Why should two sub-atomic (or elementary particle) - say electrons need to have identical static properties - identical mass, identical charge? Why can't they differ between each other by a very slight degree? Is there a theory which proves that?
Imagine an alien of size of order of Milky-way galaxy examining our solar system with a probe of size of 10's of solar system dimension and concludes that all planets are identical.
One good piece of evidence that all particles of a given type are identical is the exchange interaction. The exchange symmetry (that one can exchange any two electrons and leave the Hamiltonian unchanged) results in the Pauli exclusion principle for fermions. It also is responsible for all sorts of particle statistics effects (particles following the Fermi-Dirac or Bose-Einstein distributions) depending on whether the particles are fermions or bosons.
If the particles were even slightly non-identical, it would have large, observable effects on things like the allowed energies of the Helium atom.
Regarding to this question about Is the electron of carbon identical to that of hydrogen? How to prove it? the answer should be a little bit different from the usual answer. Electrons in the same state have the same energy amount and the same electric charge. Since they have a magnetic moment too they are separable in this parameter. And now the clue.
Electrons of different speed as well as of different bond to a nucleus are separable too. And as an add-in, electrons with relativistic velocity have shielded charges. So the answer, that electrons are not separatable is true only in a first thought.
