nLab
Feynman propagator

Context

Functional analysis

Algbraic Quantum Field Theory

Riemannian geometry

Contents

Idea

What is called the Feynman propagator over a globally hyperbolic spacetime is one of the Green functions for the Klein-Gordon operator +m 2\Box + m^2 (hence a fundamental solution to the wave equation when the mass mm vanishes).

As discussed in detail at S-matrix – Feynman diagrams and renormalization, the Feynman propagator encodes time-ordered products of quantum observables in free field perturbative quantum field theory (in the same way as the Hadamard propagator encodes normal ordered products of quantum fields). This implies that the scattering amplitude associated with a Feynman diagram in the Feynman perturbation series expansion of the S-matrix is, away from the locus of coinciding interaction points, a product of Feynman propagators, one for each edge in the Feynman diagram (the extension of distributions of this product of distributions to coinciding points is renormalization).

This is why Feynman propagators are ubiquituous in perturbative quantum field theory: they are the building blocks of perturbative scattering amplitudes.

Definition

By the discussion at S-matrix – Feynman diagrams and renormalization, the Feynman propagator ω F\omega_F is properly defined to be the linear combination of the chosen Hadamard propagator (encoding the vacuum state) and the advanced causal propagator:

Definition

(Feynman propagator on globally hyperbolic spacetimes)

Given a time-oriented globally hyperbolic spacetime Σ\Sigma there exists a unique advanced causal propagator Δ A𝒟(Σ×Σ)\Delta_A \in \mathcal{D}'(\Sigma \times \Sigma) and a Hadamard propagator ω𝒟(Σ×Σ)\omega \in \mathcal{D}'(\Sigma \times \Sigma), unique up to addition of a regular distributio (a smooth function). Given a choice of ω\omega (the vacuum state) then the corresponding Feynman propagator is the sum

ω Fω+iΔ A. \omega_F \coloneqq \omega + i \Delta_A \,.

(for Minkowski spacetime this is e.g. Scharf 95 (2.3.41), for general globally hyperbolic spacetimes this is Radzikowski 96, p. 5)

On Minkowski spacetime this may be expressed as a sum of products of distributions of a Heaviside distribution in the time coordinate with the Hadamard distribution and its opposite, and this is often taken as the definition of the Feynman propagator. But the above formula applies to general globally hyperbolic spacetimes.

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:

namesymbolAA\phantom{AA} primed
wave front set
as vacuum exp. value
of field operators
as a product of
field operators
causal propagatorΔ SΔ +Δ \Delta_S \coloneqq \Delta_+ - \Delta_-
A\phantom{A}\,\,\,-
Δ S(x,y)= [Φ(x),Φ(y)]\begin{aligned} & \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned} Peierls-Poisson bracket
advanced propagatorΔ +\Delta_+ Δ +(x,y)= Θ(x 0y 0)[Φ(x),Φ(y)]\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 propagatorΔ \Delta_- Δ (x,y)= Θ(y 0x 0)[Φ(x),Φ(y)]\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 propagatorΔ H =i2Δ S+H =Δ FiΔ +\begin{aligned} \Delta_H &= \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_+ \end{aligned} Δ H(x,y)= Φ(x)Φ(y)\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 propagatorΔ D=12(Δ ++Δ )\Delta_D = \tfrac{1}{2}(\Delta_+ + \Delta_-)
A+\phantom{A}\,\,\, +
would-be
time-ordered product
away from
coincident points
Feynman propagatorΔ F =iΔ D+H =Δ H+iΔ +\array{\Delta_F & = i \Delta_D + H \\ & = \Delta_H + i \Delta_+} Δ F(x,y)= T(Φ(x)Φ(y))\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)

On Minkowski spacetime

this section needs harmonization of notation and assumptions

Proposition

(contour integral presentation of standard Feynman propagator on Minkowski spacetime)

For pp \in \mathbb{N}, on Minkowski spacetime of dimension p+1p+1 the Feynman propagator ω F\omega_F according to prop. 1, with respect to the standard vacuum Hadamard state (this def.) is

(1)ω F(x,y) (2π) (p+1)e ik μ(xy) μk μk μ+m 2+i0d p+1k (2π) (p+1)limϵ0 +e ik μ(xy) μ(E(k)+(k 0iϵ))(E(k)(k 0+iϵ))dk 0d pk. \begin{aligned} \omega_F(x,y) & \coloneqq -(2\pi)^{-(p+1)} \int \frac{e^{-i k_\mu (x-y)^\mu}}{ k_\mu k^\mu + m^2 + i0} d^{p+1} k \\ & \coloneqq -(2\pi)^{-(p+1)} \underset{\epsilon \to 0^+}{\lim} \int \int \frac{ e^{-i k_\mu (x-y)^\mu} }{ ( E(\vec k) + (k_0 - i \epsilon) ) ( E(\vec k) - (k_0 + i \epsilon) ) } d k_0 d^{p} k \end{aligned} \,.

(compare to this computation at advanced and retarded propagators to see the point of the notation “+i0+ i 0” in the first line, and its definition by the second line).

graphics grabbed from (Kocic 16)

(e.g. Scharf 95 (2.3.44))

Proof

We may compute the line integral in the second line of (1) by completing to a contour integral in the complex plane. For (x 0y 0)>0(x^0 - y^0) \gt 0 we have that e ik 0(x 0y 0)e^{-i k_0 (x^0 - y^0)} decays as the imaginary part of k 0k_0 goes to -\infty, and hence in this case we need to close the contour in the lower half plane. Conversely, for (x 0y 0)<0(x^0 - y^0) \lt 0 we need to close the conour in the upper half plane. By the Cauchy integral formula, in the first case we pick up the residue at the pole at E(k)iϵE(\vec k) - i \epsilon with a minus sign (because the contour in this case runs clockwise), while in the second case we pick up the residue at E(k)+iϵ-E(\vec k) + i \epsilon (without an extra minus sign, because in this case it runs counter-clockwise). Hence we get

(2)ω F(x,y) ={(2π) p12E(k)e iE(k)(x 0y 0)k(xy)d pk | (x 0y 0)>0 (2π) p12E(k)e +iE(k)(x 0y 0)k(xy)d pk | (x 0y 0)<0 ={ω(x,y) | (x 0y 0)>0 ω(y,x) | (x 0y 0)<0 \begin{aligned} \omega_F(x,y) & = \left\{ \array{ (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{- i E(\vec k) (x^0 - y^0) - \vec k \cdot (\vec x - \vec y)} d^{p} \vec k & \vert & (x^0 - y^0) \gt 0 \\ (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{+ i E(\vec k) (x^0 - y^0) - \vec k \cdot (\vec x - \vec y)} d^{p} \vec k & \vert & (x^0 - y^0) \lt 0 } \right. \\ & = \left\{ \array{ \omega(x,y) & \vert & (x^0 - y^0) \gt 0 \\ \omega(y,x) & \vert & (x^0 - y^0) \lt 0 } \right. \end{aligned}

(as befits the expectation value of the time-ordered product T(ϕ(x)ϕ(y))T(\phi(x) \phi(y))).

On the other hand, by this equation for from the discussion at advanced propagator and this equation from the discussion at Hadamard propagator we have

iΔ A(x,y) ={(2π) p(e iE(k)(x 0y 0)e ik(xy)2E(k)e +iE(k)(x 0y 0)e ik(xy)2E(k)) | (x 0y 0)<0 0 | (x 0y 0)>0 ω(x,y) =(2π) p12E(k)e iE(k)(xy) 0k(xy)d pk. \begin{aligned} i \Delta_A(x,y) & = \left\{ \array{ - (2\pi)^{-p} \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) & \vert & (x^0 - y^0) \lt 0 \\ 0 & \vert & (x^0 - y^0) \gt 0 } \right. \\ \omega(x,y) & = (2\pi)^{-p} \int \tfrac{1}{2 E(\vec k)} e^{- i E(\vec k) (x-y)^0 - \vec k \cdot (\vec x - \vec y)} d^{p} \vec k \end{aligned} \,.

Adding up the two lines and comparing with (2) shows that with ω F\omega_F as defined by (1) indeed satisfies the defining equation

ω F=ω+iΔ A \omega_F = \omega + i \Delta_A

from def. 1.

As a zeta function

needs harmonization

From another perspective, the loop contributions of Feynman diagrams are typically would-be traces over inverse powers H nH^{-n} of the relativistic particle Hamiltonian.

For instance for the free scalar particle of mass mm in 4d Minkowski spacetime the 1-loop vacuum amplitude is the regularized trace over the Feynman propagator

d 4p1p 2m 2 \propto \int d^4 \mathbf{p} \; \frac{1}{\mathbf{p}^2 - m^2}

where the integral would naively be over all of 4\mathbb{R}^4, which is of course not well defined. The integrand here is typically called the Feynman propagator or propagator for short (e.g. Grozin 05, section 2.1 Kleinert 11, 8.1). See at Feynman diagram – For finitely many degrees of freedoms for how this comes about.

Several methods are considered for regularizing, hence making sense of it as a finite expression. One of these is zeta function regularization (also “analytic regularization/renormalization” Speer 71). Here one notices that the zeta function of the wave operator/Laplace operator H=p 2+m 2H = \mathbf{p}^2 + m^2 is well-defined for (s)>1\Re(s) \gt 1 by the naive trace

ζ^ H(s)Tr reg(H s) \hat \zeta_H(s)\coloneqq Tr_{reg}( H^{-s} )

and defined from there by analytic continuation on allmost all of the complex plane. The special value at s=1s = 1 (or its principal value) is the regularized Feynman propagator. See (BCEMZ 03, section 2.4.2). For the example of the above basic Feynman propagator see e.g. Grozin 05, section 2.1

Representing the (completed) zeta function here are the Mellin transform of some theta function – which in the present case is the partition function tTr regexp(tH)t\mapsto Tr_{reg} \exp(-t H) of the worldline formalism of the given theory, is what in the physics literature is known as the Schwinger parameter-formulation

Tr regH s= 0 t s1Trexp(tH)dt. Tr_{reg} H^{-s} = \int_0^\infty t^{s-1} Tr\, \exp(-t H)\,d t \,.
context/function field analogytheta function θ\thetazeta function ζ\zeta (= Mellin transform of θ(0,)\theta(0,-))L-function L zL_{\mathbf{z}} (= Mellin transform of θ(z,)\theta(\mathbf{z},-))eta function η\etaspecial values of L-functions
physics/2d CFTpartition function θ(z,τ)=Tr(exp(τ(D z) 2))\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2)) as function of complex structure τ\mathbf{\tau} of worldsheet Σ\Sigma (hence polarization of phase space) and background gauge field/source z\mathbf{z}analytically continued trace of Feynman propagator ζ(s)=Tr reg(1(D 0) 2) s= 0 τ s1θ(0,τ)dτ\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tauanalytically continued trace of Feynman propagator in background gauge field z\mathbf{z}: L z(s)Tr reg(1(D z) 2) s= 0 τ s1θ(z,τ)dτL_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tauanalytically continued trace of Dirac propagator in background gauge field z\mathbf{z} η z(s)=Tr reg(sgn(D z)|D z|) s\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s regularized 1-loop vacuum amplitude pvL z(1)=Tr reg(1(D z) 2)pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right) / regularized fermionic 1-loop vacuum amplitude pvη z(1)=Tr reg(D z(D z) 2)pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right) / vacuum energy 12L z (0)=Z H=12lndet reg(D z 2)-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)
Riemannian geometry (analysis)zeta function of an elliptic differential operatorzeta function of an elliptic differential operatoreta function of a self-adjoint operatorfunctional determinant, analytic torsion
complex analytic geometrysection θ(z,τ)\theta(\mathbf{z},\mathbf{\tau}) of line bundle over Jacobian variety J(Σ τ)J(\Sigma_{\mathbf{\tau}}) in terms of covering coordinates z\mathbf{z} on gJ(Σ τ)\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})zeta function of a Riemann surfaceSelberg zeta functionDedekind eta function
arithmetic geometry for a function fieldGoss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number fieldHecke theta function, automorphic formDedekind zeta function (being the Artin L-function L zL_{\mathbf{z}} for z=0\mathbf{z} = 0 the trivial Galois representation)Artin L-function L zL_{\mathbf{z}} of a Galois representation z\mathbf{z}, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)class number \cdot regulator
arithmetic geometry for \mathbb{Q}Jacobi theta function (z=0\mathbf{z} = 0)/ Dirichlet theta function (z=χ\mathbf{z} = \chi a Dirichlet character)Riemann zeta function (being the Dirichlet L-function L zL_{\mathbf{z}} for Dirichlet character z=0\mathbf{z} = 0)Artin L-function of a Galois representation z\mathbf{z} , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

Textbook accounts for quantum fields on Minkowski spacetime includes

An concise 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 on general globally hyperbolic spacetimes is 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)

where the issue with the underlying Hadamard propagators was settled, and reviewed for instance in

  • A. Bytsenko, G. Cognola, Emilio Elizalde, Valter Moretti, S. Zerbini, section 2 of Analytic Aspects of Quantum Fields, World Scientific Publishing, 2003, ISBN 981-238-364-6

Lecture notes (mostly for the case over Minkowski spacetime) include

  • Andrey Grozin, Lectures on QED and QCD (arXiv:hep-ph/0508242)

  • Green functions and propagators (pdf)

  • Green functions for the Klein-Gordon operator (pdf)

  • Hagen Kleinert, V. Schulte-Frohlinde, Critical properties of ϕ 4\phi^4-Theories 2001 (pdf)

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)

The zeta function regularization method originates around

  • Eugene Speer, On the structure of Analytic Renormalization, Comm. math. Phys. 23, 23-36 (1971) (Euclid)

and a comprehensive discussion is in (BCEMZ 03, section 2).

Discussion of zeta functions of Dirac operators in 2d includes

  • Michael McGuigan, Riemann Hypothesis and Short Distance Fermionic Green’s Functions (arXiv:math-ph/0504035)

Lecture notes concerning 1-loop vacuum amplitudes for the string include

  • The IIA/B superstring one-loop vacuum amplitude (pdf)

Revised on November 14, 2017 16:42:56 by Urs Schreiber (46.183.103.8)