# What is the Difference between a Lepton and a Fermion?

As the Title Says: I am Wondering what the Difference between a Lepton and A Fermion is.

I know they both have an ½ integer spin number e.g. a electron, an atom with an odd mass number such as Helium-3. But is there a fundamental difference between these two things, or as they interchangeable?

Note: Fermions are used when describing the electrons in a superconductor/superfluid, which is why I included these as 'tags' to the question.

A fermion is any particle, elementary or composite, that obeys Fermi-Dirac (as opposed to Bose-Einstein) statistics relating to how identical particles behave when you swap two of them. Due to an important but complicated result, this is taken to amount to having half-integer spin.

A lepton is one type of elementary particle with spin 1/2. The only leptons are the three generations of electrons, the three generations of neutrinos, and the antiparticles of these. (So that's 6, 9, or 12, depending on how you count.) There are also elementary fermions (the quarks) that are not leptons, so this is a much narrower class of particles.

• +1 and minor comment: although most people believe in the spin-statistics connection (as they should), perhaps it would be pedagogically favorable to qualify the first sentence with a comment about how the definition of a fermion is really a particle that obeys certain statistics, but that it is believed that all such particles have half-integer spin and vice versa. – joshphysics Dec 3 '14 at 18:42
• @joshphysics The spin statistics connection is only for particle physics tho, right? In condensed matter systems, we can, in principle, have integer spins obeying the Fermi-Dirac statistics because Lorentz invariance need not be respected. Correct me if I'm wrong. Of course, the question was as such in the particle physics context. – Dvij D.C. May 26 '20 at 11:06
• @DvijD.C. I'm not qualified to confirm that with confidence -- my condensed matter knowledge is weak, but my impression is that what you've said is correct. – joshphysics May 26 '20 at 19:00

A fermion is any particle characterized by Fermi–Dirac statistics and obeying the Pauli exclusion principle. So for example quarks are fermions, as are Helium-3 atoms. A fermion does not have to be an elementary particle. I'm not even sure that it has to be spin $\tfrac{1}{2}$, though I can't think of any fermions that aren't.

A lepton is a spin $\tfrac{1}{2}$ fermion that is elementary and does not feel the strong force.

So leptons are a subset of fermions. The only known leptons are the electron, electron neutrino, muon, muon neutrino, tau and tau neutrino, and their antiparticles.

• You're right to be cautious about saying a fermion has spin $\frac{1}{2}$, since a fermion is any particle that obeys Fermi-Dirac statistics, so any particle that has an odd multiple of $\frac{1}{2}$ as spin is a fermion. – ACuriousMind Dec 3 '14 at 18:07
• baryon resonances can have spin $3/2$ - not being in the most favourable spin configuration is where (all?) their additional mass comes from – Christoph Dec 3 '14 at 18:17
• Gravitinos are spin $\tfrac{3}{2}$ - but only if they exist ... – John Rennie Dec 3 '14 at 18:22
• one more rather obvious example: nuclear spin – Christoph Dec 3 '14 at 18:23
• Heavy holes have $\frac32$ spin, but these are quasiparticles. – Ruslan Dec 4 '14 at 14:37

The Standard Model includes 12 elementary known as fermions that respect the Pauli exclusion principle. They include six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino) (ref)

All leptons are fermions, but not all fermions are leptons.

A fermion is just a particle of half-integer spin.

Being a lepton for a particle is a matter of definition of global symmetries of the theory. This means that a lepton can in principle be both a fermion or a boson, although all known leptons are fermions (electron, muon, tau and their neutrinos). One example of bosonic lepton is the weak triplet Higgs boson of the type-2 see-saw models. In supersymmetric extensions of the Standard Model there are scalar partners of the ordinary fermions. All these scalar have spin-0 and some of them are leptons, yet another example of non-fermionic leptons. There are more ...