# nLab causal propagator

## Topics in Functional Analysis

#### Riemannian geometry

Riemannian geometry

## Concepts

Lagrangian field theory

quantum mechanical system

quantization

# Contents

## Idea

The causal propagator or Pauli-Jordan distribution (Jordan-Pauli 27) or commutator function is a distribution which gives the integral kernel for the Poisson bracket on the covariant phase space of a free local field theory (also known as the Peierls bracket).

Specifcally for the free scalar field on a spacetime $\Sigma$, its phase space is the space $ker(\Box + m^2) \hookrightarrow C^\infty(\Sigma)$ of solutions of the Klein-Gordon equation (the wave equation for vanishing mass $m$). For any point $x \in \Sigma$ we denote by $\phi(x) \colon ker(\Box + m^2) \to \mathbb{R}$ the point evaluation functional which sends $\Phi \in C^\infty(\Sigma)$ to $\Phi(x)$. An observable of the scalar field is then a functional of the form $\phi(b) \coloneqq \int b(x) \phi(x) dvol(x)$, for $b$ a bump function on $\Sigma$. On the algebra of these observables there is a canonical Poisson bracket pairing defined (also known as the Peierls bracket see at scalar field for details), taking $\phi(b_1)$ and $\phi(b_2)$ to a new observable denoted $\{\phi(b_1), \phi(b_2)\}$. While a priori this Poisson bracket is defined only on the “smeared” observables $\phi(b)$, not on the point observables $\phi(x)$, nevertheless it has a distributional integral kernel $\{\phi(x), \phi(y)\}$ such that

$\{ \phi(b_1), \phi(b_2) \} = \int b_1(x) b_2(y) \{\phi(x), \phi(y)\} dvol_{\Sigma}(x) dvol_\Sigma(y) \,.$
$\Delta(x,y) \coloneqq \{\phi(x), \phi(y)\}$

is the causal propagator or Pauli-Jordan distribution (also “commutator function”, see this prop.). This happens to be a fundamental solution/Green function to the Klein-Gordon operator $\Box + m^2$, whence a “propagator”.

For other free fields the integral kernel of their Poisson bracket is a more complicated expression, but it is typically still an expression in terms of the causal propagator of the scalar field.

What is causal about the causal propagator is that (on globally hyperbolic spacetimes such as Minkowski spacetime) its support as a distribution, is, for one of the two arguments fixed, the causal cone of that point (cor. 1 below). Moreover, the causal propagator splits, as a distribution, as a sum

$\Delta = \Delta_R - \Delta_A$

where the retarded propagator $\Delta_R$ and the advanced propagator $\Delta_A$ are such that their support is, for fixed second argument, in the past causal cone and in the future causal cone, respectively.

## Definition

### On Minkowski spacetime

Let $p \in \mathbb{N}$ and let $\mathbb{R}^{p,1}$ be $(p+1)$-dimensional Minkowski spacetime.

###### Definition

(causal propagator on Minkowski spacetime)

The causal propagator or Pauli-Jordan distribution on Minkowski spacetime $\mathbb{R}^{p,1}$ is the distribution

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

which as a generalized function is given by

\begin{aligned} \Delta(x,y) & \coloneqq -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \\ \end{aligned} \,.

Definition 1 is the expression that one obtains from a standard calculation of the Poisson bracket on the covariant phase space of the free scalar field (this prop.). But the causal propagator has various other equivalent expressions, which are useful in different contexts:

###### Proposition

(equivalent expressions for causal propagator on Minkowski spacetime)

The causal propagator on Minkowski spacetime from def. 1 has the following equivalent expressions

(1)\begin{aligned} \Delta(x,y) & \coloneqq -i (2\pi)^{-p} \int \delta( k_\mu k^\mu + m^2 ) sgn( k_0 ) e^{ - i k_\mu (x-y)^\mu } d^{p+1} k \\ & = -i (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)}\left( e^{- i E(\vec k) (x^0 - y^0) - \vec k \cdot (\vec x - \vec y)} - e^{+ i E(\vec k) (x^0 - y^0) + \vec k \cdot (\vec x - \vec y)} \right) d^p \vec k \\ & = - i (2\pi)^{-p} \int \frac{1}{E(\vec k)} sin(E(\vec k) (x^0 - y^0) ) e^{ - i \vec k \cdot (\vec x - \vec y)} d^p \vec k \\ \end{aligned} \,,

where

$E(\vec k) \coloneqq \sqrt{\vec k^2 + m^2}$

and where in the last expression the integral is to be understood as the weak limit of integrals up to ${\vert \vec k\vert}$ as ${\vert k \vert} \to \infty$ (Scharf 95 (2.3.8)).

The last expression may be computed to be equal to

(2)$\Delta(x,y) \;=\; (2 \pi)^{p-2} sgn((x^0 - y^0)) \left( \delta( -{\vert x-y\vert}^2 ) - \Theta( -{\vert x-y\vert}^2 ) \frac{m}{2 \sqrt{-{\vert x-y\vert}^2} } J_1\left( m \sqrt{-{\vert x-y\vert}^2} \right) \right) \,,$

where $J_1$ denotes the Bessel function of order 1.

Finally this may also be expressed as the contour integral

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

where the Jordan curve $C(\vec k) \subset \mathbb{C}$ runs counter-clockwise, enclosing the points $\pm E(\vec k) \in \mathbb{R} \subset \mathbb{C}$. (Compare to the analogous expression for the advanced and retarded propagators: this prop..)

graphics grabbed from Kocic 16

###### Proof

For the expression (1) decompose the original integral into its contributions from $k_0 \geq 0$ and from $k_0 \leq 0$ and then apply the changes of integration variables $k_0 = \sqrt{h}$ for $k_0 \geq 0$ and $k_0 = -\sqrt{h}$ for $k_0 \leq 0$:

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

For the derivation of (2) from the last line of (1) see Scharf 95 (2.3.8) to (2.3.18).

Finally to obtain (3), Cauchy's integral formula says that the given contour integral picks up the residues of the poles of the integrand at $\pm E(\vec k) \in \mathbb{R} \subset \mathbb{C}$:

\begin{aligned} (2\pi)^{-(p+1)} \int \oint_{C(\vec k)} \frac{e^{-i k_\mu (x-y)^\mu}}{ k_\mu k^\mu + m^2 } d k_0 d^{p} k & = (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 + E(\vec k)^2 } d k_0 d^p \vec k \\ & = (2\pi)^{-(p+1)} \int \oint_{C(\vec k)} \frac{ e^{-i k_0 (x-y)^0} e^{- i \vec k \cdot (\vec x - \vec y)} }{ ( E_\epsilon(\vec k) + k_0 ) ( E_\epsilon(\vec k) - k_0 ) } d k_0 d^p \vec k \\ & = (2\pi)^{-(p+1)} 2\pi i \int \left( \frac{ e^{-i E(\vec k) (x^0 - y^0)} e^{-i \vec k \cdot (\vec x - \vec y)} } { 2 E(\vec k) } - \frac{ e^{ + i E(\vec k) (x^0 - y^0)} e^{-i \vec k \cdot (\vec x - \vec y)} }{ 2 E(\vec k) } \right) d^p \vec k \\ & = - i (2\pi)^{-p} \int \frac{1}{E(\vec k)} sin\left( E(\vec k)(x^0 - y^0) \right) e^{-i \vec k \cdot (\vec x - \vec y)} d^p \vec k \,. \end{aligned}

That the last line here is indeed equal to the causal propagator is the statement of the last line of (1).

###### Corollary

(causal support of the causal propagator)

The distributional support of the causal propagator $\Delta$ on Minkowski spacetime (def. 1) is in the causal cone:

$supp(\Delta) \subset \left\{ (x,y) \;\vert\; -{\vert x-y\vert}^2 \geq 0 \right\} \,.$
###### Proof

By the equivalent expression (2) in prop. 1.

### On general globally hyperbolic spacetimes

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

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

The difference

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

is the causal propagator on the given spacetime.

Green functions for the Klein-Gordon operator on a globally hyperbolic spacetime:

propagator$\phantom{AA}$$\phantom{AA}$ primed wave front seton Minkowski spacetimegenerally
causal propagator$\array{\Delta \coloneqq \Delta_R - \Delta_A }$ $\array{- \\ \phantom{A} \\ \phantom{a}}$ $\array{\Delta_S(x,y) = \\ \langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$Peierls-Poisson bracket
advanced propagator$\Delta_A$$\array{\Delta_A(x,y) = \\ \Theta((y-x)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$
retarded propagator$\Delta_R$$\array{\Delta_R(x,y) = \\ \Theta((x-y)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$
Dirac propagator$\Delta_D = \tfrac{1}{2}(\Delta_A + \Delta_R)$ $\array{+ \\ \phantom{A} \\ \phantom{a}}$
Hadamard propagator\begin{aligned} \omega &= \tfrac{i}{2}\Delta + H \\ & = \omega_F - i \Delta_A \end{aligned}$\array{\omega(x,y) = \\ \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle }$normal-ordered product (2-point function of quasi-free state)
Feynman propagator$\array{\omega_F & = i \Delta_D + H \\ & = \omega + i \Delta_A}$$\array{E_F(x,y) = \\ \langle vac \vert T(\Phi(x)\Phi(y)) \vert vac \rangle }$time-ordered product

## Properties

The causal propagator yields the Peierls bracket, which is the Poisson bracket on the covariant phase space of the field scalar field. The Moyal deformation quantization of this covariant phase space yields the Wick algebra of quantum observables of the free scalar field.

## References

The causal propagator was first considered (in the context of quantum electrodynamics) in

whence often called the Jordan-Pauli distribution.

(there denoted “$-i D_m(x-y)$” and called the “Jordan-Pauli function”).