Repeated-orbital fermion exclusion

Slot convention. For $\mathrm{\ell }\ge 1$, $S_{\mathrm{\ell }}$ acts on $\{0,\dots ,\mathrm{\ell }-1\}$ as in Permutation sector projections on the n-fold tensor power. On one-based tuple positions use $\widehat{\pi }(k):=1+\pi (k-1)$ for $1\le k\le \mathrm{\ell }$. Conjugation by $k\mapsto k-1$ gives $\widehat{\pi \sigma }=\widehat{\pi }\circ \widehat{\sigma }$ and $\widehat{{\pi }^{-1}}={\widehat{\pi }}^{-1}$; thus the supplier's tensor action becomes $U_{\pi }(v_1\otimes \cdots \otimes v_{\mathrm{\ell }})=v_{{\widehat{\pi }}^{-1}(1)}\otimes \cdots \otimes v_{{\widehat{\pi }}^{-1}(\mathrm{\ell })}$. The parameters of $U_{\pi }$ and $\mathrm{sgn}(\pi )$ remain canonical zero-based permutations.

Assume the Axiom of Countable Choice and the Axiom of Choice. Let $H$ be a complex Hilbert space, let $H^{\otimes n}$ be its completed tensor powers with the unitaries $U_{\pi }$ and projections $P_{\pm }^{(n)}$ of Permutation sector projections on the n-fold tensor power. Fix $n\ge 2$ and vectors $u_1,\dots ,u_n\in H$, and let $S(u_1,\dots ,u_n):=\frac{1}{\sqrt{n!}}{\sum }_{\sigma \in S_n}\mathrm{sgn}(\sigma )u_{\widehat{\sigma }(1)}\otimes \cdots \otimes u_{\widehat{\sigma }(n)}$ be the Slater vector of Slater direct and exchange calculation. This example verifies concretely that:

(a) for every $u\in H$ the antisymmetrised two-particle tensor vanishes, $P_{-}^{(2)}(u\otimes u)=0,$ because the canonical transposition $\tau =(01)\in S_2$ of the two tensor factors satisfies $U_{\tau }(u\otimes u)=u\otimes u$ and $P_{-}^{(2)}=\frac{1}{2}(\mathrm{i}\mathrm{d}-U_{\tau })$;

(b) consequently the Slater vector with a repeated orbital is the zero vector, $S(u,u)=\frac{1}{\sqrt{2}}(u\otimes u-u\otimes u)=0(u\in H),$ and the norm-one conclusion of Slater direct and exchange calculation (a) does not apply to the degenerate list $(u,u)$: orthonormality of $u_1,\dots ,u_n$ is an essential hypothesis, not a convenience; for orthonormal $u_1,u_2$ one has $\mathrm{\parallel }S(u_1,u_2)\mathrm{\parallel }=1$;

(c) exclusion is a property of the state and not only of basis labels: for $H={\mathbb{C}}^2$ with orthonormal basis $e_1,e_2$ and the non-basis vector $u:=e_1+e_2$ the antisymmetric part still vanishes. Indeed $u\otimes u=e_1\otimes e_1+e_1\otimes e_2+e_2\otimes e_1+e_2\otimes e_2$, and $P_{-}^{(2)}$ annihilates the two diagonal tensors while exchanging the two crossed tensors with opposite sign, so $P_{-}^{(2)}(u\otimes u)=0$;

(d) in the actual tensor space $H^{\otimes 2}$, the two-particle antisymmetric sector $\mathrm{ran}{P}_{-}^{(2)}$ contains no nonzero vector of the form $P_{-}^{(2)}(u\otimes u)$: every repeated-orbital projection is zero, and zero belongs to this sector as it does to every linear subspace. Thus a nonvanishing antisymmetric two-particle vector cannot be built from two copies of the same one-particle vector. This sector is distinct from the symmetric two-particle sector of the bosonic $F(H)$ in Hilbert direct-sum construction.

The antisymmetric sector and its projection are mathematical objects; as recorded in the cited suppliers, selecting this sector for a species is a postulate and no spin-statistics statement is proved here.