© Matt Strassler [September 12, 2012]
I earlier wrote a simple descriptive and qualitative article about fermions and bosons that you might like to read. Here, I aim to follow up on the article about why particles are quanta. The qualitative facts that you need to know is:
- All elementary particles are either fermions or bosons
- Fermions (including electrons, quarks and neutrinos) satisfy the Pauli exclusion principle: no two fermions of the same type can ever be doing exactly the same thing.
- Bosons (including photons, W and Z particles, gluons, gravitons and Higgs particles) are different: two (or more) bosons of the same type may indeed do the same thing.
This last point is why lasers can be made from photons, which are bosons and can all do the same thing in lockstep, making a powerful narrow one-color beam, but not from electrons, which are fermions.
How does this distinction manifest itself in the language of the Particles and Fields articles of which this one is a pat? It turns out that all the formulas I’ve been giving you are true for bosons, and need a very small but very consequential modification for fermions. For bosons we had
- E = (n + 1/2) h ν, where n = 0,1,2,3,4,…
which means each quantum has energy h ν. That implies that boson quanta can be made to do exactly the same thing; when n is greater than 1, the boson field has a wave made from more than one quantum which are oscillating and moving in lock-step. But for fermions
- E = (n - 1/2) h ν, where n = 0 or 1.
It is still true that each quantum has energy h ν, so all of the discussion about particles and their energies, momenta and masses is still correct. But the number of quanta of an electron wave can only be zero or one; ten electrons, unlike ten photons, cannot be organized into a wave of larger amplitude. So there are no fermion waves that are made from large numbers of fermions oscillating and moving in lock-step.
[And then there’s that interesting minus sign in the zero-point energy for fermions, compared to bosons. Zero-point energy, and the two different signs, are not important in this set of articles, but will be important in other sets later.]