Solving Scalar Initial-Value Problems with Green Functions

A Green function solves the source part of a wave equation; independently specified Cauchy data supply its homogeneous part. This page combines both in the scalar initial-value problem and works through sources that are switched on suddenly or driven at resonance. The contour construction and source conventions belong to Klein–Gordon Propagators.

Required background. Klein–Gordon Propagators defines $G_R$ by $(\Box+m^2)G_R=\delta^{(4)}$; Plane-Wave Solutions explains the two Cauchy data. Use natural units $\hbar=c=1$ below.

Cauchy data plus retarded forcing

Consider

$$(\partial_t^2-\nabla^2+m^2)\phi=J,\qquad \phi(0,\mathbf x)=f(\mathbf x),\quad \partial_t\phi(0,\mathbf x)=h(\mathbf x).$$

Take $m>0$ and data and sources sufficiently regular for the displayed operations, with distributional sources interpreted by limits. Let $\Omega=\sqrt{-\nabla^2+m^2}$, understood as a spatial Fourier multiplier. For $t\geq0$ the solution is

$$\begin{aligned} \phi(t)={}&\cos(\Omega t)f+ \frac{\sin(\Omega t)}{\Omega}h\\ &+\int_0^t ds\,\frac{\sin[\Omega(t-s)]}{\Omega}J(s). \end{aligned}$$

The first two terms are fixed by the two Cauchy data. The third is Duhamel’s forcing term. Differentiating once gives no upper-limit contribution because $\sin(0)=0$; differentiating twice contributes $J(t)$ because the derivative of the sine kernel at zero is the identity. The remaining terms obey the homogeneous oscillator equation, which proves the formula.

For $t>0$, the first two terms can also be written as spatial convolutions $(\partial_tG_R)*_{\mathbf x}f+G_R*_{\mathbf x}h$. The forcing term is the spacetime convolution of $G_R$ with the source restricted to times after zero. An arbitrary homogeneous contribution cannot be discarded unless its initial data are actually zero.

Causal support uses both initial data

The retarded kernel vanishes outside the future light cone, so source points contribute only within their causal future. Compactly supported $f$ and $h$ likewise propagate within the future domain allowed by the hyperbolic equation. The operators $\cos(\Omega t)$ and $\sin(\Omega t)/\Omega$ must be considered as these complete combinations; the nonlocality of $\Omega$ alone does not negate their finite propagation property.

Conversely, projecting to the positive-frequency relation $h=-i\Omega f$ changes the admissible pair of initial data. A compact shape $f$ does not generally have compact $h$ under this constraint. The localization and frequency-sector qualifications in The Square-Root Hamiltonian remain relevant.

A source switched on at zero

Take zero initial data and one spatial Fourier pattern,

$$J(t,\mathbf x)=J_0\theta(t)\cos(\mathbf k\cdot\mathbf x), \qquad \omega=\sqrt{|\mathbf k|^2+m^2}.$$

The source is spatially extended; it is a mode example, not a localized signaling experiment. Duhamel’s integral gives

$$\phi(t,\mathbf x)= \frac{J_0}{\omega^2}[1-\cos(\omega t)] \cos(\mathbf k\cdot\mathbf x),\qquad t\geq0.$$

It has zero initial amplitude and derivative. Its stationary particular solution is $J_0\cos(\mathbf k\cdot\mathbf x)/\omega^2$, but the sudden switch leaves a persistent homogeneous oscillation. Without damping or an adiabatic preparation, this undamped spatial mode does not relax pointwise to the static solution. Localized forcing can instead allow transients to radiate away from a fixed observation region.

The static Yukawa inverse in the canonical derivation instead comes from a source prepared adiabatically in the remote past. A static inverse and a suddenly forced initial-value problem answer different questions.

Resonant forcing

Replace the temporal source by $J_0\cos(\omega t)$ for $t\geq0$ while keeping zero data and the same spatial mode. At exact resonance,

$$\phi(t,\mathbf x)= \frac{J_0t}{2\omega}\sin(\omega t)\cos(\mathbf k\cdot\mathbf x).$$

The growing factor $t$ describes coherent energy supplied by the external drive. It is not an exponentially unstable free mode or a particle-production probability. An infinite-duration linear source can eventually invalidate an approximation that neglects source backreaction or nonlinearities.

Exercises

1. Verify the resonant formula directly by differentiating its temporal factor twice.

Solution

For $F=t\sin(\omega t)/(2\omega)$, $F'=\sin(\omega t)/(2\omega)+t\cos(\omega t)/2$ and $F''=\cos(\omega t)-\omega t\sin(\omega t)/2$. Thus $F''+\omega^2F=\cos(\omega t)$, with $F(0)=F'(0)=0$.

2. Take the spatial-mode source $J_0e^{\epsilon t}$ from the remote past, with $\epsilon>0$. Evaluate its retarded response and then let $\epsilon\to0^+$ at fixed $t$.

Solution

The temporal response is

$$J_0\int_{-\infty}^t ds\, \frac{\sin[\omega(t-s)]}{\omega}e^{\epsilon s} =\frac{J_0e^{\epsilon t}}{\omega^2+\epsilon^2}.$$

Its limit is $J_0/\omega^2$, with no switch-induced oscillation. Changing the source preparation changes the homogeneous component.

3. Can a solution with a localized retarded source have nonzero field outside that source’s causal future?

Solution

Its source-generated contribution cannot. Its independently prepared homogeneous solution can already be nonzero there. A support statement about the source response is not a support statement about arbitrary initial data plus that response.

References

• R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. II, Interscience, 1962 — hyperbolic initial-value problems and domains of dependence.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes, 2006–2007, §2.7, Propagators — retarded scalar Green functions.