Pauli exclusion principle

Remarks

The space $\mathcal{H}_-^{(n)}$ is the $n$-th exterior power of $\mathfrak{h}$, and $P_-(v_1 \otimes \cdots \otimes v_n)$ is, up to the factor $1/n!$, the wedge product $v_1 \wedge \cdots \wedge v_n$ in the exterior algebra. If $v_i = v_j$, the transposition $\tau = (ij)$ fixes the product vector $w = v_1 \otimes \cdots \otimes v_n$, so $P_- w = P_-(U_\tau w) = -P_- w$, and hence $P_- w = 0$. The other parts are standard facts about exterior powers. A wedge product vanishes exactly on dependent vectors, and the wedges of increasing basis vectors form a basis.

Example: for electrons in an atom, $\mathfrak{h}$ contains the spin factor $\C^2$. Each orbital, a one-dimensional subspace of $L^2(\R^3)$, together with the spin factor gives a two-dimensional space, so by part 3 at most two electrons occupy one orbital.

The statement is about fermions only. For bosons the symmetric projection $P_+$ of $v \otimes v \otimes \cdots \otimes v$ is that vector itself, which is not $0$ when $v \ne 0$, so any number of bosons can share one state.

Going deeper

For a student

In the mathematical model, the exclusion principle is a property of exterior powers: $v \wedge v = 0$. In second quantization it appears as the relation $a^*(f)^2 = 0$ for the fermionic creation operators. A basis of the $n$-fermion space is given by Slater determinants of $n$ different one-particle basis states. The principle underlies the stability of ordinary matter and the theory of metals and white dwarf stars.