$$R \rightarrow End(V)$$

where $V$ is a part of $R-$mod. Now, there is a spectral decomposition of this $V$ as 

$$ \mathcal{V} = V \otimes_R \mathcal{O}(U)$$

where $\mathcal{V}$ is a sheaf defined over the Spec $R$ and $U \in Spec R$. So given a commutative $R$ we have a module over $R$ which sheafifies over Spec $R$. See this paper by Serre for motivation and Lemma 7.1 in Stacks Project.

Now, we take a commutative algebra $A$ and the Spec $A$ is just the linearized version of

$$A \rightarrow \mathcal{O}(X)$$

where $X$ is a space, we choose functions over this space, which satisfies the algebra. (Well, most of the time Spectrum has the same meaning, see this post.) 

We know that in quantum mechanics, the observables belong to an algebra and the Hilbert space is where the algebra works. In quantum mechanics, phase space is changed by Hilbert Space $\mathcal{H}$. (In quantum mechanics, we are mostly interested in a *-algebra and hermitian operators.) Now any observable $\mathcal{O}$ acting on the Hilbert space $\mathcal{H}$, there is a spectral decomposition that we have defined at the beginning where $\mathcal{H}$ sheafifies over $\mathbb{R}$ because we considered a single operator and $Spec A = \mathbb{R}$ in that case. So the Hilbert space decomposes over $\mathbb{R}$. Now any state $\psi \in \mathcal{H}$ is defined as a section of this sheaf $\mathcal{H}$ over Spec $R$ and thus we have an eigenspace decomposition from this section of the state vectors. Support of this section gives us measurements.

Such kind of decomposition relies on the definition of quasi-coherent sheaves as these $\mathcal{V}$ sheaves are quasic-coherent sheaves. And this is very appropriate for the physics as well.

(This will be the last post in The Quill Series as I am suspending it indefinitely as I fail to commit with a decent frequency in this category as originally thought. I will continue to post but not with this category in mind.)