The n-Category Café

Spectral Presheaves

For fixed Hilbert space $H$, let $V(H)$ be the category whose

- objects are abelian subalgebras of bounded operators $B(H)$ on $H$

- morphisms are inclusions of abelian subalgebras.

This is the “context category”.

We will prominently consider the spectral presheav on $V(H)$. This is the presheaf $$\Sigma : V(H)^\mathrm{op} \to \mathrm{Set}$$ which assigns to each abelian subalgebra $V \in V(H)$ its Gelfand spectrum $$\Sigma(V) \,,$$ i.e. the set of all algebra homomorphisms from $V$ to the complex numbers, and to each inclusion $$i_{V' V} : V' \stackrel{\subset}{\to} V$$ the obvious restriction of the spectrum $$\Sigma(i_{V' V}) : \Sigma(V) \to \Sigma(V')$$ $$\omega \mapsto \omega|_{V'} \,.$$

This spectral presheaf here is the quantum analogue of the classical phase space $P$.

This now allows to reformulate the Kochen-Specker theorem as saying that

$\Sigma$ has no global sections.

This, in turn, is equivalent to saying that

There is no morphism $1 \to \Sigma$

of presheaves, where $1$ is the terminal object in the category of presheaves over $V(H)$, i.e. the functor which assigns to each abelian subalgebra the singleton set.

This says that the quantum phase space $\Sigma$ has no classical points.

[This reformulation of Kochen Specker is apparently due to Isham and Butterfield, if I understood correctly.]

Ordinary quantum logic

In 1936 Birkhoff and vonNeumann thought of formulating the “logic of quantum systems” in terms of the lattice $$L(H)$$ of closed subspaces of the given Hilbert space $H$.

This has severe interpretational problems in that for $\mathrm{dim}(G) \gt 1$ this logic is non-distributive, which means that in general the proposition

(eggs for breakfast) and (bacon or ham for breakfast)

is no longer equal to

(eggs and bacon for breakfast) or (bacon and ham for breakfast)

[You’ll recognize that this is where New Scientist got its headline from.]

In formulas

$$E \wedge (B \vee H) \neq (A \wedge B) \vee (E \wedge H) \,.$$

Since then, there have been many further developments in quantum logic – but none of them useful for physics.

[That’s indeed what he said. I am wondering if he is aware that it would be good to work out more explicitly why and how their approach is useful for physics.]

Also, a deductive system for quantum logic is missing.

Topos of presheaves

We now want to make use of the fact that the category of presheaves over $V(H)$ $$\mathrm{Set}^{V(H)^{\mathrm{op}}}$$ is a topos with an internal logic.

A topos is a category which behaves in many ways like the category of sets: it has all finite limits and colimits and it is cartesian closed. Moreover, most importantly, is has a subobject classifier.

[If indeed you don’t know yet what a topos is, the best thing to do is to quickly have a look at John Baez’s Topos Theory in a Nutshell.]

A subobject classifier is a special object $$\Omega$$ in the topos. In the topos of sets (the category of sets), this is the 2-element set $\{0,1\}$ or $\{\mathrm{false},\mathrm{true}\}$. Morphisms from any other set to this 2-element set are in bijection with subsets of the former.

Generally, a subobject classfier is an object $\Omega$ in the topos together with a monic morphism $$\mathrm{true} : 1 \hookrightarrow \Omega$$ from the terminal object 1 into it, such that for every monic $$m : S \hookrightarrow X$$ there is a unique $$\phi : X \to \Omega$$ such that $$\array{ S &\to& 1 \\ \downarrow^{m} && \downarrow^{\mathrm{true}} \\ X &\stackrel{\phi}{\to}& \Omega }$$ commutes.

Topos logic

The subobject classifier induces an internal logic which is of intuitionistic type.

[Next there is a small logical gap in my transcript. I have the suspicion that Andreas Döring forgot to say something he planned to say. In parts this was clarified in the question session afterwards. Namely:

First of all, in any topos we define the category $$\mathrm{Sub}(A)$$ of any object $A$ of the topos as $$\mathrm{Sub}(A) := \matrm{Hom}(A,\Omega) \,.$$ The subobject $\Omega$ in the topos of presheaves over $V(H)$ is the presheaf of sieves over $V(H)$. ]

Theorem: The collection $$\mathrm{Sub}(\Sigma)$$ of subobjects of the spectral presheaf is a Heyting algebra.

A Heyting algebra is a distributive lattice in which the law of the excluded middle need not hold. Rather we have $$E \vee \not E \leq 1$$ and $$\not \not E \geq E \,,$$

Projections and propositions

Spectral theory shows that to every proposition $A \in \Delta$, where $A$ is represented by some self-adjoint operator $A \in B(H)$, there is a projector $$E[A \in \Delta] \in P(H) \,.$$ This is the part of ordinary quantum logic which we keep.

In order to relate a projector $$P = E[A \in \Delta] \in P(H)$$ to all “contexts” $V \in V(H)$ [commutative subalgebras, recall], we form the “Daseineization” $$\delta(P)_V \in P(V) \,.$$

This defines a subset of the Gelfand spectrum $\Sigma(V)$ by $$\delta(P)_V \mapsto \{ \omega \in \Sigma(V) | \omega(\delta(P)_V) = 1 \} \,.$$

[Oops, I am being thrown out of this building. Rumour has it that the Max-Planck institute in Leipzig is the only such institute where one cannot stay overnight and work. Well, I should get some sleep anyway. I’ll continue this here later.]