Symmetric, antisymmetric and mixed symmetry particles

Question

Can someone explain to me the concept of symmetric, antisymmetric, and mixed symmetry when talking about the states of identical particles?

Answer 1

You may indeed have heard that an electron is antisymmetric, whereas a photon is symmetric. What does this mean?

Suppose I have a system of several electrons. They could be orbiting a nucleus, for example. Their behaviour is described by a wave-function, $\psi$. If I swap the positions of two electrons labled $a$ and $b$ in the system, the wave-function will pick up a minus sign, $$ \psi_{ab} = - \psi_{ba}. $$ Thus we say that electrons are antisymmetric. By the spin-statistics theorem, all fermions (electrons, leptons, quarks etc) are antisymmetric.

If, on the other hand, the behaviour was simply, $$ \psi_{ab} = \psi_{ba}, $$ the particle is said to be symmetric. By the spin-statistics theorem, all bosons (photons, $W$, $Z$ etc) are symmetric.

What is the physical meaning of this? What are the implications? Well, one famous implication is the Pauli exclusion principle. Suppose I have two indistinguishable electrons (with the same quantum numbers), they cannot be labelled independently, we simply have $a=b$. What is their wave-function? It must be antisymmetric, so we have $$ \psi_{ab} = - \psi_{ba} \Rightarrow \psi_{aa} = - \psi_{aa} = 0 $$ i.e. it is impossible to have antisymmetric particles in identical quantum states.