A first idea of quantum field theory -- Propagators

## Propagators
 {#Propagators}

In this chapter we discuss the following topics:

* _Background_

  * _[Fourier analysis and Plane wave modes](#FourierAnalysis)_

  * _[Microlocal analysis and UV-Divergences](#MicrolocalAnalysisAndUltravioletDivergence)_

  * _[Cauchy principal values](#CauchyPrincipalValues)_

* _Propagators for the free scalar field on Minkowski spacetime_

  * _[advanced and regarded propagators](#AdvancedAndRetardedPropagatorsForKleinGordonEquationOnMinkowskiSpacetime)_

  * _[causal propagator](#CausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime)_

  * _[Wightman propagator](#HadamardPropagatorForKleinGordonOnMinkowskiSpacetime)_

  * _[Feynman propagator](#FeynmanPropagator)_

  * _[singular support and wave front sets](#WaveFrontSetsOfPropagatorsForKleinGordonOperatorOnMinkowskiSpacetime)_

* _[Propagators for the Dirac field on Minkowski spacetime](#DiracEquationOnMinkowskiSpacetimePropagators)_

$\,$

In the [previous chapter](#PhaseSpace) we have seen the [[covariant phase space]] (prop. \ref{CovariantPhaseSpace}) of sufficiently nice [[Lagrangian field theories]], which is the [[on-shell]] [[space of field histories]] equipped with the [[presymplectic form]] [[transgression of variational differential forms|transgressed]] from the [[presymplectic current]]
of the theory; and we have seen that in good cases this induces a bilinear pairing on sufficiently well-behaved [[observables]],
called the _[[Poisson bracket]]_ (def. \ref{PoissonBracketOnHamiltonianLocalObservables}), which reflects the [[infinitesimal symmetries]] of the [[presymplectic current]]. This [[Poisson bracket]] is of central importance for passing to actual [[quantum field theory]],
since, as we will discuss in _[Quantization](#Quantization)_ below, it is the [[infinitesimal]] approximation to the [[quantization]] of a [[Lagrangian field theory]].

We have moreover seen that the [[Poisson bracket]] on the [[covariant phase space]] of a
[[free field theory]] with [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]]
-- the [[Peierls-Poisson bracket]] --
is determined by the [[integral kernel]] of the _[[causal Green function]]_ (prop. \ref{PPeierlsBracket}).
Under the identification of linear on-shell observables with off-shell observables
that are [[generalized solution of a PDE|generalized solutions]] to the [[equations of motion]] (theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}) the convolution with this [[integral kernel]]
may be understood as _propagating_ the values of an off-shell observable through [[spacetime]],
such as to then compare it with any other observable at any spacetime point (prop. \ref{PPeierlsBracket}).
Therefore the [[integral kernel]] of the [[causal Green function]] is also called the _[[causal propagator]]_ (prop. \ref{GreenFunctionsAreContinuous}).

This means that for [[Green hyperbolic differential equation|Green hyperbolic]] [[free field theory|free]] [[Lagrangian field theory]]
the [[Poisson bracket]], and hence the infinitesimal [[quantization]] of the theory, is all encoded in the
[[causal propagator]]. Therefore here we analyze the [[causal propagator]], as well as its variant [[propagators]], in detail.

The main tool for these computations is _[[Fourier transform|Fourier analysis]]_ (reviewed [below](#FourierAnalysis)) by which [[field histories]], [[observables]] and [[propagators]] on [[Minkowski spacetime]] are decomposed as [[superpositions]] of [[plane waves]] of various [[frequencies]], [[wave lengths]] and [[wave vector]]-[[direction of a vector|direction]]. Using this, all [[propagators]] are exhibited as those [[superpositions]]
of [[plane waves]] which satisfy the [[dispersion relation]] of the given [[equation of motion]], relating [[plane wave]] [[frequency]]
to [[wave length]].

This way the [[causal propagator]] is naturally decomposed into its contribution from [[positive real number|positive]]
and from [[negative real number|negative]] [[frequencies]]. The positive frequency part of the [[causal propagator]] is called the _[[Wightman propagator]]_ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} below). It turns out (prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below) that this is equivalently the [[sum]] of the [[causal propagator]], which itself is skew-symmetric (cor. \ref{CausalPropagatorIsSkewSymmetric} below), with a symmetric component, or equivalently that the [[causal propagator]] is the skew-symmetrization of the [[Wightman propagator]].
After [[quantization]] of [[free field theory]] discussed [further below](#FreeQuantumFields), we will see that the
Wightman propagator is equivalently the [[2-point function|correlation function]] between two point-evaluation field observables (example \ref{PointEvaluationObservables}) in a _[[vacuum state]]_ of the field theory (a [[state]] in the sense of def. \ref{States}).

Moreover, by def. \ref{AdvancedAndRetardedGreenFunctions} the [[causal propagator]] also decomposes into its
contributions with [[future]] and [[past]] [[support of a distribution|support]], given by the _difference_ between
the [[advanced and retarded propagators]]. These we analyze first, starting with prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} below.

Combining these two decompositions of the [[causal propagator]] (positive/negative frequency as well as positive/negative time) yields one more
propagator, the _[[Feynman propagator]]_ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime} below).

We will see [below](#Quantization) that the [[quantization]] of a [[free field theory]] is given by a
"[[star product]]" (on [[observables]]) which is given by "[[exponential|exponentiating]]" these [[propagators]].
For that to make sense, certain pointwise products of these [[propagators]], regarded as [[generalized functions]] (prop. \ref{DistributionsAreGeneralizedFunctions}) need to exist. But since the [[propagators]] are [[distributions]] with
[[singularities]], the existence of these products requires that certain potential "[[UV divergences]]" in their [[Fourier transform of distributions|Fourier transforms]] (remark \ref{UltravioletDivergencesFromPaleyWiener} below) are absent ("[[Hörmander's criterion]]", prop. \ref{HoermanderCriterionForProductOfDistributions} below). These [[UV divergences]] are captured by what what is called the _[[wave front set]]_ (def. \ref{WaveFrontSet} below).

The study of [[UV divergences]] of [[distributions]] via their [[wave front sets]] is called _[[microlocal analysis]]_
and provides powerful tools for the understanding of [[quantum field theory]]. In particular the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem}) shows that for [[distributional solutions to a PDE|distributional solutions]] (def. \ref{DistributionalDerivatives}) of [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]], such as the [[propagators]],
their [[singular support]] propagates itself through [[spacetime]] along the [[wave front set]].

Using this theorem we work out the [[wave front sets]] of the [[propagators]] (prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski} below).
Via [[Hörmander's criterion]] (prop. \ref{HoermanderCriterionForProductOfDistributions}) this computation will serve to show
why upon [[quantization]] the [[Wightman propagator]] replaces the [[causal propagator]] in the construction of the [[Wick algebra]]
of [[quantum observables]] of the [[free field theory]] (discussed below in _[Free quantum fields](#FreeQuantumFields)_) and the [[Feynman propagator]] similarly controls the [[quantum observables]] of the [[interaction|interacting]] [[quantum field theory]]
(below in _[Feynman diagrams](#FeynmanDiagrams)_).

$\,$

The following table summarizes the structure of the system of propagators.
(The column "as vacuum expectation value of field operators" will be discussed further below in _[Free quantum fields](#FreeQuantumFields)_).

$\,$

[[!include propagators - table]]

$\,$

$\,$

**[[Fourier transform|Fourier analysis]] and [[plane wave]] modes**
 {#FourierAnalysis}

By definition, the [[equations of motion]] of [[free field theories]] (def. \ref{FreeFieldTheory}) are [[linear partial differential equations]]
and hence lend themselves to _[[harmonic analysis]]_, where all [[field histories]] are decomposed into
[[superpositions]] of [[plane waves]] via _[[Fourier transform]]_. Here we briefly survey the relevant definitions
and facts of [[Fourier transform|Fourier analysis]].

In [[formal duality]] to the [[harmonic analysis]] of the [[field histories]] themselves,
also the linear [[observables]] (def. \ref{LinearObservables}) on the [[space of field histories]], hence
the _[[distribution|distributional]] [[generalized functions]]_ (prop. \ref{LinearObservablesAreTheCompactlySupportedDistributions})
are subject to _[[Fourier transform of distributions]]_ (def. \ref{FourierTransformOnTemperedDistributions} below).

Throughout, let $n \in \mathbb{N}$ and consider the [[Cartesian space]] $\mathbb{R}^n$ of [[dimension]] $n$ (def. \ref{CartesianSpacesAndSmoothFunctionsBetweenThem}). In the application to [[field theory]], $n = p + 1$ is the
[[dimension]] of [[spacetime]] and $\mathbb{R}^n$ is either [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (def. \ref{MinkowskiSpacetime})
or its [[dual vector space]], thought of as the space of _[[wave vectors]]_ (def. \ref{PlaneWaves} below). For
$x = (x^\mu) \in \mathbb{R}^{p,1}$ and $k = (k_\mu) \in (\mathbb{R}^(p,1))^\ast$ we write

$$
  x \cdot k \;=\; x^\mu k_\mu
$$

for the canonical pairing.

+-- {: .num_defn #PlaneWaves}
###### Definition
**([[plane wave]])**

A _[[plane wave]]_ on [[Minkowski spacetime]] $\mathbb{R}^{p,1}$ (def. \ref{MinkowskiSpacetime}) is a
[[smooth function]] with values in the [[complex numbers]] given by

$$
  \array{
    \mathbb{R}^{p,1} &\longrightarrow& \mathbb{C}
    \\
    (x^\mu) &\mapsto& e^{i k_\mu x^\mu}
  }
$$

for $k = (k_\mu) \in (\mathbb{R}^{p,1})^\ast$ a [[covector]], called the _[[wave vector]]_ of the [[plane wave]].

We use the following terminology:

[[!include plane waves -- table]]

=--

+-- {: .num_defn #SchwartzSpace}
###### Definition
**([[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]])**

A [[complex number|complex]]-valued [[smooth function]] $f \in C^\infty(\mathbb{R}^n)$ is said to have _[[rapidly decreasing function|rapidly decreasing]] [[partial derivatives]]_ if for all $\alpha,\beta \in \mathbb{N}^{n}$ we have

$$
  \underset{x \in \mathbb{R}^n}{sup} {\vert  x^\beta \partial^\alpha f(x) \vert}
  \;\lt\; \infty
  \,.
$$

Write

$$
  \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)
$$

for the sub-[[vector space]] on the functions with rapidly decreasing partial derivatives regarded as a [[topological vector space]] for the [[Fréchet space]] [[structure]] induced by the [[seminorms]]

$$
  p_{\alpha, \beta}(f) \coloneqq \underset{x \in \mathbb{R}^n}{sup} {\vert  x^\beta \partial^\alpha f(x) \vert}
  \,.
$$

This is also called the _[[Schwartz space]]_.

=--

(e.g. [Hörmander 90, def. 7.1.2](Fourier+transform#Hoermander90))

+-- {: .num_example #CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}
###### Example
**([[compactly supported function|compactly supported]] [[smooth function]] are [[functions with rapidly decreasing partial derivatives]])**

Every [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R}^n)$ has rapidly decreasing partial derivatives (def. \ref{SchwartzSpace}):

$$
  C^\infty(\mathbb{R}^n)
   \hookrightarrow
  \mathcal{S}(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_prop #ConvolutionProductOnSchwartzSpace}
###### Proposition
**(pointwise product and [[convolution product]] on [[Schwartz space]])**

The [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is closed under the following operatios on smooth functions $f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)$

1. pointwise product:

   $$
     (f \cdot g)(x) \coloneqq f(x) \cdot g(x)
   $$

1. [[convolution product]]:

   $$
     (f \star g)(x) \coloneqq \underset{y \in \mathbb{R}^n}{\int} f(y)\cdot g(x-y) \, dvol(y)
     \,.
   $$

=--

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

By the [[product law]] of [[differentiation]].

=--

+-- {: .num_prop #RapidlyDecreasingFunctionsAreIntegrable}
###### Proposition
**([[rapidly decreasing functions]] are [[integrable functions|integrable]])**

Every [[rapidly decreasing function]] $f \colon \mathbb{R}^n \to \mathbb{R}$ (def. \ref{SchwartzSpace}) is an [[integrable function]] in that its [[integral]] exists:

$$
  \underset{x \in \mathbb{R}^n}{\int} f(x) \, d^n x
  \;\lt\;
  \infty
$$

In fact for each $\alpha \in \mathbb{N}^n$ the product of $f$ with the $\alpha$-power of the [[coordinate functions]] exists:

$$
  \underset{x \in \mathbb{R}^n}{\int}
    x^\alpha f(x)\, d^n x
  \;\lt\;
   \infty
  \,.
$$

=--

+-- {: .num_defn #FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}
###### Definition
**([[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**

The _[[Fourier transform]]_ is the [[continuous linear functional]]

$$
  \widehat{(-)}
    \;\colon\;
  \mathcal{S}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{S}(\mathbb{R}^n)
$$

on the [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}), which is given by [[integration]] against [[plane wave]] functions (def. \ref{PlaneWaves})

$$
  x \mapsto e^{- i k \cdot x}
$$

times the standard [[volume form]] $d^n x$:

$$
  \label{IntegralExpressionForFourierTransform}
  \hat f(k)
    \;\colon\;
  \int_{x \in \mathbb{R}^n}
    e^{- i \, k \cdot x} f(x) \, d^n x
  \,.
$$

Here the argument $k \in \mathbb{R}^n$ of the Fourier transform is also called the _[[wave vector]]_.

=--

(e.g. [Hörmander, lemma 7.1.3](Fourier+transform#Hoermander90))

+-- {: .num_prop #FourierInversion}
###### Proposition
**([[Fourier inversion theorem]])**

The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) is an [[isomorphism]], with [[inverse function]] the _[[inverse Fourier transform]]_

$$
  \widecheck {(-)}
     \;\colon\;
  \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathcal{S}(\mathcal{R}^n)
$$

given by

$$
  \widecheck g (x)
    \;\coloneqq\;
   \underset{k \in \mathbb{R}^n}{\int}
      g(k) e^{i k \cdot x}
   \, \frac{d^n k}{(2\pi)^n}
   \,.
$$

Hence in the language of [[harmonic analysis]] the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the [[superposition]] of [[plane waves]] (def. \ref{PlaneWaves}) in which the plane wave with [[wave vector]] $k\in \mathbb{R}^n$ appears with [[amplitude]] $g(k)$.

=--

(e.g. [Hörmander, theorem 7.1.5](Fourier+transform#Hoermander90))

+-- {: .num_prop #BasicPropertiesOfFourierTransformOverCartesianSpaces}
###### Proposition
**(basic properties of the [[Fourier transform]])**

The [[Fourier transform]] $\widehat{(-)}$ (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}) on the [[Schwartz space]] $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) satisfies the following properties, for all $f,g \in \mathcal{S}(\mathbb{R}^n)$:

1. (interchanging [[coordinate]] [[multiplication]] with [[partial derivatives]])

   $$
     \label{FourierTransformInterchangesCoordinateProductWithDerivative}
     \widehat{ x^a f } =  + i \partial_a \widehat f
     \phantom{AAAAA}
     \widehat{ - i\partial_a f} = k_a \widehat f
   $$

1. (interchanging pointwise multiplication with [[convolution product]], remark \ref{ConvolutionProductOnSchwartzSpace}):

   $$
     \label{FourierTransformInterchangesPointwiseProductWithConvolution}
     \widehat {(f \star g)} = \widehat{f} \cdot \widehat{g}
      \phantom{AAAA}
      \widehat{ f \cdot g } = (2\pi)^{-n} \widehat{f} \star \widehat{g}
   $$

1. ([[unitary operator|unitarity]], [[Parseval's theorem]])

   $$
     \underset{x \in \mathbb{R}^n}{\int} f(x) g^\ast(x)\, d^n x
     \;=\;
     \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \widehat{g}^\ast(k) \, d^n k
   $$

1. $$
     \label{FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor}
     \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \cdot g(k) \, d^n k
     \;=\;
     \underset{x \in \mathbb{R}^n}{\int} f(x) \cdot \widehat{g}(x) \, d^n x
   $$

=--

(e.g [Hörmander 90, lemma 7.1.3, theorem 7.1.6](Fourier+transform#Hoermander90))

The [[Schwartz space]] of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace})
serves the purpose to support the [[Fourier transform]] (def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace})
together with its inverse (prop. \ref{FourierInversion}), but for many applications
one needs to apply the Fourier transform to more general functions, and in fact to _[[generalized functions]]_
in the sense of [[distributions]] (via [this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).
But with the [[Schwartz space]] in hand, this generalization is readily obtained by [[formal duality]]:

+-- {: .num_defn #TemperedDistribution}
###### Definition
**([[tempered distribution]])**

A _[[tempered distribution]]_ is a [[continuous linear functional]]

$$
  u \;\colon\; \mathcal{S}(\mathbb{R}^n) \longrightarrow \mathbb{C}
$$

on the [[Schwartz space]] (def. \ref{SchwartzSpace}) of [[functions with rapidly decaying partial derivatives]]. The [[vector space]] of all tempered distributions is canonically a [[topological vector space]] as the [[dual space]] to the [[Schwartz space]], denoted

$$
  \mathcal{S}'(\mathbb{R}^n)
    \;\coloneqq\;
  \left(
     \mathcal{S}(\mathbb{R}^n)
  \right)^\ast
  \,.
$$

=--

e.g. ([Hörmander 90, def. 7.1.7](Fourier+transform#Hoermander90))

+-- {: .num_example #SomeNonSingularTemperedDistributions}
###### Example
**(some [[non-singular distribution|non-singular]] [[tempered distributions]])**

Every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) induces a [[tempered distribution]] $u_f \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) by [[integration|integrating]] against it:

$$
  u_f \;\colon\; g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x)\, d^n x
  \,.
$$

This construction is a linear inclusion

$$
  \mathcal{S}(\mathbb{R}^n) \overset{\text{dense}}{\hookrightarrow} \mathcal{S}'(\mathbb{R}^n)
$$

of the [[Schwartz space]] into its [[dual space]] of [[tempered distributions]]. This is a [[dense subspace]] inclusion.

In fact already the restriction of this inclusion to the [[compactly supported function|compactly supported]] [[smooth functions]] (example \ref{CompactlySupportedSmoothFunctionsAreFunctionsWithRapidlyDecreasingDerivatives}) is a [[dense subspace]] inclusion:

$$
  C^\infty_{cp}(\mathbb{R}^n)
    \overset{dense}{\hookrightarrow}
  \mathcal{S}'(\mathbb{R}^n)
  \,.
$$

This means that every [[tempered distribution]] is a [[limit of a sequence|limit]] of a [[sequence]]
of ordinary [[functions with rapidly decreasing partial derivatives]], and in fact even the [[limit of a sequence|limit]]
of a [[sequence]] of [[compactly supported function|compactly supported]] [[smooth functions]] ([[bump functions]]).

It is in this sense that [[tempered distributions]] are "generalized functions".

=--

(e.g. [Hörmander 90, lemma 7.1.8](Fourier+transform#Hoermander90))

+-- {: .num_example #CompactlySupportedDistibutionsAreTemperedDistributions}
###### Example
**([[compactly supported distributions]] are [[tempered distributions]])**

Every [[compactly supported distribution]] is a [[tempered distribution]] (def. \ref{TemperedDistribution}),
hence there is a [[linear map|linear]] [[injection|inclusion]]

$$
  \mathcal{E}'(\mathbb{R}^n)
    \hookrightarrow
  \mathcal{S}'(\mathbb{R}^n)
  \,.
$$

=--

+-- {: .num_example #DiracDeltaDistribution}
###### Example
**([[delta distribution]])**

Write

$$
 \delta_0(-) \;\in\; \mathcal{E}'(\mathbb{R}^n)
$$

for the [[distribution]] given by point evaluation of functions at the origin of $\mathbb{R}^n$:

$$
  \delta_0(-) \;\colon\; f \mapsto f(0)
  \,.
$$

This is clearly a [[compactly supported distribution]]; hence a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.

We write just "$\delta(-)$" (without the subscript) for the corresponding [[generalized function]] (example \ref{SomeNonSingularTemperedDistributions}),  so that

$$
  \underset{x \in \mathbb{R}^n}{\int} \delta(x) f(x) \, d^n x
  \;\coloneqq\;
  f(0)
  \,.
$$

=--

+-- {: .num_example #SquareIntegrableFunctionsInduceTemperedDistributions}
###### Example
**([[square integrable functions]] induce [[tempered distributions]])**

Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th [[Lebesgue space]], e.g. for $p = 2$ this means that $f$ is a [[square integrable function]]. Then the operation of [[integration]] against the [[measure]] $f dvol$

$$
  g \mapsto \underset{x \in \mathbb{R}^n}{\int} g(x) f(x) \, d^n x
$$

is a [[tempered distribution]] (def. \ref{TemperedDistribution}).

=--

(e.g. [Hörmander 90, below lemma 7.1.8](Fourier+transform#Hoermander90))

Property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) of the ordinary [[Fourier transform]] on [[functions with rapidly decreasing partial derivatives]] motivates and justifies the fullowing generalization:

+-- {: .num_defn #FourierTransformOnTemperedDistributions}
###### Definition
**([[Fourier transform of distributions]] on [[tempered distributions]])**

The _[[Fourier transform of distributions]]_ of a [[tempered distribution]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is the [[tempered distribution]] $\widehat u$ defined on a smooth function $f \in \mathcal{S}(\mathbb{R}^n)$ in the [[Schwartz space]] (def. \ref{SchwartzSpace}) by

$$
  \widehat{u}(f) \;\coloneqq\; u\left( \widehat f\right)
  \,,
$$

where on the right $\widehat f \in \mathcal{S}(\mathbb{R}^n)$ is the [[Fourier transform]] of functions from def. \ref{FourierTransformSmoothFunctionsWithRapidlyDecayingDerivativesOnCartesianSpace}.
=--

(e.g. [Hörmander 90, def. 1.7.9](Fourier+transform#Hoermander90))

+-- {: .num_example #FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}
###### Example
**([[Fourier transform of distributions]] indeed generalizes [[Fourier transform]] of [[functions with rapidly decreasing partial derivatives]])**

Let $u_f \in \mathcal{S}'(\mathbb{R}^n)$ be a [[non-singular distribution|non-singular]] [[tempered distribution]] induced, via example \ref{SomeNonSingularTemperedDistributions}, from a [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$.

Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is the [[non-singular distribution]] induced from the [[Fourier transform]] of $f$:

$$
  \widehat{u_f} \;=\; u_{\hat f}
  \,.
$$

=--

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

Let $g \in \mathcal{S}(\mathbb{R}^n)$. Then

$$
  \begin{aligned}
    \widehat{u_f}(g)
    & \coloneqq
    u_f\left( \widehat{g}\right)
    \\
    & =
    \underset{x \in \mathbb{R}^n}{\int} f(x) \hat g(x)\, d^n x
    \\
    & =
    \underset{x \in \mathbb{R}^n}{\int} \hat f(x) g(x) \, d^n x
    \\
    & =
    u_{\hat f}(g)
  \end{aligned}
$$

Here all equalities hold by definition, except for the third: this is property (eq:FourierTransformInIntegralOfProductMayBeShiftedToOtherFactor) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.

=--

+-- {: .num_example #FourierTransformOfKleinGordonEquation}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of [[Klein-Gordon equation]] [[derivative of distributions|of]] [[distributions]])**

Let $\Delta \in \mathcal{S}'(\mathbb{R}^{p,1})$ be any [[tempered distribution]] (def. \ref{TemperedDistribution}) on [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and let
$P \coloneqq \eta^{\mu \nu} \frac{\partial}{\partial x^\mu}\frac{\partial}{\partial x^\nu} - \left( \tfrac{m c}{\hbar} \right)^2$
be the [[Klein-Gordon operator]] (eq:KleinGordonOperator). Then the [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of $P \Delta$ is, in [[generalized function]]-notation (remark \ref{LinearObservablesAsGeneralizedFunctions})given by

$$
  \widehat {P \Delta}(k)
  \;=\;
  \left(
    - \eta^{\mu \nu}k_\mu k_\nu
    - \left( \tfrac{m c}{\hbar}\right)^2
  \right)
  \widehat(k)
  \,.
$$

=--

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

Let $r \in \mathcal{S}(\mathbb{R}^n)$ be any [[function with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpace}).
Then

$$
  \begin{aligned}
    \widehat {P \Delta}(r)
    & =
    P \Delta(\widehat r)
    \\
    & =
    \Delta(P^\ast \widehat r)
    \\
    & =
    \Delta(P \widehat r)
    \\
    & =
    \Delta\left( \left(-\eta^{\mu \nu}k_\mu k_\nu - \left( \tfrac{m c}{\hbar}\right)^2\right) \widehat{r}  \right)
  \end{aligned}
$$

Here the first step is def. \ref{FourierTransformOnTemperedDistributions}, the second is def. \ref{DistributionalDerivatives},
the third is example \ref{FormallySelfAdjointKleinGordonOperator}, while the last step is prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.

=--

+-- {: .num_example #CompactlySupportedDistributionFourierTransform}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of [[compactly supported distributions]])**

Under the identification of [[smooth functions]] of bounded growth with [[non-singular distributions|non-singular]] [[tempered distributions]] (example \ref{SomeNonSingularTemperedDistributions}), the [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of a [[tempered distribution]] that happens to be [[compactly supported distribution|compactly supported]] (example \ref{CompactlySupportedDistibutionsAreTemperedDistributions})

$$
  u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$

is simply

$$
  \widehat{u}(k) = u\left( e^{-  i k \cdot (-)}\right)
  \,.
$$

=--

([Hörmander 90, theorem 7.1.14](Fourier+transform#Hoermander90))

+-- {: .num_defn #FourierTransformOfDeltaDistribution}
###### Example
**([[Fourier transform of distributions|Fourier transform]] of the [[delta-distribution]])**

The [[Fourier transform of distributions|Fourier transform]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[delta distribution]] (def. \ref{DiracDeltaDistribution}), via example \ref{CompactlySupportedDistributionFourierTransform}, is the [[constant function]] on 1:

$$
  \begin{aligned}
    \widehat {\delta}(k)
    & =
    \underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- i k x} \, d x
    \\
    & =
    1
  \end{aligned}
$$

This implies by the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) that the [[delta distribution]] itself has equivalently the following expression as a [[generalized function]]

$$
  \begin{aligned}
    \delta(x)
    & =
    \widecheck{\widehat {\delta_0}}(x)
    \\
    & =
    \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \, \frac{d^n k}{ (2\pi)^n }
  \end{aligned}
$$

in the sense that for every [[function with rapidly decreasing partial derivatives]] $f \in \mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpace}) we have

$$
  \begin{aligned}
    f(x)
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    f(y) \delta(y-x)  \, d^n y
    \\
    & =
    \underset{y \in \mathbb{R}^n}{\int}
    \underset{k \in \mathbb{R}^n}{\int}
    f(y) e^{i k \cdot (y-x)}
    \, \frac{d^n k}{(2\pi)^n} \, d^n y
    \\
    & =
    \underset{k \in \mathbb{R}^n}{\int}
    e^{- i k \cdot x}
    \underset{= \widehat{f}(-k) }{
    \underbrace{
      \underset{y \in \mathbb{R}^n}{\int}
      f(y) e^{i k \cdot y}
      \, d^n y
    }
    }
    \,\, \frac{d^n k}{(2\pi)^n}
    \\
    & =
    +
    \underset{k \in \mathbb{R}^n}{\int}
    e^{i k \cdot x}
    \underset{= \widehat{f}(k) }{
    \underbrace{
      \underset{y \in \mathbb{R}^n}{\int}
      f(y) e^{- i k \cdot y}
      \, d^n y
    }
    }
    \,\, \frac{d^n k}{(2\pi)^n}
    \\
    & =
    \widecheck{\widehat{f}}(x)
  \end{aligned}
$$

which is the statement of the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).

(Here in the last step we used [[change of integration variables]] $k \mapsto -k$ which introduces one sign $(-1)^{n}$ for the new volume form,
but another sign $(-1)^n$ from the re-[[orientation]] of the integration domain. )

Equivalently, the above computation shows that the [[delta distribution]] is the [[neutral element]] for the
[[convolution product of distributions]].

=--

+-- {: .num_prop #PaleyWienerSchwartzTheorem}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]] I)**

Let $u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ be a [[compactly supported distribution]] regarded as a [[tempered distribution]] by example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}. Then its [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) is a [[non-singular distribution]] induced from a [[smooth function]] that grows at most exponentially.

=--

(e.g. [Hoermander 90, theorem 7.3.1](Paley-Wiener-Schwartz+theorem#Hoermander90))

+-- {: .num_prop #FourierInversionTheoremForDistributions}
###### Proposition
**([[Fourier inversion theorem]] for [[Fourier transform of distributions]])**

The operation of forming the [[Fourier transform of distributions]] $\widehat{u}$ (def. \ref{FourierTransformOnTemperedDistributions}) [[tempered distributions]] $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. \ref{TemperedDistribution}) is an [[isomorphism]],
with [[inverse]] given by

$$
  \widecheck{ u } \;\colon\; g \mapsto u\left( \widecheck{g}\right)
  \,,
$$

where on the right $\widecheck{g}$ is the ordinary [[inverse Fourier transform]] of $g$ according to prop. \ref{FourierInversion}.

=--

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

By def. \ref{FourierTransformOnTemperedDistributions}
this follows immediately from the [[Fourier inversion theorem]] for smooth functions (prop. \ref{FourierInversion}).

=--

We have the following distributional generalization of the basic property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}:

+-- {: .num_prop #FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct}
###### Proposition
**([[Fourier transform of distributions]] interchanges [[convolution of distributions]] with pointwise product)**

Let

$$
  u_1 \in \mathcal{S}'(\mathbb{R}^n)
$$

be a [[tempered distribution]] (def. \ref{TemperedDistribution}) and

$$
  u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$

be a [[compactly supported distribution]], regarded as a [[tempered distribution]] via example \ref{CompactlySupportedDistibutionsAreTemperedDistributions}.

Observe here that the [[Paley-Wiener-Schwartz theorem]] (prop. \ref{PaleyWienerSchwartzTheorem}) implies that the [[Fourier transform of distributions]] of $u_1$ is a [[non-singular distribution]]
$\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ is always defined.

Then the [[Fourier transform of distributions]] of the [[convolution product of distributions]] is the product of the [[Fourier transform of distributions]]:

$$
  \widehat{u_1 \star u_2}
  \;=\;
  \widehat{u_1} \cdot \widehat{u_2}
  \,.
$$

=--

(e.g. [Hörmander 90, theorem 7.1.15](Fourier+transform#Hoermander90))

+-- {: .num_remark #ProductOfDistributionsViaFourierTransformOfConvolution}
###### Remark
**([[product of distributions]] via [[Fourier transform of distributions]])**

Prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct} together with the [[Fourier inversion theorem]] (prop. \ref{FourierInversionTheoremForDistributions}) suggests to _define_ the [[product of distributions]] $u_1 \cdot u_2$ for [[compactly supported distributions]] $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula

$$
  \widehat{ u_1 \cdot u_2 }
  \;\coloneqq\;
  (2\pi)^n \widehat{u_1} \star \widehat{u_2}
$$

which would complete the generalization of of property (eq:FourierTransformInterchangesPointwiseProductWithConvolution) from prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}.

For this to make sense, the [[convolution product]] of the [[smooth functions]] on the right needs to exist, which is not guaranteed (prop. \ref{ConvolutionProductOnSchwartzSpace} does not apply here!). The condition that this exists is the [[Hörmander criterion]] on the _[[wave front set]]_ (def. \ref{WaveFrontSet}) of $u_1$ and $u_2$ (prop. \ref{HoermanderCriterionForProductOfDistributions} belwo). This we further discuss in _[Microlocal analysis and UV-Divergences](#MicrolocalAnalysisAndUltravioletDivergence)_ below.

=--

$\,$

**[[microlocal analysis]] and [[ultraviolet divergences]]**
 {#MicrolocalAnalysisAndUltravioletDivergence}

A _[[distribution]]_ (def. \ref{LinearObservablesAreTheCompactlySupportedDistributions}) or _[[generalized function]]_ (prop. \ref{DistributionsAreGeneralizedFunctions}) is like a [[smooth function]] which may have "[[singularities]]", namely points at which it values or that of its [[derivative of a distribution|derivatives]] "become infinite". Conversely, [[smooth functions]] are the [[non-singular distributions]] (prop. \ref{DistributionsAreGeneralizedFunctions}). The collection of points around which a distribution is singular (i.e. not [[non-singular distribution|non-singular]]) is called its _[[singular support]]_ (def. \ref{SingularSupportOfADistribution} below).

The [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) decomposes a [[generalized function]] into the [[plane wave]] modes that it is made of (def. \ref{PlaneWaves}). The [[Paley-Wiener-Schwartz theorem]] (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions} below) says that the singular nature of a [[compactly supported distribution]] may be read off from this [[Fourier transform|Fourier mode]] decomposition:  Singularities correspond to large contributions by Fourier modes of high [[frequency]] and small [[wavelength]], hence to large "[[ultraviolet divergence|ultraviolet]]" (UV) contributions (remark \ref{UltravioletDivergencesFromPaleyWiener} below). Therefore the [[singular support]] of a distribution is the set of points around which the Fourier transform does not sufficiently decay "in the UV".

But since the [[Fourier transform]] is a function of the full [[wave vector]] of the [[plane wave]] modes (def. \ref{PlaneWaves}),  not just of the [[frequency]]/[[wavelength]], but also of the [[direction of a vector|direction]] of the wave vector, this means that it contains _[[direction of a vector|directional]] information_ about the singularities: A distribution may have UV-singularities at some point and in some [[wave vector]] [[direction of a vector|direction]], but maybe not in other [[direction of a vector|directions]].

In particular, if the distribution in question is a [[distributional solution to a partial differential equation]] (def. \ref{DistributionalDerivatives}) on [[spacetime]] then the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem} below) says that the [[singular support]] of the solution evolves in spacetime along the direction of those [[wave vectors]] in which the Fourier transform exhibits high UV constributions. This means that these directions are the "wave front" of the distributional solution. Accordingly, the [[singular support]] of a distribution together with, over each of its points, the [[direction of a vector|directions]] of [[wave vectors]] in which the Fourier transform around that point has large UV constributions is called the _[[wave front set]]_ of the distribution (def. \ref{WaveFrontSet} below).

What is called _[[microlocal analysis]]_ is essentially the analysis of [[distributions]] with attention to their [[wave front set]], hence to the [[wave vector]]-[[direction of a vector|directions]] of [[UV divergences]].

In particular the [[product of distributions]] is well defined (only) if the [[wave front sets]] of the distributions to not "collide". And this in fact motivates the definition of the wave front set:

To see this, let $u,v \in  \mathcal{D}'(\mathbb{R}^1)$ be two [[distributions]], for simplicity of exposition taken on the [[real line]].

Since the product $u \cdot v$, is, if it exists, supposed to generalize the _pointwise_ product of smooth functions, it must be fixed locally: for every point $x \in \mathbb{R}$ there ought to be a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) $b \in C^\infty_{cp}(\mathbb{R})$ with $f(x) = 1$ such that

$$
  b^2 u \cdot v = (b u) \cdot (b v)
  \,.
$$

But now $b v$ and $b u$ are both [[compactly supported distributions]] (def. \ref{PropductOfDistributionWithASmoothFunction} below), and these have the special property that their [[Fourier transform of distributions|Fourier transforms]] $\widehat{b v}$ and $\widehat{b u}$ are, in particular, [[smooth functions]] (by the [[Paley-Wiener-Schwartz theorem]], prop \ref{PaleyWienerSchwartzTheorem}).

Moreover, the operation of [[Fourier transform]] interchanges pointwise products with [[convolution products]] (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}). This means that if the [[product of distributions]] $u \cdot v$ exists, it must locally be given by the [[inverse Fourier transform]] of the [[convolution product]] of the Fourier transforms $\widehat {b u}$ and $\widehat b v$:

$$
  \widehat{ b^2 u \cdot v }(x)
  \;=\;
  \underset{\underset{k_{max} \to \infty}{\longrightarrow}}{\lim}
  \,
  \int_{- k_{max}}^{k_{max}} \widehat{(b u)}(k) \widehat{(b v)}(x - k) d k
  \,.
$$

(Notice that the converse of this formula holds as a fact by prop. \ref{FourierTransformOfDistributionsInterchangesConvolutionOfDistributionsWithPointwiseProduct})

This shows that the [[product of distributions]] exists once there is a [[bump function]] $b$ such that the [[integral]] on the right converges as $k_{max} \to \infty$.

Now the [[Paley-Wiener-Schwartz theorem]] says more, it says that the Fourier transforms $\widehat {b u}$ and $\widehat {b u}$ are polynomially bounded. On the other hand, the [[integral]] above is well defined if the [[integrand]] decreases at least quadratically with $k \to \infty$.
This means that for the convolution product to be well defined, either $\widehat {b u}$ has to polynomially decrease faster with $k \to \pm \infty$ than $\widehat {b v}$ grows in the _other_ direction, $k \to \mp \infty$ (due to the minus sign in the argument of the second factor in the [[convolution product]]), or the other way around.

Moreover, the degree of polynomial growth of the [[Fourier transform of distributions|Fourier transform]] increases by one with each [[derivative of distributions|derivative]] (def. \ref{DistributionalDerivatives}). Therefore if the [[product law]] for [[derivatives of distributions]] is to hold generally, we need that either $\widehat{b u}$ or $\widehat{b v}$ decays faster than _any_ polynomial in the opposite of the directions in which the respective other factor does not decay.

Here the set of directions of wave vectors in which the Fourier transform of a distribution localized around any point does not decay exponentially is the _[[wave front set]]_ of a distribution (def. \ref{WaveFrontSet} below). Hence the condition that the product of two distributions is well defined is that for each wave vector direction in the wave front set of one of the two distributions, the opposite direction must not be an element of the wave front set of the other distribution. This is called _[[Hörmander's criterion]]_ (prop. \ref{HoermanderCriterionForProductOfDistributions} below).

We now say this in detail:

+-- {: .num_defn #RestrictionOfDistributions}
###### Definition
**([[restriction of distributions]])**

For  $U \subset \mathbb{R}^n$ a [[subset]], and $u \in \mathcal{D}'(\mathbb{R}^n)$ a [[distribution]], then the _[[restriction of distributions|restriction]]_ of $u$ to $U$ is the distribution

$$
  u\vert_U \in \mathcal{D}'(U)
$$

give by restricting $u$ to test functions whose [[support]] is in $U$.

=--

+-- {: .num_defn #SingularSupportOfADistribution}
###### Definition
**([[singular support of a distribution]])**

Given a [[distribution]] $u \in \mathcal{D}'(\mathbb{R}^n)$, a point $x \in \mathbb{R}^n$ is a _singular point_ if there is no [[neighbourhood]] $U \subset \mathbb{R}^n$ of $x$ such that the [[restriction of distributions|restriction]] $u\vert_U$ (def. \ref{RestrictionOfDistributions}) is a [[non-singular distribution]] (given by a [[smooth function]]).

The set of all singular points is the _[[singular support]]_ $supp_{sing}(u) \subset \mathbb{R}^n$ of $u$.

=--

+-- {: .num_defn #PropductOfDistributionWithASmoothFunction}
###### Definition
**([[product of distributions|product of a distribution]] with a [[smooth function]])**

Let $u \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]], and $f \in C^\infty(\mathbb{R}^n)$ a smooth function. Then the _product_ $f u \in \mathcal{D}'(\mathbb{R}^n)$ is the evident [[distribution]] given on a test function $b \in C^\infty_{cp}(\mathbb{R}^n)$ by

$$
  f u
  \;\colon\;
  u
  \mapsto
  u(f \cdot b)
  \,
$$

=--

+-- {: .num_prop #DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}
###### Proposition
**([[Paley-Wiener-Schwartz theorem]] II -- decay of [[Fourier transform of distributions|Fourier transform]] of [[compactly supported functions]])**

A [[compactly supported distribution]] $u \in \mathcal{E}'(\mathbb{R}^n)$ is non-[[singular support of a distribution|singular]], hence given by a [[compactly supported function]] $b \in C^\infty_{cp}(\mathbb{R}^n)$ via $u(f) = \int b(x) f(x) dvol(x)$, precisely if its [[Fourier transform of a distribution|Fourier transform]] $\hat u$ ([this def.](compactly+supported+distribution#FourierTransformOfCompactlySupportedDistribution)) satisfies the following decay property:

For all $N \in \mathbb{N}$ there exists $C_N \in \mathbb{R}_+$ such that for all $k \in \mathbb{R}^n$ we have that the [[absolute value]] ${\vert \hat v(k)\vert}$ of the Fourier transform at that point is bounded by

$$
  \label{DecayEstimateForFourierTransformOfNonSingularDistribution}
  {\vert \hat v(k)\vert}
  \;\leq\;
  C_N \left( 1 + {\vert k\vert} \right)^{-N}
  \,.
$$

=--

([Hörmander 90, around (8.1.1)](Paley-Wiener-Schwartz+theorem#Hoermander90))

+-- {: .num_remark #UltravioletDivergencesFromPaleyWiener}
###### Remark
**([[ultraviolet divergences]])**

In words, the [[Paley-Wiener-Schwartz theorem]] II (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) says that the [[singular support|singularities]] of a [[distribution]] "in position space" are reflected in non-decaying contributions of high [[frequencies]] (small [[wavelength]]) in its [[Fourier transform of distributions|Fourier mode]]-decomposition (def. \ref{FourierTransformOnTemperedDistributions}). Since for ordinary [[light waves]] one associates high [[frequency]] with the "ultraviolet", we may think of these as "ultaviolet contributions".

But apart from the [[wavelength]], the [[wave vector]] that the [[Fourier transform of distributions]] depends on also encodes the _[[direction of a vector|direction]]_ of the corresponding [[plane wave]]. Therefore the [[Paley-Wiener-Schwartz theorem]] says in more detail that a [[distribution]] is singular at some point already if along any _one_ [[direction of a vector|direction]] of the [[wave vector]] its local [[Fourier transform of distributions|Fourier transform]] picks up ultraviolet contributions _in that direction_.

It therefore makes sense to record this extra directional information in the singularity structure of a distribution. This is called the [[wave front set]] (def. \ref{WaveFrontSet}) below. The refined study of singularities taking this directional information into account is what is called _[[microlocal analysis]]_.

Moreover, the [[Paley-Wiener-Schwartz theorem]] I (prop. \ref{PaleyWienerSchwartzTheorem}) says that if the ultraviolet contributions diverge more than polynomially with high [[frequency]], then the corresponding would-be [[compactly supported distribution]] is not only singular, but is actually ill defined.

Such _[[ultraviolet divergences]]_ appear notably when forming a would-be [[product of distributions]] whose two factors have [[wave front sets]] whose UV-contributions "add up". This condition for the appearance/avoidance of [[UV-divergences]] is called _[[Hörmander's criterion]]_ (prop. \ref{HoermanderCriterionForProductOfDistributions} below).

=--

+-- {: .num_defn #WaveFrontSet}
###### Definition
**([[wavefront set]])**

Let $u \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]]. For $b \in C^\infty_{cp}(\mathbb{R}^n)$ a [[compactly supported function|compactly supported]] [[smooth function]], write $b u \in \mathcal{E}'(\mathbb{R}^n)$ for the corresponding product (def. \ref{PropductOfDistributionWithASmoothFunction}), which is now a [[compactly supported distribution]].

For $x\in supp(b) \subset \mathbb{R}^n$, we say that a unit [[covector]] $k \in S((\mathbb{R}^n)^\ast)$ is _regular_ if there exists a [[neighbourhood]] $U \subset S((\mathbb{R}^n)^\ast)$ of $k$ in the [[unit sphere]] such that for all $c k' \in (\mathbb{R}^n)^\ast$ with $c \in \mathbb{R}_+$ and  $k' \in U \subset S((\mathbb{R}^n)^\ast)$ the decay estimate (eq:DecayEstimateForFourierTransformOfNonSingularDistribution) is valid for the [[Fourier transform of distributions|Fourier transform]] $\widehat{b u}$ of $b u$; at $c k'$. Otherwise $k$ is _non-regular_. Write

$$
  \Sigma(b u)
    \;\coloneqq\;
  \left\{
    k \in S((\mathbb{R}^n)^\ast)
    \;\vert\;
    k \, \text{non-regular}
  \right\}
$$

for the set of non-regular covectors of $b u$.

The _wave front set at $x$_ is the [[intersection]] of these sets as $b$ ranges over [[bump functions]] whose [[support]] includes $x$:

$$
  \Sigma_x(u)
  \;\coloneqq\;
  \underset{
    { b \in C^\infty_{cp}(\mathbb{R}^n) }
      \atop
    { x \in supp(b) }
  }{\cap}
  \Sigma(b u)
  \,.
$$

Finally the _[[wave front set]]_ of $u$ is the subset of the [[sphere bundle]] $S(T^\ast \mathbb{R}^n)$ which over $x \in \mathbb{R}^n$ consists of $\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n$:

$$
  WF(u)
    \;\coloneqq\;
  \underset{x \in \mathbb{R}^n}{\cup}
  \Sigma_x(u)
  \;\subset\;
  S(T^\ast \mathbb{R}^n)
$$

Often this is equivalently considered as the full [[conical set]] inside the [[cotangent bundle]] generated by the unit covectors under multiplication with [[positive number|positive]] [[real numbers]].

=--

([Hörmander 90, def. 8.1.2](wavefron+set#Hoermander90))

+-- {: .num_remark #WaveFrontSetIsBundleOverSingularSupport}
###### Remark
**([[wave front set]] is the [[UV divergence]]-[[direction of a vector|direction]]-[[bundle]] over the [[singular support]])**

For $u \in \mathcal{D}'(\mathbb{R}^n)$
The [[Paley-Wiener-Schwartz theorem]] (prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}) implies that

1. Forgetting the [[direction of a vector|direction]] [[covectors]] in the [[wave front set]] $WF(u)$ (def. \ref{WaveFrontSet}) and remembering only the points where they are based yields the set of singlar points of $u$, hence the [[singular support]] (def. \ref{SingularSupportOfADistribution})

   $$
     \array{
       WF(u)
       \\
       \downarrow
       \\
       supp_{sing}(u) &\hookrightarrow& \mathbb{R}^n
     }
   $$

1. the [[wave front set]] is [[empty set|empty]], precisely if the [[singular support]] is [[empty set|empty]], which is the case precisely if $u$ is a [[non-singular distribution]].

=--

+-- {: .num_example #NonSingularDistributionTrivialWaveFrontSet}
###### Example
**([[wave front set]] of [[non-singular distribution]] is [[empty set|empty]])**

By prop. \ref{DecayPropertyOfFourierTransformOfCompactlySupportedFunctions}, the [[wave front set]] (def. \ref{WaveFrontSetIsBundleOverSingularSupport}) of a [[non-singular distribution]] (prop. \ref{DistributionsAreGeneralizedFunctions}) is [[empty set|empty]]. Conversely, a [[distribution]] is [[non-singular distribution|non-singular]] if its wave front set is empty:

$$
  u \in \mathcal{D}'\;\text{non-singular}
  \phantom{AA}
  \Leftrightarrow
  \phantom{AA}
  WF(u) = \emptyset
$$

=--

+-- {: .num_example #WaveFrontOfDeltaDistribution}
###### Example
**([[wave front set]] of [[delta distribution]])**

Consider the [[delta distribution]]

$$
  \delta_0 \in \mathcal{D}'(\mathbb{R}^n)
$$

given by [[evaluation]] at the origin. Its [[wave front set]] (def. \ref{WaveFrontSet}) consists of all the [[direction of a vector|directions]] at the origin:

$$
  WF(\delta_0)
  \;=\;
  \left\{
    (0,k)
    \;\vert\;
    k \in \mathbb{R}^n \setminus \{0\}
  \right\}
  \subset
  \mathbb{R}^n \times \mathbb{R}^n
  \simeq
  T^\ast \mathbb{R}^n
  \,.
$$

=--

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

First of all the [[singular support of a distribution|singular support]] (def. \ref{SingularSupportOfADistribution}) of $\delta_0$ is clearly $supp_{sing}(\delta(0)) = \{0\}$, hence by remark \ref{WaveFrontSetIsBundleOverSingularSupport} the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.

At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the [[Fourier transform of distributions|Fourier transform]] $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a [[constant function]] (example \ref{FourierTransformOfDeltaDistribution}) and hence does not decay in any direction.

=--

+-- {: .num_example #WaveFrontSetOfHeavisideDistribution}
###### Example
**([[wave front set]] of [[step function]])**

Let $\Theta \in \mathcal{D}'(\mathbb{R}^1)$ be the [[Heaviside step function]] given by

$$
  \Theta(b) \coloneqq \int_0^\infty b(x)\, d x
  \,.
$$

Its [[wave front set]] (def. \ref{WaveFrontSet}) is

$$
  WF(H) = \{(0,k) \vert k \neq 0\}
  \,.
$$

=--

+-- {: .num_prop #WaveFrontSetOfCompactlySupportedDistributions}
###### Proposition
**([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])**

Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two  [[compactly supported distributions]]. Then the [[wave front set]] (def. \ref{WaveFrontSet}) of their [[convolution of distributions]] (def. \ref{ConvolutionOfADistributionWithACompactlySupportedDistribution}) is

$$
  WF(u \star v)
  \;=\;
  \left\{
    (x + y, k)
    \;\vert\;
    (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
  \right\}
  \,.
$$

=--

([Bengel 77, prop. 3.1](convolution+product+of+distributions#Bengel77))

+-- {: .num_prop #HoermanderCriterionForProductOfDistributions}
###### Proposition
**([[Hörmander's criterion]] for [[product of distributions]])**

Let $u, v \in \mathcal{D}'(\mathbb{R}^n)$ be two [[distributions]]. If their
[[wave front sets]] (def \ref{WaveFrontSet}) do not collide, in that
for $v \in T^\ast_x X$ a [[covector]] contained in one of the two wave front sets then the covector $-v \in T^\ast_x X$ with the opposite [[direction of a vector|direction]] in _not_ contained in the other wave front set, i.e. the [[intersection]] [[fiber product]]
inside the [[cotangent bundle]] $T^\ast X$ of the pointwise [[sum]] of wave fronts with the [[zero section]] is [[empty set|empty]]:

$$
  \left( WF(u_1) + WF(u_2) \right) \underset{T^\ast X}{\times} X
  \;=\;
  \emptyset
$$

i.e.

$$
  \array{
     && \emptyset
     \\
     & \swarrow && \searrow
     \\
     WF(u_1) + WF(u_2) && (pb) && X
     \\
     & \searrow && \swarrow_{\mathrlap{0}}
     \\
     && T^\ast X
  }
$$

then the [[product of distributions]] $u \cdot v$ exists, given, locally, by the [[Fourier inversion theorem|Fourier inversion]] of the [[convolution product]] of their [[Fourier transform of distributions]] (remark \ref{ProductOfDistributionsViaFourierTransformOfConvolution}).

=--

For making use of [[wave front sets]], we need a collection of results about how wave front sets change as we apply certain operations to distributions:

+-- {: .num_prop #RetainsOrShrinksWaveFrontSetDifferentialOperator}
###### Proposition
**([[differential operator]] preserves or shrinks [[wave front set]])**

Let $P$ be a [[differential operator]] (def. \ref{DifferentialOperator}). Then for $u \in \mathcal{D}'$ a [[distribution]], the [[wave front set]] (def. \ref{WaveFrontSet}) of the [[derivative of distributions]] $P u$ (def. \ref{DistributionalDerivatives}) is contained in the original wave front set of $u$:

$$
  WF(P u) \subset WF(u)
$$

=--

([Hörmander 90, (8.1.11)](differential+operator#Hoermander90))

+-- {: .num_prop #WaveFrontSetOfProductOfDistributionsInsideFiberProductOfFactorWaveFrontSets}
###### Proposition
**([[wave front set]] of [[product of distributions]] is inside [[fiber]]-wise [[sum]] of [[wave front sets]])**

Let $u,v \in \mathcal{D}'(X)$ be a [[pair]] of [[distributions]] satisfying [[Hörmander's criterion]], so that their [[product of distributions]] $u \cdot v$ exists by prop. \ref{HoermanderCriterionForProductOfDistributions}. Then the [[wave front set]] (def. \ref{WaveFrontSet}) of the product distribution is contained inside the [[fiber]]-wise [[sum]] of the wave front set elements of the two factors:

$$
  WF(u \cdot v)
  \;\subset\;
  (WF(u) \cup (X \times \{0\}))
    +
  (WF(v) \cup (X \times \{0\}))
  \,.
$$

=--

([Hörmander 90, theorem 8.2.10](product+of+distributions#Hoermander90))

More generally:

+-- {: .num_prop #PartialProductOfDistributionsOfSeveralVariables}
###### Proposition
**(partial [[product of distributions|product of]] [[distributions of several variables]])**

Let

$$
  K_1 \in \mathcal{D}'(X \times Y)
  \phantom{AAA}
  K_2 \in \mathcal{D}'(Y \times Z)
$$

be two [[distributions of two variables]]. For their [[product of distributions]] to be defined over $Y$, [[Hörmander's criterion]] on the [[pair]] of [[wave front sets]] $WF(K_1), WF(K_2)$  needs to hold for the [[wave front set|wave front]] [[wave vectors]] along $X$ and $Y$ taken to be zero.

If this is satisfied, then composition of [[integral kernels]] (if it exists)

$$
  (K_1 \circ K_2)(-,-)
  \;\coloneqq\;
  \underset{Y}{\int}
    K_1(-,y) K_2(y,-)
  dvol_Y(y)
  \;\in\;
  \mathcal{D}'(X \times Z)
$$

has [[wave front set]] constrained by

$$
  \label{CompositionOfIntegralKernelsWaveFronConstraint}
  WF(K_1 \circ K_2)
  \;\subset\;
  \left\{
    (x,z, k_x, k_z)
    \;\vert\;
    \array{
      \left(
      (x,y,k_x,-k_y) \in WF(K_1)
      \,\,
      \text{and}
      \,\,
      (y,z,k_y, k_z) \in WF(K_2)
      \right)
      \\
      \text{or}
      \\
      \left(
        k_x = 0
        \,\text{and}\,
        (y,z,0,-k_z) \in WF(K_2)
      \right)
      \\
      \text{or}
      \\
      \left(
        k_z = 0
        \,\text{and}\,
        (x,y,k_x,0) \in WF(K_1)
      \right)
    }
  \right\}
$$

=--

([Hörmander 90, theorem 8.2.14](product+of+distributions#Hoermander90))

A key fact for identifying [[wave front sets]] is the _[[propagation of singularities theorem]]_ (prop. \ref{PropagationOfSingularitiesTheorem} below). In order to state this we need the following concepts regarding symbols of differential operators:

+-- {: .num_defn #SymbolOfADifferentialOperator}
###### Definition
**([[symbol of a differential operator]])**

Let

$$
  D
  \;=\;
  \underset{n \leq N}{\sum}
  D^{\mu_1 \cdots \mu_n}
  \frac{\partial}{\partial x^{\mu^1}}
  \cdots
  \frac{\partial}{\partial x^{\mu^n}}
  +
  D^0
$$

be a [[differential operator]] on $\mathbb{R}^n$ (def. \ref{DifferentialOperator}). Then its _[[symbol of a differential operator]]_ is the [[smooth function]] on the [[cotangent bundle]] $T^\ast \mathbb{R}^n \simeq \mathbb{R}^n \times \mathbb{R}^n$ (def. \ref{TangentVectorFields}) given by

$$
  \array{
    T^\ast \mathbb{R}^n
      &\overset{q}{\longrightarrow}&
    \mathbb{C}
    \\
    k &\mapsto&
  \underset{n \leq N}{\sum}
  D^{\mu_1 \cdots \mu_k} k_{\mu_1} \cdots k_{\mu_n}
   }
  \,.
$$

The _[[principal symbol]]_ is the top degree [[homogeneous function|homogeneous]] part
$D^{\mu_1 \cdots \mu_k} k_{\mu_1} \cdots k_{\mu_N}$.

=--

+-- {: .num_defn #SymbolOrder}
###### Definition
**([[symbol order]])**

A [[smooth function]] $q$ on the [[cotangent bundle]] $T^\ast \mathbb{R}^n$ (e.g. the [[symbol of a differential operator]], def. \ref{SymbolOfADifferentialOperator} ) is of _order $m$_ (and type $1,0$, denoted $q \in S^m = S^m_{1,0}$), for $m \in \mathbb{N}$, if on each [[coordinate chart]] $((x^i), (k_i))$ we have that for every [[compact subset]] $K$ of the base space and all multi-indices $\alpha$ and $\beta$, there is a  [[real number]] $C_{\alpha, \beta,K } \in \mathbb{R}$ such that the [[absolute value]] of the [[partial derivatives]] of $q$ is bounded by

$$
  \left\vert
    \frac{\partial^\alpha}{\partial k_\alpha}
    \frac{\partial^\beta}{\partial x^\beta}
    q(x,k)
  \right\vert
  \;\leq\;
  C_{\alpha,\beta,K}\left( 1+ {\vert k\vert}\right)^{m - {\vert \alpha\vert}}
$$

for all $x \in K$ and all cotangent vectors $k$ to $x$.

A [[Fourier transform|Fourier integral]] operator $Q$ is of _[[symbol class]]_ $L^m = L^m_{1,0}$ if it is of the form

   $$
     Q f (x)
     \;=\;
     \int \int e^{i k \cdot (x - y)} q(x,y,k) f(y) \, d y \, d k
   $$

with symbol $q$ of order $m$, in the above sense.

=--

([Hörmander 71, def. 1.1.1 and first sentence of section 2.1 with (1.4.1)](symbol+order#Hoermander71))

+-- {: .num_prop #PropagationOfSingularitiesTheorem}
###### Proposition
**([[propagation of singularities theorem]])**

Let $Q$ be a [[differential operator]] (def. \ref{DifferentialOperator}) of [[symbol class]] $L^m$ (def. \ref{SymbolOrder}) with real [[principal symbol]] $q$ that is [[homogeneous function|homogeneous]] of degree $m$.

For $u \in \mathcal{D}'(X)$ a [[distribution]] with $Q u = f$, then the [[complement]] of the [[wave front set]] of $u$ by that of $f$ is contained in the set of covectors on which the [[principal symbol]] $q$ vanishes:

$$
  WF(u) \setminus WF(f) \;\subset\; q^{-1}(0)
  \,.
$$

Moreover, $WF(u)$ is invariant under the [[bicharacteristic flow]] induced by the [[Hamiltonian vector field]] of $q$ with respect to the canonical [[symplectic manifold]] structure on the [[cotangent bundle]] ([here](cotangent+bundle#SymplecticStructure)).

=--

([Duistermaat-Hörmander 72, theorem 6.1.1](propagation+of+singularities+theorem#DuistermaatHoermander72), recalled for instance as [Radzikowski 96, theorem 4.6](propagation+of+singularities+theorem#Radzikowski96))

$\,$

**[[Cauchy principal value]]**
 {#CauchyPrincipalValues}

An important application of the [[Fourier transform|Fourier analysis]] of [[distributions]] is the class of distributions known
broadly as _[[Cauchy principal values]]_. Below we will find that these control the detailed nature of the
various [[propagators]] of [[free field theories]], notably the [[Feynman propagator]] is manifestly a [[Cauchy principal value]] (prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} and def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime} below),
but also the [[singular support]] properties of the [[causal propagator]] and the [[Wightman propagator]] are governed by
Cauchy principal values (prop. \ref{SingularSupportOfCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} and prop. \ref{SingularSupportOfHadamardPropagatorForKleinGordonEquationOnMinkowskiSpacetimeIsTheLightCone} below). This way the understanding of
Cauchy principal values eventually allows us to determine the [[wave front set]] of all the propagators (prop. \ref{WaveFronSetsForKGPropagatorsOnMinkowski}) below.

Therefore we now collect some basic definitions and facts on [[Cauchy principal values]].

The _Cauchy principal value_ of a [[function]] which is [[integrable function|integrable]] on the [[complement]] of one point is, if it exists, the [[limit of a sequence|limit]] of the [[integrals]] of the function over subsets in the [[complement]] of this point as these integration [[domains]] tend to that point _symmetrically_ from all sides.

One also subsumes the case that the "point" is "at infinity", hence that the function is [[integrable function|integrable]] over every [[bounded subsets|bounded]] [[domain]]. In this case the Cauchy principal value is the [[limit of a sequence|limit]], if it exists, of the [[integrals]] of the function over bounded domains, as their bounds tend _symmetrically_ to infinity.

The operation of sending a [[compactly supported function|compactly supported]] [[smooth function]] ([[bump function]]) to Cauchy principal value of its pointwise product with a function $f$ that may be singular at the origin defines a [[distribution]], usually denoted $PV(f)$.

+-- {: .num_defn #CauchyIntegralValueOfIntegralOverRealline}
###### Definition
**([[Cauchy principal value]] of an [[integral]] over the [[real line]])**

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for every [[positive real number]] $\epsilon$ its [[restriction]] to $\mathbb{R}\setminus (-\epsilon, \epsilon)$ is [[integrable function|integrable]]. Then the _[[Cauchy principal value]]_ of $f$ is, if it exists, the [[limit of a sequence|limit]]

$$
  PV(f) \coloneqq \underset{\epsilon \to 0}{\lim}
  \underset{\mathbb{R} \setminus (-\epsilon, \epsilon)}{\int} f(x) \, d x
  \,.
$$

=--

+-- {: .num_defn #CauchyPrincipalValueAsDistributionOnRealLine}
###### Definition
**([[Cauchy principal value]] as [[distribution]] on the [[real line]])**

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a [[function]] on the [[real line]] such that for all [[bump functions]] $b \in C^\infty_{cp}(\mathbb{R})$ the Cauchy principal value of the pointwise product function $f b$ exists, in the sense of def. \ref{CauchyIntegralValueOfIntegralOverRealline}. Then this assignment

$$
  PV(f)
  \;\colon\;
  b \mapsto PV(f b)
$$

defines a [[distribution]] $PV(f) \in \mathcal{D}'(\mathbb{R})$.

=--

+-- {: .num_example }
###### Example

Let $f \colon \mathbb{R} \to \mathbb{R}$ be an [[integrable function]] which is symmetric, in that $f(-x) = f(x)$ for all $x \in \mathbb{R}$. Then the principal value integral (def. \ref{CauchyIntegralValueOfIntegralOverRealline}) of $x \mapsto \frac{f(x)}{x}$ exists and is zero:

$$
  \underset{\epsilon \to 0}{\lim}
  \underset{\mathbb{R}\setminus (-\epsilon, \epsilon)}{\int}
  \frac{f(x)}{x} d x
  \; = \; 0
$$

This is because, by the symmetry of $f$ and the skew-symmetry of $x \mapsto 1/x$, the the two contributions to the integral are equal up to a sign:

$$
  \int_{-\infty}^{-\epsilon} \frac{f(x)}{x} d x
  \;=\;
  -
  \int_{\epsilon}^\infty \frac{f(x)}{x} d x
  \,.
$$

=--

+-- {: .num_example #PrincipalValueOfInverseFunctionCharacteristicEquation}
###### Example

The [[Cauchy principal value]] [[distribution]] $PV\left( \frac{1}{x}\right)$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) solves the distributional [[equation]]

$$
  \label{DistributionalEquationxfOfxEqualsOne}
  x PV\left(\frac{1}{x}\right) = 1
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{R}^1)
  \,.
$$

Since the [[delta distribution]] $\delta \in \mathcal{D}'(\mathbb{R}^1)$ solves the equation

$$
  x \delta(x) = 0
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{r}^1)
$$

we have that more generally every [[linear combination]] of the form

$$
  \label{GeneralDistributionalSolutionToxfEqualsOne}
  F(x)
  \coloneqq
  PV(1/x) + c \delta(x)
  \phantom{AAA}
  \in \mathcal{D}'(\mathbb{R}^1)
$$

for $c \in \mathbb{C}$, is a distributional solution to $x F(x) =  1$.

The [[wave front set]] of all these solutions is

$$
  WF\left(
    PV(1/x) + c \delta(x)
  \right)
  \;=\;
  \left\{
    (0,k) \;\vert\; k \in \mathbb{R}^\ast \setminus \{0\}
  \right\}
  \,.
$$

=--

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

The first statement is immediate from the definition: For $b \in C^\infty_c(\mathbb{R}^1)$ any [[bump function]] we have that

$$
  \begin{aligned}
    \left\langle x PV\left(\frac{1}{x}\right), b \right\rangle
    & \coloneqq
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1 \setminus (-\epsilon, \epsilon)}{\int}
    \frac{x}{x}b(x) \, d x
    \\
    & =
    \int b(x)  d x
    \\
    & =
    \langle 1,b\rangle
  \end{aligned}
$$

Regarding the second statement: It is clear that the wave front set is concentrated at the origin. By symmetry of the distribution around the origin, it must contain both [[direction of a vector|directions]].

=--

+-- {: .num_prop}
###### Proposition

In fact (eq:GeneralDistributionalSolutionToxfEqualsOne) is the most general distributional solution to (eq:DistributionalEquationxfOfxEqualsOne).

=--

This follows by the characterization of [[extension of distributions]] to a point, see there at [this prop.](extension+of+distributions#SpaceOfPointExtensions) ([Hörmander 90, thm. 3.2.4](Cauchy+principa+value#HoermanderLPDO1))

+-- {: .num_defn #IntegrationAgainstInverseOfxWithImaginaryOffset}
###### Definition
**(integration against inverse variable with imaginary offset)**

Write

$$
  \tfrac{1}{x + i0^\pm}
  \;\in\;
  \mathcal{D}'(\mathbb{R})
$$

for the [[distribution]] which is the [[limit]] in $\mathcal{D}'(\mathbb{R})$ of the [[non-singular distributions]] which are given by the [[smooth functions]] $x \mapsto \tfrac{1}{x \pm i \epsilon}$ as the [[positive real number]] $\epsilon$ tends to zero:

$$
  \frac{1}{
     x + i 0^\pm
  }
  \;\coloneqq\;
  \underset{ { \epsilon \in (0,\infty) } \atop { \epsilon \to 0 }  }{\lim}
  \tfrac{1}{x \pm i \epsilon}
$$

hence the distribution which sends $b \in C^\infty(\mathbb{R}^1)$ to

$$
  b \mapsto
  \underset{\mathbb{R}}{\int}
    \frac{b(x)}{x \pm i \epsilon} \, d x
  \,.
$$

=--

+-- {: .num_prop #CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}
###### Proposition
**([[Cauchy principal value]] equals integration with [[imaginary number|imaginary]] offset plus [[delta distribution]])**

The [[Cauchy principal value]] [[distribution]] $PV\left( \tfrac{1}{x}\right) \in \mathcal{D}'(\mathbb{R})$ (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}) is equal to the sum of the integration over $1/x$ with imaginary offset (def. \ref{IntegrationAgainstInverseOfxWithImaginaryOffset}) and a [[delta distribution]].

$$
 PV\left(\frac{1}{x}\right)
  \;=\;
 \frac{1}{x + i 0^\pm}
  \pm i \pi \delta
  \,.
$$

In particular, by prop. \ref{PrincipalValueOfInverseFunctionCharacteristicEquation} this means that $\tfrac{1}{x + i 0^\pm}$ solves the distributional equation

$$
  x \frac{1}{x + i 0^\pm}
  \;=\;
  1
  \phantom{AA}
  \in \mathcal{D}'(\mathbb{R}^1)
  \,.
$$

=--

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

Using that

$$
  \begin{aligned}
    \frac{1}{x \pm i \epsilon}
    & =
    \frac{
       x \mp i \epsilon
    }{
      (x + i \epsilon)(x - i \epsilon)
    }
    \\
    & =
    \frac{ x \mp i \epsilon }{(x^2 + \epsilon^2)}
  \end{aligned}
$$

we have for every [[bump function]] $b \in C^\infty_{cp}(\mathbb{R}^1)$

$$
  \begin{aligned}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{b(x)}{x \pm i \epsilon}
    d x
    & \;=\;
    \underset{
      (A)
    }{
    \underbrace{
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{x^2}{x^2 + \epsilon^2} \frac{b(x)}{x}
    d x
    }
    }
    \mp i \pi
    \underset{(B)}{
    \underbrace{
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{1}{\pi}
    \frac{\epsilon}{x^2 + \epsilon^2} b(x)
    \,
    d x
    }}
  \end{aligned}
$$

Since

$$
  \array{
     && \frac{x^2}{x^2 + \epsilon^2}
     \\
     & {}^{\mathllap{ { {\vert x \vert} \lt \epsilon } \atop { \epsilon \to 0 }  }}\swarrow
     &&
     \searrow^{\mathrlap{ {{\vert x\vert} \gt \epsilon} \atop { \epsilon \to 0 }  }}
     \\
     0 && && 1
  }
$$

it is plausible that $(A) = PV\left( \frac{b(x)}{x} \right)$, and similarly that $(B) = b(0)$. In detail:

$$
  \begin{aligned}
    (A)
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{x}{x^2 + \epsilon^2} b(x)
    d x
    \\
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
     \frac{d}{d x}
     \left(
        \tfrac{1}{2}
        \ln(x^2 + \epsilon^2)
     \right)
      b(x)
    d x
    \\
    & =
    -\tfrac{1}{2}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
     \ln(x^2 + \epsilon^2)
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -\tfrac{1}{2}
    \underset{\mathbb{R}^1}{\int}
     \ln(x^2)
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -
    \underset{\mathbb{R}^1}{\int}
     \ln({\vert x \vert})
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    -
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
     \ln( {\vert x \vert} )
     \frac{d b}{d x}(x)
    d x
    \\
    & =
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1\setminus (-\epsilon, \epsilon)}{\int}
     \frac{1}{x}
     b(x)
    d x
    \\
    & =
    PV\left( \frac{b(x)}{x} \right)
  \end{aligned}
$$

and

$$
  \begin{aligned}
    (B)
    & =
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \frac{\epsilon}{x^2 + \epsilon^2}
    b(x)
    \,
    d x
    \\
    & =
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
    \left(
      \frac{d}{d x}
      \arctan\left( \frac{x}{\epsilon} \right)
    \right)
    b(x)
    \,
    d x
    \\
    & =
    -
    \tfrac{1}{\pi}
    \underset{\epsilon \to 0}{\lim}
    \underset{\mathbb{R}^1}{\int}
      \arctan\left( \frac{x}{\epsilon} \right)
    \frac{d b}{d x}(x)
    \,
    d x
    \\
    & =
    -
    \frac{1}{2}
    \underset{\mathbb{R}^1}{\int}
      sgn(x)
    \frac{d b}{d x}(x)
    \,
    d x
    \\
    & =
    b(0)
  \end{aligned}
$$

where we used that the [[derivative]] of the [[arctan]] function is $\frac{d}{ d x} \arctan(x) = 1/(1 + x^2)$ and that $\underset{\epsilon \to + \infty}{\lim} \arctan(x/\epsilon) = \tfrac{\pi}{2}sgn(x)$ is proportional to the [[sign function]].

=--

+-- {: .num_example #FourierIntegralFormulaForStepFunction}
###### Example
**([[Fourier transform|Fourier integral]] formula for [[step function]])**

The [[Heaviside distribution]] $\Theta \in \mathcal{D}'(\mathbb{R})$ is equivalently the following [[Cauchy principal value]] (def. \ref{CauchyPrincipalValueAsDistributionOnRealLine}):

$$
  \begin{aligned}
    \Theta(x)
    & =
    \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i 0^+}
    \\
    & \coloneqq
    \underset{ \epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    \,,
  \end{aligned}
$$

where the limit is taken over [[sequences]] of [[positive numbers|positive]] [[real numbers]] $\epsilon \in (-\infty,0)$ tending to zero.

=--

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

We may think of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ uniquely extended to a [[holomorphic function]] on the [[complex plane]] and consider computing the given real [[line integral]] for fixed $\epsilon$ as a [[contour integral]] in the [[complex plane]].

If $x \in (0,\infty)$ is [[positive number|positive]], then the exponent

$$
  i \omega x = - Im(\omega) x + i Re(\omega) x
$$

<div style="float:right;margin:0 10px 10px 0;"><img src="https://ncatlab.org/nlab/files/ContoursForHeavisideFourierTransform.png" width="300">
</div>

has negative [[real part]] for _positive_ [[imaginary part]] of $\omega$. This means that the [[line integral]] equals the complex [[contour integral]] over a contour $C_+ \subset \mathbb{C}$ closing in the [[upper half plane]]. Since $i \epsilon$ has positive [[imaginary part]] by construction, this contour does encircle the [[pole]] of the [[integrand]] $\frac{e^{i \omega x}}{\omega - i \epsilon}$ at $\omega = i \epsilon$. Hence by the [[Cauchy integral formula]] in the case $x \gt 0$ one gets

$$
  \begin{aligned}
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \int_{-\infty}^\infty \frac{e^{i \omega x}}{\omega - i \epsilon} d\omega
    & =
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{2 \pi i}
    \oint_{C_+} \frac{e^{i \omega x}}{\omega - i \epsilon} d \omega
    \\
    &
    =
    \underset{\epsilon \to 0^+}{\lim}
    \left(e^{i \omega x}\vert_{\omega = i \epsilon}\right)
    \\
    & =
    \underset{\epsilon \to 0^+}{\lim}
    e^{- \epsilon x}
    \\
    & =
    e^0 = 1
  \end{aligned}
  \,.
$$

Conversely, for $x \lt 0$ the real part of the integrand decays as the _[[negative number|negative]]_ imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour $C_- \subset \mathbb{C}$ closing in the lower half plane. Since the integrand has no pole in the lower half plane, in this case the [[Cauchy integral formula]] says that this integral is zero.

=--

Conversely, by the [[Fourier inversion theorem]], the [[Fourier transform]] of the [[Heaviside distribution]] is the Cauchy principal value as in prop. \ref{CauchyPrincipalValueEqualsIntegrationWithImaginaryOffsetPlusDelta}:

+-- {: .num_example #RelationToFourierTransformOfHeavisideDistribution}
###### Example
**(relation to [[Fourier transform]] of [[Heaviside distribution]] / [[Schwinger parameter|Schwinger parameterization]])**

The [[Fourier transform of distributions]] (def. \ref{FourierTransformOnTemperedDistributions}) of the [[Heaviside distribution]] is
the following [[Cauchy principal value]]:

$$
  \begin{aligned}
    \widehat \Theta(x)
    & =
    \int_0^\infty e^{i k x} \,  dk
    \\
    & =
    i \frac{1}{x + i 0^+}
  \end{aligned}
$$

Here the second equality is also known as _complex [[Schwinger parameter|Schwinger parameterization]]_.

=--

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

As [[generalized functions]] consider the [[limit of a sequence|limit]] with a decaying component:

$$
  \begin{aligned}
    \int_0^\infty e^{i k x} \,  dk
    & =
    \underset{\epsilon \to 0^+}{\lim}
    \int_0^\infty e^{i k x - \epsilon k} \,  dk
    \\
    & =
    -
    \underset{\epsilon \to 0^+}{\lim}
    \frac{1}{ i x - \epsilon}
    \\
    & =
    i \frac{1}{x + i 0^+}
  \end{aligned}
$$

=--

Let now $q \colon \mathbb{R}^{n} \to \mathbb{R}$ be a non-degenerate real [[quadratic form]]
[[analytic continuation|analytically continued]] to a real quadratic form

$$
  q \;\colon\; \mathbb{C}^n \longrightarrow \mathbb{C}
  \,.
$$

Write $\Delta$ for the [[determinant]] of $q$

Write $q^\ast$ for the induced quadratic form on [[dual vector space]]. Notice that $q$ (and hence $a^\ast$) are assumed non-degenerate but need not necessarily be positive or negative definite.

+-- {: .num_prop #FourierTransformOfPrincipalValueOfPowerOfQuadraticForm}
###### Proposition
**([[Fourier transform of distributions|Fourier transform]] of principal value of power of [[quadratic form]])**

Let $m \in \mathbb{R}$ be any [[real number]], and $\kappa \in \mathbb{C}$ any [[complex]] number. Then the [[Fourier transform of distributions]] of $1/(q + m^2 + i 0^+)^\kappa$ is

$$
  \widehat
  {
    \left(
      \frac{1}{(q + m^2 + i0^+)^\kappa}
    \right)
  }
  \;=\;
  \frac{
    2^{1- \kappa} (\sqrt{2\pi})^{n} m^{n/2-\kappa}
  }
  {
    \Gamma(\kappa) \sqrt{\Delta}
  }
  \frac{
    K_{n/2 - \kappa}\left( m \sqrt{q^\ast - i 0^+} \right)
  }
  {
    \left(\sqrt{q^\ast - i0^+ }\right)^{n/2 - \kappa}
  }
  \,,
$$

where

1. $\Gamma$ deotes the [[Gamma function]]

1. $K_{\nu}$ denotes the [[modified Bessel function]] of order $\nu$.

Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).

=--

([Gel'fand-Shilov 66, III 2.8 (8) and (9), p 289](Cauchy+principal+value#GelfandShilov66))

+-- {: .num_prop #FourierTransformOfDeltaDistributionappliedToMassShell}
###### Proposition
**([[Fourier transform]] of [[delta distribution]] applied to [[mass shell]])**

Let $m \in \mathbb{R}$, then the [[Fourier transform of distributions]] of the [[delta distribution]] $\delta$ applied to the "mass shell" $q + m^2$ is

$$
  \widehat{
    \delta(q + m^2)
  }
  \;=\;
  - \frac{i}{\sqrt{{\vert\Delta\vert}}}
  \left(
    e^{i \pi t /2 }
    \frac{
      K_{n/2-1}
      \left(
        m \sqrt{ q^\ast + i0^+ }
      \right)
    }{
      \left(\sqrt{q^\ast + i0^+}\right)^{n/2 - 1}
    }
    \;-\;
    e^{-i \pi t /2 }
    \frac{
      K_{n/2-1}
      \left(
        m \sqrt{ q^\ast - i0^+ }
      \right)
    }{
      \left(\sqrt{q^\ast - i0^+}\right)^{n/2 - 1}
    }
  \right)
  \,,
$$

where $K_\nu$ denotes the [[modified Bessel function]] of order $\nu$.

Notice that $K_\nu(a)$ diverges for $a \to 0$ as $a^{-\nu}$ ([DLMF 10.30.2](http://dlmf.nist.gov/10.30#E2)).

=--

([Gel'fand-Shilov 66, III 2.11 (7), p 294](Cauchy+principal+value#GelfandShilov66))

$\,$

**[[propagators]] for the [[free field theory|free]] [[scalar field]] on [[Minkowski spacetime]]**

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)_

$\,$

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

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

By example \ref{FourierTransformOfKleinGordonEquation} 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 (see def. \ref{PlaneWaves})

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

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

$\,$

**[[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]] $G_\pm$ (def. \ref{AdvancedAndRetardedGreenFunctions}) of the [[Klein-Gordon operator]] on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) are induced from [[integral kernels]] ("[[propagators]]"), hence [[distributions in two variables]]

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

by (in [[generalized function]]-notation, prop. \ref{DistributionsAreGeneralizedFunctions})

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

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]] (example \ref{GreenHyperbolicKleinGordonEquation}) 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]] (def. \ref{DistributionalSections}) 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]] (def. \ref{FourierTransformOnTemperedDistributions}). If we do find a [[solution]] this way, it is guaranteed to be the unique solution by prop. \ref{AdvancedAndRetardedGreenFunctionsForGreenHyperbolicOperatorAreUnique}.

By example \ref{FourierTransformOfDistributionsIndeedGeneralizedOrdinaryFourierTransform}
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 example \ref{FourierTransformOfDeltaDistribution}.

Notice that this implies that the [[Fourier transform]] of the [[causal propagator]] (eq:CausalPropagator)

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

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

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

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.

=--

\begin{proposition}
\label{CausalPropagatorIsSkewSymmetric}
**([[causal propagator]] is skew-symmetric)**
\linebreak
Under reversal of arguments the [[advanced and retarded causal propagators]] from prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime} are related by

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

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

$$
\Delta_S(x-y) = - \Delta_S(y-x)
\,.
$$
\end{proposition}

+-- {: .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]] $\mathbb{R}^{p,1}$ (example \ref{EquationOfMotionOfFreeRealScalarField}) 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]] (prop. \ref{GreenFunctionsAreContinuous}) for the [[Klein-Gordon equation]] at [[mass]] $m$ on [[Minkowski spacetime]] (example \ref{EquationOfMotionOfFreeRealScalarField}) has the following equivalent expression, as a [[generalized function]], given as a [[contour integral]] along a [[Jordan 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](advanced+and+retarded+causal+propagators#Kocic16#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]] (example \ref{EquationOfMotionOfFreeRealScalarField}) 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 #OnMinkowskiWightmanIsDistributionalSolutionToKleinGordon}
###### Proposition
**([[Wightman propagator]] on [[Minkowski spacetime]] is [[distributional solution to a PDE|distributional solution]] to [[Klein-Gordon equation]])**

The [[Wightman propagator]] $\Delta_H$ (def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime}) is a [[distributional solution to a PDE|distributional solution]] (def. \ref{DistributionalDerivatives}) to the [[Klein-Gordon equation]]

$$
  (\Box_x - m^2)\Delta_H(x,y) = 0
  \,.
$$

=--

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

By definition \ref{StandardHadamardDistributionOnMinkowskiSpacetime} the Wightman propagator is the [[Fourier transform of distributions]] of the [[product of distributions]]

$$
  \delta(k_\mu k^\mu + m^2) \Theta(-k_0)
  \,,
$$

where in turn the argument of the [[delta distribution]] is just $-1$ times the Fourier transform of the [Klein-Gordon operator]] itself (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}). This is clearly a solution to the equation

$$
  (-k_\mu k^\mu - m^2) \, \delta(k_\mu k^\mu + m^2) \Theta(-k_0)
  \;=\;
  0
  \,.
$$

Under [[Fourier inversion theorem|Fourier inversion]] (prop. \ref{FourierInversion}), this is the equation $(\Box_x - m^2)\Delta_H(x,y) = 0$, as in the proof of prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}.

=--

+-- {: .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](advanced+and+retarded+causal+propagators#Kocic16#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

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

=--

+-- {: .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]] (example \ref{EquationOfMotionOfFreeRealScalarField}) 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

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

=--

It follows immediately that:

+-- {: .num_prop #SymmetricFeynmanPropagator}
###### Proposition
**([[Feynman propagator]] is symmetric)

The [[Feynman propagator]] $\Delta_F$ and anti-Feynman propagator $\Delta_{\overline{F}}$ (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) are symmetric:

$$
  \Delta_F(x,y) = \Delta_F(y,x)
  \,.
$$

=--

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

By equation (eq:AdvancedAndRetardedPropagatorTurnIntoEachOtherUnderSwitchingArguments) in cor. \ref{CausalPropagatorIsSkewSymmetric} we have that $\Delta_+ + \Delta_-$
is symmetric, and equation (eq:RealAndSymmetricH) in prop. \ref{SkewSymmetricPartOfHadmrdPropagatorIsCausalPropagatorForKleinGordonEquationOnMinkowskiSpacetime} says that $H$ is symmetric.

=--

+-- {: .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}.

=--

It follows that:

\begin{proposition}
\label{GreenFunctionFeynmanPropagator}
**([[Feynman propagator]] is [[Green function]])**
\linebreak
The [[Feynman propagator]] $\Delta_F$ for the [[Klein-Gordon equation]] on [[Minkowski spacetime]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) is proportional to a [[Green function]] for the [[Klein-Gordon equation]] in that

$$
  \left( \Box_x - \left( \tfrac{m c}{\hbar}\right)^2 \right) \Delta_{F}(x,y)
  =
  (+i) \delta(x-y)
  \,.
$$
\end{proposition}

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

Equation (eq:FeynmanPropagatorInCauchyPrincipalValueForm) in prop. \ref{FeynmanPropagatorAsACauchyPrincipalvalue} says that the Feynman propagator is the [[Fourier inversion theorem|inverse]] [[Fourier transform of distributions]] of

$$
  \widehat{\Delta_F}(k)
  \;=\;
  (+i)
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{
    1
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
$$

This implies the statement as in the proof of prop. \ref{AdvancedRetardedPropagatorsForKleinGordonOnMinkowskiSpacetime}, via the analogue of equation (eq:LimitOfDistributionsForFourierTransformedPropagators).

=--

$\,$

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

We now discuss the [[singular support]] (def. \ref{SingularSupportOfADistribution}) and the [[wave front sets]] (def. \ref{WaveFrontSet}) 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}
###### 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 the basic properties of the Fourier transform (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}) 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 prop. \ref{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 example \ref{FourierTransformOfDeltaDistribution}  and example \ref{RelationToFourierTransformOfHeavisideDistribution}) and hence the [[wave front set]] (def. \ref{WaveFrontSet}) of the second factor is

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

(by example \ref{WaveFrontOfDeltaDistribution} and example \ref{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 (prop. \ref{BasicPropertiesOfFourierTransformOverCartesianSpaces}) 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 prop. \ref{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 example \ref{FourierTransformOfDeltaDistribution} and example \ref{RelationToFourierTransformOfHeavisideDistribution} and hence the [[wave front set]] (def. \ref{WaveFrontSet}) of the second
factor is

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

(by example \ref{WaveFrontOfDeltaDistribution} and example \ref{PrincipalValueOfInverseFunctionCharacteristicEquation}).

With this the statement follows, via a [[partition of unity]], from prop. \ref{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)](causal+perturbation+theory#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 prop. \ref{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]] (prop. \ref{PropagationOfSingularitiesTheorem}) 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 $\widehat b(k)e^{i k_\mu a^\mu}$.

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.

=--

$\,$

**[[propagators]] for the [[Dirac equation]] on [[Minkowski spacetime]]**
 {#DiracEquationOnMinkowskiSpacetimePropagators}

We now discuss how the [[propagators]] for the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{GreenHyperbolicDiracOperator}) follow directly from those for the [[scalar field]] discussed above.

+-- {: .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]] (def. \ref{DistributionalDerivatives}) 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]] (def. \ref{DistributionalDerivatives}) of the [[Feynman propagator]] of the [[Klein-Gordon equation]] (def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}) by the [[Dirac operator]]

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

=--

$\,$

$\,$

This concludes our discussion of [[propagators]] induced from the [[covariant phase space]] of [[Green hyperbolic differential equation|Green hyperbolic]] [[free field theory|free]] [[Lagrangian field theory]]. These propagators will be the key in for [[quantization]] via [[causal perturbation theory]]. But not all [[free field theories]] have a [[covariant phase space]] of [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]], for instance the [[electromagnetic field]], a priori, does not. Therefore before turning to [[quantization]] in the [next chapter](#GaugeSymmetries) we first discuss how _[[gauge symmetries]]_  [[obstruction|obstruct]] the existence of  [[Green hyperbolic differential equation|Green hyperbolic]] [[equations of motion]].