# Why are composite fermions either bosons or fermions but not neither?

The question is getting at something subtle and not necessarily obvious to me.

It is often claimed that an atom being composed purely of fermions is either a boson or a fermion itself. I interpret this as meaning that the atom can be treated as a fundamental particle (in the sense it can be put into another schrodinger equation without worrying about it's internal structure).

So if for example, I had a box potential, and I put in many fermionic atoms inside it they would have a population described by the Fermi-Dirac distribution, and likewise bosonic atoms would be described by the Bose-Einstein distribution.

• Why is it that composite fermions (i.e., atoms) are not described by either of the two characters (Bosonic, Fermionic)?

This really amounts to saying are atoms fermions or bosons regardless of temperature, spacing, energies, etc. will they always behave in this simple model to any scale? Will they never switch character?

Intuitively, fermions are fermions because they can't overlap. It seems to me that it would be reasonable for them to act different from the fundamental fermions and bosons because the atoms have a nucleus that is localized and an electron distribution that is spread out; so, it would be less probable for the atoms to really be in the same configuration and so truly be indistinguishable. Or put another way, it would be less probable for each fermion to overlap with the corresponding fermion to cause an overall cancellation. But this is an assumption on why composite particles behave so this begs the second question.

• What is the mechanism by which composite particles act like the fundamental constituents they are made of?
• I may be wrong, but I always took this to mean the classification of the atom was determined by the overall total spin involved: en.wikipedia.org/wiki/Boson – user154420 Jun 26 '17 at 13:45
• The overall wave function is either odd or even. – Jon Custer Jun 26 '17 at 13:56
• What is meant by "It's a boson or fermion" is that the spin is either a half-integer or an integer. This follows from the SU(2) algebra of spin addition. – Omry Jun 26 '17 at 14:24
• @Omry If that is the case, what does spin have to do with the statistics? and why doesn't the fact that the spin is distributed all across the atom affect anything? – Enrique Mendez Jun 26 '17 at 14:45
• The spin statistics theorem guarantees that integer spin => boson and half-integer spin => fermion. – user154997 Jun 26 '17 at 15:06

Why are fondamental particles either bosons or fermions?

A particle specie is either a boson or a fermion depending on how the wave function changes when permuting two particles of the same specie. A general two particles wave function $\Psi(x_1,x_2)$ can be acted on by the operator $P$ which permutes the two particles. Of course, $P^2\Psi(x_1,x_2) = \Psi(x_1,x_2)$ so $1$ is an eigenvalue of $P^2$ and so the only two possible eigenvalues for $P$ are $\pm1$, $+$ for bosons and $-$ for fermions.

Why are composites also either bosons or fermions?

To answer this, I will cite the excellent answer to the question "How to combine two particles of spin $\frac{1}{2}$?". The main equation to remember is following with it's interpretation: $$(2j_a+1)\otimes(2j_b+1) = \bigoplus_{i=1}^n(2j_i+1),$$

On the left-handed side, the object describes the hilbert space of two particles, one with spin $j_a$ the other with spin $j_b$. The particles are decoupled so they have a fix total spin and can have any corresponding projection.

On the right-handed side, the object describes the hilbert space of a single composite particle which have can change total spin $j_i \in \{|j_a-j_b|, |j_a-j_b+1|, ..., j_a+j_b\}$.

Both spaces are related by a change of basis. What you need to see is that the total spin, even if it changes, is either an integer or an half-integer. If both particles are fermions or bosons, the composite particle is a boson, but else, the composite particle is a fermion.

Composite particles don't have to be either bosons or fermions - they can also form anyons, which are defined to be composite particles that are neither bosonic nor fermionic. This is a very active area of current research.