# nLab Hadamard distribution

## Topics in Functional Analysis

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

In perturbative quantum field theory on curved spacetimes, Hadamard states are certain quantum states of free fields (whose 2-point function is the corresponding Hadamard distribution on spacetime) that play the role that the vacuum state plays on Minkowski spacetime.

Here a vacuum state is supposed to be a quantum state that expresses the absence of any particle excitations of the fields. On Minkowski spacetime the vacuum state for a free field theory is the standard Hadamard state (def. 2 below). On general globally hyperbolic spacetimes there are always Hadamard states (Radzikowski 96, see def. 1 below), and they do play the role of the vacuum state in the construction of AQFT on curved spacetimes, see at locally covariant perturbative AQFT. Notably the choice of such a Hadamard state fixes the Feynman propagator, hence the time-ordered product of quantum observables and thus the perturbative S-matrix away from coinciding interaction points (the extension of these distributions to coinciding interaction points is the process of renormalization).

However, since on a general globally hyperbolic spacetime there is no globally well-defined concept of particles, there is in general no concept of vacuum state. But under good conditions (such as existence of suitable timelike Killing vectors) one may identify Hadamard states which deserve to be thought of as vacuum states (Brum-Fredenhagen 13).

Hadamard states are mathematically characterized as follows:

Given a globally hyperbolic spacetime, the causal propagator defines a symplectic form on the covariant phase space of the free scalar field. However its wave front set is too large for the would-be induced Moyal star product to exist on but a very small subalgebra of smooth functionals, due to the failure of the relevant products of distributions to exist. However the Moyal star product makes sense more generally for almost Kähler structures with a symmetric contribution added to the skew-symmetric symplectic form.

A Hadamard distribution is such a modification of the causal propagator by a symmetric component such that the resulting wave front set is just one causal half of that of the causal propagator. This allows to define the corresponding Moyal star product on the larger algebra of microcausal functionals which is large enough to contain the adiabatically switched point interaction terms $g \phi(x)^n$ (as in phi^4 theory etc.). The resulting algebra of observables is the Wick algebra of the free scalar field.

The Hadamard distribution may also be thought of as the 2-point function of a quasi-free quantum state. These states are therefore called Hadamard states.

propagators (i.e. integral kernels of Green functions)
for the wave operator and Klein-Gordon operator
on a globally hyperbolic spacetime such as Minkowski spacetime:

namesymbol$\phantom{AA}$ primed
wave front set
as vacuum exp. value
of field operators
as a product of
field operators
causal propagator$\Delta_S \coloneqq \Delta_+ - \Delta_-$
$\phantom{A}\,\,\,-$
\begin{aligned} & \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned}Peierls-Poisson bracket
advanced propagator$\Delta_+$\begin{aligned} & \Delta_+(x,y) = \\ & \Theta(x^0 - y^0) \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle \end{aligned}future part of
Peierls-Poisson bracket
retarded propagator$\Delta_-$\begin{aligned} & \Delta_-(x,y) = \\ & \Theta(y^0 - x^0) \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle \end{aligned}past part of
Peierls-Poisson bracket
Hadamard propagator\begin{aligned} \Delta_H &= \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_+ \end{aligned}\begin{aligned} & \Delta_H(x,y) = \\ & \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \end{aligned}positive frequency part of
Peierls-Poisson bracket
= normal-ordered product
= 2-point function
$\phantom{=}$ of vacuum state
$\phantom{=}$ or generally of
$\phantom{=}$ Hadamard state
Dirac propagator$\Delta_D = \tfrac{1}{2}(\Delta_+ + \Delta_-)$
$\phantom{A}\,\,\, +$
would-be
time-ordered product
away from
coincident points
Feynman propagator$\array{\Delta_F & = i \Delta_D + H \\ & = \Delta_H + i \Delta_+}$\begin{aligned} & \Delta_F(x,y) = \\ & \left\langle \; T\left(\; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \end{aligned}time-ordered product

(see also Kocic’s overview: pdf)

## Definition

Recall the following general facts about the wave equation/Klein-Gordon equation

###### Proposition

(the causal propagator])

Let $(X,g)$ be a time-oriented globally hyperbolic spacetime and let $m \in \mathbb{R}_{\geq 0}$ (the “mass”). Then the Klein-Gordon equation

$(\Box_g - m^2) \phi = 0$

(a partial differential equation on smooth functions $f \in C^\infty(X,\mathbb{R})$ ) has unique advanced and retarded Green functions $E^{R/A}$, namely continuous linear functionals

$E^{A/R} \;\colon\; C^\infty_c(X) \longrightarrow C^\infty(X)$

(from bump functions to general smooth functions) which are fundamental solutions in that

$(\Box_g - m^2) \circ E^{A/R} = \delta \phantom{AAAA} E^{A/R} \circ (\Box_g - m^2) = \delta$

and which have advanced/retarded support of a distribution when viewed (via the Schwartz kernel theorem) as distributions on the Cartesian product manifold $X \times X$

$supp( E^{A/R}) \subset \{ (x_1, x_2) \in X \times X \;\vert\; x_1 \in J^{\mp} (x_2) \} \,.$

In fact these two fundamental solutions are related by switching their arguments

$E^{A/R}(x_1, x_2) = E^{R/A}(x_2, x_1) \,.$

Finally their wave front set is

$WF(E^{A/R}) \;=\; \left\{ ((x_1, x_2), (k_1, -k_2)) \;\vert\; x_1 \in J^{\mp}(x_2) \;\text{and}\; \left( (x_1, k_1) \sim (x_2, k_2) \right) \;\text{or}\; \left( (x_1 = x_2) \,\text{and}\, k_1 = -k_2 \right) \right\} \,.$

Here the relation $(x_1, k_1) \sim (x_2, k_2)$ means that there exists a lightlike geodesic from $x_1$ to $x_2$ with cotangent vector $k_1$ at $x_1$ and $k_2$ at $x_2$.

It follows that the wave front set of their difference (the causal propagator)

$E \;\coloneqq\; E^A - E^R$

is

$WF(E) \;=\; \left\{ ((x_1 x_2), (k_1, -k_2)) \;\vert\; (x_1, k_1) \sim (x_2, k_2) \right\} \,.$
###### Definition

(Hadamard distribution)

Let $(X,g)$ be a time oriented globally hyperbolic spacetime.

A Hadamard 2-point function or Hadamard distribution for the free scalar field on $(X,g)$ is a distribution of two variables

$\omega \in \mathcal{D}'(X \times X)$

on the Cartesian product manifold such that

1. the anti-symmetric part of $\omega$ is the causal propagator $E$ (from prop. 1)

$\omega(x_1, x_2) - \omega(x_2, x_1) \;=\; i E(x_1, x_2)$
2. (wave front set spectral condition (Radzikowski 96, def. 6.1))

the wave front set is one causal half that of the causal propagator:

$WF(\omega) \;=\; \left\{ ((x_1, x_2), (k_1, -k_2)) \;\vert\; (x_1, k_1) \simeq (x_2, k_2) \;\;\text{and}\;\; k_1 \in V_{x_1}^+ \right\}$
3. (Klein-Gordon bi-solution) $(\Box_g - m^2) \omega(-,x) = 0$ and $(\Box_g - m^2)\omega(x,-) = 0$, for all $x \in X$;

4. (positive semi-definiteness) For any complex-valued bump function $b$ we have that

$\int_{X \times X} b^\ast(x) \omega(x,y) b(y) \, dvol(x) dvol(y) \;\coloneqq\; \langle \omega, b^\ast \otimes b \rangle \;\geq\; 0$
###### Remark

Before (Radzikowski 96), Hadamard distributions were characterized by their singularity structure (see … below). In (Kay-Wald 91) the spectrum condition in def. 1 was formulated, and (Radzikowski 96) proved that this is equivalent to the original definition via singularity structure.

## Properties

### Existence

###### Proposition

(existence of Hadamard distributions)

Let $(X,g)$ be a globally hyperbolic spacetime. Then a Hadamard distribution $\omega$ according to def. 1 does exist.

It is given, up to addition by a smooth function, as the difference between the advanced propagator and the Feynman propagator

In fact there exist infinitely many Hadamard distributions on any globally hyperbolic spacetime (Junker-Schrohe 02, Fulling-Narcowich-Wald 81).

## Examples

### For Klein-Gordon operator on Minkowski spacetime

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

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

Its Fourier transform is

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

The dispersion relation of this equation we write

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

where on the right we choose the non-negative square root.

###### Proposition

(mode expansion of advanced and retarded propagators for Klein-Gordon operator on Minkowski spacetime)

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

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

by (in generalized function-notation)

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

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

(2)\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. \end{aligned}

Here $\omega(\vec k)$ denotes the dispersion relation (1) of the Klein-Gordon equation.

###### Proof

The Klein-Gordon operator is a Green hyperbolic differential operator (this example) therefore its advanced and retarded Green functions exist uniquely (this prop.). Moreover, this prop. says that they are continuous linear functionals with respect to the topological vector space structures on spaces of smooth sections (this def.). 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 this prop..

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

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

Since moreover the Klein-Gordon operator is formally self-adjoint (this prop.) 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 distributional equation

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

subject to the condition that the distributional support is

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

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

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

hence amenable to Fourier transform of distributions. If we do find a solution this way, it is guaranteed to be the unique solution by this prop..

By this prop. the distributional Fourier transform of equation (3) is

(4)\begin{aligned} \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 the delta distribution from this example.

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

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

satisfies the homogeneous equation:

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

Hence we are now reduced to finding solutions $\widehat{\Delta_\pm} \in \mathcal{S}'(\mathbb{R}^{p,1})$ to (4) such that their Fourier inverse $\Delta_\pm$ has the required support properties.

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

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

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

meaning that for each bump function $b \in C^\infty_{cp}(\mathbb{R}^{p,1})$ the limit 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 (4) because on those bump functions $b(k)$ which happen to be products with $\left(-\eta^{\mu \nu}k_\mu k-\nu - \left( \tfrac{m c}{\hbar}\right)^2\right)$ we clearly have

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

Moreover, if the limiting distribution (6) exists, then it is clearly a tempered distribution, hence we may apply Fourier inversion to obtain Green functions

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

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 (6) indeed exists it is sufficient to show that the limit in (7) exists.

We compute as follows

(8)\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. \end{aligned}

where $\omega(\vec k)$ denotes the dispersion relation (1) of the Klein-Gordon equation.

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 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.

2. 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.

3. 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).

4. The result does not depend on $\epsilon$ anymore, therefore the limit $\epsilon \to 0$ is now computed trivially.

This computation shows a) that the limiting distribution indeed exists, and b) that the 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.

###### Corollary

(causal propagator is skew-symmetric)

Under reversal of arguments the advanced and retarded causal propagators are related by

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

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

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

By this prop. we have with (2)

\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).

###### Proposition

(mode expansion of causal propagator for Klein-Gordon equation on Minkowski spacetime)

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

(9)\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 $sin(\alpha)= \tfrac{1}{2i}\left( e^{i \alpha} - e^{-i \alpha} \right)$.

###### Proof

By definition and using the expression from this prop. 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 \end{aligned}

We consider a couple of equivalent expressions for the causal propagator:

###### Proposition

(causal propagator for Klein-Gordon operator on Minkowski spacetime as a contour integral)

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

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

graphics grabbed from Kocic 16

###### 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 - m^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. 4

###### Proposition

(causal propagator as Fourier transform of delta distribution on the 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

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. 4.

Prop. 6 exhibits the causal propagator of the Klein-Gordon operator on Minkowski spacetime as the difference of a contribution for 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 frequency contribution to the causal propagator is called the Hadamard propagator (def. 2 below), also known as the the vacuum state 2-point function of the 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 (10) below:

###### Definition

(Hadamard propagator or vacuum state 2-point function for Klein-Gordon operator on Minkowski spacetime)

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

(10)\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 solution to the Klein-Gordon dispersion relation.

###### Proposition

(contour integral representation of the Hadamard propagator for the Klein-Gordon operator on Minkowski spacetime)

The Hadamard propagator from def. 2 is equivalently given by the contour integral

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

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 $- \omega(\vec k)/c \in \mathbb{R} \subset \mathbb{C}$.

graphics grabbed from Kocic 16

###### 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 2.

###### Proposition

(skew-symmetric part of Hadamard propagator is the causal propagator)

The skew-symmetrization of the Hadamard propagator (def. 2) reproduces the causal propagator according to prop. 4 (which we already know to be skew-symmetric by cor. 1), up to a conventional imaginary factor (which serves to exhibit Kähler metric form, see remark 2 below):

$\Delta_S(x,y) \;=\; -i\left( \Delta_H(x,y) - \Delta_H(y,x) \right) \,.$
###### Proof

By (10) we have

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

where in the second step we applied the change of integration variables $\vec k \mapsto -\vec k$ to the second summand (which does not introduce a global sign, because in addition to $d^p \vec k \mapsto - d^p \vec k$ the orientation of the integration domain changes).

The last line is the causal propagator by (9), prop. 4.

###### Remark

(symmetric part of the Hadamard propagator)

Conversely, prop. 8 says that the Hadamard propagator for the Klein-Gordon operator on Minkowski spacetime (def. 2) is of the form

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

where

$H(x,y) \;\coloneqq\; \tfrac{1}{2}\left( \Delta_H(x,y) + \Delta_H(y,x) \right)$

is the symmetric component of $H$.

If we change $H$ by adding any symmetric non-singular distribution in two variables, the result is still called “a Hadamard propagator” for the Klein-Gordon equation on Minkowski spacetime.

## References

Textbook discussion of the Hadamard distribution for free fields in Minkowski spacetime is in

(there the Hadamard distribution is denoted “$-i D^+_m(x-y)$”).

An concise overview of the standard Hadamard propagator in Minkowski spacetime and its relation to the other pertinent propagors is given in

• Mikica Kocic, Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions, 2016 (pdf)

On general globally hyperbolic spacetimes Hadamard spectrum condition was first rigorously defined in

• B. S. Kay, Robert Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon, Phys. Rep. 207(2), 49-136 (1991)

and shown to be equivalent to the definition in terms of singular structure in

• Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)

Hadamard states with smooth dependence on the mass square (needed for dimensional regularization? methods in causal perturbation theory/perturbative AQFT) are constructed in

• Kai Keller, chapter III of Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization, PhD thesis (arXxiv:1006.2148)

Discussion of vacuum state-like Hadamard states is in

Review and further developments are in

Discussion for electromagnetism is in

• Claudio Dappiaggi, D. Siemssen, Hadamard states for the vector potential on asymptotically flat spacetimes, Rev. Math. Phys. 25, 1 (2013)

See also

• W. Junker and E. Schrohe: Adiabatic vacuum states on general spacetime manifolds: Definition, construction, and physical properties, Annales Poincare Phys. Theor. 3, 1113 (2002)

• S. A. Fulling, M. Sweeny, Robert Wald, Singularity Structure Of The Two Point Function In Quantum Field Theory In Curved Space-Time, Commun. Math. Phys. 63, 257 (1978).

• S. A. Fulling, F. J. Narcowich, Robert Wald, Singularity Structure Of The Two Point Function In Quantum Field Theory In Curved Space-Time, Annals Phys. 136, 243 (1981),

• Marcos Brum, Klaus Fredenhagen, “Vacuum-like” Hadamard states for quantum fields on curved spacetimes, Classical and Quantum Gravity, Volume 31, Number 2 (arXiv:1307.0482)

Revised on November 21, 2017 15:40:55 by Urs Schreiber (46.183.103.8)