# Orthocomplement space of $a^\dagger a$?

In this note$$^1$$ page 4 the author states that suppose $$a_i$$ acts on the whole vector space $$V$$ where $$a_i$$ is the annihilation operator of fermions. And since $$[a_i^\dagger a_i,a^\dagger_j a_j]=0,\forall i,j$$, and $$a_i^\dagger a_i$$ are hermitian operators, so we can construct a basis for them as $$\left( {a_1}^{\dagger} \right) ^{\alpha _1}...\left( a_{n}^{\dagger} \right) ^{\alpha _n}|vac\rangle$$ where $$\alpha _i=0$$ or $$1$$. Then the author states that the constructed vector space $$W$$ is a subspace of $$V$$, but I wonder that hermitian operators commuting implies that we have a common complete orthogonal basis, so how can $$W$$ be only a subspace of $$V$$?

1. "The Fermionic canonical commutation relations and the Jordan-Wigner transform" By Michael A. Nielsen, 2005

• So the question is if $W$ is a proper subspace of $V$? Commented Jul 16, 2022 at 14:05
• @Qmechanic I've added more details to the link which I think is enough. It seems that $W$ is a proper subspace of $V$ from what the author wants to convey in the note and while I don't think so, so this is my question. Commented Jul 16, 2022 at 14:24

I think you are assuming that $$\mathrm{dim}(V)=2^n$$, which is not necessarily the case. As a somewhat trivial example, imagine you have an $$8$$-dimensional space $$V$$ spanned by orthogonal vectors $$|\alpha_1,\alpha_2,\alpha_3\rangle$$ with $$\alpha_i \in \{0,1\}$$, and consider the fermionic raising/lowering operators $$a_1$$ and $$a_2$$ where e.g. $$a_1|1,\alpha_2,\alpha_3\rangle = |0,\alpha_2,\alpha_3\rangle \qquad a_2|\alpha_1,1,\alpha_3\rangle = |\alpha_1,0,\alpha_3\rangle$$
By applying $$a_1^\dagger$$ and $$a_2^\dagger$$ to the vacuum state $$|0,0,0\rangle$$, we generate the 4-dimensional subspace $$W$$ which is spanned by $$|\alpha_1,\alpha_2,0\rangle$$. The orthocomplement $$W^\perp$$ is the subspace of $$V$$ which is spanned by $$|\alpha_1,\alpha_2,1\rangle$$. As a result, we may identify $$V\simeq \mathbb C^4 \otimes \mathbb C^2$$ (i.e. $$|\alpha_1,\alpha_2,\alpha_3\rangle \simeq |\alpha_1,\alpha_2\rangle \otimes |\alpha_3\rangle$$) where the set of operators $$\{a_1,a_2\}$$ and their adjoints act non-trivially only on $$\mathbb C^4$$.