The n-Category Café

[I had broken off where Andreas Döring began explaining how to get subobjects of the spectral presheaf $\Sigma$ from a projection operator. I’ll recall that last bit.]

Given any projection operator $$P = E[A \in \Delta] \in P(H)$$ on the Hilbert space $H$, we want to interpret it “in every context” $V \in V(H)$ [recall that a “context” here is a commutative subalgebra of $B(H)$].

This is accomplished by constructing for each projector $P$ a subobject $$\delta P$$ of the spectral presheaf. This subobject is the presheaf on $V(H)$ which sends any commutative subalgebra to the following subset of its Gelfand spectrum: $$\delta(P) : V \mapsto \{ \omega \in \Sigma(V) | \omega(\delta(P)_V) = 1 \} \,,$$ where $$\delta(P)_V \in P(V)$$ is that particular projector in $V$ with the property that its image contains that of $P$ and is the smallest such image.

This map $$\delta : P(H) \to \mathrm{Sub}(\Sigma)$$ is called the daseinization map.

Notice that not every subobject of $\Sigma$ comes from a projection operator this way.

Daseinization of self-adjoint operators

[Now Andreas Döring talked about how to turn self-adjoint operators into morphisms from the spectral presheav to some other presheaf “of values”. At this point my notes become a little shaky, since he was going too fast.. But luckily this is precisely the stuff explained by Squark to me and summarized here, so I’ll just recall that:

If we know how to do something for projectors, then we can do the same, using functional caculus, for any self-adjoint operator. This gives us for each self-adjoint operator $$a \in B(H)$$ and each commutative subalgebra $V$ an operator $$\delta(a)_V \,,$$ the daseinization of $a$.

Next, let $$R^{\geq}$$ be the presheaf on $V(H)$ which sends each commutative subalgebra $V$ to the set of decreasing functions on the collection of its subalgebras.

Then the daseinization construction $\delta(a)_V$ yields a morphism of presheaves $$\delta(a) : \Sigma \to R^{\geq}$$ which is such that it sends each element $f$ in the Gelfand spectrum of $V$ to the function $$B' \mapsto (f(\delta(a)_{B'}))$$ on subalgebras $B' \subset V$.]

Pure states as morphisms

In order to assign truth values to propositions, we have to represent states within the topos of presheaves on $V(H)$.

To each pure state $$|\psi\rangle \in H$$ we will assign a truth value in the internal logic of the topos.

In classical mechanics, in the topos $\mathrm{Set}$, a pure state $\psi$ [a point in phase space] is a morphism from subsets of phase space to the subobject classifier set $\{0,1\}$ $$T^\psi : \mathrm{Sub}P \to \{0,1\} \,.$$ Hence, now, we want to find a way to get from a state $|\psi\rangle \in H$ a morphism $$T^\psi : \mathrm{Sub}(\Sigma) \to \Gamma \Omega \,.$$

[Here $\Omega$ is the presheaf of sieves which I mentioned in Part II.

I missed the definition of $\Gamma \Omega$, apparently, but it should be the collection of global sections of $\Omega$, I assume, hence $\Gamma \Omega = \mathrm{Hom}(1,\Omega)$.

Oh, dear, now I realize that I cannot really make sense of my notes on how $T^\psi$ is defined! Maybe somebody reading this here who knows this stuff can help.

Anyway, this was the punchline, then. I’ll skip to the summary and stop there. Sorry.]

Summary

In the general topos-theoretic setup, propositions about physical systems correspond to subobjects $$S \in \mathrm{Sub}(\Sigma)$$ of the spectral presheaf.

Moreover, pure states give rise to morphisms $$T^\psi : \mathrm{Sub} \Sigma \to \Gamma \Omega$$ and physical quantities to morphisms $$\delta(a) : \Sigma \to R^{\geq} \,.$$

All this can in principle be done in any other topos.

[But notice, as I had remarked in part I, that not every morphism $T^\psi : \mathrm{Sub} \Sigma \to \Gamma \Omega$ corresponds to a pure state and not every morphism $\Sigma \to R^{\geq}$ corresponds to a physical quantity. Hence, it seems to me, it would in fact not be clear how to define a “physical quantity” in an arbitrary topos with chosen objects $\Sigma$ and $R^{\geq}$.

Experts on this stuff should please correct me, but my impression is that without such a characterization, we haven’t really reached the goal yet of extracting the topos-internal “arrow theory” of physical quantities, etc.

Maybe one problem is that the current setup is rather too general in its assumptions. I am wondering why none of the older ideas by Chris Isham on “quantizing on a category” seem to enter this approach so far. They certainly appear to be relevant.

In QFT of Charged n-Particle: Algebra of Observables I argue that within this framework of “quantizing on a category” one finds a rather nice arrow-theoretic characterization of “physical observables”.

I would very much enjoy seeing a connetion of such considerations with the topos theoretic work here. But at the moment I do not yet. ]