The Third Road to Quantum Gravity

[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^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.

 

[Careful]

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.

Sure with category theory, you can do anything you want (again a tautology) Unfortunately, it does not help you with *solving* a problem.

 

[Careful]

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.

Oh yeh, I did, but you started off bad. You referred:

``The Computational Universe: Quantum gravity from quantum computation
Seth Lloyd
http://arxiv.org/abs/quant-ph/0501135´´

If you look up the word CRACKPOTISM 2005, this paper should be in the top ten. It is not only utterly naive, but it contains elementary mistakes as well.

 

[Careful]

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.

As I said you can define virtually anything in the framework of category theory: the questions are (a) what computational benifit does it give ? (b) has anything *extra* been reached with these methods already (apart from mathematical abstraction), that is does there exist a real physics problem which has been solved thanks to the use of category theory? (c) has it provided any further *physical* insight ?

 

[Kea]

Careful

I would never have mentioned Seth Lloyd if this was entirely my thread. But the thread was actually started by someone else, and I always try to be considerate of others' ideas.

 

[Kea]

(a) what computational benifit does it give? (b) has anything extra been reached with these methods already (apart from mathematical abstraction), that is does there exist a real physics problem which has been solved thanks to the use of category theory? (c) has it provided any further physical insight?

I think there is a very fair claim for an answer of yes to (c) and possibly even (b). As an answer to (b) one might mention the clarification (and hence increase in computational power) of the Racah-Wigner calculus used by spectroscopists. Without impressively concrete results, it is hard to make a claim for (a), but I think Strings, LQG and all other approaches are in the same boat here.

Perhaps to open this discussion:

When String theorists tell me that we cannot even in principle calculate the rest masses of fundamental particles, I get quite distressed. The LHC is not so far away, and as far as I can tell nobody has a good idea of what it should be able to see. This would be less worrisome if I didn't think there was more we could do, but I do.

The reason I often launch into categorical or logical jargon is because I believe physical intuition and categorical intuition have a great deal in common. To do GR, one certainly does not need category theory. To do lattice QCD, one does not need category theory. To some extent the problem with the jargon is a lack of physical terminology to go beyond these domains.

It is my personal opinion that the following two kinds of principle are both necessary and sufficient for writing down a unified theory:

1. Measurement The necessity of internalisation (the "context" or "environment" must be taken into account in determining the nature of propositions) forces an acceptance of, amongst other things, a categorical comprehension scheme. This must, as in the mathematical treatment, be an axiomatic issue.

2. Machian The String intuition of scale dualities is useful here. I also think a GR intuition is useful. Very briefly, think of the standard model (flat spacetime) as one particular domain of this generalised general covariance, which operates under a constraint of "conservation in time" which is given a priori. In general, physical geometry is determined by the logic of the propositions being asked. Alternatively, allowable propositions follow from geometrical constraints.

Clearly the real physical justification of this can only lie in the eventual computation of new physical quantities. This relies on an understanding of (something very mathematical) higher descent theory (categorical cohomology) that I do not yet have, but perhaps others do.

As an example, let us now consider the dimension raising nature of the Gray tensor product. Observe that Gray categories have already been shown to be important in understanding SU(3) confinement from a kinematical point of view. Moreover, they arise automatically out of a consideration of 1.

Thinking quantum mechanically for a moment, a representation space should be sufficiently internalised (in the categorical sense) to be able to describe states. Assuming for now that this leads to bicategory objects one is forced to take Gray tensor product for combinations of physical systems, because only this product has the universal property. This means that there is a link between particle number, or some measure of complexity for a system, and categorical dimension. Other aspects of Gray categories are also pertinent to QM, eg. weakened distributivity.

I have actually spoken quite a lot about this with people, and a common reaction is that it is all a pile of junk unless one can rigorously recover the standard model. Then again, a number of people are already working in this direction.

 

[Careful]

Careful
I would never have mentioned Seth Lloyd if this was entirely my thread. But the thread was actually started by someone else, and I always try to be considerate of others' ideas.

Science, for me, consists of reading papers, being open to ideas and working out your own stuff. Then, you *think* about it and see if they make sense or not. If they don't, because either some elementary unrepairable mistake is made (under the pleitoria of technical details), or because the idea is obviously naive and would spring to the mind of anyone with some intelligence after one hour of thought, THEN you should debunk it and warn people for it. That is your *duty* as a scientist, being considerate is the work of a politician.

 

[Careful]

**
When String theorists tell me that we cannot even in principle calculate the rest masses of fundamental particles **

What do you mean by this ? Are you just saying that the masses of elementary particles are not predicted from string theory calculations ?
This should tell you something about string theory, not about the method of physics.


** The reason I often launch into categorical or logical jargon is because I believe physical intuition and categorical intuition have a great deal in common. To do GR, one certainly does not need category theory. To do lattice QCD, one does not need category theory. To some extent the problem with the jargon is a lack of physical terminology to go beyond these domains.**

But for that, you probably do not need category theory either! You just need an entirely new *vision* (just as Einstein had).

**
1. Measurement The necessity of internalisation (the "context" or "environment" must be taken into account in determining the nature of propositions) forces an acceptance of, amongst other things, a categorical comprehension scheme. This must, as in the mathematical treatment, be an axiomatic issue.
**

A question: is the moon a part of your context when you are doing a lab experiment ? You seem to be saying that the set of propositions must be dynamically generated relative to its environment. This is certainly true in GR (and there such idea makes sense); the problem is that it is probably impossible to achieve this in a purely unitary scheme for QM (one has to impose by hand a preferred set of macrostates). Are you claiming that you are going to solve the micro-macro problem in QM through categorization ?


**The String intuition of scale dualities is useful here **

Can you explain me what this has to do with Mach (I shall disgard here that these dualities are not even rigorous at all ) ??

** Very briefly, think of the standard model (flat spacetime) as one particular domain of this generalised general covariance, which operates under a constraint of "conservation in time" which is given a priori.**

Give me your principle of generalized covariance ! Are you referring to the Kretchmann debate here (that one can write flat space physics as a generally covariant theory with constraints - through Lagrangean multipliers ?).

Sorry, but all your comments are just to vague.

** In general, physical geometry is determined by the logic of the propositions being asked. Alternatively, allowable propositions follow from geometrical constraints. **

This statement needs some clarification: you can recover the causal structure but not the local scale factors unless you go over to a fundamentally discrete scheme such as causal sets. If so, you should add that such line of thought which gives up manifoldness, imposes the almost impossible problem of recuperating it on appropriate scales (people really got *almost* nowhere in this problem). And certainly category theory is not going to solve it.


**
This relies on an understanding of (something very mathematical) higher descent theory (categorical cohomology) that I do not yet have, but perhaps others do.
**

I have given such ideas some thought (in the context of the manifoldness problem) and it occurred to me that all these constructions are too sensitive to combinatorical ``accidents´´ and hence not very useful. My view in this matter that a more robust scheme in the spirit of a ``coarse grained´´ version of metric geometry (a la Gromov) is much more useful.

**As an example, let us now consider the dimension raising nature of the Gray tensor product. Observe that Gray categories have already been shown to be important in understanding SU(3) confinement from a kinematical point of view. Moreover, they arise automatically out of a consideration of 1.**

Could you specify this more? As I said, you can almost do anyting with category theory KINEMATICALLY (this applies also to all other ``virtues´´ you mention), but the DYNAMICAL aspect is obscure to me (example: causal sets do not have a quantum dynamics yet.).

 

[Kea]

As someone who claims to be familiar with higher descent theory I am surprised you have this attitude towards it. We would be most keen to hear about your alternative program (Gromov's) on another thread. I am sorry I do not have the time at present to discuss all these points in great detail. Briefly, however:

Are you just saying that the masses of elementary particles are not predicted from string theory calculations? This should tell you something about string theory, not about the method of physics.

Quite true. But readers following this discussion are aware of my opinion that some current M-theoretic thinking is not all that different from the categorical approach, and I am more upset because I see them as allies than because I think it is all a complete waste of time.

But for that, you probably do not need category theory either! You just need an entirely new vision (just as Einstein had).

Not being Einstein will not discourage me from continuing this thread for the benefit of those who are interested.

You seem to be saying that the set of propositions must be dynamically generated relative to its environment... The problem is that it is probably impossible to achieve this in a purely unitary scheme for QM (one has to impose by hand a preferred set of macrostates). Are you claiming that you are going to solve the micro-macro problem in QM through categorization?

First of all, and this is not a small point, you use the word set casually, which in this context it is important not to do. Probably impossible does not mean impossible. Besides, you seem to have a picture of a fixed set of macrostates but one thing categories do very nicely is allow us to dodge this kind of problem. Of course, I'm not claiming that this has been solved as yet, but I will express my opinion that category theory can do it.

Can you explain me what this has to do with Mach?

I simply use the term Machian to refer to anything that relates the small scale to the large in such a way that there is a correspondence of physical observables. The principle of GGC must then be formulated with the understanding that descent topologies somehow encode observables. Since scale with duality loosely corresponds to categorical dimension, GGC takes the form of a generalised Poincare duality (I simply don't know how to express this better) in the (higher) topos cohomology.

Are you referring to the Kretchmann debate here?

No. I am not familiar with this debate.

...And certainly category theory is not going to solve it.

Really? We would appreciate it if you could substantiate such a large claim.

I'm afraid we will have to leave Gray categories to a later time. You seem to view categories as no more than an organisational tool. Even if that were true, which it is not, it may still be that is something that physics requires. This remains to be seen.

 

[Careful]

**As someone who claims to be familiar with higher descent theory I am surprised you have this attitude towards it.**

I do not claim to be actively familiar with it anymore but there were times that I considered it (a sin of youth).

**
Not being Einstein will not discourage me from continuing this thread for the benefit of those who are interested. **

Good ! You shouldn't


**First of all, and this is not a small point, you use the word set casually, which in this context it is important not to do. **

Could you clarify this (I think my use of word set is quite harmless there)?

**Probably impossible does not mean impossible. **

Oh, but I am quite confident that in this context it does ! There is no no go theorem yet (true) but I have the unmistakable evidence that it is an eighty year old wound.

**
I simply use the term Machian to refer to anything that relates the small scale to the large in such a way that there is a correspondence of physical observables. **

I guess you mean energy scales. But the use of Machian is very confusing here.

**The principle of GGC must then be formulated with the understanding that descent topologies somehow encode observables. **

In simple terms, you mean that handles glued to space represent observables (such as particles), no? I would kindly request you, for the general readership, to use the most common terminology possible (I am sure that can be done). If so, you must be informed that it is quasi impossible to obtain a non perturbative gravitational dynamics which includes such topology changing spaces (and as such it is a wild, speculative idea which has been around for at least thirty years now).


**Since scale with duality loosely corresponds to categorical dimension, GGC takes the form of a generalised Poincare duality (I simply ?**

What scale (so what is your model of spacetime, how do you put a measure stick and so on..) ?? I can see how the above idea of GGC relates to cohomology classes, but you have to tell me what this duality is about (since I see no dynamical model here).


**Really? We would appreciate it if you could substantiate such a large claim.**

I will, in due time, when you have told me what your spacetime model is (to which category do you restrict?)


** You seem to view categories as no more than an organisational tool. **
Yes

**Even if that were true, which it is not, it may still be that is something that physics requires. This remains to be seen**

Why would it not be true?

 

[Kea]

Could you clarify this (I think my use of word set is quite harmless there)?

In an elementary topos, a proposition is understood in terms of its interpretation in terms of truth values. This is an axiomatic setting outside of ordinary set theory. This is simply a fact.

In simple terms, you mean that handles glued to space represent observables (such as particles), no?

No. The point is that categories can do more subtle geometry than this. If all we were going to do was work with ordinary manifolds then I would agree: categories would not be enough. This is, however, very far from being the case.

What scale (so what is your model of spacetime, how do you put a measure stick and so on)?

One does not begin with a model of spacetime, which is clearly a highly derived concept. And yes, when I say scale I am thinking of energy scales, but then again even this is an entirely classical concept. Physically, energies are no different to quantum numbers: they need to be looked at in the context of the experiment. So, as I often say here on PF, the question what is scale is by no means trivial, and I will certainly not be answering it in a few lines. One does not work in a simple 1-dimensional category. Hence the question what category do you restrict to is completely meaningless. As I am sure you know, categorical cohomology allows different categories to act as coefficient spaces.

I can see how the above idea of GGC relates to cohomology classes...

Good! You are the first to say that.