March 3, 2007

A Topos Foundation for Theories of Physics

Posted by John Baez

Here’s a big new paper:

Isham has been working on topos theory and physics for some time now, but he recently teamed up with Andreas Döring, an expert on von Neumann algebras. Last year they got a grant from the Foundational Questions Institute to work on a topos-theoretic approach to quantum theory.

This paper explains what they’ve come up with so far. It consists of four parts:

• I. Formal Language for Physics.
• II. Daseinization and the Liberation of Quantum Theory.
• III. The Representation of Physical Quantities with Arrows.
• IV. Categories of Systems.

The first section begins with some reflections on the foundations of physics which everyone should read — it points out some really deep problems! On a lighter note, the first page has a word I’ve never seen. Does anyone know what it means?

Then things get a bit harder…

Posted at March 3, 2007 7:44 PM UTC

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Re: A Topos Foundation for Theories of Physics

Peristalithic does not seem to be in the OED. It could be a typo for peristalitic meaning having to do with peristalsis, the process by which food is moved through the digestive tract. Or, it could be a modification of peristalith, “a ring or row of standing stones surrounding a burial mound, cairn, etc.”.

Posted by: Aaron Bergman on March 3, 2007 9:15 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

If you type ‘peristalithic’ into Google, you get six links. Three refer to stones, and three refer to logic. Looking at the latter, I would guess that Isham is on a one-man quest to give ‘peristalithic’ a second, abstract meaning. It appears that he’s used it in at least two papers and one book.

What does he actually mean by it? I would guess something like ‘forever unanswerable’, in the way that the question of what the meaning of a bunch of stones laid around a grave from 3000 years ago is. But with only three data points, I’m not very confident.

Fun.

Posted by: James on March 4, 2007 1:12 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

Over at Cosmic Variance someone named Adam wrote:

Isham’s Group theory book is good (as is his QM book I mentioned earlier). His differential geometry book is good, but being at his lectures is even better.

Isham also deserves points for using the word ‘peristalithic’ in a most excellent way, in his QM book.

Does anyone here have Isham’s QM book? Does it explain his meaning of ‘peristalithic’, or just use it without definition?

Posted by: John Baez on March 4, 2007 1:27 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I’m beginning to think this word is pretty cromulent.

Posted by: John Armstrong on March 4, 2007 2:18 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I am impressed with the logodaedaly of John Armstrong and especially Chris Isham.

jim

Posted by: jim stasheff on March 6, 2007 1:33 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

Peristalith is the name of a stone circle, such as Stonehenge. Therefore, a ‘peristalithic debate’ is one that is (i) very old; and (ii) goes round in circles :-) I think that, as applied to the meaning of quantum theory, this is very appropriate :-)

Best regards
Chris

Posted by: Chris Isham on March 4, 2007 4:17 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

On part III, p.34, last paragraph, where it says

Of particular interest [are] tools for constructing theories that go beyond quantum theory and which do not use Hilbert spaces, path integrals, or any of the other familiar tools in which the continuum real and complex numbers play a fundamental role.

Few days ago I had a stimulating discussion with somebody working in general relativity.

He vividly recounted how he was trying to sensitise his students to the fact that in the history of physics, people have again and again taken various things to be obviously linear, only to later find that what looked obviously true was just the first approximation to a curved situation. The flatness of earth, the flatness of space, that of spacetime.

His conclusion was that assuming quantum mechanics to be evidently linear, no matter how closely we inspect quantum mechanical systems, is just one more fallacy in this series of linear fallacies.

Certainly he is not the first to speculate about non-linear QM, but his perspective was a rather noteworthy one, I found.

Since he also revealed himself as secretly being a “structuralist”, as far as theoretical physics is concerned, I offered him the following perspective:

Quantum theory is, apparently, the theory of representations of cobordism categories.

This perspective empowers us to deal with any potential nonlinearities in a robust way: we simply adapt the codomain of the representation and consider functors from cobordisms to categories that don’t necessarily have a forgetful functor to $\mathrm{Vect}$.

The point being, that this allows us to consistently determine what will and what will not change should we ever discover that the Hilbert space of states of a fundamental particle is, beyond some energy threshold, say, no longer conceivable as a linear space over the complex numbers.

For instance consider the body of work on correlators in topological and rational conformal 2-dimensional field theories using state sum models involving Frobenius algebras internal to certain monoidal categories.

Due to the way these are formulated internally to some context, the bulk of this work remains rather unaffected by the precise nature of this context, provided some collection of assumptions is satisfied.

As a result, there is a pretty much entirely diagrammatic way to conceive all of 2-dimensional topological and rational conformal theory, and the only point where an assumption on the linear nature of the context is used is when you want to translate such a diagrammatic correlator into an actual number.

Should there ever be evidence that, say, the quantum mechanics of the fundamental string needs to be conceived as being non-linear in some sense, we would make the necessary adjustment only at this point where we realize our diagrams by internalizing them appropriately. Most of the desireable structural aspects the quantum theory (as the sewing constraints) would carry over undisturbed, as it should.

So far there is no indication that we need to do this, so nobody has seriously looked into it.

But similar remarks actually apply also to the domain of our representation-theoretic problem: maybe we want to tamper with the nature of the category of cobordisms. This is not a linear category. Which is the reason why it can be relatively hard to deal with.

But maybe we want to reverse the above reasoning, then, and intentionally approximate something by a linear model which we know is nonlinear.

Again, this can be done by internalization, without affecting the basic tenets of what quantum theory is:

a good example is maybe what is called “topological conformal field theory”. There the category of conformal cobordisms (the domain of our representation functor) is replaced by the category which has the same objects, but where the Hom-spaces (the moduli spaces of conformal cobordisms with given boundaries) are replaced by the their homology complexes! This way the domain becomes a category enriched in complexes, $V$, and hence linear in this sense.

The entire resulting “topological conformal” quantum theory now is entirely about $V$-enriched representation theory. Remarkably, it turns out that this linear approximation still knows a lot about the true non-linear physics that we are really interested in.

In general, I believe that this is one of the powers of fully employing category-theoretic reasoning in the context of physics: it allows us to disentangle physical structures (arrow theory) from physical implementations (internalization).

Therefore it should befit us to think closely about what the arrow theory behind our physical theories really is, in particular behind quantum theory.

Posted by: urs on March 3, 2007 9:25 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

The quantum logic / quantum complexity guys like Scott Aaronson believes that certain types of quantum mechanics is ruled out by the principle that the physical world would not allow building a machine for efficiently solving all NP problems.

Specifically, solving all NP problems efficiently would mean that proving a theorem is no harder than verifying a theorem, and that would be a hard pill to swallow, on philosophical, sociological and all other -ical levels.

The specific mechanism which is disallowed is any ability to do “post-selection”, which would allow too great a control over quantum interference effects. Apparently, a very large class of non-linear possibilities are eliminated; I hesitate to say all, because I don’t understand the theorem fully.

Maybe this really is the one time that the obvious thing is right?

Posted by: genneth on April 18, 2007 10:43 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

The quantum logic / quantum complexity guys like Scott Aaronson believes that certain types of quantum mechanics is ruled out by the principle that the physical world would not allow building a machine for efficiently solving all NP problems.

I found this an intriguing idea, ever since I learned about it in TWF 235. The way Scott Aaronson applies complexity-theoretic reasoning to apparently completely unrealted contexts (like quantum mechanics, or like the anthropic principle) is certainly inspiring. His “NP-principle” has the flavor of some ideas of ‘tHooft, like the “holographic principle”: simple, elementary every-day desciption of a principle with implications for highly sophisticated theories.

Apparently, a very large class of non-linear possibilities are eliminated; I hesitate to say all, because I don’t understand the theorem fully.

Yes. I’d dare say that the proposals for non-linear generalization of quantum mechanics which are available exhaust only a tiny fraction of all possibilities (even and in particular if we concentrate on natural possibilities) that become available once we really take internalization seriously.

But if you don’t trust me on that, I won’t right now try to convince you..

Posted by: urs on April 18, 2007 12:26 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

In search for a paved way into this body of ideas I was scanning the list of references, looking for Chris Isham’s older work on “quantization on a category” # which expresses an idea that I am comfortable with. Couldn’t find such a reference. I am wondering what the reason for that is.

Posted by: urs on March 3, 2007 10:36 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I don’t understanding what you’re looking for and not finding, Urs. “Older work which expresses an idea you are comfortable with”? It sounds like you’re getting tired of radical new ideas.

There’s also this related paper:

Posted by: John Baez on March 4, 2007 4:39 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I don’t understanding what you’re looking for and not finding,

Oh, I must have expressed myself badly. I was looking for paragraphs in “A topos foundation for theories of physics” that made contact with this stuff about “quantising on a category”.

I understand the idea of “quantising on a category” – so, I thought, if I saw how the new stuff is a generalization of this, I’d have an easier time absorbing it.

Posted by: urs on March 4, 2007 12:40 PM | Permalink | Reply to this

What is the main idea?

We would certainly benefit from somebody going through the effort of providing us with a summary of a summary of Döring-Isham’s work.

I spent about an hour looking at the four parts in order to get a glimpse of the main ideas.

Certainly, in that one hour I must have missed many, many important points. My impression was, that the main idea revolved around a way to explain the difference between the concept of classical and quantum observables by internalization in different topoi.

The authors consider an object $\Sigma$ in the topos, which is to be thought of as the object of states. In the classical case this is a symplectic space, in the quantum case it is an algebra of operators (or rather the respective sheaves represented by these).

Then they consider an object $\mathcal{R}$ in which observables take their values. In the classical case this is just the real numbers. I forget what it is in the quantum case.

Then, a physical observable is modeled by a morphism $A : \Sigma \to \mathcal{R}$ in the topos. This is now something that models the assignment of values to states.

In the classical setup this is literally nothing but a real-valued function on the phase space of the system.

My impression was that much of the work (especially part II and III) revolves around understanding how quantum observables can be understood as morphisms $A : \Sigma \to \mathcal{R}$ too, when regarded in a suitable topos.

Posted by: urs on March 4, 2007 1:48 PM | Permalink | Reply to this

Re: What is the main idea?

I second Urs’ question, although I would like to rephrase is like this: Why should topos theory work?

We know that the peristalithic problems revolve around infinities in QFT. Is there any reason to expect topos theory to remove these? I’m not asking for fancy math here, but for a simple physical argument why topos theory should improve the situation.

The same question can, and should, of course be asked of any putative theory of quantum gravity.

Re: What is the main idea?

I second Urs’ question, although I would like to rephrase it like this: Why should topos theory work?

First, I don’t think that this is a rephrasing of my comment above.

Then, generally: Topos theory works for the same reason that the rest of mathematics works.

But I guess you meant to ask why topos theoretic methods should be of any use in quantum theory.

(Like: what are the predictions of Döring-Isham theory, and are they predictions of type 1, 2 or 3? ;-)

This is a justified question, I assume. By reading the introductions and conclusions to the various parts of Döring-Isham, one can at least see what the authors hope topos theory would be helpful for in quantum theory: they expect it to be helpful in understanding quantum cosmology.

If they actually demonstrate this usefulness at this point I cannot quite tell yet.

Generally, I am sympathetic to attempts to distinguish structure from implementation by doing arrow theory and internalization (whether internalization in a topos or some other category is maybe not even the primary issue).

In my first comment above I have tried to indicate why this can be and is in certain cases concretely useful in quantum theory. Really, practically concretely useful, as in: it could hardly have been done without this.

Well, as maybe somebody has noticed, most of my latest postings revolve around something like this.

Posted by: urs on March 5, 2007 8:50 AM | Permalink | Reply to this

Re: What is the main idea?

Generally, I am sympathetic to attempts to distinguish structure from implementation by doing arrow theory and internalization (whether internalization in a topos or some other category is maybe not even the primary issue).

The risk with this approach is that you abstract away the key problems. The big picture for QFT is simple enough - apply quantum mechanics to fields. The devil lies in the detailed implementation of this idea in a Hilbert space. We must renormalize because of divergences, and maybe lose the Hilbert space structure in the process. It seems to me that too abstract an approach, with arrows or topoi or whatever, risks losing track of this fundamental difficulty.

In my first comment above I have tried to indicate why this can be and is in certain cases concretely useful in quantum theory. Really, practically concretely useful, as in: it could hardly have been done without this.

Hm. You talk about correlators in CFT, but rarely mention central charge and essentially never minimal models. It seems that you may have abstracted away the physically important stuff.

Anyway, you don’t need to understand “what the arrow theory behind CFT is” to extract its most important physical lesson: that locality is compatible with background independence in one complex dimension.

arrow-theory and implementation in QFT

You talk about correlators in CFT, but rarely mention central charge

Yes! And this is precisely one of the powers of the decomposition into arrow-theory and implementation:

Roughly, this decomposition allows to decompose full 2-dimensional quantum field theory into a general topological part, given by some arrow-theory, and an analytic part, given by the concrete implementation.

To a good extent, conformal 2d QFT is topological 2d QFT internalized not in the category of ordinary vector spaces, but in the category of representations of the corresponding chiral vertex algebra. It is this codomain catageory that knows all about the complex analytic aspects.

Of course I could talk more about that implementation part of the story. This is concerned with the representations of the algebra of observables of the theory. My last series of postings is headed in that direction. More to appear in a moment.

Posted by: urs on March 6, 2007 10:44 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

From a Numb3rs rerun last Friday:
a topos theory for analyzing Bach’s sonatas
or words to that effect

and better

(personal) relationships as described in a graded tensor category

jim

Posted by: jim stasheff on March 4, 2007 3:09 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

Very interesting. I find it intriguing that in the description of quantum
mechanics using a presheaf topos, the restriction maps of a presheaf move
in the direction of less information; whereas in the intuitionist
perspective on topos theory, the restriction maps generally move in the
open set, you know more about where a hypothetical point may be located).

There seems to be at least one thing conspicuously missing, though:
where do the probabilistic predictions of experiments come in? The
daseinisation of a self-adjoint operator, as defined in section 3 of
paper III, seems only to capture the minimum and maximum values which
that observable might take on. But what about the actual probability
distribution of results? Am I missing something?

Mike

Posted by: Mike Shulman on March 5, 2007 4:08 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I’d need to keep reading in order to better grasp the ideas. But maybe there is something here:

disregarding quantum mechanics for a moment and just considering the underlying issue of passing from a symplectic space, first to the algebra of functions over it and then deforming that, using the symplectic form, to the non-commutative Weyl algebra of functions – it seems that Döring-Isham ponder a way to retain the concept of “value of a function” under this transition by passing to a richer topos.

I remember back when I first heard about topos theory, I had some discussion with somebody that vaguely revolved around the idea of understanding noncommutative geometry as geometry in a suitably exotic topos. I was advised not to try to think of it this way. But maybe that’s not so far from what Döring-Isham are considering?

Posted by: urs on March 5, 2007 4:40 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

Urs wrote:

I remember back when I first heard about topos theory, I had some discussion with somebody that vaguely revolved around the idea of understanding noncommutative geometry as geometry in a suitably exotic topos. I was advised not to try to think of it this way. But maybe that’s not so far from what Döring-Isham are considering?

Maybe you’re right. Unfortunately I’ve been way too busy to read their paper. But, Isham’s earlier work on topos theory and physics can indeed be thought of as studying noncommutative geometry as you describe. Maybe this will provide the link you’re seeking:

The Kochen–Specker Theorem is a no-go theorem which says that if you have a Hilbert space of dimension ≥ 3, it’s impossible to assign a real value $v(A)$ to each bounded self-adjoint operator $A$ such that $v(A+B) = v(A)+v(B)$ and $v(A B) = v(A) v(B)$ for all commuting pairs $A$,$B$.

As Butterfield and Isham remark, this theorem means that “to the distress of angst-ridden students, standard quantum theory precludes any such naive realist interpretation of the relation between formalism and the physical world”.

But, they get around this theorem — in a certain sense — by working in a suitable topos!

You can see the relation to noncommutative geometry. Noncommutative geometry is a world where we have an algebra of ‘functions’ that aren’t really functions on any space, because there’s no consistent way to assign them ‘values at a point’ satisfying the above rules.

I’m not sure if anyone has seriously tackled noncommutative geometry using topoi, but Butterfield and Isham’s papers can be viewed as a start.

I can’t resist two digressions.

First, note the strange clause “≥ 3” in the Kochen–Specker theorem. The theorem doesn’t hold for a 2-dimensional Hilbert space! This loophole lets you play some fun tricks with spin-1/2 particles (nowadays known as ‘qubits’).

Second, when I was an undergrad at I asked Simon Kochen to supervise my senior thesis. I wanted to work on quantum theory and logic. He asked “Do you know the spectral theorem?” I said “no”. He said, “Well, then, I’m afraid I’m not really interested…”

This pissed me off so much that I spent the summer before my senior year reading Reed and Simon’s Methods of Modern Mathematical Physics I: Functional Analysis, learning the spectral theorem. I fell in love with functional analysis, took a grad course on it from Elliot Lieb in my senior year, and went on to work with Irving Segal in grad school.

I only realized just now that Kochen may be to blame for my interest in analysis.

Posted by: John Baez on March 6, 2007 4:54 PM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

Isham’s first earlier work on topos theory and physics

When yesterday I tried to seriously read Döring-Isham (and failed to be able to fully parse, at a technical rather than heuristic level, equation (2.1) for daseinization), I realized that a necessary prerequisite for a good understanding really seems to be familiarity with “[13,14,15,16]”: Isham’s work with Butterfield that you mention.

Posted by: urs on March 7, 2007 12:14 AM | Permalink | Reply to this

Re: A Topos Foundation for Theories of Physics

I’m not sure if anyone has seriously tackled noncommutative geometry using topoi

What do you think of Cartier’s reconciliation? See page 14 of this. In the diagram there is a double arrow between a topos of $G$-sets, for some groupoid $G$, and some C*-algebra associated to $G$. The name ‘Tapia’ is written under the double arrow.

A comparison between the points of view of Grothendieck and Connes has barely been sketched out

Then we are referred to:

J. Tapia, Quelques spectres en K-théorie topologique des algèbres de Fréchet et applications à l’algèbres des fonctions de classe $C^{\infty}$ sur une variété, preprint IHES/M/91/37.

Posted by: David Corfield on March 7, 2007 3:31 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field
Weblog: The n-Category Café
Excerpt: The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
Tracked: March 5, 2007 10:18 PM

Peristalithic

That reminds me of C.F.v.Weizsäcker´s “Kreisgang” (sort of hermeneutic circle) philosophy of Physics (Aufbau der Physik 1985, engl. The Structure of Physics 2006).

I never seen him mentioned around here (or e.g. his last student Holger Lyre). Any comment on his work?

Posted by: Florifulgurator on March 8, 2007 2:43 PM | Permalink | Reply to this
Read the post Recent Developments in QFT in Leipzig
Weblog: The n-Category Café
Excerpt: A conference on new developments in Quantum Field Theory at the Max Planck Institute in Leipzig.
Tracked: March 21, 2007 1:07 PM
Read the post Topos Theory in the New Scientist
Weblog: The n-Category Café
Excerpt: Robert Matthews on Chris Isham's work on topos theory and physics.
Tracked: April 14, 2007 3:13 AM
Read the post Physical Systems as Topoi, Part I
Weblog: The n-Category Café
Excerpt: Doering on his work with Isham on topos theoretic models of quantum theory and general theories of physics.
Tracked: July 22, 2007 8:44 PM

Re: A Topos Foundation for Theories of Physics

I just started reading A topos foundation for theories of physics. I was pleased to see intuitionist logic showing up, at least partly because I’m a constructivist myself. In particular, I remember reading a thesis long ago in which someone (I forget who) analyzed the potential of using constructive math instead of classical for the conventional formulation of quantum mechanics. The thesis found situations in which the time-evolution of the wave function was not constructive; but I noticed that these cases happened to be covered by the uncertainty principle, so in once sense it didn’t matter.

Posted by: Hendrik Boom on September 6, 2007 7:33 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 A Topos for Algebraic Quantum Theory
Weblog: The n-Category Café
Excerpt: A summary of some key points of Chris Heunen's and Bas Spitter's article.
Tracked: December 7, 2007 10:04 AM

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