The Third Road to Quantum Gravity

[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^4s 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^4s. We show how doubly special relativity program can be related to some model-theoretic self-dual R^4s. 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."

 

[Careful]

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

Hehe, I have been reading Mallios and Raptis a few years ago too; it did not take me longer than 1 day to draw my conclusions Quantum gravity should IMO start from rethinking and *modifying* QM and not talking about crazy kinematical structures for a decade (or longer).

 

[Kea]

Quantum gravity should IMO start from rethinking and *modifying* QM ...

Why, yes, as Penrose likes to say. But that doesn't mean that some kinematical studies are not useful in understanding how the full theory reduces to the standard model.

 

[Kea]

What is the Basel topos?

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

As Moerdijk and Reyes explain in their introduction, the basic idea is to replace commutative rings (which get used to build spaces in Algebraic Geometry) with C^{\infty}-rings, which they define in the first chapter. The (opposite of the) category of finitely generated C^{\infty}-rings is called L, the category of loci.

The category of smooth manifolds may be embedded in L via

M \mapsto C^{\infty}(M)

Now L itself is not a topos, but by cleverly defining a Grothendieck topology on L one can take the category of sheaves Sh(L) which is of course a topos.

On page 285 the authors take the Grothendieck topology to be the one generated by the covers of L (see the book) along with some singleton families. The sheaf topos is then the Basel topos. Getting this topology right involves the notion of forcing, precisely in the sense of Cohen forcing for the independence of the Continuum Hypothesis.

As an illustration of the power of this construction the authors point out that Cartan's local point of view of Stoke's Theorem can be extended to the full theorem using Cartan's intuitionistic arguments alone.

 

[selfAdjoint]

Do you have a source for Models for Smooth Infinitesimal Analysis? My search turned up not available on Amazon, and neither abebooks nor Springer Verlag itself had any record of the book.

 

[Kea]

Do you have a source for Models for Smooth Infinitesimal Analysis? My search turned up not available on Amazon, and neither abebooks nor Springer Verlag itself had any record of the book.

That might explain why we've never come across it before, but I swear I'm holding a copy in my hand right now! ISBN 0-387-97489-X and the publisher is Springer-Verlag. The fine print says "Printed and bound by BookCrafters, Chelsea, Michegan".

 

[jarek]

A comment on Kea's sheaf argument in GR

Hi all,
just joined you. Sorry for poping up with an old post, but wanted to comment on that:

The question is: how can we describe a point in spacetime? Well, a point in spacetime isn't of any physical importance. In fact it was only by realising this that Einstein came to accept general covariance in the first place (see the book by J. Stachel, Einstein from B to Z Birkhauser 2002). What is physical are the (equivalence classes of) gravitational fields.
If we work with sheaves over a space M then a point is indeed a highly derived concept. So the physics is telling us we should use sheaves to do GR.

If you look at the hole argument which is usually invoked here, then a subset of the spacetime seems just as unphysical as a point (you give it a physical meaning only by "localizing" it with matter). Note, that I am strongly against the nightmare of modern physics called "space-time point", but the argument against it which you present might not be convincing for everybody.
Another loosely related issue: sheaves (as far as I understand, at least in some basic formulation) are functions on open sets. The topology on the space time is transported from R^4, which in turn is the metric topology of Euclidean metric. In my eyes this lacks physical justification.

-jarek

 

[Kea]

...sheaves (as far as I understand, at least in some basic formulation) are functions on open sets...

Hi jarek

Welcome to PF. You may wish to consider a little further the arguments here. At the very least, an understanding of a sheaf as a functor.

Cheers
Kea

 

[jarek]

To Kea

As far as I undertsand your line of reasoning is the following: (points unphysical according to Einstein) => (substitute point-defined objects by sheaves over M) => (abstract further and use cathegory-theoretical sheaves). I think the reason for abondoning points is not the Einstein argument - he finally resolved his hole paradox by "localizing" point-events as the intersecting points of geodesics.
-jarek

 

[Kea]

As far as I understand your line of reasoning is the following...

Let me repeat: you may wish to consider the arguments here a little further.

 

[Careful]

Let me repeat: you may wish to consider the arguments here a little further.

Could you tell me why a physicist should be interested in sheaves (no references which I probably know : I want *your* opinion ) ?? As a comment on the previous post (concerning Raptis and Mallios): it is entirely useless to speak about the kinematics before you have a clear idea how to construct the dynamics.

Cheers,

Careful

 

[selfAdjoint]

it is entirely useless to speak about the kinematics before you have a clear idea how to construct the dynamics.

So it was useless for Einstein to consider special relativity before he was in possession of the general?

Generally, careful, I find many of your obiter dicta to be careless and unproductive.

 

[Careful]

So it was useless for Einstein to consider special relativity before he was in possession of the general?
Generally, careful, I find many of your obiter dicta to be careless and unproductive.

At the time Einstein produced special relativity, he did not consider yet the thought that spacetime itself could be dynamical. Moreover, his theory of special relativity also had a dynamical side, in the sense that the laws of physics should be invariant with respect to global Lorentz transformations and guess what: such laws were known BEFORE Einstein wrote down SR (Maxwell theory), actually they were a motivation for him to do so! So your example is actually *confirming* what I claimed.

And moreover selfAdjoint, since when is it ``unproductive´´ to explain why some approaches to QG are obviously flawed ?? A first step in understanding what is meaningful, is deeply knowing what is NOT and such knowledge can only be reached through exercising yourself. If it were up to you, we would have bought an empty box in another thread. Moreover, obiter dicta means ``remarks which are not necessary to reaching a conclusion´´ : I think my remarks are always to the point and certainly conclusive.

 

[Kea]

Could you tell me why a physicist should be interested in sheaves...

The answer is simple, Careful. Topos theory. Toposes teach us how to do geometry and logic together. I claim that any approach to QG worth its salt must be able to operate in this realm. Of course, ordinary sheaf categories (Grothendieck toposes) are not enough...

 

[Kea]

...

Careful, it would be much appreciated if you took some time to go through previous discussions on this issue carefully, meaning looking up some references etc.

 

[Careful]

**The answer is simple, Careful. Topos theory. Toposes teach us how to do geometry and logic together **

geometry or *topology* (there is a difference you know) ??

At first sight I would guess you probably want to say that these functors allow you to map open sets of a topological space to a sheaf of (local) propositions (that is local(ized) operators in a local Hilbert space). Probably you also want to play around with the associated orthomodular lattice structure, no?


**I claim that any approach to QG worth its salt must be able to operate in this realm. Of course, ordinary sheaf categories (Grothendieck toposes) are not enough. **

If you mean the above, then what you say is a tautology provided you do not want QM to go down the drain (which is necessary for some part at least IMO).

So, (a) if the above is true, then you are merely formalizing things we already know for a long time and unless this brings new physics (which it doesn't) this is plain mathematics or (b) you have still some surprise under your sleeve and are going to tell us about this.

PS: concerning the covariance argument: Jarek is obviously correct. Moreover, you simply seem to say that you want spacetime to be granular in the sense that you build in a fundamental finite resolution. This is an old idea mainly launched by Sorkin in the eighties (he has written some papers on locally finite coverings, sheaves and so on, so forth)

 

[Careful]

Careful, it would be much appreciated if you took some time to go through previous discussions on this issue carefully, meaning looking up some references etc.

Sorry, but that pig does not fly. It is *your* task to explain us in a well motivated way why *physicists* should even consider what you are doing. Putting the readers nose down on a whole pile of references before you even consider discussing the idea is not only a sign of disrespect, but is also generally experienced as a weakness. I am interested in hearing about *your* insights and *physical* motivation (and all you give me is a cheap marketing slogan), so it would be much appreciated if you could just do that in *detail*. Usually, the conversation can only make progress in this way.

Cheers,

Careful

 

[jarek]

Sorry, but that pig does not fly...

Ditto! I went through all the thread before my first post, Kea. I do understand sheaf as a functor, but that's *MATH*. I simply spotted an unclear point in your *PHYSICAL* motivation. I find the topos approach intelectually appealing, that's why I'm trying to understand how to motivate this approach physically.
best,
jarek

 

[Kea]

Careful and Jarek

Firstly, allow me to say that it is quite clear that you have not read and thought about what I have already said.

Be that as it may, as soon as I get a chance I will do as you ask, and attempt to answer the question.

 

[Kea]

At first sight I would guess you probably want to say that these functors allow you to map open sets of a topological space to a sheaf of (local) propositions (that is local(ized) operators in a local Hilbert space). Probably you also want to play around with the associated orthomodular lattice structure, no?

Obviously the claim is that higher category theory allows us to go beyond this.

 

[Kea]

Let us begin with a short list of topics that have previously been mentioned, albeit briefly in some cases:

Confinement mechanisms
Mass generation
Particle Number non-conservation
Quantum Mechanics
Quantum Computing protocols
Knots in condensed matter systems
Cosmological problems
Machian principles

I'm curious as to which of these you consider to be of no physical relevance.