# Elementary charge and mass

## Main Question or Discussion Point

(question inspired by a thread on the cosmology subforum)
Is there an intuitively simple theoretical reason for the existence of elementary units of charge but not of mass?
The parallels between Gauss's law for EM and gravity, the inverse-square law for both, the fact that energy is quantized, like charge is, and that there are elementary particles seem to lead to the notion of the existence of an elementary mass unit.
Is a matter of its smallness in case it existed and teherefore difficulty to be experimentally verified, like the suspected planckian length unit(mass is sometimes expressed in terms of length)? But that seems to imply the fundamental discreteness of our universe wich is far from being clear as of now.

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Drakkith
Staff Emeritus
Is there an intuitively simple theoretical reason for the existence of elementary units of charge but not of mass?
Not as far as I know.

Jano L.
Gold Member
Is there an intuitively simple theoretical reason for the existence of elementary units of charge but not of mass?
It is a theoretical conception based on experiments. From these, we know electric charge comes in universal units of elementary charge. Theoretically, this is supported by the fact that from the Maxwell equations it follows charge is a Lorentz invariant, so its value is universal for every frame.

But so far no smallest value of rest mass has been discovered. In the theory of relativity connected with electromagnetic theory, rest mass of composite system depends also on the EM energy of the system, which can vary continuously.

Quantization of energy is a very different thing from charge coming in discrete universal units; interaction of light with atomic systems is well described with help of steps of energy ##\hbar \omega##, but omega can be any real number, depending on the exact character of the system and the conditions it is in. In other words, in the energy quantization, the discreteness is not universal, since ##\omega## can be anything.

On the other hand, the electric charge of the electron cannot change; it is always the same.

It is a theoretical conception based on experiments. From these, we know electric charge comes in universal units of elementary charge. Theoretically, this is supported by the fact that from the Maxwell equations it follows charge is a Lorentz invariant, so its value is universal for every frame.
As it is invariant mass, perhaps I should have specified I was referring to this mass.

On the other hand, the electric charge of the electron cannot change; it is always the same.
Sure, so is the electron's mass.

WannabeNewton
Energy is not always discrete. This depends on the eigenvalue spectrum (borrowing language from spectral theory, we can have in particular pure point or continuous spectrum).

Why should we expect mass to be discrete just because charge is? There are many fundamental differences between the gravitational interaction and the electromagnetic interaction.

Why should we expect mass to be discrete just because charge is? There are many fundamental differences between the gravitational interaction and the electromagnetic interaction.
True, thus my question should be easy to answer by pointing out how specifically those fundamental differences in these classical theories make electric charge discrete but "gravitational charge" not, but it isn't it seems.

Mathematically, at least in terms of Gauss's law, Poisson's equation, Coulomb's and Newton's gravitational inverse square laws, can you think of a reason?

WannabeNewton
Well there are theoretical ways of proving it, albeit not exactly experimentally justified. For example if magnetic monopoles exist (which is easily accommodated theoretically into electromagnetism) then charge quantization can be proven mathematically. What analogue would there be for the gravitational interaction in terms of mass?

Well there are theoretical ways of proving it, albeit not exactly experimentally justified. For example if magnetic monopoles exist (which is easily accommodated theoretically into electromagnetism) then charge quantization can be proven mathematically. What analogue would there be for the gravitational interaction in terms of mass?
I can certainly see no analogue for mass.

If magnetic monopoles exist, that would modify Maxwell's equations and charge quantization would be derived straight from them, yes, but that is a big if, not experimentally justified at all therefore a purely theoretical speculation.
One could just as well speculate about why if in relativity and the EM covariant formulation the magnetic and electric fields are put on an equal footing, one field is divergenceless but the other is not.

king vitamin