# Is the mass of elementary particles proportional to the Planck constant?

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.
• 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. Oct 10 at 4:32