In Aspects of Symmetry, Coleman says (p. 185)

''Most of the simple field theories with which we are familiar have the property that all of their non-singular solutions of finite total energy are dissipative. This is the case for Maxwell's equations, the Klein-Gordon equation, etc.''

However, isn't it the case that the mentioned equations for regular initial data are dispersive? For example, we take the vacuum Maxwell equations - they will have the form (for the electric/magnetic field)

\begin{equation} y_{xx}-c^{-2}y_{tt}=0. \end{equation}

If we take $y(x,0)=\sin x$ and $\partial_t y(x,0)=(1/\epsilon)\sin x$ for the electric field and mutatis mutandis for the magnetic field, we obtain the electromagnetic wave - a dispersive solution.

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    $\begingroup$ This is unclear -- what do you think the definitions of the words "dispersive" and "dissipative" are? $\endgroup$
    – knzhou
    Jun 7, 2019 at 18:56
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    $\begingroup$ I'm not OP but "dissipative" usually means a solution that does not conserve energy, like in a system with a damping term. "Dispersive" sounds more like what the 3D KG equation and the Maxwell equations act like - spreading the energy density across an ever bigger volume as time goes on, while the total remains constant. Contrast the massless Klein-Gordon equation in 1D where if your solution begins as a lump it stays as a lump and moves onwards without spreading. $\endgroup$
    – Okarin
    Jun 7, 2019 at 18:59
  • $\begingroup$ Maybe he meant something as, "Energy in any given closed subspace diminishes to zero when $t\to\infty$"? Since there's no ideal EMW, or free space, your counter-example wouldn't matter. $\endgroup$
    – acarturk
    Jun 7, 2019 at 19:38

1 Answer 1


This is the general solution of wave equation with speed $c$ or Klein-Gordon Equation with Dissipation (KGD): $$\Box y + a \dot y + b y = 0.$$ For $a=0$ and $b=0$ you have the plain d'Alembert equation where the waves travel at $c$.

For just $a=0$ you have the K-G equation where the waves' group speed is $c$ but the single waves propagate at different speeds depending on $b$ and so the waves' group is dispersed

For just $b=0$ you have Viscoelastic Maxwell Equation, a dumped oscillator where the waves propagate at $c$ but loosing amplitude, i.e. is energy is dissipated.


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