Dirac propagator



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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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Perturbative QFT



What is called the Dirac propagator is the Green functions for the wave operator/Klein-Gordon operator (hence “propagator”) on a globally hyperbolic spacetime (X,e)(X,e) which is the sum of the advanced propagator Δ A\Delta_A and the retarded propagator Δ R\Delta_R

Δ DΔ R+Δ A \Delta_D \coloneqq \Delta_R + \Delta_A

Since Δ A(x,y)=Δ R(y,x)\Delta_A(x,y) = \Delta_R(y,x), this is symmetric in its arguments, reflecting the fact that this is the integral kernel for time-ordered products away from the diagonal.

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:

namesymbolwave front setas vacuum exp. value
of field operators
as a product of
field operators
causal propagatorΔ S =Δ +Δ \begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}
iΔ S(x,y)= [Φ(x),Φ(y)]\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} Peierls-Poisson bracket
advanced propagatorΔ +\Delta_+ iΔ +(x,y)= {[Φ(x),Φ(y)] | xy 0 | yx\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned} future part of
Peierls-Poisson bracket
retarded propagatorΔ \Delta_- iΔ (x,y)= {[Φ(x),Φ(y)] | yx 0 | xy\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}past part of
Peierls-Poisson bracket
Wightman propagatorΔ H =i2(Δ +Δ )+H =i2Δ S+H =Δ FiΔ \begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned} Δ H(x,y) =Φ(x)Φ(y) =:Φ(x)Φ(y):=0 =+[Φ ()(x),Φ (+)(y)]\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned} positive frequency of
Peierls-Poisson bracket,
Wick algebra-product,
2-point function
=\phantom{=} of vacuum state
=\phantom{=} or generally of
=\phantom{=} Hadamard state
Feynman propagatorΔ F =i2(Δ ++Δ )+H =iΔ D+H =Δ H+iΔ \begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned} Δ F(x,y) =T(Φ(x)Φ(y)) ={Φ(x)Φ(x) | xy Φ(y)Φ(x) | yx\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}time-ordered product

(see also Kocic‘s overview: pdf)


What is now called the Dirac propagator was first considered in

  • Paul Dirac, Classical theory of radiating electrons, Proc. Roy. Soc. A 167 (1983) 148-169

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)

Discussion for general globally hyperbolic spacetimes includes

  • F. Friedlander, The Wave Equation on a Curved Space-Time, Cambridge: Cambridge University Press, 1975

  • Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (arXiv:0806.1036)

  • Nicolas Ginoux, Linear wave equations, Ch. 3 in Christian Bär, Klaus Fredenhagen, Quantum Field Theory on Curved Spacetimes: Concepts and Methods, Lecture Notes in Physics, Vol. 786, Springer, 2009

Review in the context of perturbative algebraic quantum field theory includes

  • Katarzyna Rejzner, sections 4.1 and 6.2.3 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (pdf)

Last revised on September 6, 2017 at 09:33:09. See the history of this page for a list of all contributions to it.