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July 23, 2007

Physical Systems as Topoi, Part III

Posted by Urs Schreiber

Yesterday night I had to interrupt my transcript of Andreas Döring’s talk. Here is the continuation.

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

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

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

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

Notice that not every subobject of Σ\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 aB(H) a \in B(H) and each commutative subalgebra VV an operator δ(a) V, \delta(a)_V \,, the daseinization of aa.

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

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

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

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

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

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

Oh, dear, now I realize that I cannot really make sense of my notes on how T ψT^\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 SSub(Σ) S \in \mathrm{Sub}(\Sigma) of the spectral presheaf.

Moreover, pure states give rise to morphisms T ψ:SubΣΓΩ T^\psi : \mathrm{Sub} \Sigma \to \Gamma \Omega and physical quantities to morphisms δ(a):ΣR . \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 ψ:SubΣΓΩ T^\psi : \mathrm{Sub} \Sigma \to \Gamma \Omega corresponds to a pure state and not every morphism ΣR \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 Σ\Sigma and R 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. ]

Posted at July 23, 2007 7:18 PM UTC

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4 Comments & 2 Trackbacks

Re: Physical Systems as Topoi, Part III

urs wrote:

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

Yeah, in Döring and Isham’s articles, ΓA\Gamma A (where AA is anything) is a default notation for a collection of global sections of AA, i.e. arrows 1–>AA. See e.g. Part II, pp. 3, 9, 21, 22 (the top line).
Posted by: mo on July 24, 2007 5:38 AM | Permalink | Reply to this

Re: Physical Systems as Topoi, Part III

Yeah, in Döring and Isham’s articles, ΓA\Gamma A (where AA is anything) is a default notation for a collection of global sections of AA

Okay, thanks.

By the way, this is one point at which I can make that little complaint I have voiced more concrete:

If we want to really think abstractly of physical systems as being given by certain diagrams internal to any topos whatsoever, we should not presuppose that the topos in question is realized as a category of presheaves.

But only in this case is ΓΩ\Gamma \Omega sensible notation.

Of course, in a way this is an incredibly minor point, and I am sure I am coming across as an odd nitpicker here. But I do think this might be worth thinking about:

as far as I am aware, the goal of the entire eterprise here should be to be able to say:

“A physical systems is a collection of objects with a collection of morphisms making certain diagrams (nn-)commute. The various objects and morphisms have this and that physical interpretation. And we make all this precise by interpreting all these diagrams in a certain chosen topos.”

For that to work out, we need to really identify the crucial properties that these objects are supposed to have, i.e. in which diagrams they are really supposed to live.

In order to find that, I believe it is crucial to free oneself from using any properties of any fixed topos. Because, clearly, this program is by construction such that everything concerning these objects and diagrams by themselves must be independent of the particular topos chosen.

The question really is:

what is a walking phase space?

what is a walking value space?

what is a walking pure state?

etc. All these should be categories such that a physical system is a functor (possibly one respecting some extra structure flying around, like closedness or the like) from these things into the chosen topos.

Spelled this way, Hom(1,Ω)\mathrm{Hom}(1,\Omega) may be a valid list of symbols to use. But ΓΩ\Gamma \Omega rather not so.

If you see what I mean.

Posted by: Urs Schreiber on July 24, 2007 6:47 PM | Permalink | Reply to this

Re: Physical Systems as Topoi, Part III

Okay, so this might be a silly train of thought but since they talk about building up using topoi, intuitionist logics, and typed languages, that at some Curry-Howard-ish level our models of computation and our physical theories are the same thing?

I’m still trying to digest the first of these papers, but I noticed that didn’t really seem to be a direction you were taking things and wanted to ask.

Posted by: Creighton Hogg on July 31, 2007 3:15 AM | Permalink | Reply to this

Re: Physical Systems as Topoi, Part III

that at some Curry-Howard-ish level our models of computation and our physical theories are the same thing?

Unfortunately, I am not very versed in these issues involving various flavors of logic and the like. I do of course realize that this is part of the main motivation behind Döring-Isham’s work, so I am probably missing the thrill and the most fun part.

but I noticed that didn’t really seem to be a direction you were taking things and wanted to ask.

Yes, in a way. I’d be happy to see quantum logic be nicely put on the same footing with different logics for other “kinds” of physics (like classical physics and, if I understand them correctly, other “kinds” of physics still undreamed of, coresponding to internalization in other topoi), but I would expect to see this as an offspin of a useful “internalization” of the physics.

So, I believe, as I already tried to indicate, it would be useful to be able to say that a “physical system” (whichever aspects one chooses to concentrate on) is such and such an abstract diagram.

Then, as we read this diagram internal to the topoi of sets, we should find it encodes classical mechanics governed by classical logic.

As we read the diagram internal to, maybe, presheaves over the category of abelian subalgebras of bounded operators, we should find it encodes quantum mechanics governed by quantum logic.

And so on.

Actually, the most interesting “and so on” which I expect to see here is “second quantum theory”, “third quantum theory” and so on:

if we can move from classical mechanics to quantum mechanics by a change of topos, I would be longing to know if this is the first step in a Story of NNth quantization.

Actually, I have a slightly more concrete picture in mind. It should be like this:

the classical actions of nn-particles (points, strings, memebranes, etc.) coupled to nn-bundles with connection (electromagnetic field, Kalb-Ramond field, 3-form field, etc) are nn-functors from nn-paths to somewhere.

As we quantize them once, we obtain the extended nn-dimensional worldvolume QFT: again an nn-functor.

I think there should actually be an nn-functor from transport nn-functors to the quantized nn-extended QFT nn-functors.

And this construction should be iteratable. In the next step we’d get the second quantized theory of the charged nn-particle (field theory, string field theory, etc.).

At least in rough outline, this seems to make good sense to me, as I have said in my remark on second quantization.

By the way, at the beginning of the existence of this Café, John was making us all think about Doctrines and how these allow us to think of quantum mechanics.

Posted by: Urs Schreiber on July 31, 2007 5:33 PM | Permalink | Reply to this
Read the post The Principle of General Tovariance
Weblog: The n-Category Café
Excerpt: Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
Tracked: December 5, 2007 7:10 PM
Read the post The Concept of a Space of States, and the Space of States of the Charged n-Particle
Weblog: The n-Category Café
Excerpt: On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
Tracked: January 29, 2008 11:58 PM

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