I’m reading Manton&Sutcliffe’s “Topological Solitons”. In that book on p. 2, they argue as follows:

In a Lorentz invariant theory, and in units where the speed of light is unity, the energy of a soliton is identified as its rest mass. In contrast, the elementary particles have a mass proportional to Planck’s constant $\hbar$ (this is sometimes not recognized, because of the choice of units).

This sentence is very confusing for me. Is this sentence simply pointing out that the mass multiplied by $\frac{c}{\hbar}$ in Klein-Gordon equation, described in the Planck units, equals that of the SI units? In other words, does this sentence relate to that the Klein-Gordon equation described in the SI units is $(\Box+(\frac{mc}{\hbar})^2)\phi=0 $, as is $(\Box+m^2)\phi=0$ in the Planck units?


Yes, OP is right. The mass term in the Klein-Gordon or the Dirac action is $\frac{mc}{\hbar}$ so that it has dimension of inverse length (in order to match the dimension of the spacetime derivative $\partial_{\mu}$ in the kinetic term). But since the classical action does not depend$^1$ on $\hbar$, this in turn means that the mass parameter$^2$ $m$ is really proportional to $\hbar$.


$^1$ The reduced Planck constant $\hbar$ is here treated as a free parameter rather than the actual physical value $\approx 1.05\times 10^{−34} {\rm Js}$.

$^2$ The physical mass $m_{\rm ph}$ is given by a pole $$(k^0)^2-{\bf k}^2~=~\left(\frac{m_{\rm ph}c}{\hbar}\right)^2 ~=~\left(\frac{mc}{\hbar}\right)^2-\Pi$$ in the connected 2-pt function/propagator. Here $\Pi$ is a self-energy term.

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    $\begingroup$ Maybe a better statement would be that the frequency gap in the wave equation and the mass of the elementary particle are related by $\hbar$ (as one might suspect, since it is quantum mechanics that relates particles to waves)? I don’t quite understand what it means for a physical constant to be “proportional” to another physical constant. $\endgroup$ Oct 10 at 4:32
  • 1
    $\begingroup$ I updated the answer. $\endgroup$
    – Qmechanic
    Oct 10 at 4:54

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