) (x^0 - y^0) /c
-
\kappa\, {\vert \vec x - \vec y\vert}
\right)
\,
d \kappa
}
}
\right)
\,.
\end{aligned}
$$

Here in the second but last step we renamed $\kappa \coloneqq {\vert \vec k\vert}$ and
doubled the integration domain for  convenience,
and in the last step we used the [[trigonometric identity]] $\sin(\alpha) \cos(\beta)\;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$.

In order to further evaluate this, we parameterize the remaining components $(\omega/c, \kappa)$ of the [[wave vector]]
by the dual [[rapidity]] $z$, via

$$
  \left(\cosh(z)\right)^2 - \left( \sinh(z)\right)^2 = 1
$$

as

$$
  \omega(\kappa)/c
  \;=\;
  \left(
    \tfrac{m c}{\hbar}
  \right)
  \cosh(z)
  \phantom{AA}
  \,,
  \phantom{AA}
  \kappa
  \;=\;
  \left(
    \tfrac{m c}{\hbar}
  \right)
  \sinh(z)
  \,,
$$

which makes use of the fact that $\omega(\kappa)$ is non-negative, by construction.
This [[change of integration variables]] makes the integrals under the braces above become

$$
  \label{TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski}
  I_\pm
  \;=\;
  \int_{-\infty}^\infty
   \sin\left(
     \tfrac{m c}{\hbar}
     \left(
       (x^0 - y^0) \cosh(z)
       \pm
       {\vert \vec x - \vec y\vert}
       \sinh(z)
     \right)
   \right)
   \, d z
   \,.
$$

Next we similarly parameterize the vector $x-y$ by its [[rapidity]] $\tau$. That parameterization depends on whether
$x-y$ is spacelike or not, and if not, whether it is future or past directed.

First, if $x-y$ is [[spacelike]] in that ${\vert x-y\vert}^2_\eta \gt 0$
then we may parameterize as

$$
  (x^0 - y^0)
  =
  \sqrt{{\vert x-y\vert}^2_\eta} \sinh(\tau)
  \phantom{AA}
  \,,
  \phantom{AA}
  {\vert \vec x - \vec y\vert}
  =
  \sqrt{ {\vert x-y\vert}^2_\eta} \cosh(\tau)
$$

which yields

$$
  \begin{aligned}
    I_{\pm}
    & =
    \int_{-\infty}^\infty
     \sin\left(
       \tfrac{m c}{\hbar}
       \sqrt{ {\vert x-y\vert}^2_\eta }
       \left(
         \sinh(\tau) \cosh(z)
         \pm
         \cosh(\tau)
         \sinh(z)
       \right)
     \right)
     \, d z
     \\
     & =
    \int_{-\infty}^\infty
     \sin\left(
       \tfrac{m c}{\hbar}
       \sqrt{ {\vert x-y\vert}^2_\eta}
       \left(
          \sinh\left( \tau \pm z\right)
       \right)
     \right)
     \, d z
     \\
     & =
    \int_{-\infty}^\infty
     \sin\left(
       \tfrac{m c}{\hbar}
       \sqrt{ {\vert x-y\vert}^2_\eta }
       \left(
          \sinh\left( z \right)
       \right)
     \right)
     \, d z
     \\
     & = 0
     \,,
  \end{aligned}
$$

where in the last line we observe that the integrand is a skew-symmetric function of $z$.

Second, if $x-y$ is [[timelike]] with $(x^0 - y^0) \gt 0$ then we may parameterize as

$$
  (x^0 - y^0)
  =
  \sqrt{ -{\vert x-y\vert}^2_\eta} \cosh(\tau)
  \phantom{AA}
  \,,
  \phantom{AA}
  {\vert \vec x - \vec y\vert}
  =
  \sqrt{ -{\vert x - y\vert}^2_\eta }
  \sinh(\tau)
$$

which yields

$$
  \label{IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski}
  \begin{aligned}
    I_\pm
    & =
    \int_{-\infty}^\infty
     \sin\left(
       \tfrac{m c}{\hbar}
       \left(
         (x^0 - y^0) \cosh(z)
         \pm
         {\vert \vec x - \vec y\vert}
         \sinh(z)
       \right)
     \right)
     \, d z
     \\
     & =
    \int_{-\infty}^\infty
     \sin\left(
       \sqrt{ - {\vert x-y\vert}^2_\eta }
       \tfrac{m c}{\hbar}
       \left(
         \cosh(\tau)\cosh(z)
         \pm
         \cosh(\tau)
         \sinh(z)
       \right)
     \right)
     \, d z
     \\
     & =
    \int_{-\infty}^\infty
     \sin\left(
       \sqrt{ - {\vert x-y\vert}^2_\eta }
       \tfrac{m c}{\hbar}
       \left(
         \cosh(z \pm \tau)
       \right)
     \right)
     \, d z
     \\
     & =
     \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
   \end{aligned}
   \,.
$$

Here in the last line we identified the integral representation of the [[Bessel function]] $J_0$ of order 0 (see [here](Bessel+function#eq:J0AsIntSinOfxCoshtdt)). The important point here is that this is a smooth function.

Similarly, if $x-y$ is [[timelike]] with $(x^0 - y^0) \lt 0$ then the same argument yields

$$
  I_\pm = -  \pi J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta} \tfrac{m c}{\hbar} \right)
$$

In conclusion, the general form of $I_\pm$ is

$$
  I_\pm
  =
  \pi
  sgn(x^0 - y^0)
  \Theta\left( -{\vert x-y\vert}^2_\eta \right)
  J_0\left( \sqrt{ - {\vert x-y\vert}^2_\eta } \tfrac{m c}{\hbar} \right)
  \,.
$$

Therefore we end up with

$$
\label{FinalResultOfComputationOf3Plus1dCausalPropagator}
\begin{aligned}
\Delta_S(x,y)
& =
\frac{1}{4 \pi {\vert \vec x - \vec y\vert}}
\frac{d}{d {\vert \vec x - \vec y\vert}}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
  \sqrt{ -{\vert x-y\vert}^2_\eta }
  \tfrac{m c}{\hbar}
\right)
\\
& =
\frac{-1}{2 \pi }
\frac{d}{d (-{\vert x-y\vert}^2_\eta)}
sgn(x^0) \Theta\left( -{\vert x-y\vert}^2_\eta \right)
J_0\left(
  \sqrt{-{\vert x-y \vert}^2_\eta} \tfrac{m c}{\hbar}
\right)
\\
& =
-\frac{1}{2 \pi }
\frac{d}{d (- \vert x-y\vert^2_{\eta})}
sgn(x^0) \Theta\left( - {\vert x - y\vert}^2_\eta \right)
J_0\left(
\tfrac{m c}{\hbar}  \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\\
& =
\frac{-1}{2\pi}
sgn(x^0)
\left(
\delta\left(
-{\vert x-y\vert}^2_\eta
\right)
\;-\;
\Theta\left( -{\vert x-y\vert}^2_\eta  \right)
\frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
J_0\left(
\tfrac{m c}{\hbar}  \sqrt{ -{\vert x-y\vert}^2_\eta }
\right)
\right)
\end{aligned}
$$

=--

+-- {: .num_prop #SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}
###### Proposition
**([[singular support]] of the [[Wightman propagator]] of the [[Klein-Gordon equation]] on [[Minkowski spacetime]] is the [[light cone]])**

The [[singular support]] of the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:

$$
  supp_{sing}(\Delta_H)
  =
  \left\{
    x \in \mathbb{R}^{p,1}
    \;\vert\;
    {\vert x\vert}^2_\eta = 0
  \right\}
  \,.
$$

=--

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

By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} the causal propagator is equivalently the [[Fourier transform of distributions]] of the [[delta distribution]] of the [[mass shell]] times the [[sign function]] of the [[angular frequency]];
and by basic properties of the Fourier transform this is the [[convolution of distributions]] of the separate
Fourier transforms:

$$
  \begin{aligned}
    \Delta_S(x)
    & \propto
    \widehat{
      \delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right) sgn( k_0 )
    }
    \\
    &\propto
    \widehat{\delta\left( \eta^{-1}(k,k) + \left( \tfrac{m c}{\hbar}\right)^2 \right)}
    \star
    \widehat{sgn( k_0 )}
  \end{aligned}
$$

By ([Gel'fand-Shilov 66, III 2.11 (7), p 294](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfDeltaDistributionappliedToMassShell), the [[singular support]]
of the first convolution factor is the [[light cone]].

The second factor is

$$
  \widehat{\Theta(k_0)}
  \propto
  \tfrac{1}{i x^0 + 0^+} \delta(\vec k)
$$

(by [this example](Dirac+distribution#FourierTransformOfDeltaDistribution) and [this example](Cauchy+principal+value#RelationToFourierTransformOfHeavisideDistribution)) and hence the [[wave front set]] of the second
factor is

$$
  WF\left(\widehat{sgn(k_0)}\right) = \{(0,k) \;\vert\; k \in S(\mathbb{R}^{p+1})\}
$$

(by [this example](wavefront+set#WaveFrontOfDeltaDistribution) and [this example](Cauchy+principal+value#PrincipalValueOfInverseFunctionCharacteristicEquation)).

With this the statement follows, via a [[partition of unity]], from [this prop.](convolution+product+of+distributions#WaveFrontSetOfCompactlySupportedDistributions).

For illustration, we now make this general statement fully explicit in the special case of [[spacetime]] [[dimension]]

$$
  p + 1 = 3 + 1
$$

by computing an explicit form for the [[causal propagator]] in terms of the [[delta distribution]],
the [[Heaviside distribution]] and [[smooth function|smooth]] [[Bessel functions]].

We follow ([Scharf 95 (2.3.36)](causal+perturbation+theory#Scharf95)).

By (eq:SymmetricPartOfHadamardPropagatorForKleinGordonOnMinkowskiSpacetime) we have

$$
  \begin{aligned}
  \Delta_H(x,y)
   & =
   \tfrac{i}{2}
   \underset{= \Delta_S(x,y)}{
   \underbrace{
     \frac{-1}{(2\pi)^p}
     \int
     \frac{1}{\omega(\vec k)/c}
     \sin\left( \omega(\vec k)(x^0 - y^0)/c \right)
     e^{i \vec k \cdot (\vec x - \vec y) }
     \, d^p \vec k
   }}
   \;+\;
   \underset{
     \coloneqq H(x,y)
   }{
   \underbrace{
    \frac{1}{(2\pi)^p}
     \int \frac{1}{2 \omega(\vec k)/c}
     \cos\left( \omega(\vec k)(x^0 - y^0)/c  \right)
     e^{i \vec k \cdot (\vec x - \vec y) }
     \, d^p \vec k
   }}
  \end{aligned}
$$

The first summand, proportional to the [[causal propagator]], which we computed as (eq:FinalResultOfComputationOf3Plus1dCausalPropagator)
in prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} to be

$$
  \tfrac{i}{2}\Delta_S(x,y)
  \;=\;
  \frac{-i}{4\pi}
  sgn(x^0)
  \left(
  \delta\left(
  -{\vert x-y\vert}^2_\eta
  \right)
  \;-\;
  \Theta\left( -{\vert x-y\vert}^2_\eta  \right)
  \frac{d}{d \left({-\vert x-y\vert}^2_\eta\right) }
  J_0\left(
  \tfrac{m c}{\hbar}  \sqrt{ -{\vert x-y\vert}^2_\eta }
  \right)
  \right)
  \,.
$$

The second term is computed in a directly analogous fashion: The integrals $I_\pm$ from
(eq:TheTwoSpecialFunctionIntegralsInTheComputationOfTheCausalPropagatorIn3Plus1DOnMinkowski) are now

$$
  I_\pm
  \coloneqq
  \int_{-\infty}^\infty
   \cos\left(
     \tfrac{m c}{\hbar}
     \left(
       (x^0 - y^0) \cosh(z)
       \pm
       {\vert \vec x - \vec y\vert}
       \sinh(z)
     \right)
   \right)
   \, d z
$$

Parameterizing by [[rapidity]], as in the proof of prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}, one finds that for [[timelike]] $x-y$ this is

$$
  \begin{aligned}
  I_\pm
  & =
    \int_{-\infty}^\infty
     \cos\left(
       \tfrac{m c}{\hbar}
       \sqrt{ {\vert x-y\vert}^2_\eta }
       \left(
          \cosh\left( z \right)
       \right)
     \right)
     \, d z
  \\
  & =
  - \pi N_0
  \left(
    \tfrac{m c}{\hbar}
    \sqrt{ {\vert x-y\vert}^2_\eta }
  \right)
  \end{aligned}
$$

while for [[spacelike]] $x-y$ it is

$$
  \begin{aligned}
    I_\pm
    & =
    \int_{-\infty}^\infty
     \cos\left(
       \tfrac{m c}{\hbar}
       \sqrt{ {\vert x-y\vert}^2_\eta }
       \left(
          \sinh\left( z \right)
       \right)
     \right)
     \, d z
     \\
    & =
    2 K_0
   \left(
     \tfrac{m c}{\hbar}
     \sqrt{ {\vert x-y\vert}^2_\eta }
   \right)
   \,,
  \end{aligned}
$$

where we identified the integral representations of the [[Neumann function]] $N_0$ (see [here](Bessel+function#N0AsIntSinOfxCoshtdt))
and of the [[modified Bessel function]] $K_0$ (see [here](Bessel+function#eq:K0AsIntSinOfxCoshtdt)).

As for the [[Bessel function]] $J_0$ in (eq:IdentifyingTheBesselFunctionInComputationOfCausalPropagatorIn3Plus1DOnMinkowski) the key point is that these are [[smooth functions]]. Hence we conclude that

$$
  H(x,y)
  \;\propto\;
  \frac{d}{d \left( {\vert x-y\vert}^2_\eta \right)}
  \left(
    -\Theta\left( -{\vert x-y\vert}^2_\eta \right)
    N_0
    \left(
      \tfrac{m c}{\hbar}
      \sqrt{ {\vert x-y\vert}^2_\eta }
    \right)
    +
    \Theta\left( {\vert x-y\vert}^2_\eta  \right)
    \tfrac{2}{\pi}
    K_0
   \left(
     \tfrac{m c}{\hbar}
     \sqrt{ {\vert x-y\vert}^2_\eta }
   \right)
  \right)
  \,.
$$

This expression has singularities on the [[light cone]] due to the [[step functions]].
In fact the expression being differentiated is continuous at the light cone
([Scharf 95 (2.3.34)](#Scharf95)), so that the singularity on the light cone is not a [[delta distribution]]
singularity from the derivative of the step functions. Accordingly it does not cancel the singularity
of $\tfrac{i}{2}\Delta_S(x,y)$ as above, and hence the singular support of $\Delta_H$ is still the whole
light cone.

=--

+-- {: .num_prop #SingularSupportOfFeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}
###### Proposition
**([[singular support]] of [[Feynman propagator]] for [[Klein-Gordon equation]] on [[Minkowski spacetime]])**

The [[singular support]] of the [[Feynman propagator]] $\Delta_H$ and of the [[anti-Feynman propagator]] $\Delta_{\overline{F}}$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded via [[translation]] [[invariant|invariance]] as a [[distribution]]
in a single variable, is the [[light cone]] of the origin:

$$
  \left.
  \array{
    supp_{sing}(\Delta_F)
    \\
    supp_{sing}(\Delta_{\overline{F}})
  }
  \right\}
  =
  \left\{
    x \in \mathbb{R}^{p,1}
    \;\vert\;
    {\vert x\vert}^2_\eta = 0
  \right\}
  \,.
$$

=--

(e.g [DeWitt 03 (27.85)](Feynman+propagator#DeWitt03))

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

By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} the Feynman propagator is equivalently
the [[Cauchy principal value]] of the inverse of the Fourier transformed Klein-Gordon operator:

$$
  \Delta_F
  \;\propto\;
  \widehat{
    \frac{1}{-k_\mu k^\mu - \left(\tfrac{m c}{\hbar}\right)^2 + i 0^+}
  }
  \,.
$$

With this the statement follows immediately from the result
([Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289](#GelfandShilov66)), see [this prop.](Cauchy+principal+value#FourierTransformOfPrincipalValueOfPowerOfQuadraticForm).

=--

+-- {: .num_prop #WaveFronSetsForKGPropagatorsOnMinkowski}
###### Proposition
**([[wave front sets]] of [[propagators]] of [[Klein-Gordon equation]] on [[Minkowski spacetime]])**

The [[wave front set]] of the various [[propagators]] for the [[Klein-Gordon equation]] on [[Minkowski spacetime]], regarded, via [[translation]] [[invariant|invariance]], as [[distributions]] in a single variable, are as follows:

* the [[causal propagator]] $\Delta_S$ (prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the lightcone:

$$
WF(\Delta_S) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \, k \neq 0  \right\}
$$

<center>
<img src="https://ncatlab.org/nlab/files/RetGreenFunction.png" width="60"> <br/> - <br/> <img src="https://ncatlab.org/nlab/files/AdvancedGreenFunction.png" width="60">
</center>

* the [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $k^0 \gt 0$:

$$
WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \; k^0 \gt 0   \right\}
$$

<center>
<img src="https://ncatlab.org/nlab/files/HadamardPropagator.png" width="60">
</center>

* the [[Feynman propagator]] $\Delta_S$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) has wave front set all pairs $(x,k)$ with $x$ and $k$ both on the light cone and $\pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0$

$$
  WF(\Delta_H) = \left\{ (x,k) \,\vert\, {\vert x\vert}^2_\eta = 0 \;\text{and} \; {\vert k\vert}^2_\eta = 0 \; \text{and} \;
    \left(
       \pm k_0 \gt 0 \;\Leftrightarrow\; \pm x^0 \gt 0
    \right)
  \right\}
$$

<center>
<img src="https://ncatlab.org/nlab/files/FeynmanPropagator.png" width="60">
</center>

=--

([Radzikowski 96, (16)](Hadamard+distribution#Radzikowski96))

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

First regarding the causal propagator:

By prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} the [[singular support]] of $\Delta_S$ is the [[light cone]].

Since the causal propagator is a solution to the homogeneous Klein-Gordon equation, the
[[propagation of singularities theorem]] says that also all [[wave vectors]] in the wave front set are lightlike.
Hence it just remains to show that all non-vanishing lightlike wave vectors based on the lightcone in spacetime
indeed do appear in the wave front set.

To that end, let $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ be a [[bump function]] whose [[compact support]] includes the origin.

For $a \in \mathbb{R}^{p,1}$ a point on the light cone, we need to determine the decay property of the Fourier transform of $x \mapsto b(x-a)\Delta_S(x)$. This is the [[convolution of distributions]] of $\hat b(k)e^{i k_\mu a^\mu}$ with $\widehat \Delta_S(k)$. By prop. \ref{CausalPropagatorAsFourierTransformOfDeltaDistributionOnTransformedKGOperator} we have

$$
\widehat \Delta_{S}(k)
\;\propto\;
\delta\left( -k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \right)
sgn(k_0)
\,.
$$

This means that the convolution product is the smearing of the mass shell by $\widehat b(k)e^{i k_\u a^\mu}$.

Since the mass shell asymptotes to the light cone, and since $e^{i k_\mu a^\mu} = 1$ for $k$ on the light cone (given that $a$ is on the light cone), this implies the claim.

Now for the [[Wightman propagator]]:

By def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} its Fourier transform is of the form

$$
\widehat \Delta_H(k)
\;\propto\;
\delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 )
$$

Moreover, its [[singular support]] is also the light cone (prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone}).

Therefore now same argument as before says that the wave front set consists of wave vectors $k$ on the light cone, but now due to the [[step function]] factor $\Theta(-k_0)$ it must satisfy $0 \leq - k_0 = k^0$.

Finally regarding the [[Feynman propagator]]:

By prop. \ref{ModeExpansionForFeynmanPropagatorOfKleinGordonEquationOnMinkowskiSpacetime} the Feynman propagator
coincides with the positive frequency Wightman propagator for $x^0 \gt 0$ and with the "negative frequency Hadamard operator"
for $x^0 \lt 0$. Therefore the form of $WF(\Delta_F)$ now follows directly with that of $WF(\Delta_H)$ above.

=--

### For Dirac operator on Minkowski spacetime
 {#ExampleForDiracOperatorOnMinkowskiSpacetime}

Finally we observe that the [[propagators]] for the [[Dirac field]] on [[Minkowski spacetime]]
follow immediately from the propagators for the [[scalar field]]:

+-- {: .num_prop #DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}
###### Proposition
**([[advanced and retarded propagator]] for [[Dirac equation]] on [[Minkowski spacetime]])**

Consider the [[Dirac operator]] on [[Minkowski spacetime]], which in [[Feynman slash notation]] reads

$$
  \begin{aligned}
    D
      & \coloneqq
    -i {\partial\!\!\!/\,}
     + \tfrac{m c}{\hbar}
    \\
    & =
    -i \gamma^\mu \frac{\partial}{\partial x^\mu} + \tfrac{m c}{\hbar}
  \end{aligned}
  \,.
$$

Its [[advanced and retarded propagators]] (def. \ref{AdvancedAndRetardedGreenFunctions})
are the [[derivatives of distributions]] of the advanced and retarded propagators $\Delta_\pm$ for the [[Klein-Gordon equation]] (prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}) by ${\partial\!\!\!/\,} + m$:

$$
  \Delta_{D, \pm}
  \;=\;
  \left(
    -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
  \right)
  \Delta_{\pm}
  \,.
$$

Hence the same is true for the [[causal propagator]]:

$$
  \Delta_{D, S}
  \;=\;
  \left(
    -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
  \right)
  \Delta_{S}
  \,.
$$

=--

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

Applying a [[differential operator]] does not change the [[support]] of a [[smooth function]], hence also not the [[support of a distribution]]. Therefore the uniqueness of the advanced and retarded propagators (prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique})
together with the translation-invariance and the anti-[[formally self-adjoint differential operator|formally self-adjointness]] of the [[Dirac operator]] (as for the [[Klein-Gordon operator]] (eq:TranslationInvariantKleinGordonPropagatorsOnMinkowskiSpacetime)
implies that it is sufficent to check that applying the [[Dirac operator]] to the $\Delta_{D, \pm}$
yields the [[delta distribution]]. This follows since the Dirac operator squares to the Klein-Gordon operator:

$$
  \begin{aligned}
    \left(
      -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
    \right)
    \Delta_{D, \pm}
    & =
    \underset{ = \Box - \left(\tfrac{m c}{\hbar}\right)^2}{
    \underbrace{
    \left(
      -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
    \right)
    \left(
      -i{\partial\!\!\!/\,} - \tfrac{m c}{\hbar}
    \right)
    }
    }
    \Delta_{\pm}
    \\
    & =
    \delta
  \end{aligned}
  \,.
$$

=--

Similarly we obtain the other [[propagators]] for the [[Dirac field]] from those of the [[real scalar field]]:

+-- {: .num_defn #HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Wightman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**

The _[[Wightman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the [[positive real number|positive]] [[frequency]] part of the [[causal propagator]] (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}),
hence the [[derivative of distributions]] of the Wightman propagator for the Klein-Gordon field (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) by the [[Dirac operator]]:

$$
  \begin{aligned}
  \left(
     -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
  \right)\Delta_{H}(x,y)
  & =
  \frac{1}{(2\pi)^p} \int \delta\left( k_\mu k^\mu + m^2 \right) \Theta( -k_0 ) ( {k\!\!\!/\,} + \tfrac{m c}{\hbar}) e^{i k_\mu (x^\mu-y^\mu) } \, d^{p+1} k
  \\
  & =
  \frac{1}{(2\pi)^p}
  \int \frac{ \gamma^0 \omega(\vec k)/c + \vec \gamma \cdot \vec k + \tfrac{m c}{\hbar}   }{2 \omega(\vec k)/c}
  e^{-i \omega(\vec k)(x^0 - y^0)/c + i \vec k \cdot (\vec x - \vec y) }
  \, d^p \vec k
  \,.
  \end{aligned}
$$

Here we used the expression (eq:StandardHadamardDistributionOnMinkowskiSpacetime) for the Wightman propagator of the Klein-Gordon equation.

=--

+-- {: .num_defn #FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime}
###### Definition
**([[Feynman propagator]] for [[Dirac operator]] on [[Minkowski spacetime]])**

The _[[Feynman propagator]]_ for the [[Dirac operator]] on [[Minkowski spacetime]] is the linear combination

$$
  \Delta_{D, F}
  \;\coloneqq\;
  \Delta_{D,H}
  +
  i \Delta_{D, -}
$$

of the [[Wightman propagator]] (def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime}) and the retarded propagator (prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators}). By prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue}
this means that it is the
[[derivative of distributions]] of the [[Feynman propagator]] of the [[Klein-Gordon equation]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) by the [[Dirac operator]]

$$
  \begin{aligned}
  \Delta_{D, F}
   & =
  \left(
     -i{\partial\!\!\!/\,} + \tfrac{m c}{\hbar}
  \right)\Delta_{F}(x,y)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{-i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     \left( {k\!\!\!/\,} + \tfrac{m c}{\hbar} \right) e^{i k_\mu (x^\mu - y^\mu)}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 + i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \,.
  \end{aligned}
$$

=--

## Related concepts

[[!include propagators - table]]

$\,$

* [[Green hyperbolic differential operator]]

## References

General discussion includes

* {#Baer14} [[Christian Bär]], _Green-hyperbolic operators on globally hyperbolic spacetimes_, Communications in Mathematical Physics 333, 1585-1615 (2014) ([doi](http://dx.doi.org/10.1007/s00220-014-2097-7), [arXiv:1310.0738](https://arxiv.org/abs/1310.0738))

* {#Khavkine14} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv:1402.1282](https://arxiv.org/abs/1402.1282))

based on

* {#Sanders12} [[Ko Sanders]], _A note on spacelike and timelike compactness_, Classical and Quantum Gravity 30, 115014 (2012) ([doi](http://dx.doi.org/10.1088/0264-9381/30/11/115014), [arXiv:1211.2469](https://arxiv.org/abs/1211.2469))

Textbook discussion for [[free fields]] in [[Minkowski spacetime]] is in

* {#Scharf95} [[Günter Scharf]],  section 2.3 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995

* {#Scharf01} [[Günter Scharf]], section 1 of  _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

An overview of the [[Green functions]] of the [[Klein-Gordon operator]], hence of the [[Feynman propagator]], [[advanced propagator]], [[retarded propagator]], [[causal propagator]] etc. is given in

* {#Kocic16} [[Mikica Kocic]], _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

Discussion on general [[globally hyperbolic spacetimes]] includes

* F. Friedlander, _The Wave Equation on a Curved Space-Time_, Cambridge: Cambridge University Press, 1975

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_,  Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009

Review in the context of [[perturbative algebraic quantum field theory]] includes

* [[Katarzyna Rejzner]], sections 4.1 and 6.2.3 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects retarded and advanced causal propagators]]

[[!redirects advanced and retarded propagators]]
[[!redirects retarded and advanced propagators]]

[[!redirects advanced and retarded propagator]]
[[!redirects retarded and advanced propagator]]

[[!redirects advanced propagator]]
[[!redirects advanced propagators]]

[[!redirects retarded propagator]]
[[!redirects retarded propagators]]

[[!redirects advanced causal propagator]]
[[!redirects advanced causal propagators]]

[[!redirects retarded causal propagator]]
[[!redirects retarded causal propagators]]

[[!redirects advanced Green function]]
[[!redirects advanced Green functions]]

[[!redirects retarded Green function]]
[[!redirects retarded Green functions]]

[[!redirects retarded and advanced Green functions]]
[[!redirects advanced and retarded Green functions]]

[[!redirects advanced Green's function]]
[[!redirects advanced Green's functions]]

[[!redirects retarded Green's function]]
[[!redirects retarded Green's functions]]

[[!redirects advanced or retarded Green function]]
[[!redirects advanced or retarded Green functions]]

[[!redirects retarded and advanced Green's functions]]
[[!redirects advanced and retarded Green's functions]]


Ady Stern

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

* [Institute page](https://webhome.weizmann.ac.il/home/stern/)

## Selected writings

On [[anyons]] in the [[quantum Hall effect]]:

* [[Ady Stern]], _Anyons and the quantum Hall effect -- A pedagogical review_, Annals of Physics Volume 323, Issue 1, January 2008, Pages 204-249 ([doi:10.1016/j.aop.2007.10.008](https://doi.org/10.1016/j.aop.2007.10.008), [arXiv:0711.4697](https://arxiv.org/abs/0711.4697))

On [[anyons]] in application to [[topological quantum computation]]:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083 (2008) ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889))

* [[Ady Stern]], Netanel H. Lindner, *Topological Quantum Computation -- From Basic Concepts to First Experiments*, Science 08 Mar 2013: Vol. 339, Issue 6124, pp. 1179-1184 ([doi:10.1126/science.1231473](https://science.sciencemag.org/content/339/6124/1179.full.pdf+html))

category: people


Adámek's fixed point theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

# Idea

_Adámek's fixed point theorem_ gives a method to construct an [[initial algebra of an endofunctor]] by "iterating" the functor.

## Construction

This theorem is attributed to [[Jiří Adámek]] but note that the same construction for set-functors already appeared in ([Pohlova,1973](#Pohlova))).

+-- {: .num_theorem #AdameksTheorem }
######Theorem (Adámek)

Let $C$ be a category with an [[initial object]] $0$ and [[transfinite composition]] of length $\omega$, hence [[colimits]] of sequences $\omega \to C$ (where $\omega$ is the first infinite [[ordinal]]), and suppose $F: C \to C$ preserves colimits of C$\omega$-chains. Then the colimit $\gamma$ of the chain

$$0 \overset{i}{\to} F(0) \overset{F(i)}{\to} \ldots \to F^{(n)}(0) \overset{F^{(n)}(i)}{\to} F^{(n+1)}(0) \to \ldots$$

carries a structure of initial $F$-algebra.
=--

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

The $F$-algebra structure $F\gamma \to \gamma$ is inverse to the canonical map $\gamma \to F\gamma$ out of the colimit (which is invertible by the hypothesis on $F$). The proof of initiality may be extracted by dualizing the corresponding proof given at [[terminal coalgebra]].
=--

This approach can be generalized to the [[transfinite construction of free algebras]].

## Related entries

* [[fixed point]]

* [[Kleene's fixed point theorem]] is precisely the [[de-categorification]] of this theorem to [[posets]]/[[preorders]].

* [[Knaster-Tarski's fixed point theorem]]

* [[Lawvere's fixed point theorem]]

* [[Lefschetz fixed point theorem]]

* [[Brouwer's fixed point theorem]]

* [[Atiyah-Bott fixed point formula]]

## References

* {#Pohlova} Věra Pohlová. "On sums in generalized algebraic categories." Czechoslovak Mathematical Journal **23**.2 (1973) 235-251 [[eudml:12718](http://eudml.org/doc/12718)]

* [[Jiří Adámek]], *Free algebras and automata realizations in the language of categories*, Commentationes Mathematicae Universitatis Carolinae **15**.4 (1974) 589-602 [[eudml:16649](https://eudml.org/doc/16649)]

* [[Jiří Adámek]], Věra Trnková, *Automata and algebras in categories* **37** Springer (1990) [[ISBN:9780792300106](https://link.springer.com/book/9780792300106)]

[[!redirects Adamek's fixed point theorem]]

[[!redirects Adamek's theorem]]

[[!redirects Adámek's theorem]]


Aeysha Khalique

* [institute page](https://sns.nust.edu.pk/faculty/aeysha-khalique/)

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

## Selected writings

On practical realization of [[quantum teleportation]]:

* Syed Tahir Amin, [[Aeysha Khalique]], *Practical Quantum Teleportation of an Unknown Quantum State*, Can. J. Phys. **95** 5 (2017) 498 [[arXiv:1508.01141](https://arxiv.org/abs/1508.01141), [doi:10.1139/cjp-2016-0758](https://doi.org/10.1139/cjp-2016-0758)]

category: people


affiliated operator

## Motivation

Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on $L^2(\mathbb{R}^n)$ which is proportional to the momentum operator in quantum mechanics.

## Idea

The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of "approximable" operators by means of operator algebra itself. One way to do this is to define the __affiliated elements__ of $C^\ast$-algebra, or the operators affiliated with the $C^\ast$-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the [[spectral theorem|spectral decomposition]] will supply the approximation.

## Details

Let $T$ be a normal unbounded operator. Thanks to the [[spectral theorem]] for unbounded operators, $T$ has a [[spectral measure]] $P_{\lambda}$. Every single spectral projection is bounded, of course, so we may look for von Neumann algebras that contain them. Since a von Neumann algebra may be characterised as the algebra of all operators that commute with some set of unitary operators, we give the following definition:

* Definition (affiliation): A closed operator $T$ is **affiliated** with a von Neumann algebra $\mathcal{M}$, written as $T \eta \mathcal{M}$, if every unitary operator $U$ in the commutant of $\mathcal{M}$ transforms $D_T$ to itself, and we have $U^*TU = T$.

We mention some interesting theorems using this concept.

* Theorem (Kadison-Ringrose 5.6.18): An operator is normal iff it is affiliated with an abelian von Neumann algebra. If A is normal, there is a smallest von Neumann algebra that A is affiliated with, this algebra is abelian.

* Definition: Given a normal operator A, the smallest (and necessarily abelian) von Neumann algebra that A is associated with is called the **von Neumann algebra generated by A**.

## Literature

* [[S. L. Woronowicz]], K. Napiórkowski, _Operator theory in $C^\ast$-framework_,  Reports on Mathematical Physics __31__, Issue 3 (1992), 353-371,
doi, [pdf](http:www.fuw.edu.pl/~slworono/PDF-y/OP.pdf)
* [[S. L. Woronowicz]], _Unbounded elements affiliated with $C^\ast$-algebras and non-compact quantum groups_, Commun. Math. Phys. __136__, 399--432 (1991) [euclid](http://projecteuclid.org/euclid.cmp/1104202358) [MR1096123](http://www.ams.org/mathscinet-getitem?mr=1096123); _$C^\ast$-algebras generated by unbounded elements_, [pdf](http://www.fuw.edu.pl/~slworono/PDF-y/GENER.pdf)
* wikipedia [affiliated operator](http://en.wikipedia.org/wiki/Affiliated_operator)

[[!redirects affiliated element]]
[[!redirects affiliated elements]]


affine algebra

Given a field $k$ an __affine algebra__ is any finitely generated [[noetherian ring|noetherian]] associative commutative unital $k$-algebra without [[nilpotent elements]]. This term is limited to a small community of algebraists and may bring confusion if quoted outside of that context. The name is from the fact that the affine algebras are the coordinate rings of [[affine varieties]]; equivalently, an affine $k$-variety is precisely the [[maximal spectrum]] of an affine $k$-algebra.

Some people drop the condition of commutativity and talk about noncommutative affine algebras. This is even less standard as in the noncommutative context there is a further hesitation to drop noetherianess as well.

Yet another confusion may arise when people informally say affine algebra for an [[affine Lie algebra]]; of course, the latter, full, term is preferred (or "affine Kac-Moody algebra").

[[!redirects affine algebras]]