# Klein-Gordon/Maxwell Equation: dissipative or dispersive?

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.

• This is unclear -- what do you think the definitions of the words "dispersive" and "dissipative" are? – knzhou Jun 7 '19 at 18:56
• 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. – Okarin Jun 7 '19 at 18:59
• 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. – acarturk Jun 7 '19 at 19:38