Elementary charge and mass

1. Oct 30, 2013

TrickyDicky

(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.

2. Oct 30, 2013

Drakkith

Staff Emeritus
Not as far as I know.

3. Oct 30, 2013

Jano L.

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.

4. Oct 30, 2013

TrickyDicky

As it is invariant mass, perhaps I should have specified I was referring to this mass.

Sure, so is the electron's mass.

5. Oct 30, 2013

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.

6. Oct 30, 2013

TrickyDicky

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?

7. Oct 30, 2013

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?

8. Oct 30, 2013

TrickyDicky

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.

9. Oct 30, 2013

king vitamin

There is no reason within currently accepted physics that every electric charge in the universe is an integer multiple of e/3. The only mechanism I'm aware of that quantizes charge is the existence of magnetic monopoles.

10. Oct 30, 2013

WannabeNewton

Well they are on equal footing only in the sense that components of the electric and magnetic fields relative to some frame can be transformed into one another when going to another frame. They are still defined differently for starters: given an EM field tensor $F_{ab}$ and an observer with 4-velocity $\xi^a$ the electric field relative to this observer is $E^{a} = F^{a}{}{}_{b}\xi^{b}$ whereas the magnetic field relative to this observer is $B^a = \frac{1}{2}\epsilon^{abcd}\xi_b F_{cd}$. The magnetic field relative to $\xi^a$ is still conceptually different from the electric field relative to $\xi^a$ as can be seen for example in how $E^a$ and $B^a$ differ physically in their effects on a coincident charged particle with 4-velocity $\eta^a$ as per the Lorentz force law, written covariantly as $\eta^b \nabla_b \eta^a = \frac{q}{m} F^{a}{}{}_{b}\eta^b$ which decomposes into an electric and magnetic part in the usual way relative to $\xi^a$.