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When speaking about electrical charges, it seems every particle either has a charge $+1$ or $-1$, in units of the electron charge. Therefore, we have a fundamental charge.

But what about mass? Is there any kind of such mass that every other mass can be seen a sum of those basic masses?

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    $\begingroup$ While quarks aren't really free particles they carry fractional charge less than the elementary charge $\endgroup$
    – Triatticus
    May 20, 2018 at 0:29
  • $\begingroup$ @Triatticus, even so, they still either have +1/3e, -1/3e. +2/3e or -2/3e charge. $\endgroup$
    – user168013
    May 20, 2018 at 0:37
  • $\begingroup$ That's because the elementary charge was defined in terms of the electron before quarks where even known. Another thing, there are also neutral fundamental particles so there is definitely a zero electric charge $\endgroup$
    – Triatticus
    May 20, 2018 at 0:57
  • $\begingroup$ @Triatticus, yes, but I said interacting. A particle with 0 charge is not involved in electromagnetic interaction. But even they are said as consisting of charged quarks. If there are other charges in quarks, the question becomes meaningless, but I am not aware of such. Actually, I would even like to know is it connected to the fact that quarks don't have mass (at least it's unclear how cumulative mass is calculated, if they have). $\endgroup$
    – user168013
    May 20, 2018 at 1:11
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    $\begingroup$ Possible duplicate of Why do we have an elementary charge but no elementary mass? $\endgroup$
    – Chris
    May 20, 2018 at 8:17

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When we come to the elementary constituents of matter, we come to the quantum mechanics regime and the special relativity space time description. In classical physics, masses are conserved and additive. This is not true in the microcosm of atoms, molecules and particles. There masses are the "length" of the special relativity four vector , $(E,p_x,p_y,p_z)$ , and are not an additive quantity and are not conserved. It is energy and momentum that are the conserved quantities. By contrast, charge is an additive conserved number characterizing elementary particles.

In elementary particle studies one has discovered elementary constituents of the proton, for example, which is composed out of three quarks and innumerable internal particle exchanges, which conserve charge and other quantum numbers. The mass of the proton is the "length" of the sum of the fourvectors of the innumerable constituents.

But what about mass? Is there any kind of such mass that every other mass can be seen as superposition of those basic masses?

This is where experimental and theoretical research are at the moment: it is a four vector addition that will define the mass of a complex system, not superposition, because mass is not a conserved quantity.

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  • $\begingroup$ Interesting that given the mass is not a separate parameter, it spreads spherically, exactly as the charge. $\endgroup$
    – user168013
    May 20, 2018 at 9:38
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Short answer: No.

Longer answer: Many of the masses in the Standard Model appear to be essentially random numbers. There's no reason to believe that all the non-zero mass particles are integer multiples of any smaller value.

This isn't experimentally falsifiable, though. For instance, we can't rule out that all masses are integer multiples of $10^{-12}~\rm eV$, since no masses are yet known to that precision.

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  • $\begingroup$ Mass is the rest energy of an object. For an electron and a proton mass is a well defined quantity but consider a hydrogen atom. Its mass is the sum of $m_e$, $m_p$, potential energy and kinetic energy. So internal dynamics of the atom show up as mass. In an excited state, the atom has a different mass! That is why the question should be narrowed to: Is the mass of ultimate constituents of matter, as we know today, quarks, leptons, Higgs, graviton, photon quantised? A reasonable answer was given above. $\endgroup$
    – my2cts
    May 20, 2018 at 9:35
  • $\begingroup$ @my2cts I was really just referring to fundamental particles. Though even allowing for composite particles, the hypothesis "all masses are integer multiples of some really, really small energy (say the energy of a photon with a wavelength the size of the observable universe)" is not really falsifiable. $\endgroup$
    – Chris
    May 20, 2018 at 9:46
  • $\begingroup$ Sure, but the OP perhaps had perhaps more in mind than only fundamental particles. $\endgroup$
    – my2cts
    May 20, 2018 at 9:51
  • $\begingroup$ Because of vacuum effects even fundamental particle can still be considered composite. $\endgroup$
    – my2cts
    May 20, 2018 at 9:52
  • $\begingroup$ @my2cts At any rate, if it doesn't even apply to fundamental particles, it certainly doesn't apply to composite particles. $\endgroup$
    – Chris
    May 20, 2018 at 9:57
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It's a question of scale. And scale matters, this is where physicists take about effective theories where they ignore higher energy effects, or equivalently, short scale effects.

At the effective level of the atom there appears roughly to be a set of basic masses: the mass of an electron, the mass of a proton. Below this scale the picture becomes murkier and questions arise about even what a particle is. Can we say that something that has a half life on the order of nano-seconds a particle?

Classically speaking, charge and mass were seen as something that continuously varies; it was an empirical finding to discover that both were discretised. If we look at the classical theories of mechanics and electromagnetism they take charge and mass variables as extensive quantities. It's only with the advent of QM that theoretical arguments for a discrete structure in the small were found. For example, Dirac is famous for discovering a quantisation condition for charge, which famously involved the existence of magnetic monopoles...and which also famously have not been discovered.

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