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.

  • 2
    $\begingroup$ This is unclear -- what do you think the definitions of the words "dispersive" and "dissipative" are? $\endgroup$ – knzhou Jun 7 at 18:56
  • $\begingroup$ 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 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 at 19:38

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