Klein–Gordon Propagators

The inverse of the Klein–Gordon differential operator is not fixed by its algebraic denominator. A pole prescription selects the temporal boundary condition, and an overall factor fixes the delta-source normalization. Deriving both together distinguishes a retarded response from the Feynman correlation used in field-theory amplitudes.

Required background. The Klein–Gordon Equation defines the operator; Plane-Wave Solutions gives its frequencies; Causality and Light Cones fixes the support interpretation. Fourier distributions and residue integration are assumed.

Fourier inversion with a declared source

Use natural units $\hbar=c=1$ throughout this page and $m>0$. For $L=\Box+m^2$, define a mathematical Green function by

$$LG(x)=\delta^{(4)}(x),\qquad G(x)=\int\frac{d^4p}{(2\pi)^4}e^{-ip\cdot x}\widetilde G(p).$$

Since $Le^{-ip\cdot x}=-(p^2-m^2)e^{-ip\cdot x}$, $\widetilde G$ must invert $-(p^2-m^2)$. The formal expression $-1/(p^2-m^2)$ is singular on shell and does not define a unique distribution. Different inverses differ by homogeneous solutions. Boundary conditions are additional information, not consequences of the mass-shell equation.

Let $\omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. Three choices are

$$\begin{aligned} \widetilde G_R(p)&=-\frac{1}{(p^0+i0)^2-\omega_{\mathbf p}^2},\\ \widetilde G_A(p)&=-\frac{1}{(p^0-i0)^2-\omega_{\mathbf p}^2},\\ \widetilde G_F(p)&=-\frac{1}{p^2-m^2+i0}. \end{aligned}$$

For $G_R$ both frequency poles lie below the real axis; for $G_A$ both lie above. For $G_F$ the positive pole lies below and the negative pole above. The notation means a distributional limit of regulated integrals; the regulators must be retained until the integrals are defined.

Evaluating the frequency integral

Consider the retarded frequency integral at fixed spatial momentum:

$$g_R(t,\mathbf p)= -\int\frac{dp^0}{2\pi} \frac{e^{-ip^0t}}{(p^0+i0)^2-\omega_{\mathbf p}^2}.$$

For $t<0$ close above, where there are no poles. For $t>0$ close below; the contour is clockwise and contributes $-i$ times the residues after the $1/(2\pi)$ normalization. The residues of the displayed integrand, including its minus sign, sum to $i\sin(\omega_{\mathbf p}t)/\omega_{\mathbf p}$. Thus

$$g_R(t,\mathbf p)=\theta(t)\frac{\sin(\omega_{\mathbf p}t)}{\omega_{\mathbf p}}.$$

The corresponding results are

$$\begin{aligned} g_A(t,\mathbf p)&=-\theta(-t) \frac{\sin(\omega_{\mathbf p}t)}{\omega_{\mathbf p}},\\ g_F(t,\mathbf p)&=\frac{i}{2\omega_{\mathbf p}} e^{-i\omega_{\mathbf p}|t|}. \end{aligned}$$

All three satisfy $(\partial_t^2+\omega_{\mathbf p}^2)g=\delta(t)$. Each is continuous at zero and its first derivative has jump $+1$. This is a simple way to audit the contour signs without repeating the contour calculation. The full spacetime functions follow by integrating $e^{i\mathbf p\cdot\mathbf x}g(t,\mathbf p)$ with $d^3p/(2\pi)^3$.

Retarded support and the Feynman normalization

The hyperbolic initial-value problem gives $G_R$ support in the future light cone and $G_A$ support in the past light cone. The difference $G_R-G_A$ solves the homogeneous equation and has equal-time data $0,\delta^3(\mathbf x)$. It is the sine evolution kernel of the scalar Cauchy problem.

The commonly used time-ordered field correlation is instead

$$D_F(x)=\int\frac{d^4p}{(2\pi)^4} \frac{i\,e^{-ip\cdot x}}{p^2-m^2+i0}.$$

With this convention,

$$LD_F=-i\delta^{(4)},\qquad G_F=iD_F.$$

Therefore $D_F*J$ is not the normalized solution of $L\phi=J$. One would need $iD_F*J$ for the Feynman boundary condition, or $G_R*J$ for retarded response. The factors of $i$ do not change support: neither Feynman object is a retarded signal kernel. Locality and Causality Warnings derives the explicit spacelike correlation and explains why it does not imply signaling. That distinction also prevents confusing an on-shell positive-frequency function with an inverse differential operator.

Static limit and the Yukawa inverse

For a time-independent source switched on adiabatically from the past, the time integral of the retarded momentum kernel is

$$\lim_{\epsilon\to0^+}\int_0^\infty dt\, e^{-\epsilon t}\frac{\sin(\omega_{\mathbf p}t)}{\omega_{\mathbf p}} =\frac{1}{\mathbf p^2+m^2}.$$

The resulting spatial inverse obeys $(-\nabla^2+m^2)G_{\rm stat}=\delta^3$. Away from the origin, rotational symmetry and decay give $G_{\rm stat}=C e^{-mr}/r$. Integrating the equation over a small sphere fixes $-4\pi r^2\partial_rG_{\rm stat}\to1$, hence

$$G_{\rm stat}(r)=\frac{e^{-mr}}{4\pi r}.$$

This is the Yukawa kernel. Its $m\to0$ limit is the Coulomb inverse $1/(4\pi r)$. The static spatial profile does not describe instantaneous formation of the field; the switching and retarded time evolution have been taken before the stationary limit.

A proper-time parameter is not laboratory time

For $\epsilon>0$, elementary integration gives

$$\frac{i}{p^2-m^2+i\epsilon} =\int_0^\infty ds\, \exp\!\left[is(p^2-m^2)-\epsilon s\right].$$

The auxiliary parameter $s$ has inverse-mass-squared units in this convention. This Schwinger parameterization is useful for Gaussian momentum integration and background-field methods. It is not the coordinate time $t$, nor by itself a probability distribution of particle travel times. Oscillatory integrations and interchange of limits still require their regulator.

Exercises

1. Verify the derivative jumps of $g_R$ and $g_F$ and their delta normalization.

Solution

For $g_R$, the derivative is zero just before zero and one just after. For $g_F$, it is $-1/2$ just before and $+1/2$ just after. Each function is continuous, so no $\delta'$ term arises; the derivative jump supplies exactly $\delta(t)$ in the second derivative.

2. Solve the impulsively forced mode $\ddot f+\omega^2f=J_0\delta(t)$ with no response for $t<0$. What changes if $g_F$ is substituted for $g_R$?

Solution

$f=J_0\theta(t)\sin(\omega t)/\omega$. It starts with continuous $f=0$ and derivative jump $J_0$. The Feynman solution $iJ_0e^{-i\omega|t|}/(2\omega)$ has the same source but is nonzero before the impulse, so it solves a different boundary-value problem.

3. Check the proper-time integral by evaluating it at finite $\epsilon$, before taking its limit.

Solution

Writing $a=p^2-m^2$, the integral is $1/(\epsilon-ia)=i/(a+i\epsilon)$. Omitting the damping factor before integration would replace a convergent integral by an undefined ordinary oscillatory integral. Across the mass shell $a=0$, the limit must be interpreted distributionally.

References

• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — propagator conventions and pole prescriptions.
• J. Schwinger, “On Gauge Invariance and Vacuum Polarization,” Physical Review 82, 664–679, 1951, doi:10.1103/PhysRev.82.664 — the proper-time method in background fields.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes, 2006–2007, §§2.6–2.7, Causality and Propagators — scalar Green functions and their time-ordering interpretation.