The Third Road to Quantum Gravity

[Kea]

Robinson Topos

Without worrying about what it is, if $\mathbf{E}$ is the Robinson topos then there is a functor

$$\mathbf{E} \rightarrow \mathbf{Set}$$

such that the image of the 'reals in $\mathbf{E}$' is the set of non-standard reals including the infinitesimals.

Ross Street says we should take non-standard analysis to be the study of the reals in $\mathbf{E}$ rather than the study of the more contrived 'reals plus infinitesimals' in the usual topos $\mathbf{Set}$.

 

[selfAdjoint]

Urs Schreiber's Dissertatiom

Kea have you seen the draft of Urs' http://www-stud.uni-essen.de/~sb0264/SchreiberDissPartI.pdf ? Loop space, categorification, gerbes and all. Urs has been discussing these topics here and there for some time, but this now is an impressive construction, and all in Urs's ultra-clear explanative style, too!

 

[Kea]

Kea have you seen the draft of Urs' dissertation?

Hi selfAdjoint

Yes, he mentioned it on the String Coffee Table, amongst other forums. Thanks for posting it. Part I, which is supposedly the outline, is 90 or so pages. John Baez will be talking about this subject at the Streetfest http://streetfest.maths.mq.edu.au/ in July.

Cheers
Kea

 

[john baez]

the real importance of the Large Hadron Collider

I have been thinking lately about where the search for Quantum Gravity may be headed in the near future and it has struck me that the LHC is going to be a major pivot point in the future of research- if they are able to observe micro black-holes with the LHC- it seems to me that such a tremendous achievement will capture the imaginations of everyone [...]

Sure. But, they won't find tiny black holes at the LHC.

The only reason anybody talks about this possibility is that some string theorists made up a far-out theory where quantum gravity effects could show up at a more or less arbitrary energy scale, and then - for publicity reasons - picked an energy scale slightly bigger than what anyone has been able to study so far, to get people excited about discovering tiny black holes at the LHC. It might happen... but it won't.

The real importance of the LHC is that string theorists have been saying for years that supersymmetry is right around the corner, detectable at an energy scale slightly bigger than what anyone has been able to study so far. (Notice a pattern?) And, lots of them claim that some effects of supersymmetry will be detected at the LHC. They might be... but we'll see.

If they're not, government funding for string theory will drop, so interest in other approaches to quantum gravity will increase.

If they are, string theory will have some actual data to chew on, and progress should accelerate.

Oh, and the Higgs. If that works as expected, the Standard Model will be confirmed - great, but ho hum. If not, things will get really exciting.

 

[john baez]

Robinson topos

Without worrying about what it is, if $\mathbf{E}$ is the Robinson topos then there is a functor

$$\mathbf{E} \rightarrow \mathbf{Set}$$

such that the image of the 'reals in $\mathbf{E}$' is the set of non-standard reals including the infinitesimals.

Ross Street says we should take non-standard analysis to be the study of the reals in $\mathbf{E}$ rather than the study of the more contrived 'reals plus infinitesimals' in the usual topos $\mathbf{Set}$.

Where can one read about the Robinson topos?

The book Synthetic Differential Geometry talks about a number of topoi with infinitesimals, following Lawvere's ideas on differential geometry (which I sketched in week200 ). But, I haven't looked at this for a long time, so I don't know exactly which topoi they consider.

Personally I don't think any of these different approaches to calculus are sufficiently different to be worth worrying about, unless one is fascinated in them for their own sake. Things would be very different if one could make real progress on some hard math or physics problems in one of these alternative approaches. People have tried - I've read papers about quantum field theory that use infinitesimals - but nothing much has some of this so far.

 

[Kea]

Where can one read about the Robinson topos?

Good question. I don't know. I heard about it from Ross Street. Apparently Princeton University Press have reissued Robinson's original 1960's book Non-Standard Analysis but I haven't yet seen it.

If one plays around with Google one can find licorice allsorts, such as
http://arxiv.org/PS_cache/quant-ph/pdf/0303/0303089.pdf
with an interesting list of references...but there are only so many hours in the day! I find myself wandering over to the Philosophy library sometimes for classic papers by topos theorists.

Kea

 

[john baez]

Robinson topos

Good question. I don't know. I heard about it from Ross Street. Apparently Princeton University Press have reissued Robinson's original 1960's book Non-Standard Analysis but I haven't yet seen it.

I'm pretty sure Robinson didn't think he was inventing a topos; I think he was working solidly within a more old-fashioned tradition in logic.

There's a similar example in Sheaves, Geometry and Logic by Mac Lane and Moerdijk. Cohen proved the independence of the axiom of choice using a technique called "forcing" to create nonstandard models of the Zermelo-Fraenkel axioms. Moerdijk and Mac Lane simplify the idea behind this by constructing a topos like the usual topos of sets, but in which the axiom of choice fails. I've never understood quite how close their construction comes to Cohen's original result - I don't think it instantly implies his result.

Maybe Street is just smart enough to realize that Robinson, like Cohen, was also subconsciously creating a new topos.

But, we're drifting from physics here. And, as you point out,

...there are only so many hours in the day!

 

[marcus]

Good question. I don't know. I heard about it from Ross Street. Apparently Princeton University Press have reissued Robinson's original 1960's book Non-Standard Analysis but I haven't yet seen it.
...

I actually bought Robinson's book in the (notorious) late 1960s and probably still have it in one of the boxes up in the attic----unless it has been donated to some library.

At least my impression of it was that it is grounded in old fashioned logic and focused on old fashioned analysis: calculus, the Reals... In line with what JB says. I read (in) it with a mixture of hope and disappointment, but didnt see how to take it anywhere.

It is interesting that Princeton UP has reissued it and that Ross Street has been talking about it.

Anais Nin has a wonderful passage that starts "Nothing is lost, but it changes..." almost a poem.

 

[Kea]

Maybe Street is just smart enough to realize that Robinson, like Cohen, was also subconsciously creating a new topos.

It's possible that the detailed definition is due to Ross Street, perhaps unpublished. He went into the details: let me reproduce a little...

Let Ev be the topos of evolving sets, that is the functor category $$[ N , \mathbf{Set} ]$$ from the ordinals into the topos Set. This is the topos that Markopoulou studied as a Newtonian causal set theory. The terminal object is the set sequence of one point sets.

Now one needs the notion of an ultrafilter $\nabla$. Firstly, a (proper) filter on a Heyting algebra is a collection of subobjects ( for sets see http://mathworld.wolfram.com/Filter.html ) such that

1. $1 \in \nabla$ and 0 is not in $\nabla$

2. $x , y \in \nabla \Rightarrow x \wedge y \in \nabla$

3. $x \leq y , x \in \nabla \Rightarrow y \in \nabla$

One of the characterisations of an ultrafilter is that the quotient of the Heyting algebra by $\nabla$ is isomorphic to 2, as a lattice. It turns out that for an ultrafilter, and a topos E, then $\mathbf{E} \backslash \nabla$ is a 2-valued topos (the terminal has 2 subobjects).

What is $\mathbf{E} \backslash \nabla$ ? It has the same objects as E. The hom set (A,B) is a colimit (over $U \in \nabla$) of $(A \times U, B)$.

This idea is used on Ev to define the Robinson topos. $\nabla$ is something called a non-principal ultrafilter on N (doesn't have a least element). The Robinson topos is an example of an elementary topos that is not a Grothendieck topos.

Kea

 

[Chronos]

A recent paper that may be of some interest:

Fibered Manifolds, Natural Bundles, Structured Sets, G-Sets and all that: The Hole Story from Space Time to Elementary Particles
J. Stachel, M. Iftime
http://xxx.sf.nchc.gov.tw/abs/gr-qc/0505138

They don't get into any heavy category theory. Stachel is a well known Einstein historian. From the conclusion:

"Therefore, the following principle of generalized covariance should be a requirement on any fundamental theory: the theory should be invariant under all permutations of the basic elements out of which the theory is constructed.

Perturbative string theory fails this test..."

If it's any consolation [or perhaps a curse] I am in complete agreement.

 

[selfAdjoint]

A new third way paper

This paper on http://www.arxiv.org/PS_cache/gr-qc/pdf/0509/0509089.pdf just appeared on the arxiv. After a lot of entertaining generalities on iconoclasm in science, he gets down to an account of Absolute Differential Calculus (ADC) a sheaf-based theory and its approach to QG. A tangy Greek salad; enjoy!

 

[Kea]

This paper on http://www.arxiv.org/PS_cache/gr-qc/pdf/0509/0509089.pdf just appeared on the arxiv.

Thanks, selfAdjoint. I see it's by I. Raptis. He's one of few people who seem to have been looking at toposes in physics for quite a while now. One of the Isham school, I think.

Must finish pulling the splinters out before I read it...

Right. That's done. Mmmm. Raptis is rather enthusiastic about the Mallios approach. He mentions its possible connection to Category Theory approaches but he doesn't seem to have learned much about categories yet. For instance, he talks about algebra replacing geometry when all category theorists know that categories do both.

Of course, I enjoyed reading it. Anyone that throws Hegel, Prometheus and Einstein into the same blender is probably a buddy of mine.

 

[Kea]

...and the correct form of the proverb is

nothing venture, nothing gain

 

[selfAdjoint]

...and the correct form of the proverb is

nothing venture, nothing gain

Probably just a typo, but it's Nothing ventured, nothing gained.

 

[Kea]

...it's Nothing ventured, nothing gained.

No! It's not. It's a 14th century English proverb, originally from the French. It states Nothing venture, nothing gain.

http://www.worldofquotes.com/proverb/French/18/

I learned this when Edmund Hillary's autobiography came out a couple of years ago.

 

[Kea]

new gerbes paper

Topological Quantum Field Theory on Non-Abelian Gerbes
Jussi Kalkkinen
36 pages
http://www.arxiv.org/abs/hep-th/0510069

and also:

Gerbes, Quantum Mechanics and Gravity
J.M. Isidro
10 pages
http://www.arxiv.org/abs/hep-th/0510075

 

[setAI]

Hot off the presses!

Geometry from quantum particles

From: David Kribs
Date: Tue, 11 Oct 2005 02:18:17 GMT (19kb)

We investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry. We provide a way for global spacetime symmetries to emerge from a background independent theory without geometry. In this, we use a quantum information theoretic formulation of quantum gravity and the method of noiseless subsystems in quantum error correction. This is also a method that can extract particles from a quantum geometric theory such as a spin foam model.

http://www.arxiv.org/abs/gr-qc/0510052

 

[marcus]

Geometry from quantum particles

http://www.arxiv.org/abs/gr-qc/0510052

This paper by Fotini Markopoulou and David Kribs was one of those listed by Smolin today, in a post mentioning some highlights from the Loops '05 conference so far.

I copied Smolin's list of QG advances here:
https://www.physicsforums.com/showthread.php?p=784856#post784856

The Kribs/Markopoulou paper is #4 in a list of 7 that he highlighted. (And still two days more to go in the conference!)

To see the Smolin's post that I exerpted in context, scroll down to comment #5 here:
http://www.math.columbia.edu/~woit/wordpress/?p=279#comments

 

[Kea]

also today ...

Spacetime topology from the tomographic histories approach II: Relativistic Case
I. Raptis, P. Wallden, R. R. Zapatrin
21 pages
http://www.arxiv.org/abs/gr-qc/0510053

One might think this subject is becoming a bit more popular!

 

[Kea]

today's catch

Topological entanglement entropy
Alexei Kitaev, John Preskill
4 pages
http://www.arxiv.org/abs/hep-th/0510092

 

[Kea]

Peirce's Existential Graphs

Over a cup of coffee recently, mccrone was telling me about the semiotics of Charles Sanders Peirce and how it fitted into a modern context of biological thinking which he was sure was of great importance to physics. I think he's probably right. In return, I naturally tried to convince him that Category Theory was the right modern language to discuss these sorts of things. Anyway, the conversation prompted me to do one of those things that is always on the to do list about half way down the page: go to the philosophy library and get out the collected works of Charles Sanders Peirce. When I saw how many volumes there were I modifed this resolution and chose just a few, including the wonderful reference 3 (see below) which became my introduction to Peirce's Existential Graphs.

Naturally, Louis Kauffman has already written a beautiful article on this subject (reference 2).

Hopefully you will know by now that diagrammatic techniques are endemic to categorical computation. What Peirce did was develop a surface diagram notation for basic logic. So for braided monoidal categories we have knots, and for logic we have Existential Graphs. Moreover, he did this over 100 years ago!

For example, how does one express the notion of not X? If X is a symbol on a page, one simply draws a circle around it. This cuts X off from anything else on the page. Two rings, one inside the other, act as an identity (this is Boolean logic). The identity can be deformed so that the two circles are joined at a point...and this naturally looks like one loop with a kink in it.
Conjunction of two terms X and Y is represented by simply writing them both down, with no extra symbols. The empty picture is the statement true. Exercise: what is the diagram for false?

This all fits into a fantastical philosophical scheme...but must go now.

References:

1. Nice webpage: http://www.clas.ufl.edu/users/jzeman/

2. L. H. Kauffman The Mathematics of Charles Sanders Peirce
in Cyber. Human Know. 8 (2001) 79-110, available at
http://www.math.uic.edu/~kauffman/Papers.html

3. Semiotic and Significs: The correspondence between C. S. Peirce and
Victoria Lady Welby ed. C. S. Hardwick, Indiana University Press (1977)

4. Collected Papers of Charles Sanders Peirce vol IV,
ed. C. Hartshorne and P. Weiss, Harvard University Press (1933)

 

[Kea]

...

Next, Peirce introduces lines of identity. That is, terms X and Y may be joined by a line. A whole lot of terms may be joined by a network of lines. It is OK for the lines to cross the circles. And now, without any further ado, the rabbit appears ... quantification can be expressed without any symbols by saying that if the outer end of a line is enclosed by an even number of circles then the term represents something definite, and if an odd number then anything at all of that type.

I must figure out how all of this can be modifed to quantum logic. We have Coecke et als diagrammatics, but that just comes from monoidal category theory and the logic seems to be a bit of an afterthought. Note that drawing a line of identity from X to X and putting it beside a line of identity from Y to Y is exactly how one represents $X \otimes Y$ in a monoidal category.

Perhaps we could alter Peirce's not not X = X rule and substitute a Heyting not not not X = not X rule, which would be an allowance of deletion of two circles but not the last two.

 

[Kea]

...

The law of the excluded middle looks like (from Kauffman)

 

[Kea]

...

and the proof of this fails in the Heyting case because at the last step the deletion of two circles about the left hand Q is not permitted.

 

[mccrone]

Hi Kea

Thanks for http://charlotte.ucsd.edu/users/goguen/pps/nel05.pdf and other links on Peirce above. I see reading from this thread that you have also taken in Hegel.

Your theme for this thread is: The third road says "get the logic right, and you'll see how computational the universe is". And the right logic is category theory – it is a general enough theory of logics to “eat” any more particular ones that may have been suggested in the past such as Peirce’s organic/semiotic/triadic approach.

I still have no feel whatsoever for the substance of category theory despite having read a bit more about it now. Perhaps I can provoke you into some jargon-free explanation which gets at its essence.

I understand set theory is based on collections of crisp, discrete, bounded, located, persistent objects. So “atoms with properties”, a mechanical view in which all action and organisation and systemhood is emergent (thus is does not need to be represented at the most fundamental level – the object and its properties).

Category theory seems to take the correct step in saying, no, reality has both locations and motions, stasis and change, form and substance, local and global – the whole gamut of standard metaphysical dichotomies. So to define a basic something, you need both an object and its actions.

A mechanical view here would say that category theory is just accounting for an object and its properties in more distinct fashion. But a holistic or background-independent view points out that all atoms exist in a void. And the void is a thing with properties. The void has crisp spacetime structure. And even the freedoms that the void permits, such as the inertial motions of particles, are essential properties of the void.

So perhaps a more organic view of category theory is that it breaks reality into its most natural dichotomy - that which is semiotically constrained and that which is semiotically not visible, thus free to happen. An object such as a particle (or a void) is produced by a system of self-constraint acting on a ground of pure potential (Peircean vagueness, Anaximander’s apeiron). A particle gains a crisp identity as all the other things it might be become constricted to near impossibility (in simple terms, a cold and expanded Universe steadily robs an electron of its chances to be a quark or tau, etc). But within every system of constraint there are also emergent freedoms. A crisply made particle (that cannot freely transmute and which now has mass and cannot fly at light speed) can now wander about in an “empty” void with weak gravity, in fairly unconstrained inertial fashion.

Peircean logic – as outlined in that Kauffman paper – is seeking to describe a figure~ground breaking in which both figure (object, or atom) and ground (context, or void) are simultaneously developed. This is indeed a background independent approach – or rather it depends on “vagueness” as the unformed, and insubstantial, ground that then divides to make crisp atoms in a crisp void. Or in category theoretic terms(?), crisp objects and their crisply permitted contextual properties, their various possibilities for action.

Or using x and not-x terminology, we would start in a realm where x-ness and its antithesis are mere unformed possibility (like perhaps order and disorder, atom and void, chance and necessity – absolutely any dichotomy that makes metaphysical sense). Then in creating the crisply not-x, we create the x. Or with equal emphatic-ness, if we create the crisply x, it creates the crisply not-x. As in relativity, the choice of reference frame – “who moved first?” – becomes arbitrary.

I think as you get deeper into Peirce, problems start to arise. For one thing, I don’t think he considers the issue of scale and so his position on hierarchies remains fuzzily developed.

However his semiotic approach as applied to modern physics might read something like this. The Universe has a “mind” – a set of interpretative habits that we know as Newtonian/relativistic mechanics. This generalised mind (a Peircean thirdness) looks into the well of quantum potential (pure vague Peircean firstness) and interprets it into particular physical events or occasions – the classical realm of particles having interactions.

The mind of the Universe never sees a naked quantum realm, only the kinds of events and regularities it has come to expect. This is the famous irreducible triadicity of semiosis. There is the interpreter and the thing in itself. And then the joint production that is the construction of particular signs – particles whizzing about hither an thither in a disinterested void.

Peircean logic contains everything and the kitchen sink. You have the monadic principle of vagueness. You have the dyadic principle of dichotomous separations (or phase transitions or symmetry breakings we might call them). And you have the triadic principle of semiosis (or hierarchical complexity).

Again, what is category theory about at root and does it really map to the whole of Peirce’s organic framework or just perhaps to the dyadic part?

Cheers – John McCrone.
---------------------------------

 

[Kea]

Peircean logic contains everything and the kitchen sink. You have the monadic principle of vagueness. You have the dyadic principle of dichotomous separations (or phase transitions or symmetry breakings we might call them). And you have the triadic principle of semiosis (or hierarchical complexity).
Again, what is category theory about at root and does it really map to the whole of Peirce’s organic framework or just perhaps to the dyadic part?

Hi John

I'm not really expecting Peirce to have all the answers to quantum gravity! But his modernity is striking. To quote another book that I picked up (D. Greenlees's Peirce's Concept of Sign), Two qualities of Peirce's philosophical thought are most apt to impress those who study it seriously: its radical originality and its incompleteness.

Although it is true that the dyadic is picked up naturally by categories in the way you describe, particular dualities become mathematically more elaborate than this, and I'm afraid one really does need a fair bit of mathematical background to see things from my, albeit very one-sided, point of view. However, to capture the whole Peircean logic and the heirarchy scheme I really think higher dimensional categories (even more complicated) are necessary, so the logic is by no means mathematically trivial!

Plenty to do.
Kea

 

[Kea]

...

The Peircean idea of using diagrams to do logic has been investigated most notably by Cockett and Seely in their prodigious works, such as the paper

Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories
J.R.B. Cockett and R.A.G. Seely
http://www.tac.mta.ca/tac/volumes/1997/n5/3-05abs.html

which it is remiss of me not to have previously mentioned.

 

[Chronos]

Actually [and at risk of exposing my naivety] it is quite simple to model higher dimensions using 2D spreadsheets with hierarchical branches. In the simplest model, all you need to do is attach two degrees of freedom [on-off bit slices] to each coordinate value from the previous table. For example, a 2D table becomes 3D when you add a z coordinate to each x-y value in the table. It then becomes 4D when you attach another 2D table to each z value. That's a simplified explanation, but not a bad way to picture how to map high dimensional surfaces, IMO.

 

[Kea]

change of topic

New:

Calabi-Yau Manifolds and the Standard Model
John C. Baez
4 pages

Abstract:
For any subgroup $G$ of $O(n)$, define a $G$-manifold to be an n-dimensional Riemannian manifold whose holonomy group is contained in $G$. Then a $G$-manifold where $G$ is the Standard Model gauge group is precisely a Calabi-Yau manifold of 10 real dimensions whose tangent spaces split into orthogonal 4- and 6-dimensional subspaces, each preserved by the complex structure and parallel transport. In particular, the product of Calabi-Yau manifolds of dimensions 4 and 6 gives such a $G$-manifold. Moreover, any such $G$-manifold is naturally a spin manifold, and Dirac spinors on this manifold transform in the representation of $G$ corresponding to one generation of Standard Model fermions and their antiparticles.

http://www.arxiv.org/abs/hep-th/0511086

This paper is currently being discussed on blogs galore, but the only interesting comments so far come from Tony Smith on Not Even Wrong http://www.math.columbia.edu/~woit/wordpress/?p=291#comments
who mentions Penrose and Rindler, the canonical reference on Twistors. That is, complex projective spaces can be taken as the choice of 4D and 6D manifolds, one for spacetime.

 

[Tonko]

What is really going on here?

I have just read:
Smooth singularities exposed: Chimeras of the differential spacetime manifold; A. Mallios, I. Raptis, http://arxiv.org/abs/gr-qc/0411121

That was probably a big exaggeration. Namely, I am still (and will probably spend next few months) struggling with learning enough of the category theory so that I can understand the terminology and technical details in abovementioned article.

What is bothering be at this time is that, even if I manage to understand the math details I still do not understand what are these guys *really* saying, even in the broadest of possible outlines.

The article is full of the most interesting and relevant quotations of Einstein and other physicists/mathematicians, regarding conceptual troubles with General Relativity, quantization, spacetime, manifolds, etc. quotations that are hard to find anywhere else, especially orgainized so pointedly.

Yet, while the authors spent considerable effort constantly exciting the reader about providing the ultimate response to the most difficult issues with singularities in physics, I felt cheated by the end.

In the end, after many repeated promises authors have not spared even a few sentences on exploring and explaining even the most elementary consequences of what (supposedly) they have done.

They removed Schwarzschild singularity as such but what does it really mean? So what does happen with the particle that falls through the horizon? What is its ultimate fate? How das banishing the singularity really affect the rest of the Universe?

Apparently, authors can not care less. IMO, all they care about is that homo... to homo to a functor to a category to a functor to, God knows what, is (presumably) well defined, mathematically that is.

Like a magic, there is a solution without a solution, as long as you can hide it behind the categories, functors and toposes.

At this point I don't know what is worse:
a)physicists pretending to do physics while really doing mathermatics or
b)mathematicians trying to solve problems that trouble physicists, apparently without having any idea of what physical world is.

Tony

 

[Kea]

I have just read:
Smooth singularities exposed: Chimeras of the differential spacetime manifold; A. Mallios, I. Raptis, http://arxiv.org/abs/gr-qc/0411121

481 pages!

Hi Tony

A hearty welcome to PF. With regards to this particular paper I quite agree with your criticism. The development of the (interesting) ideas does not seem to be physically comprehensive and the sheer volume of quotations is more than overwhelming. I certainly haven't read it myself.

I assume that you are looking around a bit. I'm afraid we can't promise you any definitive references at this point in time.

Kea

 

[Kea]

The following paper has been brought to our attention by another thread:

Model theory and the AdS/CFT correspondence
Jerzy Król
17 pages
http://arxiv.org/abs/hep-th/0506003

Abstract:
"We give arguments that exotic smooth structures on compact and noncompact 4-manifolds are essential for some approaches to quantum gravity. We rely on the recently developed model-theoretic approach to exotic smoothness in dimension four. It is possible to conjecture that exotic $R^4$s play fundamental role in quantum gravity similarily as standard local 4-spacetime patches do for classical general relativity. Renormalization in gravity--field theory limit of AdS/CFT correspondence is reformulated in terms of exotic $R^4$s. We show how doubly special relativity program can be related to some model-theoretic self-dual $R^4$s. The relevance of the structures for the Maldacena conjecture is discussed, though explicit calculations refer to the would be noncompact smooth 4-invariants based on the intuitionistic logic."

...and from the introduction:
"The purpose of this paper is to present arguments that some new mathematical tools can be relevant for such purposes. The tools in question are exotic smooth differential structures on the topologically trivial $R^4$. However, one should refer to the formal mathematical objects in perspective established by the model-theoretic paradigm rather than ascribe to the absolute classical approach where various mathematical tools are placed in the absolute 'Newton-like classical' space, and governed by the ever present absolute classical logic."

 

[marcus]

Kea is this paper of interest?

http://arxiv.org/abs/gr-qc/0511161
Spin networks, quantum automata and link invariants
Silvano Garnerone, Annalisa Marzuoli, Mario Rasetti
19 pages; to appear in the Proc. of "Constrained Dynamics and Quantum Gravity (QG05), Cala Gonone (Italy) September 12-16 2005
"The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFT). More precisely, when working with purely discrete unitary gates, the simulator is naturally modeled as families of quantum automata which in turn represent discrete versions of topological quantum computation models. Such a quantum combinatorial scheme, which essentially encodes SU(2) Racah--Wigner algebra and its braided counterpart, is particularly suitable to address problems in topology and group theory and we discuss here a finite states--quantum automaton able to accept the language of braid group in view of applications to the problem of estimating link polynomials in Chern--Simons field theory."

I can't judge the quality or relevance. If it is OK, tell me, otherwise i will delete the post so as not to intrude.

 

[Kea]

http://arxiv.org/abs/gr-qc/0511161
Spin networks, quantum automata and link invariants
Silvano Garnerone, Annalisa Marzuoli, Mario Rasetti

Thanks, Marcus. There's far too much going on to keep track of it all.

 

[Kea]

Krol papers

Two papers in the same volume of the same journal, perhaps not available online:

Exotic Smoothness and Noncommutative Spaces: the Model-Theoretical Approach
J. Krol
Found. Phys. 34, 5 (2004) 843

Background Independence in Quantum Gravity and Forcing Constructions
J. Krol
Found. Phys. 34, 3 (2004) 361

These refer to a beautiful book, which I just discovered and wish I had known about years ago, namely

Models for Smooth Infinitesimal Analysis
I. Moerdijk, G. E. Reyes
Springer-Verlag (1991)

Many of you will know the first author's name from his recent textbook on topos theory and perhaps from other excellent pedagogical papers. Krol refers to their concept of Basel topos. From the preface of the book:

"...the reader may well wonder whether we are reformulating non-standard analysis [a la Robinson] in terms of sheaves. However, one should notice that two kinds of infinitesimals were used by geometers like S. Lie and E. Cartan, namely invertible infinitesimals and nilpotent ones. Non-standard analysis only takes the invertible ones into account, and the claims to the effect that non-standard analysis provides an axiomatization of the notion of infinitesimal is therefore incorrect.

...The main novelty of our approach, with regard to both non-standard analysis and synthetic differential geometry, is precisely the construction of such mathematically natural models containing nilpotent as well as invertible infinitesimals."