nLab
causal propagator

Context

Functional analysis

Riemannian geometry

Algbraic Quantum Field Theory

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 Σ\Sigma, its phase space is the space ker(+m 2)C (Σ)ker(\Box + m^2) \hookrightarrow C^\infty(\Sigma) of solutions of the Klein-Gordon equation (the wave equation for vanishing mass mm). For any point xΣx \in \Sigma we denote by ϕ(x):ker(+m 2)\phi(x) \colon ker(\Box + m^2) \to \mathbb{R} the point evaluation functional which sends ΦC (Σ)\Phi \in C^\infty(\Sigma) to Φ(x)\Phi(x). An observable of the scalar field is then a functional of the form ϕ(b)b(x)ϕ(x)dvol(x)\phi(b) \coloneqq \int b(x) \phi(x) dvol(x), for bb a bump function on Σ\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 ϕ(b 1)\phi(b_1) and ϕ(b 2)\phi(b_2) to a new observable denoted {ϕ(b 1),ϕ(b 2)}\{\phi(b_1), \phi(b_2)\}. While a priori this Poisson bracket is defined only on the “smeared” observables ϕ(b)\phi(b), not on the point observables ϕ(x)\phi(x), nevertheless it has a distributional integral kernel {ϕ(x),ϕ(y)}\{\phi(x), \phi(y)\} such that

{ϕ(b 1),ϕ(b 2)}=b 1(x)b 2(y){ϕ(x),ϕ(y)}dvol Σ(x)dvol Σ(y). \{ \phi(b_1), \phi(b_2) \} = \int b_1(x) b_2(y) \{\phi(x), \phi(y)\} dvol_{\Sigma}(x) dvol_\Sigma(y) \,.

This distributional integral kernel

Δ(x,y){ϕ(x),ϕ(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 +m 2\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

Δ=Δ RΔ A \Delta = \Delta_R - \Delta_A

where the retarded propagator Δ R\Delta_R and the advanced propagator Δ A\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 pp \in \mathbb{N} and let p,1\mathbb{R}^{p,1} be (p+1)(p+1)-dimensional Minkowski spacetime.

Definition

(causal propagator on Minkowski spacetime)

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

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

which as a generalized function is given by

Δ(x,y) i(2π) pδ(k μk μ+m 2)sgn(k 0)e ik μ(xy) μd p+1k . \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)Δ(x,y) i(2π) pδ(k μk μ+m 2)sgn(k 0)e ik μ(xy) μd p+1k =i(2π) p12E(k)(e iE(k)(x 0y 0)k(xy)e +iE(k)(x 0y 0)+k(xy))d pk =i(2π) p1E(k)sin(E(k)(x 0y 0))e ik(xy)d pk , \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(k)k 2+m 2 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 |k|{\vert \vec k\vert} as |k|{\vert k \vert} \to \infty (Scharf 95 (2.3.8)).

The last expression may be computed to be equal to

(2)Δ(x,y)=(2π) p2sgn((x 0y 0))(δ(|xy| 2)Θ(|xy| 2)m2|xy| 2J 1(m|xy| 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 1J_1 denotes the Bessel function of order 1.

Finally this may also be expressed as the contour integral

(3)Δ(x,y)=(2π) (p+1) C(k)e ik μ(xy) μk μk μ+m 2dk 0d pk, \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(k)C(\vec k) \subset \mathbb{C} runs counter-clockwise, enclosing the points ±E(k)\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 00k_0 \geq 0 and from k 00k_0 \leq 0 and then apply the changes of integration variables k 0=hk_0 = \sqrt{h} for k 00k_0 \geq 0 and k 0=hk_0 = -\sqrt{h} for k 00k_0 \leq 0:

i(2π) pδ(k μk μ+m 2)sgn(k 0)e ik μ(xy) μd p+1k =i(2π) p 0 δ(k 0 2+k 2+m 2)e ik 0x 0ikxdk 0d pk =+i(2π) p 0δ(k 0 2+k 2+m 2)e ik 0x 0ikvecxdk 0d pk =i(2π) p 0 12hδ(h+E(k) 2)e ih(xy) 0ikxdhd pk =+i(2π) p 0 δ(h+E(k) 2)e +iE(k)(xy) 0ikxdhd pk =i(2π) p12E(k)e iE(k)(xy) 0ikxd pk =+i(2π) p12E(k)e +iE(k)(xy) 0ikxd pk =i(2π) p12E(k)e ik(xy)(e iE(k)(xy) 0e iE(k)(xy) 0) =i(2π) p1E(k)e ik(xy)sin(E(k)(xy) 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 ±E(k)\pm E(\vec k) \in \mathbb{R} \subset \mathbb{C}:

(2π) (p+1) C(k)e ik μ(xy) μk μk μ+m 2dk 0d pk =(2π) (p+1) C(k)e ik 0x 0e ik(xy)k 0 2+E(k) 2dk 0d pk =(2π) (p+1) C(k)e ik 0(xy) 0e ik(xy)(E ϵ(k)+k 0)(E ϵ(k)k 0)dk 0d pk =(2π) (p+1)2πi(e iE(k)(x 0y 0)e ik(xy)2E(k)e +iE(k)(x 0y 0)e ik(xy)2E(k))d pk =i(2π) p1E(k)sin(E(k)(x 0y 0))e ik(xy)d pk. \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 Δ\Delta on Minkowski spacetime (def. 1) is in the causal cone:

supp(Δ){(x,y)||xy| 20}. 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)(X,g) be a time-oriented globally hyperbolic spacetime and let m 0m \in \mathbb{R}_{\geq 0} (the “mass”). Then the Klein-Gordon equation

( gm 2)ϕ=0 (\Box_g - m^2) \phi = 0

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

E A/R:C c (X)C (X) 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

( gm 2)E A/R=δAAAAE A/R( gm 2)=δ (\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×XX \times X

supp(E A/R){(x 1,x 2)X×X|x 1J (x 2)}. 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). E^{A/R}(x_1, x_2) = E^{R/A}(x_2, x_1) \,.

The difference

EE RE A 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:

propagatorAA\phantom{AA}AA\phantom{AA} primed wave front seton Minkowski spacetimegenerally
causal propagatorΔΔ RΔ A\array{\Delta \coloneqq \Delta_R - \Delta_A } A a\array{- \\ \phantom{A} \\ \phantom{a}} Δ S(x,y)= vac|[Φ(x),Φ(y)]|vac\array{\Delta_S(x,y) = \\ \langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }Peierls-Poisson bracket
advanced propagatorΔ A\Delta_AΔ A(x,y)= Θ((yx) 0)vac|[Φ(x),Φ(y)]|vac\array{\Delta_A(x,y) = \\ \Theta((y-x)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }
retarded propagatorΔ R\Delta_RΔ R(x,y)= Θ((xy) 0)vac|[Φ(x),Φ(y)]|vac\array{\Delta_R(x,y) = \\ \Theta((x-y)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }
Dirac propagatorΔ D=12(Δ A+Δ R)\Delta_D = \tfrac{1}{2}(\Delta_A + \Delta_R) + A a\array{+ \\ \phantom{A} \\ \phantom{a}}
Hadamard propagatorω =i2Δ+H =ω FiΔ A\begin{aligned} \omega &= \tfrac{i}{2}\Delta + H \\ & = \omega_F - i \Delta_A \end{aligned}ω(x,y)= vac|Φ(x)Φ(y)|vac\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 propagatorω F =iΔ D+H =ω+iΔ A\array{\omega_F & = i \Delta_D + H \\ & = \omega + i \Delta_A}E F(x,y)= vac|T(Φ(x)Φ(y))|vac\array{E_F(x,y) = \\ \langle vac \vert T(\Phi(x)\Phi(y)) \vert vac \rangle }time-ordered product

(see also Kocic’s overview: pdf)

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.

Textbook discussion for free fields in Minkowski spacetime is in

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

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

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

For more see the references at wave equation.

Revised on November 17, 2017 16:03:19 by Urs Schreiber (46.183.103.8)